Impulse control systems for servomechanisms with nonlinear friction

Transcription

1 University of Wollongong Thesis Collections University of Wollongong Thesis Collection University of Wollongong Year 2006 Impulse control systems for servomechanisms with nonlinear friction Stephen Van Duin University of Wollongong Van Duin, Stephen, Impulse control systems for servomechanisms with nonlinear friction, PhD thesis, School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, This paper is posted at Research Online.

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3 IMPULSE CONTROL SYSTEMS FOR SERVOMECHANISMS WITH NONLINEAR FRICTION A thesis submitted in fulfilment of the requirements for the award of the degree DOCTOR OF PHILOSOPHY From UNIVERSITY OF WOLLONGONG By STEPHEN VAN DUIN, BE (mech.) Hons SCHOOL OF ELECTRICAL, COMPUTER AND TELECOMMUNICATIONS ENGINEERING

4 THESIS CERTIFICATION I, Stephen van Duin declare that this thesis submitted in fulfilment of the requirements for the award of Doctor of Philosophy in the School of Electrical, Computer and Telecommunications Engineering, University of Wollongong, is wholly my own work unless otherwise referenced or acknowledged. The document has not been submitted for qualification at any other academic institution. Stephen van Duin 30 th August 2006

5 i TABLE OF CONTENTS LIST OF FIGURES... LIST OF TABLES... LIST OF ABBREVIATIONS... ABSTRACT... ACKNOWLEDGEMENTS... v xii xiii xiv xv CHAPTER 1: INTRODUCTION BACKGROUND FRICTION MODELS Classical Friction Models Dynamic Friction Modelling Modified Bristle Model a new model Position Dependent Friction FRICTION COMPENSATION Problem Avoidance Non-Model Based Friction Compensation Model Based Friction Compensation IMPULSE CONTROLLERS DISCUSSION AND SUMMARY OF THE LITERATURE REVIEW PROBLEM DEFINITION OUTLINE OF THE THESIS CHAPTER 2: EXPERIMENTAL EQUIPMENT INTRODUCTION FRICTION TEST BED Friction Mechanism Control Interface and Digital Signal Processing Control Scheme Electrical Circuit Response Mechanical System Parameters Static Friction and Pre-Sliding Constant Estimation of Inertia, Viscous Friction and Coulomb Friction HIRATA ROBOT Electrical Circuit Response and Control Scheme Mechanical System Parameters CONCLUSIONS... 53

6 ii CHAPTER 3: MODELLING AND SIMULATION INTRODUCTION SYSTEM MODELLING Static Modelling of the Friction Test Bed Dynamic Modelling of the Friction Test Bed MODELLING OF THE IMPULSE CONTROLLER Loop Topology Hybrid PID + Impulse Control for Improved Stability CONCLUSIONS CHAPTER 4: IMPULSE CONTROLLER DESIGN INTRODUCTION MATHEMATICAL ANALYSIS MINIMUM PULSE WIDTH VARIABLE PULSE WIDTH IMPULSE CONTROL Controller Design Regulated Pulse Height Simulation of the Variable Pulse Width Controller VARIABLE PULSE HEIGHT IMPULSE CONTROL Controller Design Simulation of the Variable Pulse Height Controller PULSE SHAPING FOR IMPROVED PRECISION Impulse Shape Impulse Shape Impulse Shape Impulse Shape Comparison of Pulse Shapes LIMIT CYCLE OFFSET FOR IMPROVED POSITIONING Motivation Limit Cycle Offset Controller Design Simulation of the Limit Cycle Offset Function SUMMARY AND CONCLUSIONS CHAPTER 5: PERFORMANCE ANALYSIS AND IMROVEMENT OF THE IMPULSE CONTROLLER USING THE FRICTION TEST BED INTRODUCTION IMPROVING THE IMPULSE CONTROLLER FOR A REAL SYSTEM Velocity Reversal Compensation Direction Dependent Friction Values

7 iii 5.3 EXPERIMENTAL EVALUATION OF PULSE SHAPES 1 TO Position Pointing Using Pulse Shapes 1 to Low Speed Position Tracking Using Pulse Shapes 1 to Impulse Height Blending for Pulse Shapes 3 and Vibration Analysis HIGH SPEED POSITION TRACKING LIMIT CYCLE OFFSET IMPULSE CONTROL VERSUS TANGENTIAL DITHER CONCLUSIONS CHAPTER 6: PERFORMANCE ANALYSIS USING THE HIRATA SCARA ROBOT INTRODUCTION EXPERIMENTAL EVALUATION OF PULSE SHAPES 1 TO Position Pointing Using Pulse Shapes 1 to Low Speed Position Tracking Using Pulse Shapes 1 to LIMIT CYCLE OFFSET CIRCULAR TRACE EXPERIMENTS USING A AND B AXES HIGH SPEED POSITION TRACKING CONCLUSIONS CHAPTER 7: CONCLUSIONS AND FUTURE WORK CONCLUSIONS FUTURE WORK REFERENCES APPENDIX A - LIST OF SYMBOLS APPENDIX B - EXPERIMENTAL EQUIPMENT DATA B.1 FRICTION TEST BED B.1.1 Friction Test Bed Direct Drive Motor Specifications B.1.2 Friction Test Bed Direct Drive Digital Amplifier Specifications B.1.3 Coulomb Friction Estimation second method B.2 HIRATA ROBOT B.2.1 Position Data and Calibration B.2.2 Physical Layout of the Robot B.2.3 Movement Specifications B.2.4 Inverse Kinematics B.2.5 Motor and Drive Ratings B.2.6 Electrical Circuit Response and Control Scheme B.2.7 Mechanical System Parameters B.3 ACCELEROMETER B.4 SIMPLIFICATION OF EQUATION

8 APPENDIX C - EXPERIMENTAL FRICTION TEST BED DESIGN DRAWINGS C.1 FRICTION TEST BED DRAWING. NO C.2 FRICTION TEST BED DRAWING. NO C.3 FRICTION TEST BED DRAWING. NO C.4 FRICTION TEST BED DRAWING. NO C.5 FRICTION TEST BED DRAWING. NO C.6 FRICTION TEST BED DRAWING. NO C.7 FRICTION TEST BED DRAWING. NO C.8 FRICTION TEST BED DRAWING. NO iv

9 v LIST OF FIGURES Figure 1.1 A simple set-up for stick-slip motion [1] Figure 1.2 A simulation of stick-slip motion [1] Figure 1.3 Memoryless friction models. The friction force is given by a memoryless function except possibly at zero velocity. Figure a) shows Coulomb friction and Figure b) Coulomb plus viscous friction. Stiction plus Coulomb friction are shown in Figure c), and Figure d) shows the Stribeck friction [1]... 6 Figure 1.4 Full fluid lubrication, regime IV of the Stribeck curve [5] Figure 1.5 The generalized Stribeck curve, showing friction as a dynamic function of velocity for low velocities [5]... 8 Figure 1.6 Asperity contact behaving like springs [15]... 9 Figure 1.7 Figure 1.8 Figure 1.9 Figure 1.10 Figure 1.11 Spring force profile during stick-slip motion at two velocities; spring force decreases when velocity increases [15] Static Friction (breakaway force) as a function of dwell time, schematic; with stick slip cycle shown. Dwell time is the time in static friction, shown as T 2 in Figure 1.7 [15] The friction interface between two surface is thought of as a contact between bristles. For simplicity the bristles on the lower part are shown as being rigid [1] Bristle model; Figure a) shows the deflection of a single bristle. Figure b) shows the resulting static friction model for a single instance in time Friction compensation using: 1) Problem avoidance, 2) Non-model based control, and 3) Model based control Figure 1.12 Direction and effect of Dither [5] Figure 1.13 Model Based Friction Compensation [5] Figure 1.14 Experimentally determined displacement as a function of pulse width and pulse height [13] Figure 2.1 Three dimensional drawing of the friction test bed Figure 2.2 Exploded view of the friction mechanism Figure 2.3 Communication flow diagram for the friction test bed

10 vi Figure 2.4 Combined electrical and mechanical system block diagram Figure 2.5 Electrical circuit block diagram Figure 2.6 Reduced block diagram Figure 2.7 Figure 2.8 Figure 2.9 Pre-sliding displacement and breakaway friction for: (a) counter clock wise rotation; and (b) clock wise rotation Position dependent static friction for: (a) counter clockwise rotation; (b) clockwise rotation; and (c) magnified counter clockwise rotation. 47 a) Velocity response to step torque input for clockwise and counter clockwise motion, and b) resulting friction curve using Least Squares Method Figure 2.10 Mechanical system time constant Figure 2.11 Photograph of the Hirata SCARA robot AR-i Figure 3.1 Simplified Simulink model of the friction test bed open loop Figure 3.2 Simulation model and measured results for step torque inputs Figure 3.3 Figure 3.4 Modified model with a friction function which includes mean static friction, Stribeck effect, Coulomb and viscous friction for both clockwise and counter clockwise rotation Static friction model including stiction and Stribeck effect. a) A general friction model given by [1], and b) Simplified model for simulation purposes Figure 3.5 Tuning the PID controller of the friction test bed using 5 to 10% maximum overshoot Figure 3.6 Classic staircase stick-slip motion using PID control Figure 3.7 Block Simulink model of the new Bristle dynamic model Figure 3.8 The sticking behaviour for the simplified standard model without damping (σ 1 = 0) Figure 3.9 Olsson [1] simulation Figure 3.10 The sticking behaviour for the standard model with velocity dependent damping. The friction increases until the velocity is reached when it drops abruptly to zero Figure 3.11 Olsson [1] simulation

11 vii Figure Breakaway behaviour for the model with σ 1 = Figure 3.13 Olsson [1] simulation Figure 3.14 Figure 3.15 Olsson simulation of varying break-away force as a function of the rate of increase of the applied force: for the default parameters (+); v s = (*); and σ 0 = 10,000 [1] Diagram indicating the typical nested loop structure used in servo systems Figure 3.16 Friction test bed Simulink simulation model with the impulse controller and no velocity loop Figure 3.17 Block diagram of the friction test bed experimental system controller Figure 3.18 Hybrid controller output Figure 4.1 Simulated dynamics of a single torque impulse Figure 4.2 Expected pulse shapes Figure 4.3 Experimentally measured displacement (friction test bed) for both positive and negative impulses using successive pulse widths 1.5 ms and 2 ms Figure 4.4 Simplified variable width impulse controller simulation model Figure 4.5 Simulation of a servomechanism position pointing task using; a) PID only, and b) PID + impulse control. The third plot of each set of graphs uses a very fine axis resolution for position Figure 4.6 Figure 4.7 Figure 4.8 Simulation of a servomechanism low speed position tracking task using; a) PID only, and b) PID + impulse control Simulation of a servomechanism low speed sinusoidal position tracking task using; a) PID only, and b) PID + impulse control Graphical representation of the pulse height as a function of the error e(k) Figure 4.9 Simplified variable height impulse controller simulation model Figure 4.10 Simulation of a servomechanism position pointing task using; a) variable width PID + impulse, and b) variable height impulse + PID. 90 Figure 4.11 Simulation of a servomechanism low speed position tracking task using; a) variable width PID + impulse, and b) variable height PID + impulse

12 viii Figure 4.12 Figure 4.13 Figure 4.14 Figure 4.15 Simulation of a servomechanism low speed position tracking task using; a) variable width PID + impulse, and b) variable height PID + impulse Simulated rectangular impulse F t where fp = 125% of F s : Denoted Shape Simulated rectangular impulse having a negative trailing pulse with amplitude 90% F s : Denoted Shape Simulated stepped pulse with 2 ms startup force having an amplitude 125% F s followed by secondary pulse 3 ms having an amplitude 130% F C : Denoted Shape Figure 4.16 Simulated stepped pulse followed by a trailing negative pulse 90% F s : Denoted Shape Figure 4.17 Experimental displacements for varying pulse widths for shapes 1, 2, 3, and Figure 4.18 Figure 4.19 Exploded view of Figure 4.17 for first two pulse widths showing the variation in minimum displacement (precision) for shapes 1, 2, 3, and Typical impulse showing net available torque (yellow area) after subtracting friction; a) standard rectangular pulse: Shape 1, and b) modified shape to offset friction: Shape Figure 4.20 Simulated displacements as a function of pulse width Figure 4.21 Simulation of the impulse controller limit cycling around the position reference set-point where the final torque output is a pulse with minimum width and mean peak to peak oscillation is d Figure 4.22 Figure 4.23 Conceptual example of reducing the steady state error using Limit Cycle Offset with the limit cycle shifted up by d2-d1 and the new error that is guaranteed to fall within the dead-zone Simulation model of the modified impulse controller with Limit Cycle Offset Figure 4.24 Simulation of the limit cycle offset function used with the PID + impulse controller and Shape Figure 5.1 Figure 5.2 Figure 5.3 Modified Simulink model to include xpc analog out and DSP blocks (red) Integral windup observed at zero velocity and velocity reversal when using a) PID only control, and b) PID with integral reset Modified impulse controller Simulink model with direction

13 ix dependent friction parameters, non linear pulse width gain, regulated pulse height and limit cycle offset function Figure 5.4 A sample step input and position response using pulse Shape 1. Mean final oscillating displacement µ d = 1.440e-4 radians for a sample of 10 repeated experiments Figure 5.5 A sample step input and position response using pulse Shape 2. Mean final oscillating displacement µ d =1.001e-4 radians for a sample of 10 repeated experiments Figure 5.6 A sample step input and position response using pulse Shape 3. Mean final oscillating displacement µ d = 0.957e-4 radians for a sample of 10 repeated experiments Figure 5.7 A sample step input and position response using pulse Shape 4. Mean final oscillating displacement µ d = 0.901e-4 radians for a sample of 10 repeated experiments Figure 5.8 Figure 5.9 Figure 5.10 Figure 5.11 Magnified linear position ramp response showing an experimental comparison between the impulse controller with pulses shape 1 to Integral Absolute Error (IAE) for impulse shapes 1, 2 3 & 4 for a low speed position tracking task Experimental speed regulated sinusoidal position tracking using PID and PID + impulse controllers Magnified velocity reversal showing an experimental comparison of precision between the impulse controllers with Shapes 1, 2, 3 & Figure 5.12 Separating Shape 1 and Shape 4 from Figure Figure 5.13 Integral Absolute Error (IAE) for impulse shapes 1, 2 3 & 4 for a sinusoidal position tracking task Figure 5.14 Modified pulse shape as a function of increasing error e(k) Figure 5.15 Figure 5.16 Photograph of the friction test bed with the attached Kistler type 5134 accelerometer for the measure of system vibration Spectral analysis of system vibration using FFT for pulse shapes 1 to Figure 5.17 Tracking response for the friction test bed using PID and PID + impulse controllers for varying position ramps (0.02 rad/s to 0.35 rad/s) Figure 5.18 Figure 5.19 Mean value of the absolute error for each of the position tracking ramps shown in Figure 5.17 for the period 7 10 seconds Steady state limit cycle for the PID + impulse controller using pulse shape 3. The mean peak to peak displacement µ d is the non-elastic

14 x part of limit cycle Figure 5.20 Figure 5.21 Figure 5.22 Using the Limit Cycle Offset function to reduce the final steady state error PID control with velocity reversal compensation and gains Kp=70, Ki=130 and Kd= PID + tangential dither with amplitude A o =4.2 Nm and frequency ω o =250Hz Figure 5.23 PID + impulse control using pulse shape Figure 5.24 Spectral analysis of system vibration using FFT for PID, PID + impulse and PID + dither control Figure 6.1 A sample step input and position response using pulse shapes 1 to 4. Mean final oscillating displacement µ d measured in radians for a sample of 10 repeated experiments Figure 6.2 Figure 6.3 Magnified linear position ramp rad/s showing an experimental comparison between the impulse controller with Shape 1 and Shape Integral Absolute Error (IAE) for impulse shapes 1, 2 3 & 4 for a low speed position tracking task Figure 6.4 Sinusoidal trace and comparison of Shape 1 and Shape Figure 6.5 Figure 6.6 Figure 6.7 Integral Absolute Error (IAE) for impulse shapes 1, 2 3 & 4 for a sinusoidal position tracking task Steady state limit cycle for the PID + impulse controller using pulse shape 1. The mean peak to peak displacement µ d is the non-elastic part of limit cycle Using the Limit Cycle Offset function to reduce the final steady state error using pulse shape Figure 6.8 Reference control signals for the A and B axes (ω=31.4 mrad/s) Figure 6.9 Circle trace with a 100 mm diameter using the PID only controller Figure 6.10 Circle trace with a 100 mm diameter using the PID + impulse controller with Shape Figure 6.11 Circle tracking errors for PID and PID + impulse controllers Figure 6.12 Circle tracking errors for PID + impulse controllers using pulse shape 1 and shape

15 xi Figure 6.13 Figure 6.14 Tracking response for the A axis using PID and PID +impulse controllers for varying position ramps (0.02 rad/s to 0.35 rad/s) Mean value of the absolute error for each of the position tracking ramps shown in Figure 6.13 for the period 7 10 seconds Figure B.1 (a) Continuous velocity when torque is reduced after breakaway, and (b) Coulomb friction prevents continuous velocity when torque is reduced after breakaway Figure B.2 Hirata robot physical layout and work volume [1] Figure B.3 Hirata robot workspace showing the rotational angles for the A and B axes used to calculate the inverse kinematics Figure B.4 Electrical circuit block diagram for the Hirata robot Figure B.5 a) Velocity response to step torque input for A-axis Hirata robot, and b) resulting friction curve using Least Squares Method Figure B.6 Mechanical system time constant Figure B.7 Simulink model for the experimental friction test bed Figure B.8 Subsystem friction force Figure B.9 Subsystem dz/dt Figure B.8 Subsystem g(v) Figure B.9 Subsystem Sigma

16 xii LIST OF TABLES Table 1.1 Comparison of impulse controller strategy with known experimentation Table 2.1 Electrical and mechanical time constants for the friction test bed Table 2.2 Friction test bed mechanical system parameters Table 2.3 Electrical and mechanical time constants for the Hirata robot Table 2.4 Hirata robot mechanical system parameters Table 3.1 Default parameter values for the simplified standard model [1] Table 3.2 Default parameters for the standard model [1] Table 4.1 Simulation parameters for the new friction model Table 5.1 Measured mean peak to peak displacement of the steady state limit cycle for shapes 1 to Table 6.1 PID gains for the Hirata robot s A and B axes Table 6.2 Measured mean peak to peak displacement of the steady state limit cycle for shapes 1 to Table B.1 Direct Drive motor specifications Table B.2 Direct drive digital amplifier specifications Table B.3 Hirata robot encoder resolution Table B.4 Hirata robot movement specifications Table B.5 Hirata robot motor and drive ratings Table B.6 Accelerometer Type Kistler 8694M1 specifications Table B.7 Accelerometer amplifier Type Kistler 5134 specifications

17 xiii LIST OF ABBREVIATIONS DC Direct Current DSP Digital Signal Processing emf Electro-Magnetic Force FFT Fast Fourier Transform IAE Integral of the Absolute Error PC - Personal Computer PCD Pitch Circle Diameter PD Proportional Derivative PDTV Position Dependent Torque Variation PID Proportional Integral Derivative PWMH Pulse Width Modulated Sampled Data Hold SCARA Selective Compliant Assembly Robot Arm Sgn Sign function Stiction Static Friction TCP Tool Centre Point

18 xiv ABSTRACT At low velocities, friction is highly non linear and difficult to control. In practical mechanisms, friction may also be position dependent and highly variable. This can lead to tracking errors, limit cycles, and a phenomenon referred to as stick-slip, when a periodic cycle of alternating motion and rest, limits the mechanism s velocity and position accuracy. Impulse control is a friction compensator that does not require an accurate friction model. It achieves precise motion of a servomechanism by applying small impacts which overcome static friction with a controlled breakaway. The size of the impact and its duration determine how much the mechanism moves. By controlling the pulse, the positional accuracy of the mechanism can be improved. The work presented in this thesis results in new impulse controllers which: 1) improve the precision of a servomechanism without mechanical modification for the tasks of position pointing and low speed position tracking; 2) eliminate phenomena such as stick-slip, quadrant glitch, and limit cycling; 3) minimise system vibration and low speed position tracking ripple. The new controllers are tested by simulations, and experimentally verified on two different mechanical systems. One of these is a test bed built specifically for friction control experiments, and the other is a SCARA robot manipulator.

19 xv ACKNOWLEDGEMENTS This thesis was carried out as part of an Engineering Manufacturing funded project within the School of Electrical, Computer and Telecommunications Engineering at the University of Wollongong. I would like to thank these organisations for their support which made this thesis possible. I would like to give a special thanks to my supervisor Professor Chris Cook and my cosupervisors Dr. Zheng Li and Dr. Gursel Alici for their assistance and guidance throughout my candidature. I would also like to thank Assoc. Prof. Friso de Boer who was initially my supervisor and was a great help and mentor in the initial stages of my thesis. I would like to thank my fellow researchers, Dr. Marta Fernandes, Mr. Laurence Bate, Mr. Jeff Moscrop and Mr. Simon Webb as colleagues working together in the same area in which ideas could be discussed clearly, openly and freely. A special thank you is reserved for Dr. John Simpson, who not only helped me with all engineering aspects of my thesis, but as a friend always took the time to answer my questions and offer support. I would like to thank all the general staff of the school who helped me reach my goal. A special thanks to Mr. Brian Webb for the machining of the Friction Test Bed. Finally, a very warm and special thanks to my dear wife Leesa, and daughters Tyla and Skye, who never once complained about the inconvenience and self indulgent undertaking of my Ph.D. thesis over so many years. Their sacrifice was undoubtedly greater than mine.

20 1 CHAPTER 1 INTRODUCTION 1.1 Background Friction exists in all mechanical systems and is highly non-linear at very low velocities. For the control engineer, this non-linearity affects the overall performance of servomechanisms by introducing typical errors such as position regulation and tracking lags [1]. In order to stay competitive, precision machine tool manufacturers are continually searching for low cost ways to improve the positional accuracy of their machines. Precise position and velocity control is frequently the key to the improvement of product quality and production efficiency [2]. In many cases, the non-linear friction characteristics of a mechanism can be reduced or eliminated by modifying the mechanical design of the plant itself; however, this approach is not always cost effective and in many instances not practical or possible. Alternatively, another way to improve the precision of a machine is to develop better computer controllers, whereby the power electronics used to drive the servomechanism are modified in order to counteract the friction phenomena and improve its overall

21 2 performance. A major advantage of this method is that friction compensation can be easily adapted to an existing mechanical plant with no mechanical modification. This thesis examines impulse control to overcome non-linear friction at low velocities. The aim of this thesis is to improve the precision of a mechanism by developing impulse controllers in order to obtain a notable improvement in the precision of the machine. For a practical demonstration of this improvement, two servomechanisms are used. One is a friction test bed specifically constructed for friction compensation research, and the other is an industrial SCARA robot. Classical continuous control schemes, such as PID control, have been widely accepted in industrial linear control because of their robustness to parameter uncertainty, ease of use and familiarity to the control engineer. In linear control systems, the integrator of a classical PID controller ensures precise tracking of a step or ramp trajectory in the steady state. However, in the presence of friction, the integrator term inevitably causes the so-called stick-slip motion in servomechanisms [3-6] and inherently forces the mechanism to limit cycle when tracking or positioning around zero velocity [3, 4, 7]. Figure 1.1 A simple set-up for stick-slip motion [1] Stick-slip can be tested using a spring-mass system shown in Figure 1.1. A force applied to the unit mass will counteract the spring force and a small pre-sliding

22 3 displacement will result. When the force exceeds the static friction, the unit mass begins to slide and the friction force reduces dramatically causing the spring to relax. The mass slows and the friction force rises again causing the unit mass to stick. The phenomenon then repeats itself. In Figure 1.2 stick slip motion occurs and is represented by the classic staircase position response. Figure 1.2 A simulation of stick-slip motion [1]. For a PID controller, the Integrator (I) works against the spring and triggers the stickslip motion. To account for this limitation, several researchers [2, 8-10] have modified the traditional PID control scheme to include Stiff PD, Integral Dead Band, Dither and Impulse Control to handle the non-linear effects of friction. Impulse control is a non-model based way to reduce the effects of the stick slip at low velocities. By supplying an impulse of controlled energy to the mechanism, the static friction can be momentarily overcome [11]. Typically, the power electronics supply a controlled pulse of energy to the servomechanism in order to overcome the restricting static friction. By applying a short pulse of sufficient force, plastic deformation occurs between the asperities of mating surfaces resulting in a permanently controlled

23 4 movement. If the initial pulse causes insufficient movement, the impulsive controller applies successive pulses until the position error is reduced to the smallest available incremental movement. The pulse shape also affects the smallest controlled motion of a mechanism. A number of investigators have devised impulsive controllers which achieve precise motion in the presence of friction by controlling the height or width of a pulse [2, 12-17]. The detail of how and why the pulse shape is changed is discussed further in Section 1.4. In this thesis, new impulsive controllers are developed for frictions having elastic and dynamic behaviour. It is shown that if the material has spring like characteristics, an impulse higher than the static friction is not enough to guarantee plastic displacement. In other words, the machine will move back to the pre-pulse position after the impulse is applied. The experimental results also show that the shape of the impulse can affect the controllability of a servomechanism. By modifying the pulse shape, the overall precision of the controller can be improved. This allows a smaller minimum movement of the servomechanism to be achieved without modification to the mechanical plant. 1.2 Friction Models Friction models have several purposes. For model based controllers, the friction model is used to predict the likely behaviour of the system friction so that compensation can be used. For this type of compensation, the model may be limited by computational complexities; particularly near zero velocity where friction is highly non-linear. System models are also used to provide a physical insight into the friction dynamics and are a valuable tool for controller design or simulation.

24 5 In this thesis, various friction models have been used to gain a better understanding of the friction dynamics. This knowledge is used for the design of a new impulse controller. The friction models have also been used to assist the design of prototype controllers using simulation Classical Friction Models Leonardo Da Vinci [18] was the first to model friction in In his model, the friction force is proportional to the load, opposes the motion of the moving surface and is independent of contact area. Da Vinci s work was rediscovered by Amontons (1699) [19] and then further developed by Coulomb (1785) [20] with the main idea being that friction was dependent only on the sign of the velocity v and opposes motion. This was represented by the first of the static friction models: F = F sgn(v) (1.1) C This description was termed Coulomb friction and can be represented graphically by Figure 1.3(a). The Coulomb friction model does not specify the friction force at zero velocity and this force may be taken as either zero or any value between F C and F C [1]. In the 19 th Century, Reynolds (1886) [21] developed the theory of hydrodynamics leading to expressions for the friction force relating to the viscous effects of lubricants. The term viscous friction is used for this friction force component, which is normally described as: F = F v (1.2) v and graphically represented together with Coulomb friction in Figure 1.3b.

25 6 Figure 1.3 Memoryless friction models. The friction force is given by a memoryless function except possibly at zero velocity. Figure a) shows Coulomb friction and Figure b) Coulomb plus viscous friction. Stiction plus Coulomb friction are shown in Figure c), and Figure d) shows the Stribeck friction [1]. In 1833, the idea of stiction, which is short for static friction, was introduced by Morin [22] (Figure 1.3(c)). Here the friction force is that of a tangential constraint [23], and at rest is higher than the Coulomb level. The tangential stiffness k t, is a function of asperity geometry, material elasticity and applied normal force [24]. To the first approximation, it is actually the breakaway displacement that is constant; and the stiffness is then given by: F b k t = (1.3) xb where F b is the breakaway force and x b is the maximum deformation of the asperities before breakaway [5]. A model, which includes static friction Fs and an applied force Fe, can be described by: Fe F = Fs sgn( Fe) Fc sgn( v) + Fv v if if if v v v = 0 = 0 0 and and F F e e < F F s s (1.4)

26 7 So far, these classical models include components that are either linear in velocity or constant. Stribeck (1902) [25] extended the classical models by showing that if accurate measurements of friction for steady velocity are taken, then at low velocities a negative viscous friction will appear and vary continuously with increasing velocity. This phenomenon is termed the Stribeck effect and the resulting friction force at the low velocities above the Coulomb friction is termed Stribeck friction [1] and is represented in the curve of Figure 1.3(d). This effect is most prevalent when a lubricating additive has been used between the mating surfaces (see Figure 1.4). Lubrication is necessary in most machines to separate the mating surfaces to reduce friction and wear. For the control engineer, the use of lubricant means that the dynamics of the mechanism will change and a new friction model is needed. Figure 1.4 Full fluid lubrication, regime IV of the Stribeck curve [5]. Armstrong-Hélouvry et al [5], applied four regimes of lubrication to the generalised Stribeck curve as shown in Figure 1.5. Here, friction is a function of velocity. Each of the four regimes is described as follows: Regime I: Static Friction and Pre-sliding Displacement, is the result of contact at asperity junctions whereby the junctions have two important behaviours; they deform elastically like springs, giving rise to pre-sliding displacement; and both the boundary

27 8 Figure 1.5 The generalized Stribeck curve, showing friction as a dynamic function of velocity for low velocities [5]. lubrication film and the asperities deform plastically, giving rise to rising static friction. Here there is a displacement (pre-sliding displacement) which is approximately a linear force and spring-like up to a critical force at which breakaway occurs [5]. At this point, the springs are broken and the surfaces plastically move. A number of authors [6, 24, 26-29] have measured pre-sliding displacements of 2-5 microns in steel. These displacements, however, may be much larger elsewhere in the mechanism; for example in robots where the arm itself acts as a lever and small movements at the motor result in large displacements at the Tool Centre Point (TCP). A schematic representation of the elastic nature is represented in Figure 1.6(a). Here the pre-sliding displacement is spring like as in Figure 1.6(b), and finally breaks, as in Figure 1.6(c). The point at which breakaway occurs is denoted Static Friction, Fs, and is easily identifiable in Figure 1.5 as the maximum friction force before permanent large scale movement occurs. Regime II: Boundary Lubrication. This is a region of very slow velocity. Here there is insufficient movement to adequately draw lubricant between the mating surfaces and shearing only occurs at the boundary layer. The boundary layer is a solid layer of oxide

28 9 Figure 1.6 Asperity contact behaving like springs [15]. which forms on the surface asperities of steel and other engineering materials [5]. If a lubricant is used, then additives within the bulk oil react with the surfaces to form the boundary layer. The shear strength of the boundary layer varies, but lubricants are formulated to have shear strength less than that of the asperity junctions. This regime is still referred to as having solid to solid contact. Regime III: Partial Fluid Lubrication. In this region, fluid lubricant is drawn into the load bearing surfaces and a viscous film begins to form. A negative viscous friction curve (Stribeck effect) is observed and it is this region which is most likely to be unstable. Regime IV: Full Fluid Lubrication. The contact zones are completely separated by a thick film of lubricant. This is the region of viscous friction, F v which increases linearly

29 10 with velocity as shown in Figure 1.5. A model that allows for these dynamic influences is given in [5] and [1] and is described by the parameterised equation: F = sgn( v) F + ( F F ) e + F δ v/ v s c s c v v (1.5) where δ and vs are empirically determined parameters. This model is shown graphically in Figure 1.3(d) Dynamic Friction Modelling On a broad scale, the properties of friction are both well understood and documented. Armstrong-Hélouvry et al [5] have surveyed some of the collective understandings of how friction can be modelled to include the complexities of mating surfaces at a microscopic level. Most macroscopic friction behaviour can be captured with the classical models. However, the models described so far are applicable for a static case only and neglect friction parameters that are likely to change with respect to time and rate of force application. As experiments historically become more sensitive [6, 23, 30, 31], it has become apparent that at a microscopic level, there are dynamics within the surface processes which influence the friction. Rabinowicz [32] devised an experiment that uncovered two temporal phenomena in the stick-slip process. The temporal phenomena are [5]: (1) the level of static friction is a function of the time a junction spends in the stuck condition, i.e. dwell time (rising static friction); and (2) a time delay or phase lag between a change in velocity and the corresponding change in friction (frictional memory or frictional lag). Figures 1.7 and 1.8 help illustrate the stages of the stick slip cycle and how these stages affect rising static friction and frictional memory. Each figure reveals a dependence of

30 11 Figure 1.7 Spring force profile during stick-slip motion at two velocities; spring force decreases when velocity increases [15]. Figure 1.8 Static Friction (breakaway force) as a function of dwell time, schematic; with stick slip cycle shown. Dwell time is the time in static friction, shown as T 2 in Figure 1.7 [15]. static friction on elapsed time as well as the rate of force application. In terms of time dependency, the general rule is that the longer the system has been at rest in the stuck condition, the higher the level of the static friction (up to a maximum). In a similar

31 12 study using a PUMA robot by Armstrong-Hélouvry [29], it was found that if the system was at a standstill for more than a few minutes, the level of static friction would begin to rise. It was suggested that this is caused by the lubricant being slowly squeezed out from between the two mating surfaces. Similarly, in investigations of stick-slip oscillations, the length of time between successive breakaway events was thought to be affecting static friction in the same way as above. Hence, it was suggested that the observed reduction in stick-slip under increasing velocity was due to the reduction in time between sticks causing a corresponding reduction in static friction [5]. In contrast to these findings, experiments by Richardson and Nolle [33] and Johannes et al [34] have shown that the force application rate seems to be a more dominant factor in the cause of reduced static friction under stick-slip conditions and increasing velocity. When considering the second of the temporal phenomena, frictional memory (or frictional lag) has been observed by a number of researchers [4, 23, 29, 35, 36] adding to the experiments of Rabinowicz [32]. Here, a time or phase lag between the observed friction and change in velocity can been found. The lag is typically in the order of milliseconds to seconds and its impact on stick-slip motion may be substantial. Frictional memory is suggested to be the result of the state in the frictional contact, where there are dynamics associated with the viscous friction properties of the lubricant [5]. To account for the lack of theoretically motivated models for the components of dynamic friction, a number of empirical models have emerged [26, 29, 35, 37]. Olsson [1] provides a good comparison of each of the models and their deficiencies. The Dahl

32 13 model [26] captures most of the variable breakaway friction phenomena by describing the spring like behaviour during stiction, but fails to include the Stribeck effect. Armstrong-Hélouvry [29] on the other hand, proposes a seven parameter model that is in part, one model for stiction and another for sliding friction. For computational purposes, this model is efficient for simulations, but is limited by the need to have a mechanism that determines when to switch between the sub-models. Olsson [1] describes the necessity for a model that captures all of the important dynamic phenomena without sacrificing simulation efficiency. The result is a new dynamic model described next in Subsection Modified Bristle Model a new model Canudas de Wit et al [37] contribute to dynamic modelling by presenting a new model that more accurately captures the dynamic phenomena of rising static friction [32], frictional lag [32], varying breakaway force [33, 34], dwell time[38], pre-sliding displacement [24, 26, 27] and Stribeck effect [1]. The friction interface is thought of as a contact between elastic bristles as shown in Figure 1.9. When a tangential force is applied, the bristles deflect like springs which give rise to the friction force [37]. This can be seen in Figure 1.10(a). If the effective applied force F e exceeds the spring force of the bristles, some of the bristles will be forced to slip, and permanent plastic movement occurs between each of the mating surfaces. The set of equations governing the dynamics of the bristles are given by [1]: dz dt v v g = (1.6) () v z ( vv) 2 ( C s C ) 1 s ( ) ( ) g v = F + F F e (1.7) σ 0

33 14 F = 0z + σ 1 dz dt () v + F v σ (1.8) v ( v) σ σ e 1 1 ( vv d )2 = (1.9) where v is the relative velocity between the two surfaces and z is the average deflection of the bristles, σ 0 is the bristle stiffness and σ 1 is the bristle damping. The term v s is used to introduce the velocity at which the Stribeck effect begins while the parameter v d determines the velocity interval around zero for which the velocity damping is active. Figure 1.9 The friction interface between two surface is thought of as a contact between bristles. For simplicity the bristles on the lower part are shown as being rigid [1]. a) b) Figure 1.10 Bristle model; Figure a) shows the deflection of a single bristle. Figure b) shows the resulting static friction model for a single instance in time. Fs is the average static friction while Fc is the average Coulomb friction. Figure 1.10(b) shows the general static friction curve as a result of the system model. In reality, the friction model and friction curve are dynamic and Figure 1.10(b) shows a special case.

34 15 For very low velocities, the viscous friction Fv is negligible, but is included for model completeness. F s, F C, and F v are all estimated experimentally by subjecting a real mechanical system to a series of step torque inputs. The parameters σ 0, σ 1, v s and v d are also determined by measuring the steady state friction force when the velocity is held constant [37] Position Dependent Friction The friction models discussed so far have been developed for mating surfaces having simple geometry. In reality, the mating surfaces of most surfaces are more complex and include mechanisms with numerous sliding and rolling parts where simple models cannot fully describe the complexity of the friction phenomena [39]. Armstrong-Helouvry [29], Murkerjee et al [40] and Canudas de Wit et al [41] show this is particularly true for machines having transmissions with spatial in-homogeneities. Gears are a common example where friction varies as function of the position. The geometry of the gear and its position determines the resulting variation in friction. Similarly, for linear motion mechanisms, the friction measured at one position of machine travel may be vastly different from that measured at another. Variations in surface geometry, lubrication, wear, machine stiffness and temperature are some of the causes of position dependent friction. Both Armstrong-Helouvry [29] and Popovic et al [39] have shown that friction variations can be consistent and repeatable if measured under controlled conditions. It was found that this was the case for both static and sliding friction. For static friction, a series of Breakaway experiments were used to measure the static friction at numerous positions. The experiments used a binning technique to capture sets of breakaway friction data at various intervals along the travel of the mechanism.

35 16 The size of the interval used to collect sets of data, is referred to as the bin interval. A small bin interval with a large sample size provides a more accurate data set for the region of interest and thus requires an experiment that takes considerable time to bin the data. To determine the sliding friction values, the mechanism was run at steady state velocities and the motor torque recorded [39]. By analysing the position domain, as opposed to the time domain, Popovic and Goldenberg [39] were able to use Fourier transforms to determine the spectral components of the varying static friction. Their experiments, using a PUMA 560 robot, showed that, in this case, the largest spectral components were attributed to gear meshing and the sliding brushes of the commutator. In addition to position dependent friction variation, it has also been observed experimentally on a number of occasions [29, 40, 41], that there are different friction values for opposing directions of motion. This variance is not only restricted to static friction, but has also been observed for Coulomb and viscous friction. Armstrong- Helouvry et al note this may be due to the anisotropies in material geometry [5]. 1.3 Friction Compensation To compensate for friction, it must be understood what friction regime the mechanism is operating within and what type of control is required. Armstrong-Helourvry et al [5] outlines five specific tasks a servomechanism may be required to perform. Task I, the regulator, is encountered with positioning and pointing systems. Typical examples include, telescopes, disk drives, antennas, machine tools and robots [5]. In a positioning task, the machine will be expected to reach zero velocity and so the system spends most of its time near or within the stiction regime. For PID controllers, the

36 17 integral action will inevitably cause stick-slip oscillation about the reference position and is referred to as hunting [5]. This task is extensively discussed in [12, 42-45]. Task II, Tracking with Velocity Reversals, is closely related to Task I where a position trajectory is to be followed over a time interval. A change in motion direction (velocity reversal) ensures the machine comes to rest before continuing. Due to static friction, the motion through zero velocity is not smooth and for two axis machines, results in quadrant glitch. Here, one of the axes remains momentarily in the stuck position while the other continues at higher velocities. In many papers, a sinusoidal position reference is studied to ensure there are varying velocities in the stick-slip region and two definite velocity reversals [41, 44, 46]. Task III, Tracking at Low Velocities, is the task most associated with stick-slip. This task differs to Task II in that the mechanism is tracking position in a single direction and perhaps at constant velocity. This task is used to study if stick-slip limit cycling exists for a mechanism and for what system parameters it will be stable [5]. Armstrong- Helouvry et al [5] cites controller design for tracking at low velocities being studied by Gilbert and Winston [47], Walrath [46] and Kubo [48]. Task IV, Tracking at High Velocities, occurs in machine tools where the process demands high velocity work. This is typical in machine tools, robots and tracking mechanisms and is usually required in addition to Tasks I to IV. This task is very much different from the previous tasks because viscous friction is the primary resisting friction and stability is not usually a problem. Instead, tracking error is a typical problem and the servomechanism is typically tuned for higher velocities and so performs poorly for lower velocities. Similarly, a controller tuned for low velocities will be highly under-damped and is likely to present stability problems when tracking at higher velocities.

37 18 There are several friction compensation techniques available, but most of these fall into one of the following three categories: 1) Problem Avoidance, 2) Non-Model Based Control, and 3) Model Based Control. A categorisation of some common compensation techniques can be seen in Figure Problem Avoidance Friction problem avoidance is not truly a compensation technique. In this method, one attempts to replace the given system with one which is easier to control [5]. Problem Friction Compensation 1. Problem Avoidance 2. Non - Model Based Control 3. Model Based Control Lubrication Machine Design Stiff PD or high gain LQR/Hh etc. Learning Control Adaptive Control Non - Adaptive Reduced Loading Velocity Joint Torque Intergral Limiting Control Control Fuzzy Logic Neural Networks Dual Mode Control Constant Movement Control Dither Control Impulse Control Figure 1.11 Friction compensation using: 1) Problem avoidance, 2) Non-model based control, and 3) Model based control.

38 19 avoidance is usually the first strategy employed to defeat friction problems because it directly modifies the physical friction properties. Armstrong-Helouvery et al [5] cites examples [49-51] where stick-slip has been reduced or eliminated by decreasing the mass, increasing the damping or increasing the stiffness of a mechanical system. Lubrication choice may be an effective way to reduce some of the non-linear friction phenomena. For example, the use of polar hydrocarbons, in the form of way oils, has been reported to eliminate stick-slip in machine slide-ways [52, 53]. However, improving the characteristics of friction at lower velocities through lubrication choice can have adverse effects at higher velocities. In contrast, for improved controllability machine modification does not necessarily always aim to reduce the overall level of friction in its entirety. Armstrong-Helouvry et al [5] cites an example where friction materials, such as Rulon, are used to provide a high Coulomb friction with a reduced excess of static friction over Coulomb friction The friction materials consume substantial energy, but the reduced static friction improves the overall controllability of the mechanical system. Bearing selection can have significant effect on stick-slip friction. For instance, roller bearings generate much less friction than slide-way bearings at comparable loads and speeds. Most of the friction generated by roller bearings is through sliding, where either the rolling elements elastically deform and rub on the raceway, or there is rubbing of the rollers and cage or seals [54]. Unfortunately, in high loading applications, such as machine tools and linear drives, the use of way bearings may be the only option.

39 20 Modifying the machine stiffness is another problem avoidance technique. Very high machine stiffnesses are desired when designing high accuracy apparatus. This may necessitate the removal of compliant transmission elements such as gears and belts, or may require the stiffening of shafts and slide-ways. Dieterich, (1979) [55], Ruina, (1980) [56], and Polycarpu and Soom, (1992) [23] all use very stiff experimental equipment for studying friction. Finally, by reducing the inertia of a servomechanism, it is possible to create a stabilising effect on stick-slip systems. In some cases though, the inertia reduction can cause controllability problems. This is particularly the case where loading forces are significantly large and applied at destabilizing frequencies [5]. As mentioned in Section 1.1, it is not always practical to modify the mechanical design of a servomechanism. Even if machine redesign is possible, the effects of friction still may not be reduced to a controllable level. In these instances, other friction compensation techniques will need to be employed as discussed in the following sections Non-Model Based Friction Compensation Non Model Based Control, is a friction compensation technique where the controller does not predict friction (feed-forward compensation) or implicitly model friction in its control architecture. Many of these controllers can be categorised as High Gain controllers where the system gains are maximised for a stabilizing effect. Here, stickslip can be eliminated by either high derivative (velocity) feedback or high proportional (position) feedback and is most successful for machines which have high rigidity and low encoder noise [5].

40 21 As mentioned in Section 1.1, the Integral action (I) of a PID controller is introduced to minimize steady-state errors at the expense of limit cycling at low or zero velocities. One method used to overcome this problem is to impose an Integral dead-band where the input to the Integral block is limited. In addition to this, Integral reset is used. Here, the integral windup from prior motion can inhibit breakaway [5] at velocity reversals. To prevent this, the integrator is reset when the reference velocity changes sign. Suzuki and Tomizuka, (1991) [57] have shown this can lead to significant tracking errors at higher velocities. As a consequence, they propose a controller that applies a pulse to overcome the static friction. Dither is a high frequency open loop signal injected into the mechanical system to modify its behaviour. The dither signal is used either tangentially (machine axis) or normal (orthogonal to the machine axis) as shown in Figure Tangential dither can be excited using the servomechanism response while normal dither is typically induced Figure 1.12 Direction and effect of Dither [5].

41 22 by an additional mechanism. The effects of both are considerably different. Tangential dither has the effect of averaging or smoothing the non-linear friction. For example, a system is discontinuous and described by the relationship yt () sgn( ut ()) = (1.10) However, when a tangential dither of amplitude α and frequency ω is added to the input, the averaged output becomes: t t 2πω ( α ω ) (1.11) yt () = sgn ut () + sin( t ) dt Normal excitation, on the other hand, aims to reduce the friction coefficient. Godfrey (1967) [9] reports a reduction of the coefficient of friction from 0.15 to 0.06 in lubricated steel contact with the addition of 1 khz vibrations. Finally, Impulse Control is a non-model based friction controller and one that this thesis is concerned with. Similar to dither, it is applied to the controller output (PID output) such that the resultant torque command will, at some point, be greater than the static friction value to ensure breakaway occurs. Besides this similarity, there are many differences. The most important of these include: 1) Impulse control relies on the position error signal to calculate the impulse output and in this sense, is closed loop [15] In contrast, dither is open loop and constant in output. 2) The energy resulting from the impulse shape alone is sufficient to cause the mechanism to follow the desired motion [15] The motion from dither only facilitates larger movement from an additional controller output. 3) The motion from impulse control can be brought to a complete stop - Motion resulting from dither does not completely stop [12].

42 23 4) Bearing wear is accelerated at a greater rate when using dither [12]. 5) Power is conserved when using impulse control by applying only what is needed - power is wasted using dither [12]. For both impulse and dither control, the vibrations caused by the inherent nature of the controller output may result in undesirable machine marks for fine machining processes (e.g. grinding) at ultra low speed. So far, a brief summary of impulse control has been given. In Section 1.4, a more complete description of impulse control is included Model Based Friction Compensation For completeness, this section briefly discusses the final class of friction compensation called Model Based Friction Compensation. Figure 1.13 Model Based Friction Compensation [5]. If a model of friction accurately describes the mechanism in question, it is possible to compensate for friction by predicting it and applying an equal and opposite force/torque

43 24 command. This is referred to as feed-forward compensation and the model known as a friction predictor as seen in Figure For a feed-forward force/torque command to be effective the source of the predominant friction is assumed to be stiffly coupled to the motor. This is typically the case, since the predominant friction source is usually the motor or transmission units [5]. A number of studies [15, 29, 41, 44, 46-48, 57-59] have successfully used feed-forward compensation to improve the performance of servomechanism control. The success of the Model Based friction compensation depends on the accuracy of the model used. Subsequently, both the task (see Section 1.3) and the friction parameters must be well understood so that the relevant model is used. This is of particular importance when considering systems at very low or zero velocities where a good dynamic friction model should be used. Armstrong-Helouvry et al [5] notes one of the major difficulties in applying friction compensation is the difficulty in modelling friction at very low velocities [5]. 1.4 Impulse Controllers As briefly discussed in Section1.1 and 1.3.1, impulse control is a non-model based friction compensator that negates the need for an accurate friction model. It achieves precise motion of a servomechanism having friction by applying small impacts that overcome stiction with a controlled breakaway. The size of the impact and the duration, determine how much the mechanism moves. The pulse shape affects the smallest controlled motion of a mechanism. A number of investigators have devised impulsive controllers which achieve precise motion in the

44 25 presence of friction by controlling the height or width of a pulse. Yang and Tomizuka [12] applied a standard rectangular shaped pulse. The height of the pulse f p is a force about 3 to 4 times greater than the static friction to guarantee movement. The pulse width t p is calculated from: d = bt 2 sgn( f ) (1.12) p p Where d is the required displacement and b is an arbitrary initial estimate. If an error remains, an adaptive algorithm calculates a new value of b based on the previous displacement. A new pulse is applied and the whole process is repeated until the error tends to zero. Alternatively, Popovic et al [13, 60, 61] developed a fuzzy logic pulse controller that determines both the optimum pulse amplitude and pulse width simultaneously using a set of membership functions. Popovic et al (2000) made an extensive study of the Figure 1.14 Experimentally determined displacement as a function of pulse width and pulse height [13].

45 26 mechanism s response to narrow open loop torque pulses and the results are shown in Figure Using this knowledge, the pulse height and duration is adaptively changed according to the mechanism s state. Hojjat and Higuchi [14] limited the pulse width to a fixed duration of 1 ms and vary the amplitude by applying a force about 10 times the static friction. They reliably achieved a remarkable 10 nm per impulse motion and speculate that repeatable 1 nm per impulse motions may be possible [62]. In all other cases, the role of the impulse controller is to initiate breakaway leading to transition of another controller which regulates macroscopic movements [5]. Armstrong (1988) [15] and Armstrong-Helouvry, (1991) [15, 29] use a calibrated lookup table of impulse magnitude and duration for an impulse controller designed for force control. The impulse controller applies pulses that are only 10-20% greater in magnitude than the static friction. The result is a controller capable of manipulating an object with crush strength 1/60 th the level of the static friction. In a survey of friction controllers by Armstrong-Hélouvry et al [5], it was commented that underlying the functioning of these impulsive controllers is the requirement for the mechanism to be in the stuck or stationary position before subsequent impulses are applied. Each small impacting pulse is followed by an open loop slide and by arriving at a complete stop for improved predictability of the following impulses. A number of authors [12, 14, 15] rely on the mechanism coming to a complete stop before consecutive impulses are applied. Li and Cook [2] however, have devised an

46 27 impulse control scheme that controls the width of the pulse by holding it for a determined duration. The Pulse Width Modulated Hold (PWMH) technique generates a pulse in each sampling period whose width is proportional to the error signal. The PWMH is introduced only to deal with the non-linear effects of friction. In all other cases, a normal discrete PID controller with a zero order hold is used to provide the driving force [2]. The advantage of this structure is that the system no longer needs to arrive at a complete stop before the following pulse is calculated and applied. Simpson et al [63] and Simpson [64] contributes to Li and Cook [2] by demonstrating how the impulse controller can be used for position tracking tasks. Here the sampling width of the impulse controller is a fraction of the discrete PID controller such that each proceeding pulse width is calculated based on a position error sampled at a higher rate. Simpson [64] has successfully shown a significant improvement in the circular position tracking of a high friction industrial SCARA robot (as used in this thesis) using this control scheme. 1.5 Discussion and Summary of the Literature Review The literature review above has shown that non-linear friction is a complex phenomenon that mechanisms need to overcome. For machines with high friction and particularly high static friction, the control of a servomechanism for precise pointing and position tracking is a non trivial task. Classical PID controllers are widely used for the control of machine tools in industry but the integral action of the controller will heighten limit cycling near or at zero velocities. This limit cycling creates the stick-slip phenomena which can be observed in

47 28 machines having a static breakaway friction (stiction) greater than the Coulomb friction [65]. A description of the various friction models gives an insight of the complexities that mating surfaces undergo during sliding at a microscopic level. A number of friction models have been presented which give a good analysis tool for understanding both the static and dynamic nature of friction at high, low, and very low velocities. The bristle model has been described in detail because of its ability to model the dynamic nature of friction at very low velocities and to capture such phenomena as rising static friction, frictional lag, varying breakaway force, dwell time, pre-sliding displacement and the Stribeck effect. Position dependent friction has been separately discussed because of the important role it plays in a mechanism s response. The varying nature of friction over the travel of a machine will influence the dynamics of the servomechanism and the performance of a controller used to control these dynamics. By measuring the position dependent friction variables through an average binning technique, an inventory of friction parameters can be recorded and used for the controller design. Three main branches of friction compensation for servomechanisms have been discussed. These include: 1) Problem Avoidance, 2) Non-Model Based Control, and 3) Model Based Control. A brief description of each branch is included for background to this thesis and some compensation techniques have been included as examples. This thesis is concerned with impulse control which falls under the classification of Non-

48 29 model based friction compensation. It does not rely on a friction model directly in the control scheme but instead uses the model to facilitate the controller design. Finally, the research contribution for various types of impulse controllers experimentally used to date, has been discussed. The discussion gives an understanding of the types of impulse control, and in Section 1.6, an analysis of the shortcomings of current controllers are further discussed. Section 1.7 outlines the strategy used in this thesis to overcome some of these shortcomings. 1.6 Problem Definition From the literature review above, it has been shown that friction is an inherent characteristic of mechanisms with mating surfaces. One of the main objectives of the control engineer is to optimise the process they are controlling. This includes, the optimisation of machine precision while maintaining stability, power efficiency, wear etc. Over the past two decades, advances in digital technology have made it possible to design superior control schemes using digital computing. Similarly, advances in the power electronics have allowed these designs to be easily used with a servomechanism. For the impulse control, this means the shape of the pulse can be accurately manipulated to suit the dominant friction and the task being performed. The literature review shows there has been limited study on the dynamic effects that shaping the pulse has on servomechanisms. So far, the shape of an individual pulse has mostly been restricted to a calculated constant height and constant width. The height is typically a value greater

49 30 than the average static friction to ensure breakaway occurs. In all of the cited work, the height is maintained constant for the full pulse width which results in a standard rectangular shaped pulse. Table 1.1 gives an overview of the current cited work on impulse control design and objectives of this thesis. Hojjat and Higuchi [14] appear to have the only experiment that uses very thin pulses to achieve extremely precise motion. Although these experiments use the standard rectangular pulse, Hojjat and Higuchi have experimented with very thin pulse widths and a pulse height much greater than the static friction. The result is very precise motion in the scale of nanometres. The literature however, does not provide a method of determining the minimum pulse width or what the minimum pulse limitation is. Table 1.1 Comparison of Impulse Control strategy with known experimentation. Strategy Yang, Suzuki and Tomizuka Hojjatt and Higuchi Armstrong, Armstrong -Helouvry Reference Popovic, Gorinevsky & Goldenberg Cook and Li Simpson This thesis Variable width Variable height Pointing task Position tracking task Continuous motion Micro and macroscopic movement Pulse shaping

50 31 The objective of this thesis is to use a standard dynamic model for very low velocities to determine the benefits of changing the shape of an individual pulse to match the dynamic friction characteristics. The most dominant of these characteristics at very low velocities include, pre-sliding displacement, varying breakaway and Stribeck effect. There will also be a physical restriction to the pulse shape generated because of machine dynamics, friction response, and a limitation on digital servo response. The impulse controller of Li and Cook [2] and Simpson et al [66] has some practical restrictions that prevent it from achieving optimum performance, particularly for very fine pointing and position tracking. However, their experiments show an improvement in machine precision when compared to a conventional continuous controller. This thesis builds upon the fundamentals of Li and Cook [2] to include pulse shaping in order to obtain an improvement in precision for any servomechanism. A further objective of this thesis is to improve the precision of this controller for very low velocities whilst improving the transition between impulse control and continuous control for higher velocities. Finally, an innovative approach to optimising the precision of an impulse controller is explored. The smallest limit cycle for a pointing application is determined and used to increase the pointing precision of any servomechanism. It is proposed that the mean difference of the limit cycle from the reference position will be offset or shifted so that the mean difference in error can be reduced.

51 Outline of the Thesis This chapter has given an introduction to this thesis and includes a discussion of the literature relevant to the area of friction, friction modelling and friction compensation. It also provides a problem formulation giving the objectives of this thesis. The second chapter describes in detail the experimental equipment used in this thesis. Two mechatronic systems have been used to evaluate the thesis objectives. The first system is a purpose built experimental friction test rig which allows both the friction characteristics and the motion controller parameters to be easily modified. The second system is a four axis industrial SCARA robot which is used to verify the results. At low velocities, both mechatronic systems have highly non linear friction characteristics. For each system the electrical and mechanical system parameters are characterised and are subsequently used in Chapters 3-6. Chapter 3 develops a simulation model of the two systems. The simulation uses a friction sub-model which assists the understanding of friction dynamics at very low velocities. This understanding is further used to design an impulse controller with variable pulse shape. The simulation is performed using Matlab s Simulink toolbox and the impulse controller is implemented using Simulink block code. The block code is further used in Chapters 4-6 with a Matlab xpc real time workshop digital signal processing board. All digital control of the two test rigs used in this thesis is done using the xpc hardware and software. Chapter 4 further develops the impulse controller design. This chapter begins with an analysis of the effect pulse width and pulse height has on the servomechanism

52 33 dynamics. Firstly, a variably adjusted pulse width is studied followed by a variably adjusted pulse height. The next section continues by introducing hybrid impulse control. Here, the motivation for combining a continuous controller with the impulse controller is discussed. This is followed by an implementation strategy and an evaluation of its performance. Chapter 4 continues with a series of experiments that determine the minimum pulse width that will guarantee permanent movement. Because the shape of the pulse is affected by the system s electrical circuit and mechanical response, a practical limit is placed on the amplitude of the pulse over very short durations and restricts the amount of energy that can be contained within a very thin pulse. Consequently, there exists a minimum pulse width that is necessary to guarantee plastic movement. This information is used to design a series of four pulse shapes suited to the optimum control of stiction for both pointing and position tracking. Finally, a new variant to the impulse controller is developed. The controller adjusts the pulse width to match the mean error of the minimum limit cycle. The motivation of this type of controller is discussed with an evaluation of its performance. In the fifth chapter, the friction test bed is used to evaluate the new impulse controller designs developed in Chapter 4. Pointing tasks and position tracking tasks are used to evaluate the controller s performance. A series of experiments are discussed and an evaluation and comparison of results made.

53 34 Chapter 6 repeats the experiments of Chapter 5 using the industrial SCARA type Hirata robot. The performance of the controllers are discussed and compared to results obtained from Chapter 5. Chapter 7 presents the thesis outcomes, conclusions and suggestions for further work.

54 35 CHAPTER 2 CHARACTERISATION OF EXPERIMENTAL EQUIPMENT 2.1 Introduction The low speed and absolute position control schemes developed in this thesis were implemented and tested on two real servo mechanisms; the first being a friction test bed designed specifically for this thesis and the second, an industrial four axis Hirata robot. The purpose of the friction test bed is to replicate in a single mechanism an adjustable range of non linear friction parameters typically found in machine tools. To focus the research on friction compensation, the design of the test bed eliminates other types of non-linear torque disturbances which may be associated with some machine tool drive trains. For comparative purposes, the Hirata robot is used to validate the experimental work obtained from the friction test bed. Both mechanisms exhibit the characteristics of non-linear friction at low velocities. This chapter describes the design, setup and identification of both systems electrical and mechanical parameters that are used in Chapters 3 and 4. The friction test bed is described first, followed by the Hirata robot.

55 36 In Section 2.2, an overview of the friction test bed experimental setup is discussed including the selection criteria for a DC brushless direct drive servo motor, motor drive, support bearings, and the design methodology of the friction mechanism itself. In Subsection 2.2.2, the controller interface is discussed. MATLAB s xpc target oriented server was used to provide digital control for both the friction test bed and Hirata robot s servomechanism drives. Subsections and 2.2.5, describe how the electrical and mechanical system parameters are experimentally determined for the friction test bed. These include the measurement of the machine s electrical and mechanical time constants, inertia, mass, Coulomb, viscous, static friction components, and pre-sliding displacement constants. These are used in the force estimation and impulse controller design in Chapter 3, and simulations in Chapters 4 and 5. In Section 2.3 the Hirata robot is discussed. The mechanical layout of the robot is described as well as the controller and drive system. The Hirata robot s electrical and mechanical time constants, mass, Coulomb, viscous and static friction constants are calculated using the same least square method used for the friction test bed. 2.2 Friction Test Bed The friction test bed uses a single axis direct drive mechanism with no flexible coupling between the motor and load (friction mechanism). Eliminating a flexible coupling in the test bed design avoids the presence of other non-linear torque disturbances unrelated to friction, such as compliance, backlash, belt cogging, gear cogging etc. The electric

56 37 Tail Support Bearing Motor Shaft Drive to Load Coupling Direct Drive Motor Friction Mechanism Figure Three dimensional drawing of the friction test bed. motor used is a Kollmorgen D.061 direct drive rotary DC brushless type with a hollow shaft. It provides relatively high torque at low speed and its armature design reduces the effect of electric motor polling. The motor and drive constants and other additional data are given in Appendix B.1 Digital torque control of the motor is achieved using a 3 phase ServoStar 606-CR10 direct drive servo amplifier. The digital drive allows position, velocity or current control of the motor. The position data is obtained from an inbuilt incremental encoder with a precision of 256 encoder counts per revolution up to a maximum 524,288 encoder counts per revolution. Each pulse is counted by an external encoder Computer Board (National Instruments PCI-QUAD04).

57 38 Clamping Mechanism 100 mm diameter Lubricated Friction Disk Interchangeable Metallic Friction Blocks Linear Bearings for Caliper Equal Force Distribution Figure Exploded view of the friction mechanism Friction Mechanism Friction is artificially generated within the test bed using the friction mechanism shown in Figure 2.1. The design of the mechanism allows some friction parameters to be varied through mechanical adjustment. An exploded three dimensional view of the friction mechanism is shown in Figure 2.2. Clamping of a helical spring compresses a calliper onto a 100 mm diameter friction disc. Two static metallic friction blocks interface between the calliper and the drive shaft s rotating disc, and provide the frictional sliding surfaces. Each friction block has an effective contact surface area of 200 mm 2 and contacts the disc at a mean radius of 40 mm. The calliper is mounted on

58 39 two linear bearings orthogonal to the disc surface. These bearings allow the calliper to laterally float so that an equal distribution of force is applied to either side of the rotating disk. The friction surfaces are lubricated with a manually applied film of Mobil XHP 222 multi service grease. The non-drive end of the direct drive shaft is supported by a single spherical self aligning ball bearing SKF and the shaft material is mild steel. Engineering computer aided drawings of the friction test bed are included in Appendix C Control Interface and Digital Signal Processing The controller and digital signal processing are discussed in this section. MATLAB xpc target oriented server was used to control the friction test bed and Hirata robot. MATLAB xpc Target is run using a personal computer (PC) and allows Simulink and Stateflow models to be easily compiled and then executed in real time. This allows the programmer to rapidly prototype controllers using simple block code and allows a variety of control and signal processing strategies to be implemented. Each Simulink file is compiled into an executable program using MATLAB s xpc toolbox. The executable code is transferred to a secondary target PC where a real time kernel is run. The target PC also allows for data acquisition of the system states directly back into MATLAB software so that an easy comparison between the real system and simulation can be made offline. The general flow of communication and hardware for the friction test bed is shown in Figure 2.3. A dedicated Digital Signal Processor (DSP) provides a -10 Volt to 10 Volt analogue output to the motor s control drive. The DSP used is a National Instruments PCI-MIO- 16E4 and is connected to the xpc target PC via serial communication. For the

59 40 TCP/IP Serial Com. xpc DSP Host PC Target PC Encoder Counter DDR Drive Digital Friction Test Bed Mechanism Figure Communication flow diagram for the friction test bed. experiments within this thesis, the digital drive was set to current control mode, which means the output voltage from the DSP becomes a torque command to the servomotor Control Scheme The control scheme for the digital servo drive and motor are discussed in this section. For an armature inductance L a, and an armature resistance R a, the differential equation for the armature circuit relating the armature voltage V a and armature current I a is given in Laplace transform notation by: V a =( L a s + R a )I a (s) + e b (2.1)

60 41 where the back emf e b is given by the motor s electrical constant K e and the angular velocity of the motor shaft ω(s), is given by: e b (s) = K e ω(s) (2.2) In current control, the drive s output voltage is adjusted so that the armature current I a follows the reference current I ref as closely as possible. If the drive s current controller is given by G(s) then the armature voltage is given by: ( I ( s) I ( )) V ( s) = G( s) s (2.3) a The reference current I ref, is a function of a voltage gain K V so that the DSP analogue output voltage V DSP, is given by: ref a I ref (s) = K V V DSP (s) (2.4) The torque developed by the motor T m, is given by: where K m is the mechanical torque constant of the motor. T m (s) = K m I a (s) (2.5) The equation for torque equilibrium of a mechanical system can be expressed as a function of the torque supplied by the motor T m, and the torque dissipated by viscous friction ƒ v, inertia J and any other external torque disturbances T d. This is described by the following function: T ( Js + f ) ( s) T ( ) ( s) = ω s (2.6) m v m + The torque disturbance T d, amongst other disturbances, consists of Coulomb friction as in Section 1.2. Combining Eqs. 2.1 to 2.6 gives the electrical and mechanical block diagram as shown in Figure 2.4. d

61 42 K e V DSP (s) K V I (s) ref G(s) V (s) 1 I (s) T (s) 1 ω a a K (s) m m m L a S + R a s(js + f v ) T d (s) Figure 2.4 Combined electrical and mechanical system block diagram Electrical Circuit Response It is desirable for model simplicity to treat the electrical system response as a constant gain whilst the mechanical system acts essentially as a low pass filter. For this simplification to be justified, the electrical circuit response must be considerably faster than the mechanical system. The motor drive frequency response is provided in the drive specification manual as 16 khz and the motor armature resistance was give as R a La = 2.9 Ω and the inductance as L a = 6.8 mh. This gives a time constantτ = = 2.3ms. R The mechanical time constant is calculated later in Subsection and is listed in Table 2.1. This was more than 120 times slower than the electrical time constant and therefore, the motor response resembles a low pass filter. Consequently, I ref I a and the torque developed by the motor can be expressed as: T m (s) K m I ref (s) (2.7) The induced back emf e b is a function of the motor s angular velocity ω m and will also change considerably more slowly than the current control circuit. The electrical system can therefore be reduced to the following block diagram: a

62 43 V DSP (s) K V I ref (s) G(s) V a (s) 1 I a (s) L a s+r a Figure 2.5 Electrical circuit block diagram. Table 2.1 Electrical and mechanical time constants for the friction test bed. Electrical Circuit Mechanical System Time Constant (ms) 2.3 ms 295 ms Corner Frequency (Hz) 69 Hz 0.54 Furthermore, the complete electrical system behaves as a constant gain K E and can be represented by: K E KmK V = (2.8) The system block diagram can then be reduced to the following: V DSP (s) T m (s) 1 ω m (s) K E Js + f v T d (s) Figure 2.6 Reduced block diagram Mechanical System Parameters This section measures the friction test bed s inertia and friction parameters that are later used in Chapter 3 for the design of the impulse controller, and for system simulation in Chapters 4 and 5.

63 44 As discussed in Subsection 2.2.3, the drives input V DSP will give an armature current and associated motor torque such that: T m = K E V DSP (2.9) For the friction test bed, the electrical motor torque predominantly consists of: 1) the torque required to accelerate the system and overcome inertia, and 2) the torque required to overcome friction. Several experiments were conducted to determine the static friction, pre-sliding spring constant, inertia, and Coulomb and viscous friction constants Static Friction and Pre-Sliding Constant As discussed in Section 1.2, static friction F s is in the region of zero velocity and no significant motion occurs until the applied motor torque exceeds the static friction where T stic = F s.r, and r is the radius at which the sliding friction surfaces mate on the friction disk. This model is given by Olsson [1] and describes the resisting torque T by the function: T T (ϖ ) = Tm Tstic sgn ( T ) m if ω 0 if ω = 0 otherwise and T < T (2.10) m stic Static friction counteracts the motor torque and prevents the mechanism from moving. Once the motor torque exceeds the static friction, the friction force can no longer prevent motion, and breakaway occurs. On a smaller scale, pre-sliding motion can be observed as the mating surfaces elastically deform and a small displacement results. When removing the motor torque, the elastic movement is reversed and the mechanism resumes its original position. Armstrong-Helouvry et al [5] state there is a (pre-sliding) displacement which is an approximately linear function of the applied force, up to a

64 45 critical force, at which breakaway occurs. The pre-sliding spring constant k sys is adapted from Eq. 1.3 and is given by: k sys T θ m = if T m Tstic m (2.11) where θ m is the maximum deformation before full breakaway occurs. To measure the static friction, the friction test bed motor torque was increased by a linear ramp function T m (t) =C t, where t is time and C is a proportionality constant. The difference between pre-sliding velocity and break away velocity is significant and so the torque at which the static friction is overcome can be easily determined. The static friction was determined to be the measure of torque when the mechanism s velocity significantly increases. Figure 2.7 (a) and (b) gives a sample experiment measuring the pre-sliding displacement and breakaway torque T stic when initiated from one position for (a) (b) T stic T stic Breakaway Pre-Sliding Displacement Pre-Sliding Displacement Breakaway Time (Sample points) Time (Sample points) Figure 2.7 Pre-sliding displacement and breakaway friction for: (a) counter clock wise rotation; and (b) clock wise rotation.

65 46 both clockwise and counter clockwise directions for the friction test bed. In this instance, the motor torque required to overcome static friction is different for opposing directions. In Subsection 1.2.4, the binning technique used by Armstrong-Helouvry [29] and Popovic et al [39] was introduced to measure the position dependent friction. Similarly, this thesis used a series of position dependent experiments to measure the average static friction over the rotational range of the friction test bed. The range of position dependent friction covers one full rotation (2π radians) of the friction disk. By using the breakaway experiment shown in Figure 2.7, the mechanism was allowed to breakaway. At the same time, the static friction T stic and respective angular position are recorded. Upon breakaway detection, the torque was immediately removed before a consecutive ramp applied. The angular distance increment between static friction recordings was approximately 2.5e-3 radians and over a 24 hour experiment, the mechanism rotated almost 4 revolutions with 10,000 individual static friction sample points. Figure 2.8 (a) and (b) shows in polar plot form, the position dependent static friction T stic (θ) for counter clockwise and clockwise directions. Figure 2.8 (c) magnifies a segment of the friction disk and shows the repeatability of results for each of the four time dependent rotations. Both the counter clockwise and clockwise directions have differing static friction characteristics. In particular, the counter clockwise recordings have a distinctive static friction variation in the form of an oscillation with 12 maxima and minima per quadrant. This variation is attributed to the electric motor design. The mean pre-sliding

66 47 T stic for 4 revolutions 12 oscillations per quadrant Figure 2.8 Position dependent static friction for: (a) counter clockwise rotation; (b) clockwise rotation; and (c) magnified counter clockwise rotation. displacement θ m and mean static friction for both counter clockwise and clockwise directions are included in Table 2.2 of Subsection Estimation of Inertia, Viscous Friction and Coulomb Friction Using the conventional friction model described by Eq. 2.6, the disturbance torque T d is primarily a function of Coulomb friction f C. When combined with viscous friction f v and inertia J, the mechanical system can be described by the equation: KV = J ω + fω+ f sgnω (2.12) E DSP v C

67 48 where ω is the angular velocity of the motor shaft. Experiments were conducted to estimate the inertia, viscous and Coulomb friction using the Least Squares Estimation [67]. By applying a series of increasing torque steps to the servomechanism, the motor shaft was accelerated from zero to steady state velocity where the friction is approximately linear. The angular velocities of the shaft were recorded and are shown in Figure 2.9. The oscillations within each velocity response are the result of unavoidable friction variation over one full rotation inherent in any practical mechanism. Using vector notation, let θ be a vector containing the parameters J, f v, f c. Let φ be a vector of the input function ω, and Φ be a matrix with each row being the transpose vector φ T recorded at each sampling point from i = 0, 1, 2, 3 N. T is the torque output vector recorded at each sampling point. The vector notation can be shown as: θ J f, f c = v ω φ = ω, sgn( ω) T φ (0) T (0) Φ=, T = (2.13) φ T ( k) Tk ( ) Let ˆ T be the estimated torque vector such that ˆ T = Φ θ. Let the estimation error beε where ε = T Tˆ. The parameters of the vector θ can then be determined in such a way that 2 ε is minimal. The solution to the Least Squares problem is given by the standard solution [67]: T ( ) 1 θ = Φ Φ Φ T T (2.14)

68 49 Figure 2.9 a) Velocity response to step torque input for clockwise and counter clockwise motion, and b) resulting friction curve using Least Squares Method. Using Eq and 2.14, the parameters inertia J, Coulomb friction f C and viscous friction f v were calculated and included in Table 2.2 for both clockwise and counter clockwise directions. The response of the mechanical system was measured by applying a step torque input and observing the velocity response of the system. Figure 2.10 shows the step response for a step torque input of 3.8 Nm. The velocity response has been filtered with a zerophase forward and reverse digital filter. After filtering in the forward direction, the

69 Step Torque Input, 3.8 Nm - Filtered 50 Hz (zero-phase filter) Velocity (rad/s) % τ m Time (s) Figure 2.10 Mechanical system time constant. filtered sequence is then reversed and run back through the filter. The result has precisely zero phase distortion and magnitude modified by the square of the filter's magnitude response. Table 2.2 Friction test bed mechanical system parameters. Counter Clockwise Rotation Clockwise Rotation Inertia J (Nm.s 2 /rad) Coulomb Friction f c (Nm) Viscous Friction f v (Nm.s/rad) Mean Static Friction f s (Nm) Pre- Sliding Constant K sys (rad/nm) Mech. Time Constant τ m e e (ms) If the response is assumed to be that of a first order system, in one time constant the exponential response curve reaches 63.2 % of the final value [68] and is given by: ω ( ) = 1 e 1 = (2.15) τ m

70 51 The system s mechanical time constant τ m, is included in Table 2.2 and is compared to the electrical circuit time constant given in Subsection Hirata Robot The Hirata ARi350 SCARA (Selective Compliance Assembly Robot Arm) robot has four axes named A, B, Z and W. The main rotational axes are A-axis (radius 350mm) and B-axis (radius 300mm) and they control the end effector motion in the horizontal plane. The Z-axis moves the end effector in the vertical plane with a linear motion, while the W-axis is a revolute joint and rotates the end effector about the Z-axis. A drawing of the Hirata robot, and the dimensions of each axis and working area, are shown in Appendix B.2 and a photograph of the robot is shown in Figure For the purpose of this thesis, only the A and B axis of the Hirata robot are controlled. Both the A and B axes have a harmonic gearbox between the motor and robot arm. Their gear ratios are respectively 100:1 and 80:1. Each axis has characteristics of high non-linear friction which are measured in this next section. Figure 2.11 Photograph of the Hirata SCARA robot AR-i350.

71 Electrical Circuit Response and Control Scheme The original robot controller has been replaced with the same DSP system used for the friction test bed described in Section Both the drives of the A and B axes have been replaced with Baldor drives and their specifications are given in Appendix B.2. The electrical control scheme described in Section is also used and Eq.s 2.1 to 2.6 can be reapplied. The electrical circuit time constant τ is calculated in Appendix B.2.6 and is included in Table 2.3 below. Table 2.3 Electrical and mechanical time constants for the Hirata robot. A-axis Electrical B-axis Electrical A-axis Mechanical B-axis Mechanical Time Constant (ms) 1.2 ms 1.5 ms 260 ms 290 ms Corner Frequency (Hz) 130 Hz 105 Hz 0.61 Hz 0.54 Hz Mechanical System Parameters The mechanical system parameters for the Hirata robot were calculated using the same methods used for the test bed in Section The step torque response curves for the A-axis of the Hirata robot are included in Appendix B.2.3. The least squares method was used to calculate the inertia, viscous and Coulomb friction values for both axes and the results are included in Table 2.4 below. Table 2.4 Hirata robot mechanical system parameters. Inertia J Coulomb Friction f c Viscous Friction f v Mean Static Friction f s Pre- Sliding Constant K sys Mech. Time Constant τ m (Nm.s 2 /rad) (Nm) (Nm.s/rad) (Nm) (rad/nm) (ms) A-Axis e B-Axis e-4 290

72 Conclusions The experimental equipment used in this thesis has been described in this section. The control system, electrical circuit response and mechanical parameters have all been analysed for both the friction test bed and Hirata robot (A and B axes). For both systems, the electrical circuit response is considerably faster than the mechanical system and allows the output voltage from the DSP (input voltage to the servo drive) to be treated as a motor torque with a suitable scaling factor. The inertia, viscous friction and Coulomb friction were all estimated using a least squares method and is used in Chapter 3 for system modelling. The mean static friction and pre-sliding displacement were also measured from experimental results.

73 54 CHAPTER 3 MODELLING AND SIMULATION 3.1 Introduction A simulation platform is necessary to evaluate impulsive friction compensation schemes. This requires a thorough understanding of how friction can be effectively modelled and how it affects the system as a whole. Additional benefits of the modelling process include the development of a thorough knowledge of the test bed dynamics and a validation that the estimated parameters from Chapter 2 are accurate. Subsequently it will be possible to examine the effect of minor changes on the system, such as different impulse and PID parameters, as well as the effect of major changes in architecture, such as implementing alternative impulse controllers. Because the control schemes outlined in Chapter 2 use a MATLAB DSP platform, the simulation architecture developed in this chapter can be directly applied to the experimental system using MATLAB Simulink models via xpc. 3.2 System Modelling In order to build up a realistic model for the purpose of simulating the system, several stages are employed.

74 55 Stage One involves creating a model that replicates the plant dynamics for open loop control and closed loop control using a classical PID linear controller. As a result, experimentally derived gains will be used in the simulation to replicate actual performance. Stage Two includes a simulated model of the friction characteristics for a range of speeds. The purpose of this stage is to obtain simulated stick-slip behaviour around zero velocity. The model will be refined to include other nonlinear effects such as the Stribeck effect, pre-sliding displacement and varying static friction at very low or zero velocities. Stages one and two make up the base-line test bed simulation to which the impulse controller can be applied. Stage Three will involve the implementation of the impulse controller that are covered more fully in Chapter 4 and experimentally applied in Chapters 5 and 6. Figure 3.1 Simplified Simulink model of the friction test bed open loop Static Modelling of the Friction Test Bed This section describes the basic modelling of the friction test bed to cover stage one using MATLAB s Simulink. Figure 3.1 shows a preliminary model of the servomechanism with the system parameters assigned for one rotational direction only. The model uses the mean parameters characterised in Chapter 2 and the torque step responses have been simulated and compared in Figure 3.2. Each of the experimentally

75 56 Figure 3.2 Simulation model and measured results for step torque inputs. measured curves has a definite oscillation during the response. Each oscillation is the result of a variation in position dependent friction between the mating surfaces and is directly related to one full rotation of the friction disk. As expected, the period of the oscillation reduces as the velocity output increases. This simplified model is limited to higher velocities where viscous and Coulomb friction are most dominant. Stiction and the Stribeck effects are accounted for by replacing the standard Coulomb plus viscous friction block with a friction function that more accurately describes the friction behaviour at zero or near zero velocity. Figure 3.3 shows the friction function simulation model with two state inputs. Modifying the static friction model of Section 1.2.1, stiction F s and an applied force F e can be described by:

76 57 F Fe Fs sgn( Fe) = FC sgn( v) + FStribeck v FC sgn( v) + Fv v if if if if v v v v = 0 = and and and and F F e e < v v S v> v S F s F s (3.1) where v S (Stribeck velocity) represents the boundary of the nonlinear friction region for velocities near zero, v is the angular velocity, and F C and F v are the Coulomb friction and viscous friction respectively. For computational reasons, a mathematical dead-zone is used to define zero velocity. This model is designed to use both the clockwise and counter clockwise directional frictional parameters. Figure 3.4(a) shows the general stiction and Stribeck curve while Figure 3.4(b) shows the friction function s linearised model to include a definite Stribeck velocity v S inflection point. The parameter v S, or velocity threshold, is extrapolated from the Least Squares Error curve determined in Section Figure 3.3 Modified model with a friction function, which includes mean static friction, Stribeck effect, Coulomb and viscous friction for both clockwise and counter clockwise rotation.

77 58 Figure 3.4 Static friction model including stiction and Stribeck effect. a) A general friction model given by [1], and b) Simplified model for simulation purposes. Figure 3.3 also shows the system within closed loop control using a standard continuous PID controller. The PID controller s transfer function c(s) is given by [68]: us () KI cs () = KP KDs es () = + s + (3.2) where u(s) is the controller s output and e(s) is the controller s input or the error signal. K P, K I and K D are the proportional, integral and derivative terms [68]. For simulation purposes, Simulink contains a continuous PID controller block which, when compiled using xpc, is converted to a discrete PID controller. Following the approach given in [69], the PID controller can be expressed in the discrete Z domain using Z transforms in the form: uz ( ) a + a z + a z cz ( ) = = ez ( ) 1 z (3.3) The constants a 0, a 1 and a 2 are given by: a 0 KIτ s KD KIτ s 2KD KD = KP + +, a1 = KP +, and a2 = (3.4) 2 τ 2 τ τ s s s where τ s is the sample time.

78 59 The PID controller was tuned on the experimental test bed using the first of the Ziegler- Nichols tuning rules [68] which determine values for the proportional gain K P, integral gain K I and derivative gain K D based on the transient response characteristics of the Friction Test Bed. There are two methods called Ziegler-Nichols tuning rules. Both methods, aim to obtain 25% maximum overshoot in the step response [68]. This thesis is concerned with the study of servomechanisms used in precision machinery, and in particular, within machine tools where a minimum position overshoot is needed to meet process demands. Therefore, the Ziegler-Nichols rules were used as an initial estimate before iteratively adjusting the PID gains to achieve a maximum of 5 to 10% overshoot. Figure 3.5 shows the tuned PID controller unit step response for the Friction Test Bed. Using the combination of PID gains, inertia and friction parameters, the simulation model can be used to simulate stick-slip motion. The updated model includes several new additions, which modify the system dynamics and allow stick-slip to occur. Figure 3.5 Tuning the PID controller of the friction test bed using 5 to 10% maximum overshoot.

79 60 Armstrong-Hélouvry [16] suggests that stick-slip is governed by four phenomena: - The mass-spring (system control) dynamics of the system; - The non-linear, low velocity friction; - The interdependence of the static friction and dwell time; - The time lag between a change in system state and the corresponding change in friction. It is further noted by Armstrong-Hélouvry [16] that experimental work by Khatib and Burdick [70] observed unstable stick-slip motion even when no integral control term was used. The phenomenon is explained by negative viscous friction or Stribeck effect [16]. Figure 3.6 shows both the model and experimental results when applying a position reference that has a constant and very low speed. Stick-slip is observed and the resulting motion takes the classical staircase form. Both the model and experimental measurements replicate each other well. For the results of this figure the experimental encoder resolution was 1,144 counts per radian equating to an error uncertainty of 8.74e-4 radians compared to a vertical axis position resolution of 2e-3 radians. For the simulated result, each model parameter was measured to within at least three significant figures. The error uncertainty of the simulation is therefore within one thousandth of each graduation of the vertical position axis Dynamic Modelling of the Friction Test Bed The simulation model developed in Subsection uses a static friction sub-model which approximates friction at low velocities. For very low velocities, the friction characteristics can vary significantly and a model capturing the dynamic phenomena, such as pre-sliding displacement, rising static friction, varying breakaway force and

80 61 Figure 3.6 Classic staircase stick-slip motion using PID control. frictional lag, is required. This section simulates the new bristle model given by Canudas de Wit et al [71], as discussed in Section and it completes stage two of the modelling process. The results of this section are used extensively in Chapters 4 and 5 to help develop an improved impulse controller. Revisiting the equations given in Section [71] we have: dz dt v = v (3.5) g () v z ( vv) 2 ( C s C ) 1 s ( ) ( ) g v = F + F F e (3.6) σ 0 F = 0 z + σ 1 dz dt () v + F v σ (3.7) v

81 62 1 ( v v ) () v = σ e 2 d σ (3.8) 1 where v is the relative velocity between the two surfaces and z is the average deflection of the bristles (Figure 1.10). σ 0 is the bristle stiffness and σ 1 is the bristle damping. The term v s is used to introduce the velocity at which the Stribeck effect begins while the parameter v d determines the velocity interval around zero for which the velocity damping is active. F s is the average static friction while F C is the average Coulomb friction. The parameters σ 0, σ 1, v s and v d are determined by measuring the steady state friction force when the velocity is held constant [71]. Figure 3.7 Block Simulink model of the new Bristle dynamic model. To simulate the dynamic friction model, a sub-system block of the dynamic equations is designed. The subsystem block is partially expanded in Figure 3.7 and fully expanded in Appendix B.5. To ensure that the friction model is correct, a number of simulations were performed to replicate the work of Olsson [1] and Canudas de Wit et al [71]. Solid-to-solid friction without damping and viscous friction with the simplified standard model is examined

82 63 using the parameters set by [1] and shown in Table 3.1. The simplified version of the standard model has no velocity dependent damping and so Eq. 3.7 is replaced with: F = 0 z + σ 1 dz + Fvv dt σ (3.9) Figure 3.8 and Figure 3.9 compare the sticking behaviour for the simplified standard Canudas de Wit et al model with F v = 0 and σ 1 = 0. The mass has an initial velocity v(0) = 0.05 and z(0) = Both model outputs have a strong resemblance. Table 3.1 Default parameter values for the simplified standard model [1]. Figure 3.8 The sticking behaviour for the simplified standard model without damping (σ 1 =0). Figure 3.9 Olsson [1] simulation.

83 64 Using the standard model given by Canudas de Wit et al [71], the parameters are adjusted to account for the nonlinearity of friction around zero velocity. The sticking behaviour with velocity dependent dampening is introduced and Eq. 3.9 is replaced with Eq. 3.7 to include the damping coefficient σ 1 (v). The coefficient may increase with a decreasing velocity so that the decrease in dz/dt is compensated by an increase in σ 1 (v). Thus, the damping coefficient is modified according to Eq Using the standard model default parameters within Table 3.2, the friction force now continues to increase almost until the motion stops and then it drops sharply to zero [1]. This behaviour of motion and the friction force is shown in Figure 3.10 below. These are very similar to the results from Olsson [1] (Figure 3.11). Table 3.2 Default parameters for the standard model [1]. Figure 3.10 The sticking behaviour for the standard model with velocity dependent damping. The friction increases until the velocity is reached when it drops abruptly to zero.

84 65 Figure 3.11 Olsson [1] simulation. Next, the behaviour for break away is studied. Figure 3.12 shows how the applied force F e is linearly ramped up from zero. The friction follows the applied force up to a critical value before break away takes place. The drop in the friction force occurs in two phases. The first drop is due to the vanishing damping, which is determined by v d. The second drop is due to the Stribeck effect, which is determined by vs. Olsson [1], suggests that if high damping for zero velocity is desired by using Eq. 3.8, v d must be chosen in relation to v s. 3 Resisting Friction 2 Force (N) Velocity of Mass Velocity dy/dt Time (sec) 5 Figure 3.12 Breakaway behaviour for the model with σ 1 = 10.

85 66 Figure 3.13 Olsson [1] simulation. Figures 3.8 to 3.13 show that the simulation models developed for this thesis are comparable to the dynamic models of Canudas de Wit et al [71] and Olsson [1]. For completeness, Figure 3.14 is included to demonstrate the models ability to mathematically define varying static friction as a function of force application rate. Varying break away phenomena is important for the impulse control and its effect on the resulting dynamic motion of an individual impulse. Figure 3.14 Olsson simulation of varying break-away force as a function of the rate of increase of the applied force: for the default parameters (+); v s = (*); and σ 0 = 10,000 [1].

86 Modelling of the Impulse Controller To initiate stage 3 of the simulation model, the impulse controller is included in the control scheme. The design and derivation of the impulse controller is covered in Chapters 4 and 5. However, the location of the impulse controller in the system model is first discussed in this section to complete the general layout of the system control scheme topology. Figure 3.15 Diagram indicating the typical nested loop structure used in servo systems. Figure 3.16 Friction test bed Simulink simulation model with the impulse controller and no velocity loop Loop Topology One of the points to be considered in impulse control is the positioning of the controller in the loop structure of the system. Commonly used in servo systems is the nested or

87 68 cascade loop structure employing the position loop which surrounds the velocity loop as shown in Figure This thesis follows the work of [12] and [2] where the impulse controller is placed in the position loop (Figure 3.16). The discontinuous output of the impulse controller is an additional torque that is added to the PID controller. Figure 3.17 Block diagram of the friction test bed experimental system controller Hybrid PID + Impulse Control for Improved Stability Figure 3.17 shows the block diagram of the conventional linear controller + impulsive controller. This hybrid controller has been suggested by Li et al [2] whereby the PID torque requests and the impulsive controller torque requests are added together. Each successive pulse width is calculated at an impulse sampling rate. Unlike the impulse controllers devised by [12, 14, 15], it is unnecessary to stop at the end of each sampling period. Consequently, the controller can be used for both position and speed control. The controller can be divided into two parts; the upper part is the continuous driving force for large scale movement and control of external force disturbances. The lower part is an additional proportional controller k pwm with a pulse width modulated sampleddata hold (PWMH), and is the basis of the impulsive controller for the control of stickslip.

88 69 The sample rate used for the friction test bed is 2 khz. Impulses are applied at one twentieth of the overall sampling period (ie at 100Hz) to be compatible with the mechanical system dynamics. Figure 3.18 shows a typical output of the hybrid controller for one impulse sampling period τ s. The pulse with height f p is added to the PID output. Because the PID controller is constantly active, the system has the ability to counteract constant disturbances applied to the servomechanism. The (I) controller can also make the average position tracking error zero during stick-slip motion. If the stick-slip is cancelled, the position tracking error will be zero. The continuous part of the controller is tuned to react to large errors and high velocity, while the impulse part is adjusted for final positioning where stiction is most prevalent. Force fp PID Output τ s Figure 3.18 Hybrid controller output. For large pulse widths lasting the full sample period τ s, the combined control action of the PID controller and the impulsive controller will be continuous. Conversely, for small errors, the proportional controller output becomes too small to have any substantial effect on the servomechanism dynamics other than counteracting large disturbances. The high impulse sampling rate combined with the small error ensures

89 70 that the integral (I) part of the PID controller has insufficient time to rise and initiate limit cycling. This general description of the impulse shape with reference to its height and width, is fully discussed in Chapter Conclusions Three stages of system modelling have been outlined in this chapter. Firstly, a basic model has been built and presented. The basic model is simulated using the friction test bed system parameters experimentally determined in Chapter 2. A comparison of the open loop velocity responses indicates that the system model using the estimated parameters closely replicates the actual experimental system response curves. This comparison validates the system parameters that continue to be used in Chapters 4 and 5. Secondly, the basic Coulomb plus viscous friction Simulink block is replaced with a memoryless friction model function. The friction model includes Coulomb, viscous and static friction using directional friction coefficients. The model has exhibited stick-slip phenomena as expected. A comparison between the simulation and actual experimental results at low speeds shows a very similar result. This indicates that the friction model for low speed is accurate and can be confidently used as a tool to design the impulse controllers in Chapter 4. To complete stage 2 of the modelling process, the static friction model is replaced with the dynamic friction model given by Canudas de Wit et al [71] and Olsson [1]. Several

90 71 simulations are compared to the model simulations of Olsson [1] to show the accuracy of the models used in this thesis. The model parameters and structure are progressively changed to finally capture the true dynamic nature of friction at very low and zero velocities to include; rising static friction, frictional lag, varying breakaway force, dwell time, pre-sliding displacement and Stribeck effect. Finally, stage 3 of the simulation model is introduced. Here the loop topology is considered and the placement of the impulse controller is decided. The control scheme topology in this thesis uses a closed position loop with the impulse controller output added to the PID output.

91 72 CHAPTER 4 IMPULSE CONTROLLER DESIGN 4.1 Introduction This chapter covers the design and development of an improved impulse controller that will be used in Chapter 5 on the experimental friction test bed and in Chapter 6 on the Hirata robot. Section 4.2 begins the design process with a mathematical analysis for a standard single rectangular torque pulse. The analysis is necessary to help understand the general dynamics of the impulse controller and the effect that varying the pulse height or pulse width has on the servomechanism displacement. Sections 4.3 explains the notion that there is a minimum pulse width that will guarantee net displacement and that below this, the impulse controller is ineffective. Sections 4.4 and 4.5 present the designs of two types of impulse controllers: 1) variable pulse width impulse control, and 2) variable pulse height impulse control. Each

92 73 controller is given a series of simulated tasks with the results being compared. Included in the comparison is the simulation of a conventional PID control system. Section 4.6 discusses the development of a new way of improving the performance of the impulse controller through pulse shaping. To help understand the complexities of the low velocity dynamics for each pulse shape, the friction models developed in Chapter 3 will be used to simulate and test each design. Finally, Section 4.7 introduces a new concept for improving the impulse controller position pointing precision by limit cycle offsetting. The impulse controller takes advantage of the small position oscillation which occurs at the steady state set point. This section demonstrates that by shifting the limit cycle either up or down from the reference set point, the servomechanism resolution can be improved by a measurable amount. For each of the experimental graphical results given in this chapter, the maximum error uncertainty was measured to be 8.74e-4 radians compared to a maximum vertical axis resolution of 1e-4 radians. Motivation for the Impulse Controller - The primary purpose of an impulse controller is to counteract the non-linear effects of friction and to overcome some of the negative phenomena experienced by conventional linear controllers. An impulse controller can have several roles. These include one or all of the following in order of increasing complexity:

93 74 1. Move the system from or through the non-linear friction region. The impulse controller should guarantee at least some amount of positive net displacement; or, initiate movement from a stuck position leading to the transition of another controller which regulates higher velocity control; 2. Move the system a precise fixed displacement, independent of any other controller; 3. Move the system to a precise position set point whilst avoiding stick slip; 4. Facilitate moving the system at very low velocities (within the Stribeck region). Figure 4.1 Simulated dynamics of a single torque impulse. 4.2 Mathematical Analysis In order to begin the design of an improved impulse controller, it is necessary to understand the mathematical relationship between impulse and displacement. When considering a simple rectangular torque pulse (Figure 4.1), applied to a machine

94 75 that is influenced by the static and coulomb friction, the resulting angular displacement θ of the system can be described by the classical Newtonian expressions translated into rotational terms, T = Jω, ω = ω ωt and θ = ω t + ω t (where ω 0 is the initial 0 2 angular velocity). Using Newton s second law: T ω = T p c (4.1) J where T p is the toque supplied by the pulse, T C is the opposing Coulomb friction and J is the mass moment of inertia. The total distance moved from a single pulse is: total ( ) ( ) θ = θ + θ (4.2) 1 2 where θ 1 is the distance moved while accelerating and θ 2 is the distance moved while decelerating. The distance moved by the applied pulse θ 1, is: = 0 tp + 2 p t (4.3) θ ω ω p where t p is the pulse duration (width). Substituting Eq. 4.1 into Eq.4.3 gives: Tp TC θ1 = ω0 tp + t 2 J 2 p (4.4) In the case where the system comes to a complete stop between impulses, time taken is the residual time t r and is given by: t r ω ω ω 2 1 = (4.5) where ω 1 is the velocity at the instant the applied impulse is removed and ω 2 is the final velocity at the end of the sample period. Figure 4.1 shows ω 0, ω 1 and ω 2 with respect to time. Substituting Eq. 4.1 into Eg.4.5 gives: t r ω ω 2 1 = T C J (4.6)

95 76 The angular displacement becomes: t r ω ω θ + = (4.7) Substituting Eq. 4.6 into Eq. 4.7 gives: T C J ω ω ω ω θ + = (4.8) If the system comes to rest, ω 2 = 0 and Eq. 4.8 reduces to: C T J ω θ = (4.9) Expressing the angular velocity in terms of the distance moved θ 1, gives: t p θ ω ω = (4.10) Substituting Eq. 4.4 into Eq gives the following: p C p p p T T t t J t ω ω ω + = (4.11) and, substituting Eq into Eq. 4.9 gives: p c p p p C T T t t J t T J ω ω θ + = (4.12)

96 77 Finally, the total angular displacement θ total = θ 1 + θ 2 from a singular rectangular impulse can be expressed by: θ 2 T T 2 ω t + t θ 1 θ 2 p C 2 0 p p Tp TC 2 J 2 J ω0 tp + t p ω0 total = 2 J + t p 2 TC (4.13) If the velocity of the system before the pulse is applied is zero (ω 0 = 0), then Eq becomes: θ total 2 Tp TC 2 2 t p Tp TC 2 J 2 J = t p + 2 J t p 2 TC (4.14) which can be simplified to: θ total 2 ( ) T T T t = 2 J T p p C p C (4.15) For a more thorough simplification between Eq to 4.15, see Appendix B.4. Eq is now in agreement with the equation presented by Yang and Tomizuka [12] who make note that the displacement is linearly proportional to the square of the pulse width. This has particular importance when designing an impulse controller with a variable width structure (Section 4.4). Yang and Tomizuka [12] broaden the mathematical model by including viscous damping in addition to constant Coulomb friction. The total displacement θ total due to a singular impulse then becomes:

97 78 θ total Tt p p JT Tp Tt v p J C = ln ( e 1) Tv Tv TC (4.16) where T v is the viscous friction torque. For very low velocities, where the impulse controller takes control, the viscous friction becomes insignificant and the angular displacement can be approximated using the simplified Eq Minimum Pulse Width The precision of the system is governed by the smallest incremental movement available from a single pulse. In Section it is discussed that for every practical mechanism, there is a minimum amount of energy required to break the friction contact and that below this a mechanism will only result in pre-sliding elastic movement. Because the shape of the pulse is affected by the system s electrical circuit response, a practical limit is placed on the amplitude of the pulse over very short durations. This restricts the amount of energy that can be contained within a very thin pulse. The Figure 4.2 Expected pulse shapes.

98 79 simulated responses of the expected pulse shapes for the friction test bed are shown in Figure 4.2. Similarly, the mechanical response of the system with high inertia means that the mechanism behaves like a low pass filter. Consequently, there exists a minimum pulse width that is required to guarantee plastic movement. For the experimental friction test bed, the minimum pulse width guaranteeing plastic displacement is determined experimentally to be 2 ms and therefore the pulse width is adjusted between 2 and 10 ms. Any pulse smaller than 2 ms results in elastic movement of the mating surfaces in the form of pre-sliding displacement. In this regime, short impulses can produce unpredictable displacement or even no displacement at all. For example, Figure 4.3 shows the displacement of the experimental friction test bed for five consecutive positive impulses followed by five negative impulses. The experiment compares impulses of width 2 ms and 1.5 ms. For impulses of 2 ms duration, the displacement is represented by the consistent staircase movement. For a slightly lesser Figure 4.3 Experimentally measured displacement (friction test bed) for both positive and negative impulses using successive pulse widths 1.5 ms and 2 ms.

99 80 width of 1.5 ms, the displacement is unpredictable with mostly pre-sliding movement being elastic that results in zero net displacement. In some cases, the mechanism may spring back greater than the forward displacement resulting in a larger error. Wu et al [72] use the pre-sliding displacement as a means to increase the precision of the controller by switching the impulse controller off and using a continuous ramped driving torque to hold the system in the desired position. The torque is maintained even after the machine is at rest. This is difficult in practice as pre-sliding movement must be carefully controlled in the presence of varying static friction so that inadvertent breakaway followed by limit cycling is avoided. In the next section, two impulse controller structures are investigated; 1) variable pulse width (fixed height) impulse control, and 2) variable pulse height (fixed width) impulse control. 4.4 Variable Pulse Width Impulse Control Controller Design For this type of impulse controller, the pulse width is variable while the pulse height T p remains constant. For large errors, the impulse width approaches the full sample period τ s, and for very large errors, transforms into a continuous driving torque. When this occurs, the combined control action of the PID controller and the impulsive controller is continuous.

100 81 When the error is below a threshold, the impulsive controller begins to segment into individual pulses of varying width and becomes the primary driving torque. One way of achieving this is to make the pulse width determined by: kpwm e( k) τ s =, if k pwm ek ( ) T p T p = τ s otherwise (4.17) In Eq. 4.17: p p ( ( )) T = T sign e k (4.18) where e(k) is the error input to the controller, T p is a fixed pulse height greater than the highest static friction and τ s is the overall impulse controller sampling period. For the friction test bed, the pulse height is set 10% greater than the maximum measured static friction. The sampling period τ s is 10 ms and the pulse width can be incrementally varied by 1 ms intervals. The pulse width cannot be infinitely variable because there are practical limits on the sampling time. The pulse width gain k pwm, is experimentally determined by initially matching the mechanism s observed displacement to the calculated pulse width t p using Eq The gain is then iteratively adjusted until the net displacement for each incremental pulse width is approximately proportional to the error input. For mechanisms using a small sampling period τ s, the displacement from a varying pulse width becomes approximately linear, conversely for mechanisms requiring large sampling period τ s, the larger movement is nonlinear and the displacement becomes the square of the pulse width (see Eq.3.15). For controllers using a large sample period and greater displacement per incremental pulse width it may be necessary to calculate the pulse width as a function of the square root of the error. This

101 82 makes the whole system linear so that linear control methods can be applied. The simulation model of the variable width controller is shown in Figure 4.4. Figure 4.4 Simplified variable width impulse controller simulation model Regulated Pulse Height Using the hybrid PID + impulse controller, Simpson et al [17, 66, 73] and Simpson [64] assume that the PID output will automatically adjust itself to a value approximating the Coulomb friction so that, when an impulse is added, the resulting torque request will always exceed the static friction. With this assumption, motion is always guaranteed. However, in a practical system with a fixed sample time, the PID torque request may conflict with the impulse part when the two have different signs. Similarly, for very low velocities, the PID torque request may not approximate the Coulomb friction value as expected. For the hybrid friction controller, the combined PID + impulse torque command may be insufficient to overcome the absolute value of static friction resulting in stick-slip. In order to guarantee that the torque request of the combined PID + impulse controller is greater than the absolute value of the static friction, a new approach is taken by regulating the impulse height. At each sample period, the pulse height f p is recalculated

102 83 so that the PID + impulse controller output always exceeds the static friction. One way of achieving this is to make the output of the impulse controller equal to the static friction T s minus the PID output T PID using the following conditions: T p = T K Ψ T, if TPID Ts K Ψ s PID T = 0 otherwise (4.19) p where K Ψ is the percentage gain used to overcome the maximum measured static friction for each direction of travel Simulation of the Variable Pulse Width Controller To compare the performance of the variable pulse width controller (PID + impulse) against conventional PID control, three simulated tasks were used. These include. Task 1 Position pointing to a reference set point; Task 2 Low speed tracking using a linear ramp position reference; and Task 3 Low speed tracking using a sinusoidal position reference. Task 1 The purpose of this task is to demonstrate the potential precision of the servomechanism when using impulse control. Figure 4.5 shows a comparison of two simulations; one using PID control only, and the other using PID + impulse control. The magnitude of the set reference point chosen ensures that the impulse controller requires the PID action to provide most of the required driving torque. As the error decreases, the impulse controller segments into a discontinuous driving torque to move the servomechanism towards zero steady state error. In order to compare the precision of each of the controllers, parts of the graphs are magnified around the steady state set point. Also note the time scale for both sets of

103 84 a) b) Figure 4.5 Simulation of a servomechanism for a position pointing task using; a) PID only, and b) PID + impulse control. The third plot of each set of graphs uses a very fine axis resolution for position. graphs is different (10 and 2 seconds respectively) which demonstrates the impulse controller s ability to reach steady state considerably faster. The PID only simulation exhibits a large self-excited oscillation or limit cycle near steady state, that if allowed will continue indefinitely. This phenomena is typical where an integral (I) action of the PID controller is used to control non-linear friction. In contrast, the PID + impulse controller reduces the steady state error by using successive pulses and when compared to PID only, shows a significant improvement in precision. Task 2 This task compares both the PID and PID + impulse controllers when used to move a mechanism at a constant low speed using position tracking. In this instance a rad/s position ramp is used to analyse how well each controller can maintain constant average velocity in the Stribeck friction region (i.e. ω ω Stribeck ).

104 85 a) b) Figure 4.6 Simulation of a servomechanism for a low speed position tracking task using; a) PID only, and b) PID + impulse control. Figure 4.6 compares both controllers using the simulation. For the PID controller, stickslip is observed and the resulting motion takes the classical staircase form. In comparison, the PID + impulse controller successfully follows the position trace with a significant increase in precision. Task 3 Finally, each controller is compared when used to follow a sinusoidal position reference having an amplitude of 0.01 radians and a period of 10 seconds. This type of reference input has two velocity reversals that force the mechanisms to momentarily come to rest. More importantly, the varying low speed reference requires the impulse controller to output varying pulse widths in order to maintain the correct speed. Figure 4.7 compares both controllers using the model simulation. Once again the precision of the PID + impulse controller outperforms the PID only controller.

105 86 a) b) Figure 4.7 Simulation of a servomechanism for a low speed sinusoidal position tracking task using; a) PID only, and b) PID + impulse control. Summary - For tasks 1 to 3 the impulse controller increased the precision of the servomechanism for both position pointing and low speed position tracking. The design of the controller shows that the mechanism does not need to come to rest in order for the impulse controller to be applied. Furthermore, the pulse width is a function of the error and is continually readjusted at a high sampling rate. Consequently, it is unnecessary to have a precise knowledge of the position dependent friction variables for the impulse controller to work effectively. Finally, the pulse height of the impulse torque request is regulated so that the combined PID torque request plus impulse torque request is always greater than the static frictiont + T Ts. The result is guaranteed movement when p PID > the impulse controller is applied, regardless of the torque requested by the PID controller.

106 Variable Pulse Height Impulse Control Controller Design For this type of impulse controller, the pulse height T p is adjusted by the controller while the pulse width remains constant. To guarantee movement of the mechanism, a minimum pulse height is chosen to ensure the pulse torque is greater than the static friction. Likewise, a minimum pulse width must also be determined using the same reasoning as presented in Section 4.3. To ensure the finest resolution in terms of impulse, the smallest possible fixed pulse width is chosen. The variable pulse height therefore controls the amount of usable angular impulse where the angular impulse I is given by: I = T t (4.20) As determined in Chapter 2, the magnitude of static friction (breakaway torque) is position dependent. When considering a variable pulse height controller, the minimum pulse height T p min must be chosen to be greater than the maximum position dependent static friction, i.e. T p min > T s. For the friction test bed, Tpmin is 10% greater than the maximum measured static friction, as chosen in Subsection Similar to the control structure of the variable width controller (Section 4.4), when the error is below a threshold, the impulsive controller begins to vary the pulse height and becomes the primary driving torque. One way of achieving this is to make the pulse height T p determined by: kphm e( k).( Tp max Tp min) Tp = + Tp min (4.21) τ s In Eq. 4.21: p p ( ( )) T = T sign e k (4.22)

107 88 where K phm is experimentally determined using an iterative approach whereby the resulting displacement is made proportional to the error for the range of pulse heights. The gain is iteratively adjusted until the net displacement for a selective set of incremental pulse heights is approximately equal to the error input. The maximum pulse height T p max, in a practical sense, is restricted by the electrical circuit response. For the mechatronic systems used in this thesis, T p max is set at the maximum torque output from the DSP. The pulse height function is graphically represented in Figure 4.8 and the simulation model of the variable height impulse controller is shown in Figure Simulation of the Variable Pulse Height Controller To evaluate the performance of the variable pulse height controller, the three simulation tasks used in Subsection are replicated. In this case, the variable pulse height controller is compared to the variable pulse width controller for each simulation. T p MAX Pulse Height Tp Slope = K phm Saturation T p MIN Static friction (Fs) Position error e(k) Figure 4.8 Graphical representation of the pulse height as a function of the error e(k).

108 89 Figure 4.9 Simplified variable height impulse controller simulation model. Task 1 - Figure 4.10 shows a comparison of two simulations; one using a variable width impulse controller (from Section 4.4.2), and the other using a variable height impulse controller. The comparison shows the variable height controller has difficulty maintaining controlled movement at the final stages of positioning. As a result, the variable height controller limit cycles and the final controller output produces a pulse height greater than the minimum height with alternating sign. Task 2 This task compares the variable width controller with the variable height controller when used to move at a constant slow speed of rad/s. Figure 4.11 compares both controllers using simulation. For the position response, both controllers well outperform the PID only simulation (Figure 4.6 (a)); however, when comparing both the impulse controllers, the difference is negligible. This can be explained by the observation that for low speed, both controllers are using the same minimum pulse width and minimum pulse height respectively. The discontinuous torque part of the controller is therefore almost identical and the position trace remains the same.

109 90 a) b) Figure 4.10 Simulation of a servomechanism for a position pointing task using; a) variable width PID + impulse, and b) variable height PID + impulse. a) b) Figure 4.11 Simulation of a servomechanism for a low speed position tracking task using; a) variable width PID + impulse, and b) variable height PID + impulse.

110 91 Task 3 - For this task the controller tracks a sinusoidal position trace which requires the servomechanism to move at varying speeds. Consequently, the controller output is required to vary at the same rate. Figure 4.12 shows a comparison for the two impulse controllers. The tracking error is greater for the variable height impulse controller (Figure 4.12 (b)) when compared to the variable pulse width controller (Figure 4.12 (a)). a) b) Figure 4.12 Simulation of a servomechanism for a low speed position tracking task using; a) variable width PID + impulse, and b) variable height PID + impulse. This task demonstrates that the precision of the variable pulse height controller deteriorates at higher speeds. This can be explained by the knowledge that the frequency of the variable height controller pulse is applied at a set rate (τ s = 10 ms). When the velocity of the reference becomes large, the fixed sampling rate τ s limits the degree to which the controller can match the system s mechanical response. Consequently, the error accumulates during the time between each successive pulse. The proceeding pulse

111 92 is then required to compensate by providing very large pulses (typically saturating the controller). This forces the machine to accelerate quickly during the on period of the narrow pulse in order to counteract the deceleration during the much longer off period of the narrow pulse. The higher acceleration and decelerations result in position tracking ripple and are shown in Figure 4.12 (b) where the slope of the sinusoidal trace is greatest. Summary Pulse height regulation was again used so that the minimum combined PID torque request plus impulse torque request is always greater than the static frictiont + T Ts. The result is guaranteed movement when the impulse controller p PID > is applied, regardless of the torque requested by the PID controller. For tasks 1 to 2 the variable pulse height impulse controller significantly outperformed the PID only controller and was similar to the variable pulse width impulse controller. For Task 3 the variable pulse height controller again outperformed the PID only control but was outperformed by the variable pulse width controller. For this reason, only the variable pulse width controller is further refined and used in Chapters 5 and Pulse Shaping For Improved Precision The motivation for pulse shaping is to develop new ways of improving the performance of the impulse controller for precision tasks. This section describes a series of designs used for shaping the pulse of the impulse controller. Each shape is both simulated using the new model with Eqs 3.5 to 3.8 and validated on the experimental system. The first of the four shapes tested (Shape 1) is the conventional rectangular pulse and serves as a comparison for the other shapes.

112 Impulse Shape 1 The most simple and common shape used for impulse design is the rectangular pulse applied with the same sign as the error signal (Figure 3.18). Armstrong-Hélouvry [16], Hojjat et al [14], Li et al [74], Popovic et al [13], Yang et al [12], Li et al [74] and Simpson et al [64] all use impulsive controllers with this standard shape. Table 4.1 Simulation parameters for the new friction model. Parameter F s F C σ 0 σ 1 F v v s v d Value * , Figure 4.13 Simulated rectangular impulse F t where f p = 125% of F s : Denoted Shape 1. Figure 4.13 shows the simulation of a standard shaped pulse being applied for the duration of 4 ms with a pulse amplitude 25 percent larger than the average static friction. For practical reasons, a time constant of the power electronics response is also included for model completeness so that the corners of the resulting pulse shape are rounded on the rising and falling edges. The new friction model given by Canudas de Wit et al [71] is included in Figure 4.13 to demonstrate the dynamics of the friction. In order to graphically compare the friction

113 94 force with the pulse input, the sign of the friction force has been reversed. Once the static friction force is overcome and motion begins, the resisting friction reduces dramatically giving rise to the Stribeck effect before settling at the Coulomb friction. The Coulomb friction is maintained until the mechanism slows below the Stribeck velocity threshold when the static friction resumes. At this point, the mechanism comes to a complete stop and holds the stuck position. The resulting torque affecting the Figure 4.14 Simulated rectangular impulse having a negative trailing pulse with amplitude 90% F s : Denoted Shape 2. velocity of the mechanism is time dependent and is given by the controller output minus the opposing friction force for the duration of impulse sampling time τ s for any instant in time t for which the pulse is applied Impulse Shape 2 Armstrong-Helouvry et al [62] describe the motion as a small bang followed by an open loop slide. During the open-loop slide the only resistance to motion is friction. In principle, by applying a negative pulse immediately after the positive pulse width, the

114 95 deceleration of the mechanical system can be increased and the minimum displacement achievable by the impulsive controller would be improved. Figure 4.14 shows that a negative pulse can be applied for the duration τ s -. The amplitude of the negative pulse T p-neg is made less than the average static friction, i.e. F s -T p-neg > 0, so that if motion stops between sampling the mechanism will be prevented from moving in the opposite direction to the error. Because the static friction is not overcome at zero velocity, the resisting friction absorbs the remaining negative pulse until the next positive pulse is applied. In this case, the mechanism comes to rest almost 4 ms faster than that of Figure 4.13 where there is no negative pulse. The result is a smaller incremental movement being achieved Impulse Shape 3 To overcome stiction, it is necessary to have an initial driving force greater than the static friction. Immediately after motion begins, the opposing friction reduces dramatically and, if motion continues, will be maintained at the Coulomb friction value. In both Figures 4.13 and 4.14, most of the effective energy of the pulse commences immediately after the static friction dissipates. Although at this stage, the friction resistance is at or below the Coulomb friction, the shape of a standard rectangular pulse will continue to be at the more excessive amplitude used to overcome stiction. This results in the controller using more force to sustain movement than is necessary. By reducing the pulse immediately after the stiction is reduced, to an amplitude below the static friction but greater than the Coulomb friction, a much smaller incremental movement can be achieved through the lower velocities seen in Figure The maximum velocity in this case has been halved, yet the motion remains linear and smooth. The resultant force with respect to time has been reduced substantially and the

115 96 minimum displacement of the mechanism is improved even further. Integration of the curve showed almost 200 percent improvement in position resolution. Figure 4.15 Simulated stepped pulse with 2 ms start-up force having an amplitude 125% Fs followed by secondary pulse 3 ms having an amplitude 130% F C : Denoted Shape 3. The acceleration and deceleration of the mechanism is also reduced over the same impulse sample period. This in turn reduces the amount of shock the mechanical system sustains allowing smoother control and the likelihood of less wear on the mechanical components Impulse Shape 4 Combining both the negative decelerating pulse of shape 2 and the reduced amplitude pulse of shape 3, the controller precision can be improved even further. The effect on the velocity curve can be seen in Figure As for shape 3, the acceleration of the mechanism remains smoother by decreased velocities of approximately fifty percent of that seen for the comparative standard rectangular pulse denoted shape 1. The negative pulse part of the sample period decelerates the mechanism as quickly as possible to

116 97 reduce the overall displacement. Integration of the curve again shows an improvement in position resolution. Figue 4.16 Simulated stepped pulse followed by a trailing negative pulse 90% F s : Denoted Shape Comparison of Pulse Shapes A series of simulations are used to assess each of the pulse shapes as the pulse width is increased. For each of the four pulse shapes, the displacement versus pulse width is plotted and can be seen in Figure When comparing each successive curve (shape 1 to 4), the minimum controlled displacement reduces for each pulse width from 2 to 10 ms. For progressively increasing pulse widths, the displacement increases until the positive part of the pulse approaches the maximum pulse width of 10 ms. Here the two curves converge; this is due to the practical limitations of applying the negative pulse. As the positive part of the pulse is increased, the remaining time for the negative pulse to be applied within the remaining sampling period diminishes. This means that the curve from shape 1 at time 10 ms is identical to that of shape 2 at time 10 ms. This is also true when comparing shapes 3 and 4.

117 98 Figure 4.17 Simulated displacements for varying pulse widths for shapes 1, 2, 3, and 4. Figure 4.18 Exploded view of Figure 4.17 for first two pulse widths showing the variation in minimum displacement (precision) for shapes 1, 2, 3, and 4. The smallest incremental movement for a pulse width of 2 ms is shown in Figure 4.18, an exploded view of Figure The difference indicates that shaping of the impulse can reduce the smallest controlled displacement significantly when compared to the

118 99 conventional rectangular shape. In this instance, the resolution has increased by more than a factor of three. a) b) Figure 4.19 Typical impulse showing net available torque (yellow area) after subtracting friction; a) standard rectangular pulse: Shape 1, and b) modified shape to offset friction: Shape 3. Summary - In this section, a new set of impulse shapes have been shown to influence the dynamics of a mechanism with friction at very low velocity. It is shown that by adjusting the shape of the impulse, the energy applied to the mechanism can be adjusted so that the non linear effects of friction can be counteracted more efficiently allowing finer control of the displacement caused by a pulse. One of the benefits is that the motion is much smoother in terms of acceleration and deceleration. Consequently, a reduction in vibration is obtained which leads to a reduced wear and fatigue of mechanical components. The vibration is quantified in Chapters 5 and 6 using the two servomechanisms. Controlling the pulse also creates a much more even distribution of usable energy with respect to time. A conceptual comparison of the net energy distribution with respect to time is given in Figures 4.19 (a) and (b). Results from modelling and experimental verification show that the lower velocities generated by a carefully designed pulse shape, directly affect the resolution of the displacement created from each incremental pulse width. More extensive experimental verification is given in Chapters 5 and 6.

119 Limit Cycle Offset for Improved Positioning Motivation This section aims to improve the precision of the variable pulse width PID + impulse controller using a new approach, which for the purpose of this thesis is termed Limit Cycle Offset. Fig 4.20 Simulated displacements as a function of pulse width. In Section 4.3, it is shown that there is a minimum pulse width that will guarantee net displacement and that below this the impulse controller is ineffective. As an example, Figure 4.20 shows the simulated displacements of varying pulse widths which have been labelled d1, d2, d3 dn respectively, where d1 is the minimum pulse width which will generate non elastic movement and defines the system s resolution. Using the variable pulse width PID + impulse controller (Section 4.4) for a position pointing task, the torque request will incrementally move the mechanism towards the reference set point in an attempt to reach steady state. Around the set point, the system

120 101 will inevitably begin to limit cycle when the error e(k) is approximately the same magnitude as the system resolution (displacement for the minimum pulse width d1). Figure 4.21 Simulation of the impulse controller limit cycling around the position reference set-point where the final torque output is a pulse with minimum width and the mean peak to peak oscillation is d1. Limit cycling will occur for all general servomechanisms using a torque pulse because every practical system inherently has a minimum pulse width that defines the system s resolution. Figure 4.21 simulates a typical limit cycle with a peak to peak oscillation equal to the displacement of the minimum pulse width d1. In an attempt to reach steady state, the error and its alternating sign, will cause the mechanism to limit cycle indefinitely. For the cycle to be extinguished, the controller must be disabled. For example, the limit cycle in Figure 4.21 is extinguished by

121 102 disabling the impulse controller at t=0.18s and the resulting error is approximately half the displacement of the minimum pulse width d1. One way to automatically extinguish the limit cycle is to include a dead-zone that disables the controller output when the error is between an upper and lower bound of the reference point (Figure 4.21). The final error is then dependent on the amount of offset the limit cycle has in relation to the reference point. Figure 4.21 shows a unique case where the ± amplitude of the limit cycle is almost evenly distributed either side of the reference set point; i.e. the centre line of the oscillation lies along the reference set point. In this instance, disabling the controller would create an error e(k) equal to approximately d1. This however, would vary in practice and the centreline is likely to 2 be offset by some arbitrary amount. The maximum precision of the system will therefore be between d1 and zero Limit Cycle Offset Using a standard impulse controller without the limit cycle offset function, the final error will arbitrarily lie between d1and zero and determined by chance. By controlling the offset of the limit cycle centreline, it is possible to guarantee that the final error lies within the dead-zone, and therefore increases the precision of the system. As a conceptual example, Figure 4.22 shows the system limit cycling either side of the reference point by the minimum displacement d1. By applying the next smallest pulse d2, then followed by the smallest pulse d1, the limit cycle can be shifted by d2 d1. The effect is that the peak to peak centreline of the oscillation has now been shifted away from the reference point.

122 103 Position reference position d2 - d1 new error Time Figure 4.22 Conceptual example of reducing the steady state error using Limit Cycle Offset with the limit cycle shifted up by d2-d1 and the new error that is guaranteed to fall within the deadzone. However, at least one of the peaks of the oscillation has been shifted closer to the set point. If the controller is disabled when the mechanism is closest to the reference set point, a new reduced error is created. For this to be realised, the incremental difference in displacement between successively increasing pulses must be less than the displacement from the minimum pulse width; for example d2 d1 < d Controller Design For the limit cycle to be offset at the correct time, the impulse controller must have a set of additional control conditions which identify that a limit cycle has been initiated with the minimum width pulse. The controller then readjusts itself accordingly using a switching bound and finally disables itself when within a new specified error deadzone. One way to achieve this is to adjust the pulse width so that it is increased by one increment when satisfying the following conditions: if switching bound > e(k) dead-zone

123 104 then otherwise kpwm e( k) τ s = + 1 T p kpwm e( k) τ s = (4.23) T p where the switching bound is given by: d1 switching bound < 2 (4.24) and the dead-zone is given by: dead-zone = The steady state error e(k) becomes: (d2 - d1) 2 (4.25) e(k) steady state deadzone 2 (4.26) Simulation of the Limit Cycle Offset Function To demonstrate the limit cycle offset function, the modified controller is simulated using a simple unit mass with the new friction model using Eqs 3.5 to 3.8. The simulation model of the controller is shown in Figure 4.23 and the area surrounded by the red hatched line contains the block conditions for the limit cycle offset. The area surrounded by the green hatched line contains the pulse shaping block conditions for generating the pulse shapes developed in Section 4.6. A simulated step response is shown in Figure 4.24 to demonstrate how the modified controller works. Here the mechanism moves towards the reference set point and begins limit cycling. Because at least one of the peaks of the limit cycle immediately lies within the switching bound, the controller shifts the peak to peak oscillation by d2 - d1 by applying the next smallest pulse, and then followed by the smallest pulse. In this example, the first shift is insufficient to move either peak into the set dead-zone so the

124 105 Figure 4.23 Simulation model of the modified impulse controller with Limit Cycle Offset. Parameter F s F C σ 0 σ 1 F v v s v d Value * , Figure 4.24 Simulation of the limit cycle offset function used with the PID + impulse controller and Shape 3.

125 106 controller follows with a second shift. At time 0.1 seconds, the controller is disabled; however, the elastic nature of the friction model causes the mechanism s position to move out of the dead-zone. As a result, the controller is reactivated (time 0.12s) and the controller follows with a third shift. In this instance, the mechanism reaches steady state 1 at t=0.2s, and the final error is ek ( ) (dead zone). A final analysis of the result 2 shows that the new controller has reduced the error by a significant amount more than a standard impulse controller. This reduction correlates directly to the system s new resolution. Summary The Limit Cycle Offset function of the impulse controller has been shown to improve the resolution of the system when used for a simulated position pointing task. The simulation friction model uses the mean friction parameters so that the displacement for a constant pulse width results in repeatable displacement. In Chapters 5 and 6, the Limit Cycle offset controller is experimentally applied to the friction test bed and Hirata robot to evaluate its practical usefulness where position dependent friction must be considered. 4.8 Summary and Conclusions This chapter has begun by defining the primary role of the impulse controller and the motivation behind its use. The objectives of this research include; improving the precision of the impulse controller, elimination of friction phenomena such as stick-slip, quadrature glitch; and the more efficient use of impulse energy in order to reduce mechanical vibration and speed regulation ripple.

126 107 The mathematical analysis of the impulse controller provides an understanding of how impulse I = T t relates to the displacement of a servomechanism. The total displacement of a single pulse is broken into two distinct modes; 1) the displacement caused during acceleration, and 2) the displacement caused during deceleration. The final mathematical model is shown to match that of Yang and Tomizuka [12] and shows that the displacement is linearly proportional to the square of the pulse width. This simple understanding is noted to have particular importance when designing an impulse controller with variable width. There is a practical limit on the amplitude of a pulse over very short durations and there is a minimum pulse width that is required to guarantee plastic movement. For the experimental friction test bed, the minimum pulse width guaranteeing plastic displacement was determined to be 2 ms and therefore the pulse width is adjusted between 2 and 10 ms. Experimental testing demonstrated that the displacement from the minimum pulse was very repeatable; however, any pulse smaller than 2 ms resulted in unpredictable elastic movement of the mating surfaces in the form of pre-sliding displacement. Two types of impulse control are investigated; 1) a variable pulse width impulse controller, and 2) a variable pulse height impulse controller. In the first, the controller is designed to have a varying pulse width while the pulse height remains constant by regulating its output based on the value of the PID torque request. The pulse height is chosen to ensure that it exceeds the maximum measured static friction over the mechanisms travel. For small errors, the PID output is too small

127 108 to have any substantial immediate effect on the servomechanism dynamics other than counteracting disturbances. Here, the impulse controller begins to segment into individual pulses of varying width and becomes the primary driving torque. For very large position errors, the pulse width transforms into a continuous driving torque and the PID part adjusts itself accordingly. An example controller design is given and a comparison has been made between the impulse control and standard PID controller using the simulation of three tasks including position pointing, constant speed tracking and variable speed tracking. Simulations have shown a significant improvement in the control and precision of the system when compared to conventional PID only control for all three tasks. It is also noted that it is unnecessary to have a precise knowledge of the position dependent friction variables for the impulse controller to work effectively. The second structure investigated is an impulse controller having a variable pulse height and a constant pulse width. This structure also uses a minimum pulse width and minimum pulse height to guarantee plastic movement. Within the controller design, the pulse height is variable between the minimum pulse height (10% greater than maximum static friction) and a maximum pulse height (voltage saturation of the DSP). The controller is modified so that the pulse height is regulated according to the output of the PID torque request. With regulation, the impulse torque request is calculated so that the combined PID torque request plus impulse torque request is always greater than the static frictiont + T Ts. The result is guaranteed movement when the impulse p PID > controller is applied, regardless of the torque requested by the PID controller.

128 109 Using the same previous three tasks, simulation has shown the controller has again outperformed the PID only controller; however, a comparison between each of the impulse controllers have shown that the variable width controller has a similar precision in final positioning for the pointing task. Conversely, it has outperformed the variable height impulse controller for variable speed tracking, particularly when the speed of the reference is close to the Stribeck velocity threshold. This is because the pulse of the variable height controller is applied at a set rate. When the velocity of the reference becomes large, the fixed sampling rate limits the degree to which the controller can match the system s mechanical response. Consequently, the error accumulates during the time between each successive pulse. The proceeding pulse is then required to compensate by providing very large pulses (typically saturating the controller). This forces the machine to accelerate quickly during the on period of the narrow pulse in order to counteract the deceleration during the much longer off period of the narrow pulse. The higher acceleration and decelerations result in position tracking ripple at higher velocities. Two new impulse controller designs are developed. For the first, pulse shaping is introduced as a way to increase the precision of the impulse controller. Four different shapes are simulated and compared using the dynamic new friction model given by Canudas de Wit et al [71]. The first of the four shapes tested (Shape 1) is the conventional rectangular pulse and served as a comparison for the new shapes. The simulation of pulse Shape 1 has shown that the friction reduces dramatically after the start of the pulse as a result of the Stribeck effect. This is followed by the stabilising of the friction force at the Coulomb friction value. Consequently, most of the impulse is contained within the latter part of the pulse after stiction is overcome. The mechanism

129 110 therefore accelerates freely. To arrest the acceleration and reduce the total net displacement the pulse shape is redesigned to have a negative component that is used to decelerate the mechanism. This new design is termed Shape 2. Simulation has shown that the net displacement of the single pulse can be reduced and that a higher resolution per pulse can be obtained. The third design (Shape 3) is based on the realisation that a smaller net displacement can be achieved by reducing the pulse height to a value still greater than the Coulomb friction but less than the static friction immediately after stiction is overcome. The net impulse (applied pulse minus friction) is approximately constant for the full width of the pulse and results in reduced displacement. By combining the advantages of pulse Shape 2 and pulse Shape 3, an additional new pulse shape is designed (Shape 4). When comparing each shape, using simulation, the impulse resolution is increased when changing from shape 1 to shape 4. Another major benefit of pulse shaping is that the motion is much smoother in terms of acceleration and deceleration. Consequently, a reduction in vibration is obtained which is likely to reduce the wear and fatigue of mechanical components. Finally, the second new impulse controller design has been introduced. Here, the concept of Limit Cycle Offset is used as another way to increase the precision of the impulse controller. The controller uses the limit cycle created by the minimum width pulse when near the reference set point. By applying alternate width pulses, the limit cycle is shown to be shifted up or down with respect to the position reference set point. Once either of the limit cycle s peaks enters a designated dead-zone, the controller is

130 111 disabled. The result is a reduced error that is less than the original steady state error. Simulations have shown that the Limit Cycle Offset impulse controller has improved the positional pointing accuracy of the system when using constant friction parameters for the friction model. Chapter 5 explores the practical application of this controller for the friction test bed having position dependent friction variables.

131 112 CHAPTER 5 PERFORMANCE ANALYSIS AND IMPROVEMENT OF THE IMPULSE CONTROLLER USING THE FRICTION TEST BED 5.1 Introduction This chapter uses the friction test bed to experimentally evaluate the impulse controllers developed in Chapter 4. As discussed in Chapter 2, Matlab s xpc is used to provide the control with DSP hardware using Simulink block code. Figure 5.1 shows the Simulink model, with the mechanical system transfer function and friction model, being replaced with the xpc digital signal processing computer board blocks (red) and the experimental servomechanism (grey). Figure 5.1 Modified Simulink model to include xpc analog out and DSP blocks (red).

132 113 Section 5.2 discusses the modification and refinement of the impulse controller so that it is more compatible with the real mechanism. The improvement begins with the inclusion of velocity reversal compensation and directional friction parameters. In Section 5.3, each of the impulse control pulse shapes 1 to 4 are experimentally evaluated and validated. Several experiments are carried out to replicate the simulations of Chapter 4 using the three tasks of position pointing, low speed tracking using a linear ramp position reference and low speed tracking using a sinusoidal position reference. In Section 5.4, the concept of Limit Cycle Offset is successfully evaluated using the friction test bed. Finally, the impulse controller is compared to tangential dither in order to distinguish between the two, and further validate the superior performance of the impulse controller. For each of the experimental graphical results given in this chapter, the maximum error uncertainty was measured to be 1.57e-5 radians compared to a maximum vertical axis resolution of 2e-5 radians. 5.2 Improving the Impulse Controller for a Real System There are several practical considerations that must be considered when controlling friction using a real servomechanism. Most of the simulation models in this thesis have been constructed and evaluated using mean friction values as opposed to actual position dependent variables. With the friction test bed, the parameters of static, Coulomb and viscous friction are highly variable and position dependent (Section ). Despite this, the work presented in this chapter proves that the impulse controller remains

133 114 effective using the mean values. However, the mean parameters are also different for opposing directions of motion (i.e. clockwise and counter clockwise rotation). To account for this variation, the controller requires some modification in order to improve its performance Velocity Reversal Compensation Friction has been shown to be discontinuous at zero velocity. When a servomechanism changes velocity, the static friction value jumps from T s to T s. For the standard PID controller, there is a dead time while the controller winds down from T C to T s. The result is a prominent position flat spot or glitch whenever the reference velocity changes sign. (a) (b) Figure 5.2 Integral windup observed at zero velocity and velocity reversal when using a) PID only control, and b) PID with integral reset. Figure 5.2(a) shows a linear ramp whose slope changes sign at t=5s. At t=0s, the integral part of the PID controller is forced to wind up to a value greater than the static friction so that breakaway can occur. Similarly, at t=5s, the controller must wind down and then wind up with the opposite sign. In both instances, the integral term delays the motion. The standard solution is to reset the integral term of the PID controller when a velocity reversal is detected using the reference signal [62]. The purpose of the reset

134 115 function is to unwind the integrator by replacing its output with the Coulomb friction parameter having the sign of the direction travelled. One way of achieving this is to modify the u(k) term such that: if sgn( ω ( k)) sgn( ω ( k 1)), then u k) sgn( ω ) ref ref ( T C ref = (5.1) where the velocity reference ω ref is the derivative of the position reference signal. Figure 5.2(b) shows the improved tracking performance of the controllers using velocity reversal compensation for the same reference used in Figure 5.2(a). At t=0s and t=5s, the integral part of the PID controller is reset to the mean Coulomb friction value having the sign of the reference velocity. The result is a significant reduction in delay Direction Dependent Friction Values For simulation purposes, the impulse controller uses the maximum experimentally determined friction values to account for position dependent variation. However, it has also been observed experimentally on a number of occasions [16, 40, 75], that there are different friction values for opposing directions of motion. This variance is not only restricted to static friction, but has also been observed for Coulomb and viscous friction. This is also the case for the friction test bed (Section ). To improve the servomechanism s performance, the controller is modified so that a set of switching functions is used to detect the sign of the error so that the relevant set of direction dependent friction values can be applied. The modification to the controller is illustrated in Figure 5.3 and the area of interest is bounded by a red hatched line. The effect of the direction dependent friction values is a more precise controller for both directions of motion.

135 Figure 5.3 Modified impulse controller Simulink model with direction dependent friction parameters, non linear pulse width gain, regulated pulse height and limit cycle offset function. 116

136 Experimental Evaluation of Pulse Shapes 1 to 4 In Chapter 4, the concept of shaping the impulse was introduced as a means to increase the performance of the impulse controller. This section extends the simulation work to demonstrate the practical application of pulse shaping for mechanisms with varying friction. Similar to the simulations used in Section 4.4.2, each of the four pulse shapes are evaluated using the three tasks of position pointing, low speed tracking using a linear ramp position reference and low speed tracking using a sinusoidal position reference. Figure 5.4 A sample step input and position response using pulse Shape 1. Mean final oscillating displacement µ d = 1.440e-4 radians for a sample of 10 repeated experiments Position Pointing Using Pulse Shapes 1 to 4 Using a step input position reference of radians, the PID + impulse controller is used to move the mechanism to the reference set point. Figure 5.4 shows an

137 118 experimental sample response using pulse Shape 1. The experiment is repeated ten times for each pulse shape and the peak to peak displacement of the steady state limit cycle is measured (neglecting the pre-sliding elastic movement). The experiments are repeated for each pulse shape and a sample response is shown for pulse shapes 3, 4 and 5 in Figures 5.5, 5.6 and 5.7 respectively. The mean peak to peak displacement µ d of the steady state limit cycle is determined for each sample set of experiments and a comparison is given in Table 5.1. Table 5.1 Measured mean peak to peak displacement of the steady state limit cycle for shapes 1 to 4. Mean peak to peak displacement µ d (radians) Pulse Shape 1 (sample =10) Pulse Shape 2 (sample =10) Pulse Shape 3 (sample =10) Pulse Shape 4 (sample =10) 1.440E E E E-4 Figure 5.5 A sample step input and position response using pulse Shape 2. Mean final oscillating displacement µ d =1.001e-4 radians for a sample of 10 repeated experiments.

138 119 Figure 5.6 A sample step input and position response using pulse Shape 3. Mean final oscillating displacement µ d = 0.957e-4 radians for a sample of 10 repeated experiments. Figure 5.7 A sample step input and position response using pulse Shape 4. Mean final oscillating displacement µ d = 0.901e-4 radians for a sample of 10 repeated experiments.

139 120 Summary The results given in Table 5.1 clearly indicate that the peak to peak displacement of the steady state limit cycle reduces from shapes 1 to 4 respectively. This reduction correlates to an increase in precision of the servomechanism for the same conditions Low Speed Position Tracking Using Pulse Shapes 1 to 4 The ability of each pulse shape to experimentally track a changing position is assessed by subjecting the friction test bed to; 1) a position ramp with constant low speed; and 2) a sinusoid with variable low speed. For the first set of experiments, the different pulse shapes are used to control the position tracking of a low speed position ramp with a continuous slope of rad/s. Figure 5.8 shows a superimposed and magnified result for a sample experiment using each pulse shape. Figure 5.8 Magnified linear position ramp response showing an experimental comparison between the impulse controller with pulses shape 1 to 4.

140 121 Figure 5.9 Integral Absolute Error (IAE) for impulse shapes 1, 2 3 & 4 for a low speed position tracking task. The difference in amplitude of the oscillations between the traces of each pulse shape represents the tracking error and consequent mechanical vibration experienced by the mechanism. By using the Integral Absolute Error (IAE) criterion, the error of each trace can be calculated and a performance measure between each pulse shape can be established. The standard solution to this problem is give by [68]: et () dt (5.2) 0 The experiment is repeated and the performance of each of the pulse shapes 1 to 4 compared. Figure 5.9 shows the comparison using the IAE criterion for ten repeated samples for each pulse shape. The results indicate that there is a progressive increase in

141 122 the precision of speed regulation as you change from shape 1 through to shape 4 respectively. For the second set of experiments, the different pulse shapes are used to control the position tracking of a low speed position sinusoidal trace with an amplitude of radians and a period of 4 seconds. As shown in Figure 5.10, the sinusoidal trace gives Figure 5.10 Experimental speed regulated sinusoidal position tracking using PID and PID + impulse controllers. Figure 5.11 Magnified velocity reversal showing an experimental comparison of precision between the impulse controllers with Shapes 1, 2, 3 & 4.

142 123 Figure 5.12 Separating Shape 1 and Shape 4 from Figure two velocity reversals per cycle and makes it necessary for the mechanism to track the input at velocities within the stick slip regime. Figure 5.10 shows the inadequacy of a standard PID controller to effectively trace the reference input. As a comparison, the combined PID + impulse controller is included to demonstrate the effectiveness of the impulse control scheme. The magnified result for the position tracking of pulse shapes 1 to 4 are shown in Figure At this section of the trace, the mechanism is tracking a velocity well within the non linear friction region and the effects of elastic pre-sliding motion can be observed. For clarification, Figure 5.12 shows the position trace using a sample experiment for shapes 1 and 4 only. A difference in vibration between each shape in the form of position oscillation can be clearly seen. By using the Integral Absolute Error (IAE) criterion, the error of each trace can be calculated and a performance measure between each pulse shape established. The

143 124 Figure 5.13 Integral Absolute Error (IAE) for impulse shapes 1, 2 3 & 4 for a sinusoidal position tracking task. experiment is repeated and the performance of each of the pulse shapes 1 to 4 compared. Figure 5.13 shows the comparison using the IAE criterion for each pulse shape after ten repeated samples. The results again indicate that there is a progressive increase in the precision of speed regulation from shape 1 through to shape 4 respectively. Summary In this section, the new set of impulse shapes has been shown to influence the dynamics of a mechanism with friction at very low velocity. It is shown that by adjusting the shape of the impulse, the energy applied to the mechanism can be adjusted so that the non linear effects of friction can be counteracted more efficiently. By using the Integral Absolute Error (IAE) criterion, the error of each trace could be calculated and a performance measure between each pulse shape could be established. The results show that for low speed position tracking using either a constant speed or variable speed reference, the precision of the servomechanism is increased from pulse shapes 1 to 4

144 125 respectively. Therefore, it is experimentally shown that pulse shaping is a successful method that can be used to improve servomechanism accuracy Impulse Height Blending for Pulse Shapes 3 and 4 For further refinement of the impulse controller, an improvement is made to the impulse controller to suit the pulse shapes 3 and 4. With these shapes, the leading section of the pulse shape is used to overcome static friction while the last section is used to overcome the Coulomb friction. For the condition where the position error e(k) is large, the impulse width becomes the full sample period τ s. In the case of the friction test bed, the design of the pulse shapes 3 and 4 uses a 1ms pulse to overcome static friction. When the error is very large, the pulse width saturates into the full sample width τ s, and the 1ms pulse is added to the PID + impulse controller torque requests. The result is a superfluous 1ms pulse which gives an unnecessary vibration on top of the controller output. Figure 5.14 Modified pulse shape as a function of increasing error e(k).

145 126 To counteract this, the controller is modified so that when the pulse width saturates, the pulse height linearly increases at a rate proportional to the error until it reaches the height of the 1ms pulse (Figure 5.14). The result of this modification is a controller output which provides a much smoother transition from the discontinuous to continuous mode Vibration Analysis One of the objectives of this thesis is to demonstrate that through improved impulse controller design, the vibration associated with intermittent pulse action can be minimised. This section uses vibration analysis to determine if there is any improvement in reducing the factors which contribute to machine wear and fatigue. For each of the experiments discussed in Subsection 5.3.2, the servomechanism vibration was recorded using an accelerometer. A Kistler type 5134 accelerometer is attached to the friction brake assembly as shown in the photograph of Figure Figure 5.15 Photograph of the friction test bed with the attached Kistler type 5134 accelerometer for the measure of system vibration.

146 127 A spectral analysis of the vibration measurements was performed using a Fast Fourier Transform (FFT). The spectral distribution for each pulse shape is shown in Figure 5.16 and gives the predominant frequencies and amplitude of vibration. Figure 5.16 Spectral analysis of system vibration using FFT for pulse shapes 1 to 4. Results indicate that there is a significant reduction in servomechanism structure vibration moving from shape 1 to shape 4. This weakening of vibration is expected to be directly correlated to a likely reduction in mechanical wear and fatigue of components, and a reduction in undesirable machining process marks. 5.4 High Speed Position Tracking So far, the hybrid friction controller has been used to improve the precision of low speed position tracking. This section investigates how the hybrid PID + impulse controller performs for higher velocities exceeding the Stribeck threshold. For this region of velocities, the highly non linear static and negative viscous friction components are substantially reduced and the Coulomb and viscous friction become the dominant resisting friction. For these velocities, the conventional linear PID controller is well suited. Subsequently, the addition of an impulse torque request may be deleterious to the servomechanism s performance in the region of higher velocity.

147 128 Figure 5.17 shows a series of varying ramp responses from 0.02 rad/s up to 0.35 rad/s using the friction test bed. The range of speeds ensures that the mechanism is operating in both the nonlinear and linear friction regions. Figure 5.18 compares the mean value of the absolute value of the error for each speed from 7 to 10 seconds respectively. For Figure 5.17 Tracking response for the friction test bed using PID and PID + impulse controllers for varying position ramps (0.02 rad/s to 0.35 rad/s). Figure 5.18 Mean value of the absolute error for each of the position tracking ramps shown in Figure 5.17 for the period 7 10 seconds.

148 129 the velocities below the Stribeck velocity threshold, the hybrid PID + impulse controller significantly outperforms the conventional PID controller. However, as the velocity increases, the mean errors of both the PID and PID + impulse controllers begin to converge (ω 0.08rad/s) and above the Stribeck region, the PID controller becomes slightly more precise. This increase in precision for the PID controller can be expected since for this higher range of velocities, a conventional linear PID controller will sufficiently counteract the linear fiction without the need of an additional torque. Although the loss of performance is minimal, these experiments show that combining the impulse action for high range velocities can be unnecessary and in some instances counterproductive. One way to avoid this slight loss of performance using hybrid controller is to disable the impulse torque request at higher velocities so that the PID part works autonomously. 5.5 Limit Cycle Offset Section 4.7 has presented the concept of Limit Cycle Offset as a means to improve the precision of a servomechanism controller. Simulation has demonstrated how the modified controller can theoretically improve the system precision for a model using non-position dependent friction variables. Conversely, this section evaluates the limit cycle offset function using the experimental friction test bed having position dependent variables. Figure 5.19 shows a steady state limit cycle for a position pointing step response of radians using a PID + impulse controller with a pulse shape 3. Pulse shape 3 has been chosen in place of shape 4 in order to simplify the PID + impulse torque output plot. The mean peak to peak displacement of the smallest non-elastic part of the limit

149 130 cycle is µ d and was measured in Subsection to be 0.957e-4 radians (mean µ d for ten samples). Figure 5.19 Steady state limit cycle for the PID + impulse controller using pulse shape 3. The mean peak to peak displacement µ d is the non-elastic part of limit cycle. The experiment was repeated using the limit cycle offset function with the same position step reference of radians. In this example, the impulse controller moves the mechanism in a single direction towards the reference set point until the sign of the error alternates and the steady state limit cycling begins. Figure 5.20 shows the limit cycle offset function being activated at t=0.58s. At this time, the amplitude of the non elastic part of the limit cycle is identified as lying between the switching bounds. The switching bounds and dead-zone are set according to the methodology given in Section 4.7. Once the offset function is activated, the controller adjusts itself by forcing the proceeding pulse to be one increment wider before returning to the smallest pulse width. This results in the limit cycle being shifted down into the dead-zone region where the

150 131 impulse controller is automatically disabled at t=0.6s. At this time, the final error is guaranteed to be less than the error dead zone; otherwise the cycle repeats itself. Figure 5.20 Using the Limit Cycle Offset function to reduce the final steady state error. One limitation of the limit cycle offset function is that it is only applicable for position pointing tasks where the reference set point is a fixed constant value. Conversely, for position tracking, the offset function must be removed to avoid unintentional disabling of the impulse part of the controller. 5.6 Impulse Control versus Tangential Dither The injection of high frequency AC signals, or dither signals, has been used in systems with friction. Tangential dither signals can be added to the output of the control loop

151 132 and Canudas de Wit et al [76] describes how dither signals can be added to the torque signal to give a new control torque T o such that: T o = T + T + A sin ω t (5.3) PID P o o where ω o is chosen to be greater that the system bandwidth, and A o is the amplitude of the dither torque used to exceed the static friction. The basic difference between the dither approach and the impulsive control is that the former is an open loop approach and the latter is a closed loop control method. Figure 5.21 PID control with velocity reversal compensation and gains Kp=70, Ki=130 and Kd=1.2. In order to evaluate the performance of a tangential dither, a reference position trace having a sinusoidal trace + step input + position ramp was used. For comparison reasons, the friction test bed was tested using PID control, PID + dither control and PID + impulse control (shape 4). Figure 5.21 shows the result of PID control. For PID control, stick-slip is evident and the servomechanism performs relatively poorly in all three position tracking tasks.

152 133 Figure 5.22 PID + tangential dither with amplitude A o =4.2 Nm and frequency ω o =250Hz. Figure 5.23 PID + impulse control using pulse shape 4. A tangential dither signal whose amplitude A o is 4.2Nm and period ω o is 250Hz is added to the PID output. Figure 5.22 shows the experimental result with stick-slip being eliminated. The figure shows that the injection of a dither signal gives a significant improvement on position tracking performance when compared to PID control.