FlexLab and LevLab: A Portable Lab for Dynamics and Control Teaching


 Barrie Lawson
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1 FlexLab and LevLab: A Portable Lab for Dynamics and Control Teaching Lei Zhou, Mohammad Imani Nejad, David L. Trumper Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 39, USA Abstract This paper presents the design, modeling and control for an educational lab for control and mechatronics teaching in the department of Mechanical Engineering at MIT. The FlexLab is a system of flexible cantilever beam with permanent magnets attached to it, while the LevLab demonstrates a magnetic suspension system. Both labs are implemented on one printed circuit board, with actuators, sensors, power amplifiers arranged on it. This system works together with myrio from National Instruments. The design file for the system open source on the website of Precision Motion Control Lab at MIT. We would like to share the lab with others who might like to use it in their dynamics and control teaching. Keywords: educational lab, mechatronics, flexible beam, magnetic levitation. Introduction One major challenge in engineering course is offering student the chance of handson experiences, which is essential to obtain a fundamental understanding of phenomenas and design skills. However, in many circumstances, limited lab resources and concentrated lab time for classes may limit the depth of knowledge that student can learn. To address this, instructors for control system at MIT generated the idea of "portable educational labs". We believe that if we can make the lab portable and lend the hardware to student, the student can have chance to play with the system, discover the subtleties outside the structured lab time, and therefore achieve a better understanding of the knowledge. This will requires the educators to design a lowcost, portable, robust system that can demonstrate the intrinsic characteristics that make it applicable to a wide range of educational pursuits. Figure : Photograph of the FlexLab system. This paper introduces the design, modeling and control of the FlexLab/LevLab board, which is one of our our portable educational lab series. The design files of the lab can be downloaded at the website of Precision Motion Control Lab at MIT. The FlexLab is a system of flexible cantilever beam with permanent magnets attached to it, while the LevLab demonstrates a magnetic suspension system. Both labs are implemented on one printed circuit board, with actuators, sensors, power amplifiers arranged on it. This system works together with myrio from National Instruments [].. Hardware configuration Figure shows the hardware configuration of the FlexLab system, which is implemented as one printed circuit board (PCB). This system works together with NI myrio via the standard connector. The same system can be used for magnetic suspension experiment, known as LevLab. Figure shows the magnetic suspension with this system. In the design of the FlexLab, a inch long flexible cantilever beam made out of.5mm thick phosphore bronze sheet is anchored on the PCB. Two pair of disk shape permanent magnets are attached at the middle and at the end of cantilever beam, as shown in Figure. When the tip displacement of the can Preprint submitted to Mechatronics Figure : Photograph of the LevLab experiment November 7, 4
2 +5 V +5 V R m C m 8 V ref +  R R Coil R s +  R 4 R 3 V ref sum of sensor output(v) measured fitted V in Figure 3: Circuit diagram of the coil power amplifiers. The op amps are the power amplifiers. Here V re f is V. V re f is.5 V. Controlling signal in injected at V in. The amplifier gain is selected to be 4. Therefore R = 4R, and R 3 = R 4. R s is the sensing resistor for current measurement. R m and C m are placed for stability. In our design R m = Ω, and C m =. µf gap(mm) Figure 4: Calibration data of the hall effect sensors for position measurement. tilever is small, this system presents a linear, lightly damped massspring system. A complete electromechanical feedback system is built around this cantilever beam system, all integrated on the PCB. The controller for the system is implemented in the NI myrio. On the PCB, two spiral coils are embedded in the PCB via routing. Each coil has 3 turns, with 4 layers in total and 8 turns per layer. By driving current through the coils, a force can be generated to the cantilever beam through the interaction between the actuating current and the magnetic field from the permanent magnet. The coils are driven by linear power operational amplifiers to get rid of switching noise. In our design, TCA37 from ON semiconductor [] is selected to drive the coils. In order to make better use of the power supply from the NI myrio, single side power supply (  5V) is selected for the power operational amplifiers. The power amplifiers are differentially configured to provide the coil with both positive and negative currents. Figure 3 shows a circuit diagram of the coil power amplifier. Here the controlling signal is injected at V in, which is generated by the D/A converter of the myrio. The amplifier gain is selected to be 4. Therefore R = 4R, and R 3 = R 4. R s is the sensing resistor for current measurement. R m and C m are placed for stability. In our design R m = Ω, and C m =. µf. Two hall effect sensors are configured at both sides of every magnet on the beam to measure the position of the permanent magnet, which is also the local displacement of the cantilever beam. The signals from the two hall effect sensors around one magnet are added together and goes into the A/D converter of the myrio. The controller for the system is implemented digitally in the myrio. Figure 4 shows the calibration data of the hall sensors, with its xaxis being the vertical position of the magnet (gap between the PCB board and the magnet) while its yaxis is the sensor output. Data shows the position measurement by the hall sensors is approximately linear. The same system can be used for magnetic suspension experiment. If we take off the flexible beam, and correctly design the controller for the system, a spherical permanent magnet can be Figure 5: Principle of magnetic force generation. magnetically levitated underneath the coil in PCB. We call this system LevLab, as is shown in figure. 3. Modeling and system identification In this section the modeling and system identification for both FlexLab and LevLab systems are introduced. Here lists the assumptions that our model is based on: The motion of the permanent magnets have only one degree of freedom; Hall effect sensors and voltage control power amplifiers are linear and have no dynamics; Small deviation from the operating point is assumed; Inductance of the coil is small. 3.. Electromechanical interaction and the force generation In order to model the dynamics of the system, we first need the expression for the magnetic force that the coil applies to the magnet. This force is Lorentz force that generated by the field from magnet and the current in the coil. Figure 5 shows the principle of the magnetic force generation. In Figure 5, the permanent magnet field is modeled as a dipole pattern. When a current i is flowing in the coil, a Lorentz
3 force is generated to the coil. The Lorentz force can be calculated by F = J B, therefore the force in the horizontal plane is generated by the vertical component of the magnetic flux B x, while the force in the vertical direction is generated by the horizontal component of the magnetic flux. The Lorentz forces in the horizontal direction are canceled due to symmetry, as a result, the total force that the magnet applies to coil is in the vertical direction. We can get the form of the magnetic force as: f magnetic = C i. () Here the C is a constant that is determined by the strength of magnet, the number of turns of the coil and the geometry. x is the distance between the coil and the permanent magnet. We will use this expression for the magnetic force in the modeling for both FlexLab and LevLab systems. 3.. LevLab modeling and system identification In this section the modeling and identification of the LevLab system is presented. In this electromagnetic system, energy is transfered from electrical domain to mechanical domain. Figure 6 shows a diagram of the magnetic levitation system. equation, we define the following constants: K i = C [N/A] : Force constant x (5a) K s = C i [N/m] : Negative sti f f ness x (5b) 3 By substituting these constants and removing the variation sign δ we reached the dynamic equation for the magnet s movement is mẍ = K i i + K s x. (6) Notice that here the x and i represents the small variations from the operating point. For the electrical domain, the mechanical system influence the electrical domain via back emf of v em f = K i ẋ. Here the constant K i is the force constant shown in Equation (6). Since the inductance of the coil is very small, the electrical dynamics is much faster than that of mechanical domain. Therefore it is reasonable for us to ignore the dynamics of the circuit by assuming the inductance value L =. According to the Kirchhoff s circuit law, the equation for the circuit can be written as: e K i ẋ = ir. (7) Substituting Equation (7) into Equation (6), the equation for the system can be achieved: Figure 6: Diagram for DOF magnetic levitation. In the mechanical domain, the dynamics of the levitated permanent magnet is: mẍ = mg f magnetic. () Note that here positive direction of x is pointing down, which corresponds to increasing the distance between the magnet and the coil. By substituting the expression for magnetic force, we reached the dynamic equation for the magnet: mẍ = K s x + K i ( e R K i ẋ) (8) R Choosing state variables = x and = ẋ, the input signal to the power amplifier to be the control input u, and the output signal of the hall effect displacement sensors to be the system output y. We can rewrite (8) into a state space representation as: = K s m K i mr y = [ g sensor + ] g amp K i mr u (9) () Using Laplace transformation we can reach the transfer function of the system: mẍ = mg C i. (3) Linearizing this equation about operating point x and i and we reached the linear differential equation: mδx = mg C i x C x δi + C i δx. (4) x 3 The constant part of the magnetic force is balanced with the magnet s weight, which is mg = C i x. In order to simplify the 3 K i mr Y(s) U(s) = g sensorg amp s + K i mr s K s m () Here g amp [V/V] is the gain of the power amplifier, while g sensor [V/m] is the gain of the hall effect displacement sensor. The LevLab system parameters are being identified by measuring the system dynamics. Table presents the design parameters of the LevLab system. Figure 7 shows the experimentally measured and fitted Bode plot of the LevLab plant plotting together, with the signal to the power amplifier being the input,
4 Table : Design parameters of the LevLab system st mode Magnet mass m.5 [kg] Power amplifier gain g amp Hall effect displacement sensor gain g sensor.5 [V/V].8 [V/mm] i i st mode Resistance of the coil R 6 [Ω] Figure 8: FlexLab system and mode shape. Magnitude Measured Model Figure 7: Measured and modeled plant bode plot of the LevLab. b F k k m m b Figure 9: Massspring model of the FlexLab system. Based on Figure 9, the differential equations of the system dynamics can be written as: m x = F k b + k ( ) + b ( ) m x = F k ( ) b ( ) F (3a) (3b) and taking the sensor signal as the output. Notice that this Bode measurement is taken with the system under closedloop control, since the system is inherently unstable. The fitted transfer function is: Y(s) U(s) =.68 (s + 6)(s 6). () By substituting values in Table into Equation () and comparing it with (), we can calculate that K s = 5.34 [N/m], and K i =. [N/A]. Therefore we can calculate the electrical damping value as b e = K i mr =.53 [Ns/m] FlexLab modeling and system identification This section introduces the modeling and identification of the FlexLab system. The FlexLab system uses the same PCB with the LevLab. In our control teaching labs, the FlexLab system can use only one set of magnets arranged on the tip of the cantilever to demonstrate a lightly damped nd order system, and the coil in the middle can be used to add disturbance to the system. The FlexLab system can also work with two sets of magnets, which can let the students to explore the complete vibration dynamics of the system. Here we present the modeling and the identification of the complete system dynamics of the FlexLab system. Figure 8 shows a diagram of the FlexLab system. Here the two coils are driven with currents i and i respectively, and magnetic forces are generated between the coils and the magnets that introduced in Section 3.. When driving the system into resonances, the system will present different mode shapes. A fourthorder massspringdamper model is being used to model the FlexLab system. Figure 9 shows the linear system. 4 Here the forces F and F are the forces the coils acting on the magnet. The calculation of these magnetic forces is discussed in Section 3.. By linearizing these force about an operating point as discussed in Section 3., the forces can be expressed to be linear with the supplied voltage to the power amplifier, with a coefficient of K i g amp /R. Define the input voltages of the two coils to be u and u, and selecting state variables x = [ ] T, the system can be written in the state space form, as: d dt = y y k +k m b +b m k m b m k b m m k m b m m + m = g sensor g sensor K i g amp R u K i g amp R u (4) (5)
5 Magnitude Magnitude Model Measured 4 Model Measured Figure : Bode plot of the FlexLab system /u. Figure : Bode plot of the FlexLab system /u. Magnitude permanent magnets on the beam. In this mode, the system demonstrates a nd order system after linearization. A step response of the system is shown in Figure 3. It is shown that this system is very lightly damped inherently. The system demonstrated a natural frequency of 38 rad/s ( Hz) and a damping ratio of ζ = Model Measured response(v).5 Figure : Bode plot of the FlexLab system /u. Note that in Equation (4) the stiffness values are the sum of mechanical stiffness of the cantilever beam and the negative stiffness between the magnet and the coil, while the damping values are a combination of both air drag damping and electrical damping due to back emf. The identification of the parameters for the FlexLab is similar to that for the LevLab, except that this is a twoinputtwooutput system. Figure through Figure present the experimentally measured and modeled Bode plot for the FlexLab system, with different input and output selection. We can see that the system zeros are appearing in the collocated measurement, and have no zero in noncollocated measurement. By matching the pole and zero positions to the measured Bode plot, we identified the system parameters as m =.3 kg, m =.4 kg, k = 8 N/m, k = 9 N/m, b =.5 Ns/m, b =. Ns/m. The modeled Bode plot of the system with these identified values are plotted in blue in Figure through Figure. Good match between the model and the measured data confirms our modeling. The FlexLab system can also operate with only one pair of time(s) Figure 3: Step response of the FlexLab system with tip magnet only. 4. System control design When operating, the FlexLab and LevLab system need to operate under closed loop control to be stable and achieve better performance. The feedback control system design and analysis are discussed in this section. 4.. Stabilization of the LevLab In this section the control for the magnetic suspension for the LevLab system is introduced. As discussed in Section 3., there is one right plane pole in the plant transfer function of the LevLab system, which makes the system unstable in openloop. As a result, feedback control is needed to stabilize it. The controller s design for the LevLab is based on the Lev Lab plant dynamics, which is depicted in Figure 7. In our design, series compensation to stabilize this magnetic suspension
6 Magnitude Magnutude Plant Controller Loop Return Ratio 3 4 Plant Loop without notch Loop return ratio controller Figure 4: Bode plot of loop shaping control design for the LevLab system. Figure 5: Bode plot of loop shaping control design for the FlexLab system. system. This is the approach that is generally used in practice as it only assumes the measurement of the magnet s vertical position. Leadlag form of the PID controller is used for both stabilization and providing better disturbance rejection ability. The controller form is selected to be: G c (s) = K p ( + T i s )ατs + τs + (6) For the LevLab system, the lead network is chosen to have a polezero separation factor α =. The loop is designed to cross over at rad/s, thus we can calculate the lead time constant τ =.6 s to place the phase maximum at the desired crossover frequency. The integral gain for the system /T i is chosen to be 4 rad/s. Figure 4 shows a Bode plot of the loop shaping control design for the LevLab system. As a result, the system can reach a 35 Hz ( rad/s) crossover frequency with a 4 o phase margin. 4.. Position control of FlexLab In Section 3.3 we identified the system dynamics of the FlexLab, where we can see that the system has very lightly damped resonances. To better control the position of the magnet on the cantilever beam, feedback control is needed to add active damping to the system. In this experiment, the actuator at the tip of the cantilever is being used as the control input, while the actuator in the middle can be used to inject disturbance. A lead compensator is used to add damping to this position control system. In order to crossover at a higher frequency, a notch filter is used to suppress the second resonance and maintain stability. The Bode plot of the plant, controller, and the corresponding loop return ratio is shown in Figure 5. The plot shows that the notch of the controller can effectively hit down the notch in the plant. With the designed controller, a crossover frequency of 5 Hz is reached, with a phase margin of 4 degrees. Figure 6 shows a closedloop step response of the magnet position and the corresponding control effort signal of the FlexLab system. 6 Closed loop response(v) Control Effort(V) time(s) time(s) Figure 6: Bode plot of loop shaping control design for the FlexLab system Selfresonance control for FlexLab The FlexLab system can also be used to demonstrate selfresonance control, which is an important building block for a taping mode atomic force microscope probe [3]. Here let us consider the FlexLab with only the magnets on the cantilever tip, which makes the system a secondorder system after linearization. By selfresonance control we can regulate the system at its selfresonance and control its oscillation amplitude. The details of the selfresonance control is introduced in [4]. For a secondorder linear system with poles at σ ± jω n, the system will have a response as y(t) = Ae σt sin(ω n t + φ). (7) Ideally, when the real part of the system poles become, the system is marginally stable, and will demonstrate sustain oscillation. However, in practice, it is not possible to set the real part of the pols to exactly and make a perfect marginally stable system. The closedloop poles will eventually have a positive or negative real part, and the closedloop system will become either stable or unstable. To make the cantilever beam in the
7 Figure 7: Block diagram of FlexLab selfresonance amplitude control. FlexLab a pure oscillator, a closedloop amplitude controller, which measures the oscillation amplitude and adds positive or negative damping to the system to control the amplitude of the self resonance. With one pair of magnets attached to the cantilever tip, the natural frequency of the system is measured to be ω n = 38 rad/s ( Hz). It is possible to show that the oscillation amplitude envelope of the FlexLab changes as A(t) = A e ζω nt. (8) Here A is the initial amplitude of the oscillation, and ζ is the system damping ratio. Although the relationship between A and ζ is nonlinear, we can linearize it around a set point and apply linear control theory to design the controller for the oscillation amplitude control. A firstorder Taylor series linear approximation of Equation (8) about t = yields A(t) = A A ζω n t. (9) With the oscillation frequency fixed at ω n, the oscillator dynamics with the oscillation amplitude change A being the output and the damping ratio ζ being the input can be reached by applying a Laplace transform to Equation (9): Magnitude (db) Phase (deg) Frequency (rad/s) Plant Controller Loop return ratio Figure 8: Bode plot for FlexLab oscillation amplitude control design. In the control loop shown in Figure 7, the dynamics from ζ to A can be modeled as a transfer function () with a lowpass filter, where the lowpass filter represents the dynamics of the amplitude estimation functions. In our implementation the filter is designed as A(s) ζ(s) = A ω n. () s The above transfer function shows that the incremental change in the envelope of oscillation A(s) is proportional to the integral of the damping ratio ζ. We can design a controller to control the oscillation amplitude of the FlexLab system in real time based on the plant transfer function in (). Figure 7 shows a block diagram of the control system implementation. The oscillatory magnet position signal is measured by the hall effect sensors and being acquired into the embedded controller (NI myrio) via A/D conversion. A absolute value function (rectifier) and a lowpass filter is used for the measured signal to get an estimation of the envelope amplitude of the oscillation signal. The amplitude signal A is then compared with a reference amplitude A re f, and the error goes through the controller and generated the control effort signal ζ, which is the controlling damping ratio (positive or negative) added to the system. The damping signal is generated by multiplying ω n and taking derivative to the signal (added another pole for filtering), and injected to the FlexLab system. 7 LPF(s) = s/ +. () The reference amplitude of oscillation is set as A r e f = A = V. Figure 8 shows the Bode plot for the oscillation amplitude controller design. Here the plant transfer function demonstrate the dynamics from control effort ζ to amplitude estimation Â. Note that the minus sign in Equation () is reversed for design purpose. A PI controller with a form of C(s) = K p ( + /T i s) is selected for the system. By targeting at a crossover frequency at ω = rad/s and a phase margin of 4 o, the controller parameters are selected as K p =.35 and T i =.. The control loop shown in Figure?? is implemented on the FlexLab system with myrio controller. Figure 9 shows a waveform of the system output under selfresonance amplitude control under step changes of the reference amplitude A re f, and Figure and shows the corresponding envelope amplitude estimation and the control effort signal ζ. Note that the signals are asymmetric due to the nonlinearity, with the plant transfer function () depend on the current amplitude A.
8 response(v) system analysis and control at MIT. We have attempted to describe this system and its related experiments in sufficiently detail so that it can be readily duplicated by others who might like to use it in their control teaching efforts. The design file of the system is available to be download at the website of Precision Motion Control Lab at MIT. We welcome comments, questions, or suggestions for improvement of this lab design and exercise time(s) 6. Acknowledgments The authors would like to thank National Instruments for funding this project. Figure 9: Waveform of FlexLab selfresonance amplitude control. Envelope amplitude(v) References. NI myrio Hardware at a Glance. Tech. Rep.; National Instruments; 4.. On semiconductors.. A Output Current, Dual Power Operational Amplifiers. TCA37 Datasheet; Trumper DL, Hocken RJ, AminShahidi D, Ljubicic D, Overcash J. Highaccuracy atomic force microscope. In: Control Technologies for Emerging Micro and Nanoscale Systems. Springer; : Roberge JK. Operational amplifiers. Wiley; Time(s) Figure : Amplitude signal of FlexLab selfresonance amplitude control Control effort ζ Time(s) Figure : Control effort signal ( ζ) of FlexLab selfresonance amplitude control. Here ζ = ζ + ζ, where ζ is the system damping ratio, ζ is the controlling damping ratio added through control, and ζ is the total damping ratio. 5. Conclusion In this paper, a compact mechatronics system that can work as either a flexible cantilever beam or a magnetic levitation system is reported. This system will be used in teaching feedback 8
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