ESTIMATION AND CONTROL OF A FLEXIBLE LINK Tito Carreno*

Transcription

1 ESTIMATION AND CONTROL OF A FLEXIBLE LINK Tito Carreno* * Department of Mechanical Engineering - Instituto Superior Técnico, Technical University of Lisbon (TULisbon) Av. Rovisco Pais, Lisboa, Portugal; tcarreno@netcabo.pt Abstract: The main goals of this work are to achieve the modeling of a flexible link and its active damping control. At first, the problem of the flexible links is presented. A solution through the use of piezoelectric actuators and sensors is proposed. The modeling of this structure is then done trough Finite Element Modeling (FEM) and Assumed Modes Modelling (AMM). The models obtained are further improved through the implementation of a model updating technique and a software implementation is developed. The final models are then used for the simulation of a PPF control strategy. Afterwards, an experimental implementation of this strategy is achieved. The results of this work are presented and some conclusions are drawn FLEXIBLE LINKS Introduction Robotic manipulator arms are and have been the main industrial robotic component. They are the most used solution in a large number of applications due to their capability of adaptation to a number of different task and their study has been a paradigm of robotics. Traditionally, they are big, heavy structures which require a considerable effort to operate becoming therefore expensive and energy consuming, while operating payloads that are largely inferior to their own weight. It is important to make robotic manipulator arms more light and agile in order to perform task more quickly and economically while allowing a greater mobility and flexibility of integration in the industrial environment. To achieve the desired weight reduction implies a stiffness loss which produces greater elastic deformation movements and vibration, thus deteriorating manipulator performance. It is from this problem that the interest in flexible link control is born: through the use of control schemes the link performance can be recovered while still maintaining the desirable structural lightness. This involves trajectory planning and tracking, so that the manipulator can move its end-point accurately from point A to point B in space, impedance (force) control, so that the payload is correctly manipulated and vibration control, in order to achieve the least oscillating performance possible. The leap between rigid link to flexible link brings a variety of problems. Rigid links have a finite number of degrees of freedom as opposed to flexible links which have an infinite number of degrees of freedom. This brings the need for other control techniques than the traditional ones. It also brings a loss in controllability and observability since joint measurement/actuation becomes insufficient. New ways of actuating and measuring are need. A field that has been seeking solutions to this problem is the field of smart materials and structures. SMART MATERIALS AND STRUCTURES A smart material is a material whose properties can be dramatically changed through neighborhood changes or external stimuli. For example, it can change its color when the temperature distribution around it changes. Nevertheless, for a material to be considered smart, the change on the material s property must be high when compared to the change that was applied. For instants, oil flows better when warm, however, some fluids can behave as solids once a slightly different stress distribution is applied. A smart structure is one that combines both smart and traditional materials (said to be dumb) with a high degree of integration and a control system to control it. PIEZOELECTRIC MATERIALS One kind of smart materials that has been very much used in structural control is piezoelectric materials. These materials display the so called piezoelectric effect: when subject to mechanical stress they develop an electric potential. This is called the direct piezoelectric effect but the inverse effect is also present: when subjected to an electric field, piezoelectric materials produce mechanical stress and/or strain. The constitutive laws for a piezoelectric continuum can be written in the following way (Martins, 2007): In which ε ij and σ ij are the strain and tensions, Epz and pz are Young s modulus and Poisson s ratio, d ij are the piezoelectric coupling terms, ξ i is the material permissivity, E i is the electrical field and D i the charge per volume. 1

2 Piezoelectric materials can occur naturally, although man-made materials are the most used. The most common of these is Lead Zirconate Titanate, or PZT, a ceramic. There is a number of reasons why piezoelectric materials are the most used smart material in structural control. They often have a high structural stiffness allowing the actuation to depend mainly on the electrical voltage, they have a wide frequency range of interaction with dynamic systems and the more recent developments in composite materials have brought more desired characteristics. The monolithic piezoceramics used in the past offered some problems: they were very brittle and typically lead-based (heavy). Modern composite piezoceramics have piezoceramic fibers immersed in a polymeric matrix, conjugated the pleasant properties of both and allowing for anisotropy and the development of directional actuators and sensors. (Crawley, et al., 1987) were among the first authors to approach the use of piezoelectric materials in the design of smart structures. They demonstrated their effectiveness both analytically and experimentally, in a variety of configuration and materials and shown that the size of the structure is not an important factor. The actuators and sensors used in this work are PZT Macro-fiber composites there is still work to be developed in the control of tridimensional flexible links with bending on both axes and torsion. The purpose of this work is to achieve the active damping of such a flexible link. Modelling Being distributed parameter systems, flexible links have been traditionally modeled through spatial discretization methods, since obtaining analytic functions describing the behavior of the whole structure is a complex task. Two main approaches are normally used: either the assumed modes method (AMM) or the finite element method (FEM). These two techniques serve a similar purpose: they approximate a continuous, infinite degree of freedom system by one with a finite number of the degrees of freedom. The basic difference between them is that while in AMM the solution is obtain through a finite series of admissible functions for the whole domain, in FEM the functions are used in a series of smaller subdomains, the finite elements. This brings the main advantage of FEM over AMM: versatility. The task of defining admissible functions for the whole domain in AMM is often complex and sometimes impracticable. In FEM simpler functions, often piecewise polynomials can be defined in a small subdomain, and then assemble together to form the model. That way, through FEM one can solve a variety of problems without having to find new functions, distributing the elements in a different way instead. However, when the geometry is simple and without sudden changes in system properties, FEM yields worse performances than AMM. This is due to the very high number of degrees of freedom needed by FEM to achieve good accuracy, making it computionally heavy. Fig. 1. The experimental setup THE EXPERIMENTAL SETUP The experimental setup in this work consists of a flexible link, essentially a polycarbonate tube with MFC PZT actuators and sensors bonded to its surface. It has a steel plate, which can be exchagend by a steel disk in one end and is clamped in an aluminum base on the other end. Its two sensors are placed at a 45º angle with the longitudinal axis and the three actuators are aligned with. Actuator 1 has is fibers also aligned with the tube, while actuators 2 and 3 have the fibers at a 45º degree angle with the longitudinal axis of the link. Figure tal is a photograph of the link. The link is controlled with the use of a real-time host-target system with DAC and ADC boards. PURPOSE OF THE WORK Most of the work done in flexible links has to do with the planar case. In the surveys the authors mention that 2 (Theodore, et al., 1995) have made a comparison between the two methods for the modeling of flexible manipulators. They recommend AMM for single-link or constant cross-section manipulators and FEM for variable cross-section or multi-link manipulators. Also, (Vakil, 2008) makes the recommendation of FEM for manipulator design and AMM for control and simulation. In this work, both approaches are used. Two models are built: a FE model, using the software ANSYS and a AM model, using the Matlab toolbox Mecanismo (Martins, 2007). After this, both models are updated through an optimization algorithm in order to better approximate experimental results. This technique is called model updating. MODEL UPDATING Model updating is the improvement of a parametric model by changing its parameters to optimally approximate experimental data. A program consisting of an interface between both modeling softwares (Ansys and Mecanismo) and Matlab was developed specially for this purpose. The objective of the optimization was to find the parameters that when supplied to the model

3 yield the closest frequency response to the experimental system possible. The program algorithm can be resumed as follows: 1. Write the first set of parameters in a file, so that the modeling software (Ansys or Mecanismo) can read it. 2. Calculate the natural frequencies of the system with these new parameters, and write them in a file so that Matlab can read them. 3. Calculate the error between these and the experimental results 4. If a stopping criteria is met, stop. Otherwise continue from step 2. The stopping criteria were: - Parameter variation smaller than a given tolerance (the program has stagnated) - Objective function value variation smaller than a given tolerance (convergence) - Violation of a constraint above a given tolerance The search space was reduced by the use of constraints that limited the parameter variation to an interval. The optimization method used was Matlab function fmincon from the optimization toolbox. MODEL PREPARATION As a last step in the modeling phase, the models were finalized to better allow its use for the next phase, the control. The models from Ansys were of very high order and hard to deal with. Therefore a model order reduction needed to be used. There are a variety of techniques to achieve this objective. In this work, the solution found was the identification of the Ansys frequency responses through the Levy method, yielding a reduced order model, more suitable for the use in control and simulation. The Mecanismo models were linearized through the use of the function linearize of the control systems Matlab toolbox. Control of flexible links The control of flexible links is a wide field of study. A thorough state-of-the-art survey is out of scope in this document. These can be found in (Dwivedy, et al., 2006), (Book, 1990), (Piedbouef, et al., 1993) and (Benosman, et al., 2004). One of the control techniques most relevant to this work is collocated control. COLLOCATED CONTROL Collocated control is a control through the use of collocated actuators and sensors. These are said to be collocated if the measurements and actuations are done in the same place in the structure. Though a strictly collocated is not the case of some systems, such as the one in this work, collocated systems have some interesting proprieties that bear relevance to nearly collocated systems (as the one in this work). These systems can be thought of collocated in a finite bandwidth if some properties are verified. For example, it is shown in (Preumont, 1997) that it is characteristic of the transfer function between an actuactor/sensor collocated pair of an undamped structure that: - The amplitude goes to +- infinito at the resonance frequencies, in which there is a pair of pure complex conjugated poles. Phase suffers a lag of 180º. - Between each consecutive resonance frequency there is an anti-resonance where the amplitude tends to zero, and a pair of pure complex conjugated zeros occur. Phase suffers a 180º lead. This property, which shall be referred to as pole/zero alternating, is important for the following reasons: - The undamped system has its poles and zeros in the limit of stability. If some damping, however small, is added, the poles and zeros will move further to the left semi-plane, that displacement being larger as the frequencies of the poles/zeros increase; - Thanks to this property, a root locus analysis shows that the Fig. 2. Root locus of a collocated system branches of the closed loop also remain in the left semi-plane. Therefore, the system does not have problems with instability or non-minimum phase behavior. If the actuator/sensor pair is not collocated however, the pole/zero alternating is not guaranteed and therefore neither is stability. Furthermore, even if the actuator/sensor pair is collocated, pole-zero flipping can occur, if we have compensator zeros cancelling system poles or by altering system parameters. If pole zero flipping occurs, the closed loop may become unstable since the root locus branches now go through the right semi-plane. Also, one must bear in mind that guaranteed stability does not mean guaranteed performance. The latter is strongly dependent of good controllability and observability, which in turn depend on good actuactor and sensor placement. Another advantage of the pole/zero alternating is that it allows the design of controllers separately for each actuator/sensor pole as an independent SISO system. The fact that the system in this work is not strictly collocated does not mean that there are not benefits in having the sensors and actuators in the surface of the link. The uncertainty in the transfer function is reduced by doing so, causing the robustness to be improved. 3

4 % ADDITIONAL CONSIDERATIONS There are other important properties are important in the control of flexible structures. A system should have enough roll-off to accommodate signal condition dynamics, and the inevitable phase lag. Roll-off is the decay of the gain with frequency. It is a desirable property for it leads to less sensibility to noise, disturbances and it helps with spillover. Spillover is a very important phenomenon in the control of flexible structures. Spillover is due to the fact that unlike the real system, the model upon which the controller was designed has a finite number of degrees of freedom, and therefore a finite number of vibration modes. Spillover occurs when the controller excites nonmodeled modes, possibly causing instability, because the output becomes contaminated with modes that the controller cannot deal with. Hence, the term spillover, because it as if the control energy has spilled over the modeled bandwidth. 4. Calculate the closed-loop damping ratio ξ CL from equation (4). 5. Calculate the required PPF filter damping ratio ξ f from equation (3). ( 5 ) ( 3 ) ( 4 ) In which, w pz = w p / w z is the pole/zero spacing. Fig. 3. Optimal pole location according to McEver POSITIVE POSITION FEEDBACK The strategy chosen for the control of the flexible link in this work was Positive Position Feedback (PPF). This technique was introduced by (Goh, et al., 1985). The choice of this control strategy was due to the fact that the actuators and sensors used in this work were strain piezoelectrics and also the attractive properties of this approach. PPF control is robust against uncertainty in structural properties, does not require an analytical model of the system, has guaranteed stability given some conditions (Fanson, et al., 1990), and the design of controllers is simple. This has led to several applications ((Dosch, et al., 1992), (Fagan, 1993), (Agnes, 1997), (McEver, 1999), (DeGiulio, 2000), (Shan, et al., 2005)). The terminology gives away the control loop: the position is fed back with positive sign. A PPF controller is a essentially a second order filter: PPF filters can be designed individually for each mode for a given actuator/sensor pair. The filter is tuned by selecting the appropriate parameters: the filter gain, damping ratio and natural frequency. This tuning has been widely done in literature, most approaches being by trial and error. McEver address this problem by devising an algorithm that yields the optimal parameters in one try. This is achieved by observing that the optimum closed loop pole locations are the ones that lead to the maximum damping, and that that is equivalent to having the poles of the system and the controller in a way that they have the same damping. After this simple conclusion, through some mathematical manipulation the following algorithm is derived : 1. Choose PPF filter gain g, positive if w pz > 1, negative if w pz < Calculate w fp from equation (5). 3. Choose the closed-loop pole spacing α to be 1. This algorithm was conceived to collocated systems, where there is always pole-zero alternating. This is not the case in the present work. However, calibrating the filter using a frequency 10% bigger than that of the pole as the frequency of a fictitious zero (in a collocated mode the zero normally occurs within that range of the pole) has revealed to be a good starting point, but unfortunately does not eliminate the trial and error process. The control in this work was first implemented in Simulink, using the models obtained in the modeling phase to simulate de system and then implemented in the actual experimental setup. Results Some of the results are presented from this work ,25 9,95 21,69 19,07 Conclusions 1,39 0,01-0,08-0,74 0,62 13,66 D thk E rho NU12 Ep1 Ep2 L rho_ste_st Lx ax tx 3,33-8,54-15,25 Fig. 4. Variation of the parameters model in FEM, with plate 4

5 % Fig. 8. Simulation of the control. FEM model with plate. Blue - uncontrolled; Red - controlled Fig. 5. Frequency response. FEM model, with plate. Blue - experimental; Red -Model , ,43-28,95-8,26 50,00 48,75-99,98 13,66-61,53 123,33 D thk E rho L rho_st E_St Lx ax tx -500 Fig. 6. Variation of the parameters - model in AMM, with plate Fig. 9. Simulation of the control. Frequency response. FEM Model with plate. Blue - uncontrolled; Red - controlled Fig. 7. Frequency response. AMM model, with plate. Blue - experimental; Red - Model 5 Fig. 10. Experimental results. Blue - uncontrolled; Red - controlled

6 Conclusions Finite element modeling provided better results than the assumed mode method, but it is a lengthy process. The final model is also cumbersome and needs to be reduced to be used in the control and simulation, and the AMM doesn t need this step. Also, in the scope of this work, it is probably more interesting to have a good approximation of dynamics of the system than of the exact natural frequencies of the system. In a model-base control strategy however, it is of paramount importance. Although PPF control confirmed the attractive properties described in the literature, the tuning of a PPF filter for each mode and for each sensor/actuator pair can become time-consuming. In nearly collocated systems this is still a trial and error procedure. The control strategy revealed good results: the vibration was considerably reduced, even though only at the resonant frequencies. The rest of the bandwidth remains approximately with the same damping. This suggests an approach where PPF filters are used in a master/slave configuration together with more sophisticated controllers. That being said, PPF has shown it can be successfully used in the control of tridimensional links subjected to bending in two axes and torsion to increase the damping of the system s natural frequencies. Martins J. Modelling Identification and Control of Flexible Structures McEver M. Optimal Vibration Suppression Using On-line Pole/Zero Identification Piedbouef J. [et al.] Modelling and Control of Flexible Manipulators - Revisited // IEEE Preumont A. Vibration Control of Active Structures - An Introduction - [s.l.] : Kluwer Academic Publishers, Shan J., Liu H. e Sun D. Slewing and Vibration Control of a Single-link Flexible Manipulator by Positive Position Feedback // Mechatronics Theodore R. e Ghosal A. Comparison of the Assumed Modes and Finite Element Methods for Flexible Multilink Manipulators // The International Journal of Robotics Research Vakil M. Dynamics and Control of Flexible Manipulators References Agnes G. Performance of Nonlinear Mechanical, Resonant-Shunted Piezoelectric, and Electronic Vibration Absorbers for Multi-Degree-of-Freedom Structures Baz A., Poh S. e Fedor J. Independent Modal Space Control with Positive Psoition Feedback // Transactions of the ASME Vol Benosman M. e Le Vey G. Control of flexible manipulators: A survey // Robotica Book W. Modeling, Design and Control of Flexible Manipulator Arms: A tutorial review // Proceedings of th 29th Conference on Decision and Control Crawley E. e Luis J. Use of Piezoelectric Actuators as Elements of Intelligent Structures // AIAA Journal DeGiulio A. A Comprehensive Experimental Evaluation of Actively Controlled Piezoceramics with Positive Position Feedback for Structural Damping Dosch J., Inman D. e Garcia E. A self-sensing piezoelectric actuator for collocated control // Journal of Intelligent Materials, Systems and Structures : Vol. 3. Dwivedy, S. e Eberhard P. Dynamic Analysis of Flexible Manipulators, a Literature Review // Mechanism and Machine Theory Fagan G. An Experimental Evaluation into Active Damage Control Systems Using Positive Position Feedback for AVC Fanson J. e Caughey T. Positive Position Feedback control for large space structures // AIAA Journal Goh C. e Caughey T. On the stability problem caused by finite actuator dynamics in the control of large space structures // International Journal of Control