Practical implementation of flatness based tracking and vibration control on a flexible robot

Transcription

1 Practical implementation of flatness based tracking and vibration control on a flexible robot Jan Polzer, Dirk Nissing Faculty of Mechanical Engineering Department of Measurement and Control (Prof. Dr.-Ing. H. Schwarz) University of Duisburg Duisburg, Germany Keywords: nonlinear control, flatness, vibration damping, tracking control, hydraulic actuators Abstract Using robots for very wide operating ranges or for heavy loads requires taking the elastic deformations of the links into account. Thereby tracking a trajectory at the end effector is a demanding task. This paper deals with the practical implementation of a control concept which allows trajectory tracking and vibration damping. For the testbed PSfrag anreplacements elastic robot arm is chosen. The tracking control is realized using a flatness based approach and for the vibration damping the virtual spring damper control concept is used. The experimental results are included. 1 Introduction Figure 1: Hydraulically driven flexible robot For robots without elastic deformations, the creation of an analytic model and the control of the plant is not a difficult task. But if heavy loads are to be moved or wide operating ranges are considered then elastic deformations have to be taken into account. The surveyed testbed is a robot with an elastic link manufactured of spring steel actuated by hydraulic differential cylinder as shown in figure 1. The aim of the control concept is the tracking of a reference trajectory without oscillation of the end effector. To attain this goal the control concept proposed by [9] is implemented. Although the model of the differential cylinder is not flat, a flatness based approach for the tracking control is successful if some of the states are measured. To eliminate the vibrations in the flexible beam, the non-model based control concept of a virtual spring damper is used. The input/output behaviour of the actuator is controlled to act like a spring damper system. In the second section the testbed and its modeling will be explained. Section 3 focuses on flatness and hydraulic cylinders. The chosen control concept is shown in section 4. The focus of section 5 is on the results of the pratical implementation. The last section gives a conclusion and an outlook for PSfrag replacements further research. 2 Testbed modeling 2.1 Hydraulic differential cylinder The robot s revolute joint is driven by a hydraulic translatory drive within a closed kinematic loop to transform the translation of the drive into a rotation of the joint. The cylinder s schematics are shown in figure 2 [9]. Figure 2: Schematic view of the hydraulic cylinder

2 PSfrag replacements Under the assumption that: there are no gravitational effects, the chamber volumes are constant, there is no leakage, the servo valve has proportional behaviour and the states are defined as : piston position, : piston velocity, : oil pressure chamber A, : oil pressure chamber B a nonlinear model can be set up as in [9]: Figure 3: Substitute model for the flexible beam (6) (1) In practice the beam parameters have to be identified from meassured data. They were found to be, and [7]. 3 Differential algebra and flatness with if if The input voltage is applied to the servo valve. The friction force is a combination of viscous friction, static friction and coulomb friction : Its time derivative can be approximated by: 2.2 Substitute model for the beam The best way to quantify the control concept s quality, is to measure the positon of the end effector. In practice this is quite difficult and expensive. To visualize the dynamic behaviour of the end effector, a model of the flexible link can be used. Because of significant problems in obtaining or validating an analytical model for the flexible beam [7] suggests using an identified substitute model. This substitute model describes the essential dynamics and vibration behaviour of the flexible beam. The substitute model consists of a mass, a spring and a damper, as pictured in figure 3. This assumption is warrantable, because experimental results show that only the first natural frequency is of significance [1]. The equations of motion are established by the impulse theorem or the second order Lagrange equations [7]. The resulting equations that describe the vibrations are: (2) (3) (4) (5) Differential algebra was introduced through the mathematician Ritt in the 1950 s. In the 1980 s Fliess applied the field of the differential algebra to nonlinear control theory. Flatness [5, 8] is a rather new field of research in differential algebra. For an introduction to differential algebra see [2, 4]. The definitions which are used in this paper are: Definition 1 [3, 4] A nonlinear input/output system is a differential field extension which is differential algebraic. The finite set is called the input and is called the output. Definition 2 Equivalence [5]: Two systems and are called equivalent or equivalent by endogenous feedback iff any element of (or resp.) is algebraic over (or resp.). Two dynamics and are said to be equivalent iff the correspnding systems, and, are so. The term of an equivalent system is very important in this context: Definition 3 Flatness [5]: A system is called (differentially) flat iff it is equivalent to a purely differential transcendental system. A differential transcendence basis of with the property is called linearizing or flat output of the system. As described in [8] the flatness of a system implies that every system state and every system input can be calculated directly from and its time derivatives. The measured output of the hydraulic differential cylinder is not flat, but by measurment of the states and the main idea of flatness enables trajectory tracking [9]. The force can be calculated with the help of and but in this approach is also measured to maintain accuracy.

3 off-line calculation: reference trajectory,,,,,, nonlinear precontroller vibration damping,, cylinder & valve 2nd order controller Figure 4: Flatness-based control of a differential cylinder with additional vibration damping 4 Controller design The flatness based (pre-)controller is used to bring the system s output into the region arround the reference trajectory. Additional the vibration damping is included. And finaly a 2nd oder position controller is added to improve accuracy. The complete control concept is shown in figure 4. actuator. The basic concept of the vibration damping strategy by a virtual spring damper element will be briefly explained for a simple flexible beam. For more details see [1]. The hy- PSfrag replacements 4.1 Flatness based controller For the derivation of the flatness based controller see [9]. The main difference here is the inclusion of the time derivative of the cylinder force. Distinguishing between the cases for and yields (with, ): The control law (7) and (8) is in some sense inconvenient, since for the calculation of the case or has to be distinguished. The denominator of is always greater than zero [9]. Additionally, the numerator is the same for both cases, so first the numerator is evaluated and, depending on its sign, the corresponding denominator is chosen. (7) (8) Figure 5: Flexible robot with a spring damper element draulic cylinder is treated as a virtual passive spring-damperelement. The schematics are illustrated in figure 5. It is not necessary to equip the actuator with a real mechanical element. The piston position is measured by a pulse generator and the force acting on the piston rod is measured by a force sensor. The robot s arm can then be driven by variation of the spring base. To satisfy performance requirements the parameters and have to be adjusted experimentally or by employing knowledge gained from a system model. Here and are used [6]. By arranging the equations of motion of a translatory system, a desired piston velocity for the cylinder is obtained, which has to be controlled by a velocity controller for the cylinder [1]. (9) 4.2 Vibration damping The main aspect of vibration damping is to reduce the vibration energy in the system which can be done actively by the

4 5 Implementation and experimental results The flexible robot shown in figure 1 is to be controlled. The cylinder which is responsible for actuating the rotational degree of freedom is supposed to follow a reference trajectory without any vibrations at the end effector. For the reference trajectory the sine function PSfrag replacements (10) is chosen. First the results of the trajectory tracking with the flatness based precontroller are shown. To improve the accuracy, a 2nd order proportional position controller is added in a feedback loop. It can be seen that the trajectory tracking of the piston position works very well but the end effector oscillates. To eliminate these vibrations the vibration damping is included in the control concept. This yields good trajectory tracking and a very fast abating oscillation. reason for the vibration of the flexible beam. These vibrations generate an oscillating force between the actuator and 5.1 Trajectory tracking To follow the trajectory (10), the flatness based controller is the flexible beam ( ) which is measured and plotted in implemented as described in section 4.1. This control concept keeps the position of the piston rod in the neighbourhood figure 8. With this plot it becomes obvious that the beam osof the reference trajectory. The error of the piston rod s position is slowly increasing over time. The main sources of this error are inaccuracies in modeling and parameter estimation. Additional error results from the fact that the proposed design does not include the feedback of the error between the desired and the actual value for the position. To /N compensate for these effects, an additional 2nd order proportional position controller is implemented with apsfrag natural replacements frequency of, a damping rate of and a gain (fig. 4). In figure 6 the reference non- m non- Figure 7: Comparison between and non- error of the piston position Figure 8: Force acting on piston rod frag replacements m non; reference trajectory Figure 6: Piston position, non- and reference trajectory trajectory and the measured non- trajectory are shown. Both trajectories seem to coincide. The position error of the piston rod can be seen in figure 7. After s the maximal error reduces to mm, which refers to an absolute deviaton of of the working range. The elasticity of the beam has not yet been taken into account. This is the cillates. To demonstrate the dynamic behaviour of the end effector the measured position/velocity of the piston rod and the substitute model in eq. (6) are used to simulate the velocity of the end effector. Without any vibration this trajectory should be sinusoidal. As it can be seen in figure 9, the trajectory of the non- end effector velocity oscillates significantly. The end effector velocity error is shown in figure 10. It follows that the end effector oscillates and the position (of the end effector) is actually not tracked at all, although the tracking of the piston rod position is rather good. 5.2 Vibration damping To overcome this drawback a vibration damping concept is included as presented in figure 4. The reduction of the vibration energy is done actively by damping the system. Therefore the piston rod has to be moved. Hence the control concepts position control and vibration damping are contradic-

5 m frag replacements non- The reference trajectory is negative at these times. Presumably the reason for this is a non-symmetric relation between velocity and friction force. The vibration damping works well as can be seen in the plot of the force in figure 8. Due to the change in the direction of motion the force cannot be constant. Again, the velocity of the end effector is simulated with the substitute model eq. (6). This simulation uses the measured position and velocity of the piston rod and again the fast reduction of the vibration can be seen in figure 9 as well as in figure Conclusion frag replacements Figure 9: Velocity of the end effector:, non- and reference trajectory m non- Figure 10: Comparison between and non- error of the end-effector velocity tory. That means if the position is held with very high precision then the vibration damping is bad and vice versa. So the gain of the 2nd order proportional controller has to be reduced to make a vibration damping possible. A good compromise between position control of the piston rod and vibration damping with and is found with V/m. As can be seen in figure 6 and figure 7, the accuracy of the piston rod position is reduced as a result of the vibration damping. After the maximal error of the piston rod is about. Remarkably this error function has two different peaks. The maximal errors occurs at the times with (11) At these times the reference trajectory has positive values. The other peaks have a smaller amplitude and occur at the times (12) This paper deals with practical implementation of flatness based control for non-flat systems. Although the differential cylinder may not be considered as a flat system, a flatness based approach to control the piston position of the differential cylinder is well suited for trajectory tracking. Additionally, a vibration damping and a 2nd order proportional position controller is included. Measurements prove the suitability of the control concept used for trajectory tracking and vibration damping. An increase in accuracy can be achieved by an optimization of the relation between velocity and friction force. Acknowledgements This research was supported by the Deutsche Forschungsgemeinschaft under grant DFG WE 1836/1-1. The authors are grateful to the DFG. References [1] W. Bernzen. Active vibration control of flexible robots using virtual spring-damper-systems. Journal of Intelligent and Robotic Systems, 24:69 88, [2] M. Fliess. Nonlinear control theory and differential algebra. In C. I. Byrnes and A. Kurszanski, editors, Modeling and Adaptive Control. Springer, Berlin/Germany, [3] M. Fliess. Automatique et corps différentiels. Forum Mathematik, 1: , [4] M. Fliess and S. T. Glad. An algebraic approach to linear and nonlinear control. In H. L. Trentelmann and J. C. Willems, editors, Essays on Control: Perspectives in the Theory and its Applications, volume 14 of Progress in Systems and Control Theory, pages Birkhäuser, Boston/USA, [5] M. Fliess, J. Lévine, P. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: Introductory theory and examples. Int. J. Control, [6] D. Nissing. A vibration flexible robot: Identification and parameter optimization. In American Control Conference, 2000, Chicago, Illinois, USA, (Accepted paper).

6 [7] D. Nissing and J. Polzer. Parameter identification of a substitution model for a flexible link. In Identification Symposium 2000 (SYSID 2000), Santa Barbara, CA, USA, (Accepted paper). [8] R. Rothfuß, J. Rudolph, and M. Zeitz. Controlling a chemical reactor model using its flatness. In Proc. 13th IFAC World Congress, volume F, pages , San Francisco/USA, Proc. 13th Triennial World Congress, IFAC. [9] T. Wey, M. Lemmen, and W. Bernzen. Hydraulic actuators for flexible robots: A flatness based approach for tracking and vibration control. In Proc European Control Conference, ECC, Karlsruhe/Germany, 1999.