Practical implementation of flatness based tracking and vibration controlon a flexible robot

Jan Polzer, Dirk NissingFaculty of Mechanical Engineering

Department of Measurement and Control(Prof. Dr.-Ing. H. Schwarz)

University of Duisburg47048 Duisburg, Germany

polzer,nissing @uni-duisburg.de

Keywords: nonlinear control, flatness, vibration damping,tracking control, hydraulic actuators

Abstract

Using robots for very wide operating ranges or for heavyloads requires taking the elastic deformations of the linksinto account. Thereby tracking a trajectory at the end effec-tor is a demanding task. This paper deals with the practicalimplementation of a control concept which allows trajectorytracking and vibration damping. For the testbed an elasticrobot arm is chosen. The tracking control is realized usinga flatness based approach and for the vibration damping thevirtual springdamper control concept is used. The experi-mental results are included.

1 Introduction

For robots without elastic deformations, the creation of ananalytic model and the control of the plant is not a difficulttask. But if heavy loads are to be moved or wide operat-ing ranges are considered then elastic deformations have tobe taken into account. The surveyed testbed is a robot withan elastic link manufactured of spring steel actuated by hy-draulic differential cylinder as shown in figure 1. The aim ofthe control concept is the tracking of a reference trajectorywithout oscillation of the end effector. To attain this goal thecontrol concept proposed by [9] is implemented. Althoughthe model of the differential cylinder is not flat, a flatnessbased approach for the tracking control is successful if someof the states are measured. To eliminate the vibrations inthe flexible beam, the non-model based control concept of avirtual springdamper is used. The input/output behaviour ofthe actuator is controlled to act like a springdampersystem.In the second section the testbed and its modeling will be ex-plained. Section 3 focuses on flatness and hydraulic cylin-ders. The chosen control concept is shown in section 4. Thefocus of section 5 is on the results of the pratical implemen-tation. The last section gives a conclusion and an outlook for

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Figure 1: Hydraulically driven flexible robot

further research.

2 Testbed modeling

2.1 Hydraulic differential cylinder

The robots revolute joint is driven by a hydraulic translatorydrive within a closed kinematic loop to transform the trans-lation of the drive into a rotation of the joint. The cylindersschematics are shown in figure 2 [9].

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Figure 2: Schematic view of the hydraulic cylinder

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]:

(1)

withifif

(2)

The input voltage is applied to the servo valve. The fric-tion force is a combination of viscous friction , staticfriction and coulomb friction :

(3)

Its time derivative can be approximated by:

(4)

2.2 Substitute model for the beam

The best way to quantify the control concepts quality, is tomeasure the positon of the end effector. In practice this isquite difficult and expensive. To visualize the dynamic be-haviour of the end effector, a model of the flexible link canbe used. Because of significant problems in obtaining or val-idating an analytical model for the flexible beam [7] suggestsusing an identified substitute model. This substitute modeldescribes the essential dynamics and vibration behaviour ofthe flexible beam. The substitute model consists of a mass

, a spring and a damper , as pictured infigure 3. This assumption is warrantable, because experi-mental results show that only the first natural frequency isof significance [1]. The equations of motion are establishedby the impulse theorem or the second order Lagrange equa-tions [7]. The resulting equations that describe the vibrationsare:

(5)

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Figure 3: Substitute model for the flexible beam

(6)

In practice the beam parameters have to be identifiedfrom meassured data. They were found to be

, and[7].

3 Differential algebra and flatness

Differential algebra was introduced through the mathemati-cian Ritt in the 1950s. In the 1980s Fliess applied the fieldof the differential algebra to nonlinear control theory. Flat-ness [5, 8] is a rather new field of research in differentialalgebra. 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 dif-ferential field extension which is dif-ferential algebraic. The finite setis called the input and is called theoutput.

Definition 2 Equivalence [5]: Two systems andare called equivalent or equivalent by endogenous feed-back iff any element of (or resp.) is algebraic over

(or resp.). Two dynamics and aresaid to be equivalent iff the correspnding systems,and , are so.

The term of an equivalent system is very important in thiscontext:

Definition 3 Flatness [5]: A system is called (differ-entially) flat iff it is equivalent to a purely differen-tial transcendental system . A differential transcen-dence 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 ev-ery system state and every system input can be calculated di-rectly from and its time derivatives. The measured outputof the hydraulic differential cylinder is not flat, but by mea-surment of the states and the main ideaof flatness enables trajectory tracking [9]. The force canbe calculated with the help of and but in this approach

is also measured to maintain accuracy.

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, ,

, ,

,

2nd order

controller

off-line calculation:reference trajectory

, , ,

vibrationdamping

cylinder &valve

nonlinearprecontroller

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 sys-tems output into the region arround the reference trajectory.Additional the vibration damping is included. And finaly a2nd oder position controller is added to improve accuracy.The complete control concept is shown in figure 4.

4.1 Flatness based controller

For the derivation of the flatness based controller see [9]. Themain difference here is the inclusion of the time derivativeof the cylinder force . Distinguishing between the casesfor and yields (with ,

):

(7)

sgn sgn

sgn sgn

(8)

sgn sgn

sgn sgn

The control law (7) and (8) is in some sense inconvenient,since for the calculation of the case or hasto be distinguished. The denominator of is always greaterthan zero [9]. Additionally, the numerator is the same forboth cases, so first the numerator is evaluated and, dependingon its sign, the corresponding denominator is chosen.

4.2 Vibration damping

The main aspect of vibration damping is to reduce the vibra-tion energy in the system which can be done actively by the

actuator. The basic concept of the vibration damping strategyby a virtual springdamper element will be briefly explainedfor a simple flexible beam. For more details see [1]. The hy-

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Figure 5: Flexible robot with a springdamper element

draulic cylinder is treated as a virtual passive spring-damper-element. The schematics are illustrated in figure 5. It is notnecessary to equip the actuator with a real mechanical ele-ment. The piston position is measured by a pulse gen-erator and the force acting on the piston rod is mea-sured by a force sensor. The robots arm can then be drivenby variation of the spring base . To satisfy performance re-quirements the parameters and have to be adjusted exper-imentally or by employing knowledge gained from a systemmodel. Here and are used[6]. By arranging the equations of motion of a translatorysystem, a desired piston velocity for the cylinder

(9)

is obtained, which has to be controlled by a velocity con-troller for the cylinder [1].

5 Implementation and experimental results

The flexible robot shown in figure 1 is to be controlled. Thecylinder which is responsible for actuating the rotational de-gree of freedom is supposed to follow a reference trajectorywithout any vibrations at the end effector. For the referencetrajectory the sine function

(10)

is chosen. First the results of the trajectory tracking with theflatness based precontroller are shown. To improve the accu-racy, a 2nd order proportional position controller is added ina feedback loop. It can be seen that the trajectory tracking ofthe piston position works very well but the end effector oscil-lates. To eliminate these vibrations the vibration damping isincluded in the control concept. This yields good trajectorytracking and a very fast abating oscillation.

5.1 Trajectory tracking

To follow the trajectory (10), the flatness based controller isimplemented as described in section 4.1. This control con-cept keeps the position of the piston rod in the neighbourhoodof the reference trajectory. The error of the piston rods po-sition is slowly increasing over time. The main sources ofthis error are inaccuracies in modeling and parameter esti-mation. Additional error results from the fact that the pro-posed design does not include the feedback of the error be-tween the desired and the actual value for the position. Tocompensate for these effects, an additional 2nd order pro-portional position controller is implemented with a naturalfrequency of , a damping rate of anda gain (fig. 4). In figure 6 the reference

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m

damped

non-damped;referencetrajectory

Figure 6: Piston position damped, non-damped and refer-ence trajectory

trajectory and the measured non-damped trajectory areshown. Both trajectories seem to coincide. The position er-ror of the piston rod can be seen in figure 7. After s themaximal error reduces to mm, which refers to an abso-lute deviaton of of the working range. The elasticityof the beam has not yet been taken into account. This is the

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m

damped

non-damped

Figure 7: Comparison between damped and non-damped er-ror of the piston position

reason for the vibration of the flexible beam. These vibra-tions generate an oscillating force between the actuator andthe flexible beam ( ) which is measured and plotted infigure 8. With this plot it becomes obvious that the beam os-

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/N

damped

non-damped

Figure 8: Force acting on piston rod

cillates. To demonstrate the dynamic behaviour of the endeffector the measured position/velocity of the piston rod andthe substitute model in eq. (6) are used to simulate the veloc-ity of the end effector. Without any vibration this trajectoryshould be sinusoidal. As it can be seen in figure 9, the trajec-tory of the non-damped end effector velocity oscillates sig-nificantly. The end effector velocity error is shown in figure10. It follows that the end effector oscillates and the position(of the end effector) is actually not tracked at all, althoughthe tracking of the piston rod position is rather good.

5.2 Vibration damping

To overcome this drawback a vibration damping concept isincluded as presented in figure 4. The reduction of the vibra-tion energy is done actively by damping the system. There-fore the piston rod has to be moved. Hence the control con-cepts position control and vibration damping are contradic-

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m

damped

non-damped

Figure 9: Velocity of the end effector: damped, non-dampedand reference trajectory

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m

damped

non-damped

Figure 10: Comparison between damped and non-dampederror of the end-effector velocity

tory. That means if the position is held with very high pre-cision then the vibration damping is bad and vice versa. Sothe gain of the 2nd order proportional controller has to bereduced to make a vibration damping possible. A good com-promise between position control of the piston rod and vi-bration damping with andis found with V/m. As can be seen in figure 6 andfigure 7, the accuracy of the piston rod position is reduced asa result of the vibration damping. After the maximal er-ror of the piston rod is about . Remarkably this errorfunction has two different peaks. The maximal errors occursat the times

with (11)

At these times the reference trajectory has positive values.The other peaks have a smaller amplitude and occur atthe times

(12)

The reference trajectory is negative at these times. Presum-ably the reason for this is a non-symmetric relation betweenvelocity and friction force. The vibration damping workswell as can be seen in the plot of the force in figure 8.Due to the change in the direction of motion the forcecannot be constant. Again, the velocity of the end effector issimulated with the substitute model eq. (6). This simulationuses the measured position and velocity of the piston rod andagain the fast reduction of the vibration can be seen in figure9 as well as in figure 10.

6 Conclusion

This paper deals with practical implementation of flatnessbased control for non-flat systems. Although the differentialcylinder may not be considered as a flat system, a flatnessbased approach to control the piston position of the differ-ential cylinder is well suited for trajectory tracking. Addi-tionally, a vibration damping and a 2nd order proportionalposition controller is included. Measurements prove the suit-ability of the control concept used for trajectory tracking andvibration damping. An increase in accuracy can be achievedby an optimization of the relation between velocity and fric-tion force.

Acknowledgements

This research was supported by the Deutsche Forschungs-gemeinschaft under grant DFG WE 1836/1-1. The authorsare grateful to the DFG.

References

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