found that it is possible to achieve excellent results but with unrealistically high torques. This problem

Transcription

1 David Wang Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario, Canada N2L-3G1 M. Vidyasagar Center for Artificial Intelligence and Robotics Bangalore , India Transfer Functions for a Single Flexible Link Abstract This article examines some issues in the transfer function modeling of a single flexible link. Using the assumedmodes approach to represent the elastic deformation, one can find the transfer function between the torque input and the net tip deflection. It is shown here that when the number of modes is increased for more accurate modeling, the relative degree of the transfer function becomes ill defined. This can greatly affect the performance of a controller designed using this model. It is then shown that this problem occurs regardless of the method used to represent the elastic deformation. An alternate modeling approach is proposed that uses the rigid body deformations minus the elastic deformations as the output. This solves this problem and results in a transfer function with a well-defined relative degree of two. Simulation results are presented that illustrate the advantages of using the proposed alternate transfer function. 1. Introduction In recent years, the problem of controlling flexible manipulators has been studied extensively. Initial studies have concentrated on the single flexible link. Although most of the previously proposed control strategies for the single flexible link required only a state space model (Cannon and Schmitz 1984; Hastings and Book 1985; Hastings and Book 1986; Usoro, Nadira and Mahil 1984), other control strategies require a transfer function for the system. A transfer function model for a single flexible link with a tip mass was obtained in Rakhsha and Goldenburg (1985) using a Newton-Euler approach. In Alberts et al. (1986), a transfer function was derived for a beam with a distributed damping added by using a viscoelastic layer surface treatment. In Wang and The International Journal of Robotics Research, Vol. 10, No. 5, October 1991, C 1991 Massachusetts Institute of Technology. Vidyasagar (1986), an initial study was made into ~-optimal control of the transfer function. It was found that it is possible to achieve excellent results but with unrealistically high torques. This problem was solved in Krishnan (1988), and excellent experimental results were attained. An alternate transfer function was presented in Kotnik and co-workers (1988) that used the acceleration at the tip of the flexible link as the output. These are but a few of the papers studying the modeling of single flexible links. A good summary of the various techniques is presented in Kanoh et al. (1986). Many of the transfer functions proposed thus far use the torque to the flexible link as the input and the net tip displacement of the flexible link as the output. This article examines problems with using this transfer function. The transfer function is often derived using the popular assumed-modes approach, which represents the elastic deflection i4,(x, t) as where the c~;(x) are the mode shapes of the beam. Assuming no damping, this results in a rational transfer function of the form Note that the relative degree of the transfer function is two. A problem arises when the number of modes n included in (1) is increased to improve the accuracy of the model. This article shows that as rz is increased, the coefficient a,, of the transfer function 540

2 in (2) approaches zero, and simultaneously, the next leading coefficient an - approaches infinity. This shows that in the limit, the relative degree of the transfer function is ill defined. If the relative degree of the transfer function is not well defined, the performance of a controller designed using this model can be affected. In addition, any attempt to identify the transfer function model will be affected. It is then shown that this problem will be encountered even if the elastic deflections are modeled using techniques other than the assumed-modes approach. To remedy this problem, an alternate output variable is proposed in this article, namely the rigid body deflection minus an elastic deflection, instead of the net tip deflection. This results in a transfer function with a well-defined relative degree of two. The presentation of this article is as follows. In section 2, a state space model for a single flexible link is presented using the assumed-modes approach. From this state space model, the transfer function between the torque applied to the link and the net tip deflection is derived in section 3, which also discusses the problems in determining the relative degree of this transfer function. It is shown that this problem is independent of the method used to model the elastic deformation. Section 4 of this article proposes an alternate transfer function that has a well-defined relative degree even as the number of modes approaches infinity. Some simulated comparisons are carried out in section 5 between the HZoptimal controllers designed for both the usual transfer function and for our proposed transfer function. Finally, the last section presents some conclusions and recommendations. 2. State Space Modeling In this section, a state space equation is derived for a single flexible link in which one end is connected to a motor and is driven by a torque T, and the other end is free (Fig. 1). The link is assumed to move horizontally. The approach used is the assumed-modes method. The damping in the beam is ignored. Using this state space model, the transfer function between the torque and the net tip deflection will be found in the next section. Increasing the order of the transfer function to improve accuracy is achieved by including more modes of oscillation in the state space model. Although this model is identical to that in Siciliano et al. (1986) and Wang and Vidyasagar (1988), it is worthwhile to present the derivation of the model again, as many parts of the formulation are referred to in later sections. Fig. 1. Single flexible link. It is assumed that the height of the beam is much greater than the width, thus constraining the beam to vibrate in the horizontal direction. All deflections of the beam are assumed to be small. The shear deformation and rotary inertia effects are ignored for simplicity. The beam has a moment of inertia Ib, a linear mass density p, and a length h. The angular rotation of the beam is denoted as 0(t). Points on the beam have their position fixed by the variable x, which is the distance of that point from the hub of the motor driving the beam. The elastic deformation at x is given as w(x, t), and y(x, t) is the net movement of that point. In other words, Thus the net deflection at x is the sum of a rigid body deflection and an elastic deformation. A frame x -y is attached rigidly to the point where the beam is attached to the motor hub (Fig. 2), with the x - axis tangent to the beam. In other words, the frame x -y rotates such that the slope of the beam at x 0 is zero. This implies that the boundary conditions for w(x, t ) are clamped free and that w(x, t) can be expanded using the assumed-modes approach (Meirovitch 1967) as where the eigenfunctions (pi(x) are the clamped-free eigenfunctions and are given by 541

3 hold for any representation of the elastic deflection w(x, t) and are not specific to the assumed-modes approach. A state space model for this system can now be derived. The input is considered to be the torque T, whereas the output is considered to be the net endpoint deflection The position P(x) of a point on the beam at a distance x from the hub is given by Therefore Fig. 2. Coordinate frames. The ki are the solutions to Now, the kinetic energy K and the potential energy V are given by and cl is a constant that normalizes the eigenfunctions so that We choose the clamped-free eigenfunctions as opposed to the pinned-free eigenfunctions that most other researchers use. Other researchers who use the pinned-free eigenfunctions (Cannon and Schmitz 1984; Hastings and Book 1986) are describing the vibrations relative to a frame situated such that the x -axis intersects the center of mass of the beam (Fig. 3). However, it is shown in Krishnan (1988) that, for a given hub inertia Ih, the poles, zeros, and gain of the transfer functions obtained using both methods match closely. In other words, the results hold whether the clamped-free or the pinned-free eigenfunctions are used in the expansion of (4). In the next section, it will be shown that these results where 1,, is the hub inertia. One simplifying assumption is now made. It is customary to assume that w is very small (Cannon and Schmitz 1984; Hastings and Book 1986). In that case, the term in the kinetic energy function can be approximated by Now the Lagrangian L K - V can be found as Fig. 3. Pinned-free boundary conditions. where wa are the resonance frequencies of the clamped-free eigenfunctions (Timoshenko 1955). By definition, the q;s can be considered as the generalized coordinates of the system so that one can apply the Euler-Lagrange equations. The equa- 542

4 tions of motion are therefore where Bessel s equality Substituting (15) into (16) and (17) gives is used. Thus approaches zero as n ~ 00. The state space equations the form can now be derived in The matrices x, A, B, and C are displayed for the case n 3, but the same pattern holds for all n: C [h 0 01(h) 0 ~(h) 0 <~(/!) 0] 543

5 ~ matrices as The resonance modes of this model have been confirmed experimentally. This can be shown using the Markov parameters for the transfer function. Because the relative degree of this system is now zero, the transfer function may now be inverted to find 3. Problems in Transfer Function Modeling In many control strategies such as H2-optimization (Wang and Vidyasagar 1986), one requires the transfer function for the system rather than the state space equation. In this section, the transfer function for the single flexible link is found using the state space equations derived in the previous section. It is shown that for the output defined in (8), the transfer function that most researchers (Cannon and Schmitz 1984; Krishnan 1988; Wang and Vidyasagar 1986) use does not have a well-defined relative degree as the number of modes approaches infinity. It will then be shown that this is true regardless of the method used for modeling the elastic deflection w(x, t). In the previous section, the state space equation of the single-link beam was derived giving a set of A, B, and C matrices. The input-output transfer function is therefore where zi are the zeros of the transfer function, pi are the poles of the transfer function, k is the gain of the transfer function. Let us define v - t to be the relative degree of the transfer function. Given the A, B, and C matrices, it is possible to find the poles, zeros, gain, and relative degree of the transfer function. The poles of the transfer function are just the eigenvalues of A. Finding the zeros, the gain, and the relative degree is a little more complicated. The relative degree v is the smallest value of 8 such that CA ~~ B is nonzero. Thus 1J is found by increasing 8 from 0 until the term CA fi- B is nonzero. The gain k is then equal to CA &dquo;- B. One way to find the zis is to take v derivatives of y(t), resulting in the transfer function The equivalent state space representation is Now the zeros zi of the original transfer function in (24) are the poles of (27), excluding the v poles at zero. Thus the zeros zi are the eigenvalues of A - A~B(CA~ B) C, excluding the v eigenvalues at zero. Therefore we have shown that, given the A, B, and C matrices of the state space model in (23), we can find the poles, zeros, gain, and relative degree of the transfer function. Let us return to the transfer function for the single flexible link. Until now, researchers (Cannon and Schmitz 1984; Krishnan 1988; Wang and Vidyasagar 1986) have assumed that the relative degree of the transfer function is two. It is now shown that this is not a good assumption. To show this, we calculate the relative degree v by increasing 5 from 1. First, for 5 1, CB for the transfer function is obviously zero from (22). Choosing 5 2 gives Note that CAB is the an in (2). However, the expansion of x with respect to the eigenfunctions 4> is because the q5is are a complete and orthonormal set (Meirovitch 1967). Thus letting x h gives and This can be written in terms of the A, B, and C 544

6 Thus the CAB in (29) is a number very close to zero for large n. As an example, consider the beam studied in Wang and Vidyasagar (1988), in which 7/, is assumed to be 0.05 kgm2. With n 2, CAB is 0.88, but with n 5, CAB is This implies that assuming the relative degree of the transfer function to be two is not good, as CAB is a term that is small and approaches zero as n ~ x. In addition, note that the gain of the transfer function can change sign as n increases. This is also not desirable. Now the next logical step is to continue to increase 5 until CA s- B is nonzero. For 5 3, it is easily shown that CA2B 0, but 54 gives This is a,, - 1 in (2). However, it is straightforward to show from equation (5) that as n increases, CA 38 increases by a factor of approximately kn. For example, using the beam studied in Wang and Vidyasagar (1988) with //, 0.05 kgm2, we have CA;B 14.9 x 103 for n 2 and CA3B 52.2 x 104 for n 5. In fact, as n ~ x, CA ~B ~ x. Thus although CA~B is nonzero, it results in a transfer function with a gain that approaches x as n ~ x. This is also not acceptable. There is an alternate method of showing that the relative degree of our plant is not two. To do this, another definition of relative degree (Byrnes and Isidori 1984) is introduced that is equivalent to the above definition for linear systems but is also valid for nonlinear systems. This alternative definition is needed later. This definition will also allow us to show that the modeling problems hold in general and not just for the assumed-modes approach. The relative degree of a system is DEFINITION 1. the number of times that the output y must be differentiated before the input appears explicitly. In other words, the relative degree v is defined to be the fix, ic) but smallest value of 8 such that y where y p. f ~,(x} whenever ~t < 6. From (8) and (4), it is seen that two. Although it seems counterintuitive that y does not depend explicitly on the torque T, it should be remembered that the effect of T on the beam must travel as a wave to the end point (Cannon and Schmitz 1984), and this takes a finite time. We will now show that these results can be extended to any representation of the deflection w(x, t). This is done by examining the element of length dx at the free end of the single flexible link (Fig. 4). From Timoshenko (1955), it is known that the shear force V,(x, t) is given by and that V,,.(h, t) is zero because of the boundary conditions. The acceleration of the element dx is given by y(t), as y(t) is the net tip deflection. Thus one can use D Alembert s principle (Wang and Vidyasagar 1986) and consider - p dxy(t) to be an inertial force. Summing the forces in the direction perpendicular to x in Figure 4 shows that This reduces to which is equivalent to (35). There is no assumption here on how w(h, t) is modeled. Thus regardless of the method used to model w(h, t), the relative degree of the transfer function is not two. In particular, the set of assumed modes used in (4) has no effect on the results. This can be easily confirmed numerically for the case of pinned-free eigenfunctions (Krishnan 1988). Thus we have shown using two methods that the linear model of (22) has no well-defined relative degree as 11 ~ x. It may be conjectured that the explanation for this strange result is that the term By substituting (18) and (19) into (34) and using (32), one finds after straightforward computations that Note that the right side of (35) does not include the torque. This implies that the relative degree v is not Fig. 4. Free body diagram of end point of link. 545

7 f wz dm, which was neglected in equation (13), affects the transfer function of the beam. However, it is now demonstrated that even if this nonlinear term is included, the relative degree is not well defined. If f w2 dm is included, then equations (18) and (19) become Now note that T appears, explicitly suggesting that v 3. However, if 6 and w(h, t) are zero, then y3 does not depend explicitly on T. Hence even with the accurate nonlinear model, the relative degree is not well defined. It can therefore be concluded that when the output variable is defined to be the net tip deflection in (8), the relative degree of the resulting transfer function is ill defined. For the finite expansion of (4), the relative degree becomes ill defined as the number of modes of oscillation included in the modeling approaches infinity. This problem exists regardless of the method used to model the elastic deflection. 4. Proposed Transfer Function Modeling and From (39) and (40) and using (32), we obtain which is not a function of the input T. Thus by Definition 1, the relative degree of the transfer function is still not two. From (41), one may calculate y3 a3ylo~t3. This gives the nonlinear equation In the previous section it is shown that for the transfer function between the input torque and the net tip deflection, there is no well-defined relative degree. An alternate method that does give a well-defined transfer function is presented in this section. This works even if f w2 drra is not neglected in (13). Instead of using the output variable defined in (8), we now define the output variable to be This variable is physically realizable, as sensors can be designed to measure 0(t) and w(h, t ) separately (Wang and Vidyasagar 1987). Equation (43) is obtained by subtracting the elastic deformation from the rigid body motion rather than taking the sum of the two, as is done in (8). Note that designing a controller to make y(t) ~ Yref and w(h, t) -j 0 is equivalent to designing a controller to make y (t) ~ y~e~ and w(h, t) ~ 0. Neglecting f w2 dm in (13) gives the following for y (t) B &dquo;t&dquo;t } Thus by Definition 1, the relative degree of the input-output transfer function is two, regardless of how many modes are included. Also, it is easy to show that this system is linear and time invariant. In addition, the gain of the transfer function is always positive and is equal to 2hlI~T. If the term f w2 dm is not neglected in (13), then 546

8 . Fig. This again implies that our system is of relative degree two. Note that in general 6 is much smaller than the Wjs. Also, 4i is a maximum when qi is zero, and vice versa. Finally, w is assumed small. From these assumptions, one may show that (45) reduces to (44). 5. Simulation Results In this section, a comparison is made between the transfer function using the y(t) of (8) as the output and the transfer function using the y (t) of (43) as the output. For both models, the H2-optimal controller is found, and their step and torque responses are simulated. The method of finding the H2-optimal controller is described in detail in Wang and Vidyasagar (1987) and will not be repeated here. Essentially, the control strategy optimizes the cost function over the set of all stabilizing controllers. Here e(t) is the tracking error (Fig. 5), and a is an exponential weighting factor. The factor a is used to reduce the steady-state error. It is well known that the optimal controller will be physically unrealizable and that suboptimal controllers with a bandwidth of We are used. The optimal controller is approached as w~. ~ 00. In the following simulations, the reference input to the system (see Fig. 5) is chosen to be This function approaches a step function exponentially and is used to reduce torque requirements (Wang and Vidyasagar 1987). In addition, a is chosen to be 1, and Ú)e is chosen to be 1000 for both models. The H2-optimal controller designed using y(t) as the output gives the step response shown in Figure 6 and the torque response shown in Figure 7. Note the unrealistically high values of the torque response. In fact, the maximum input torque to the plant is approximately 104 Nm. Fig. 6. Step response for usual output. When the proposed output y (t) is used, the step and torque responses shown in Figures 8 and 9, respectively, are obtained. Note the dramatic improvement in the maximum input torque which is now only approximately 60 Nm. The physical reason for this improvement is that the gain k is very small when y(t) is used as the output of the transfer function. Thus to attain a step response similar to the one for the proposed output variable y (t), the controller must exert a much larger torque. This problem is made worse when the number of modes n in (4) is increased. The gain k of the transfer function using y(t) as the output becomes smaller, resulting in even higher torque requirements. However, for the proposed transfer function using y (t) as the output variable, the maximum torque will remain virtually unchanged regardless of the order n, because its gain k stays constant at approximately 2hIIh. This can easily be confirmed numerically. To improve the step response for the proposed output variable y (t), the exponential weighting a Fig. 5. Feedback configuration. 7. Torque response for usual output. 547

9 Fig. 8. Step response for proposed output. Fig. 11. Torque response with increased weighting. can be increased to 2. This gives the step and torque responses shown in Figures 10 and 11, respectively. The step response is greatly improved. Fig. 9. Torque r-esponse for proposed output. Fig. 10. Step response with increased weighting. 6. Conclusions It has been shown that if the output is defined to be the net tip deflection, the relative degree of the transfer function for the single flexible link is not well defined, as the number of modes approaches infinity. This problem was shown to exist for all modeling methods that used an output of the form of (8). However, if the output is defined to be the rigid body deflection minus the elastic deflection, then a transfer function with a relative degree of two does exist, irrespective of how many modes are used. Simulation results indicate that large improvements can be made in the torque requirements of the controllers with this change in output variable. Future research will focus on extending these results to a single flexible link with a tip payload. It is expected that similar results will still hold. Outputs other than the one in (43) will also be studied to see if further improvements are possible. The extension of these results to multilink manipulators has already led to some interesting results. In Wang and Vidyasagar (1989), it has been shown that outputs of the form (8) led to nonlinear differential equations that are not input-output feedback linearizable, whereas the proposed output of (43) does allow input-output feedback linearizability. Further studies of multilink manipulators will be done. It has already been shown that for some flexible links, one can control the transfer function of section 4 with a passive controller. This has been confirmed experimentally (Wang and Vidyasagar 1990). 548

10 Future research will also examine these properties and extend them to multilink manipulators. Acknowledgment This research was supported by the Manufacturing Research Corporation of Ontario, Ontario, Canada. References - Alberts, T. E., Dickerson, S. L., and Book, W. J (Anaheim, Calif., December). On the transfer function modeling of flexible structure with distributed damping. In Robotics: Theory and Applications. ASME, New York, pp Byrnes, C., and Isidori, I (Las Vegas, December). A frequency domain philosophy for nonlinear systems with applications to stabilization and adaptive control. In Polis, M. P. (ed.): Proc. 23rd IEEE Conf. on Decision and Control. IEEE, New York, pp Cannon, R. H., and Schmitz, E Initial experiments on the end-point control of a flexible one-link robot. Int. J. Robot. Res. 3(3): Hastings, G., and Book, W. J (Boston, June). Experiments in the optimal control of a flexible manipulator. In Proceedings ACC, pp Hastings, G., and Book, W. J (San Francisco, April). Verification of a linear dynamic model for flexible robotic manipulators. In Proc. IEEE Conf. on Robotics and Automation. IEEE Computer Society Press, Silver Spring, Md., pp Kanoh, H., Tzafestas, S., Lee, H. G., Kalat, J (Athens, Greece, December). Modelling and control of flexible robot arms. In Proc. 25th IEEE Conf. on Decision and Control. IEEE, New York, pp Kotnik, P. T., Yurkovich, S., and Ozguner, U Acceleration feedback control for a flexible manipulator arm. J. Robot. Sys. 5(3): Krishnan, H Bounded input discrete-time control of a single-link flexible beam. Master s thesis, Department of Electrical Engineering, University of Waterloo, Waterloo, Canada. Meirovitch, L Analytical Methods in Vibrations. New York: Macmillan. Rakhsha, F., and Goldenburg, A. A (St. Louis, April). Dynamics modelling of a single-link flexible robot. In Proc. 25th IEEE Conf. on Robotics and Automation. IEEE Computer Society Press, Silver Spring, Md., pp Siciliano, B., Book. W. J., and De Maria, G (Athens, Greece, December). An integral manifold approach to control of a one link flexible arm. In Proc. 25th IEEE Conf. on Decision and Control. IEEE, New York, pp Timoshenko, S Vibration Problems in Engineering. Princeton, N.J.: D. Van Nostrand Company. Usoro, P. B., Nadira, R., and Mahil, S. S A finite element/lagrange approach to modelling lightweight flexible manipulators. In Sensors and Controls for Automated Manufacturing and Robotics. ASME, New York, pp Wang, D., and Vidyasagar, M (Anaheim, Calif., December). Modelling and control of a flexible beam using the stable factorization approach. In Robotics: Theory and Applications. ASME, New York, pp Wang, D., and Vidyasagar, M (Raleigh, N.C., April). Control of a flexible beam for optimum step response. In Proc. IEEE Conf. on Robotics and Automation. IEEE Computer Society Press, Silver Spring, Md., pp Wang, D., and Vidyasagar, M (Philadelphia, April). Modelling of a 5-bar-linkage manipulator with one flexible link. In Proc. IEEE Conf. on Robotics and Automation. IEEE Computer Society Press, Silver Spring, Md., pp Wang, D., and Vidyasagar, M (Tampa Bay, Fla., December). Feedback linearizability properties of multilink manipulators. In Proc. 28th IEEE Conf. on Decision and Control. IEEE, New York, pp Wang, D., and Vidyasagar, M (Cincinnati, May). Passive control of a single flexible link. In Proc. IEEE Conf. on Robotics and Automation. IEEE Computer Society Press, Silver Spring, Md., pp