Proceedings of the 40th IEEE Conference on Decision and Control Orlando, Florida USA, December 2001
ThP12-3
Mechanical Systems with Unilateral Constraints: Controlled Singularity Approach Joseph Bentsman 1. and Boris M. Miller 2.* *Department of Mechanical and Industrial Engineering, University of Illinois at UrbanaChampaign,1206 West Green Street, Urbana, IL 61801. E-mail:
[email protected] **Institute for Information Transmission Problems, Russian Academy of Sciences, B. Karetny 19, 101447, Moscow, Russia, e-mail:
[email protected]
during some time. Usually, in controlling mechanical system with impacts, for example when providing stabilization [9], one tries to avoid such modes of operation. However, in optimal control problems such paths can not be neglected, because they could provide the optimal solution.
Abstract
Admitting the introduction of the impulsive control actions during singular phases of the dynamical system motion, such as changes in dimension, and discontinuities in the state, a well-posed representation of the discontinuous limiting motion of mechanical systems with unilateral constraints in terms of differential equations with measure is developed. This representation is mapped into a detailed multi-scale system description via a space-time transformation. The resulting framework, which includes the equations of the detailed and limit dynamics connected via the space-time transformation, is used to describe a difficult to model class of systems - mechanical systems with impulsive control actions introduced by a collision-type system interaction with nonstationary controllable constraints.
The conventional mathematical view of the contact force considers it as some effect arising when the system hits the boundary of an obstacle and associates this force with some measure-valued action localized at the constraint boundary. The appearance of such measures was demonstrated by Moreau [19], [20], who showed that the correct representation of the constrained motion requires the use of differential equations with measure. More generally, in the cases of uncertain contact properties this motion has to be described by measure differential inclusions [18]. It is, therefore, natural to consider the optimal control problem for constrained mechanical systems as the problem with measure-valued control inputs. Theory of such problems has been extensively developed during recent years starting from the works of Warga [24], [25], and continuing by a number of authors [3], [4], [13], [14], and many others. The key point of all these investigations is that the reaction of a nonlinear control system to an impulsive input can not be uniquely defined, since it is impossible to uniquely define the nonlinear function of an argument given by a 5-type function. Therefore, the common approach to these problems has been to allow some control components to be unbounded in their norms, but to impose simultaneously the integral constraint on them. Thereby, some of the control components occur bounded in L1 norm and can be taken as closely as desired to the impulsive, or generalized, functions like the 5-function. W h e n one chooses some sequence of ordinary control signals which converges in the weak • - topology to some generalized input, the appropriate sequence of solutions belonging to a compact (in weak • - topology) set of paths can converge everywhere, except maybe at the j u m p points, to some discontinuous function. The latter function can be considered, then, to be a reaction of a nonlinear dynamical
1 Introduction and Motivation
The motion of a real dynamical system subject to the unilateral constraints can be considered as having natural and singular phases, where the first one corresponds to the motion in the constraint-free domain, and the second one relates to the phase of the interaction with the constraints. The latter phase gives rise to jumps of velocity, non-uniqueness of solution, etc. [10],[19]. The conventional description of the velocity jumps presumes the use of either the so-called collision mapping, or the restitution law, both of which give an expression for the velocity after the impact in terms of the velocity and position before the impact [5], [6], [23]. The collision mapping, however, is rather restrictive, since it cannot describe the b o u n d a r y of a time-varying and controllable, or active, constraint when the control forces during the phase of contact are impulsive, and also the cases when the system can stick to the constraint and be sliding along the constraint surface 1Supported by Graingcr Center and NSF grant CMS 0000458. 2Supported in part by Russian Basic Research Foundation Grant 99-01-01-088. 0-7803-7061-9/01/$10.00 © 2001 IEEE
3692
of an auxiliary controllable system, obtained from the original system via a specially constructed space-time transformation and related to the discontinuous motion representation via a limiting procedure. This approach to the description of the contact impulsive forces, first indicated by the authors in [1], [15], can be summarized as follows: i) the contact force is considered to be the result of a small violation of a constraint which takes place when the system hits the boundary of a constraint; ii) this force resists the penetration of the system into the domain, inhibited by the constraint, and causes a very fast (almost instantaneous) change of the sign of normal velocity component to the opposite one; iii) since we admit the controlled motion in the inhibited domain, we suggest that this motion can be described by the nonstandard controlled singularly perturbed differential equation with infinitely growing right-hand-side, such that the solution of this equation behaves in the limit as a stepwise function with respect to the components of the generalized velocity.
system to the impulsive, or generalized, control input. Generally, the reaction to such input is not unique due to the integral funnel nature of the solution. There is, however, a regular method of choosing the approximation sequence of ordinary control signals, referred to as the method of discontinuous time change, which provides the convergence of the appropriate sequence of trajectories to any a priori selected discontinuous (generalized) solution within the funnel. The idea of this method was suggested by Warga [24] and leads to a certain time-scale transformation, so that system response to the impulsive input can be described in terms of the auxiliary control system, defined on some time intervals corresponding to each jump point. The discontinuous solution of the system can, then, be obtained via an inverse time transformation, which is a discontinuous one. This method permits one to prove the existence of the optimal generalized solution, derive its representation in terms of a nonlinear differential equation with measures, and obtain the necessary optimality conditions in the maximum principle form [13], [14].
Staying within these guidelines, we apply the new approach to unilaterally constrained mechanical system based on the ideas of dynamical systems with controllable singularities [2]. Unlike [2], only part of the state, velocity, is allowed to be discontinuous. We describe the motion as having two controlled phases: natural and singular [16]. The second one arises when the system hits the constraint and continues the motion in the inhibited domain. We prove that if the obstacle in some sense repulses the system and the elasticity of the obstacle tends to infinity the resulting limiting motion can be described as a generalized solution of an equation with measure.
In the case of the collision of rigid bodies, however, the contact forces arise, in fact, due to small violation of the constraints, which are not perfectly rigid ones. Therefore, in reality, there is a very fast, but continuous, phase of motion which only looks like discontinuous with respect to the velocities in the natural time scale. If one considers this phase in the enlarged timespace scale generated via some space-time transformation, one can obtain a more detailed description of the collision. In [11], [12], and [17] this approach is used to derive the equations of collision mapping for robotic manipulator, however the description of the discontinuous system behavior is not rigorously obtained.
The paper has the following structure. Section II presents the detailed multi-scale description of mechanical systems with unilateral constraints that combines natural and singular system motions. Section III introduces the space-time transformation that stretches, both in space and time, the vicinity of the point of collision with the constraint and derives the equation, the limit form of which describes the discontinuous system behavior in the singular phase. Section IV develops the description of the limit form of the entire behavior of system with controllable constraints in terms of a differential equation with measure and rigorously introduces the solution of this equation. Finally, conclusions are given in Section V.
Unlike the "passive constraints" case considered in [11], [12], and [17], in the case of the controllable constraints, introduced in [1] and [15], and further developed in the present work, the impulsive control action should be admitted during the contact. To provide such a possibility, we modify the description of the collision mapping; however, its application to the real constrained system has to be rigorously supported. The latter task turns out to be nontrivial, since the results from the impulsive control theory (e.g. [21]) are not directly applicable to the case of constrained mechanical systems due to the fact that the contact forces are inherent to the constrained system itself and should be considered as the internal ones, as opposed to the impulsive control inputs, which are external to the system. This results in the gap between the controlled mechanical systems and the optimal control theory. This gap, however, could be bridged, if one would focus on the physical hature of the contact forces during collision. For this purpose, we propose to describe the system motion in the singular phase in the enlarged space-time scale in terms
2 N a t u r a l a n d S i n g u l a r M o t i o n P h a s e s of Mechanical Systems with Unilateral Constraints Consider the motion with the so-called elastic constraints, with elasticity characterized by some parame-
3693
We suppose t h a t for any given 0 _< it < oo the joint system (1),(2) has the unique solution for any given measurable controls u(.), w(.). Our objective is to determine the behavior of the joint system for it --~ oc, and to find out if there exists the appropriate limit for its solution. If the limit exists one can treat it as the
ter it. These constraints are assumed to be imperfect, i.e. a d m i t t i n g motion, albeit inhibited, within the area occupied by the constraint. Further on, consider decomposition of this motion into two phases, the one corresponding to the motion in the area free of constraints and the second one describing the motion in the area inhibited by constraints, referred to as the n a t u r a l and the singular phases, respectively.
generalized solution of a dynamical system with unilateral constraints, which would then be described by the limiting form of the joint system - a differential equation with m e a s u r e . .
Let the controllable dynamical system be described by the set of variables xp(t) E R n, x,(t) E R n, where vector Xp is referred to as the set of generalized coordinates and x~ as the set of generalized velocities, so t h a t x~(t) = Ycp(t). Suppose t h a t there is some constraint given in the form of the inequality
c(~,(t),
1 The principal feature of the systems considered is that their generalized coordinates are continuous while their velocities admit the jumps. At the same time the generalized coordinates are exactly the ones responsible .for the appearance of the contact .forces. Therefore, in order to properly describe system behavior one needs to carry out a more involved analysis in the vicinity of the constraint violation points. General ideas of this approach are presented in [2].
Remark
t) 0"
Gt. (xp (~-),~-) Gt (xp (~-),~-)
(8) . (xF (~-),~-)
The condition of Theorem 1 means that the force F has the property to repulse the system from the inhibited domain. We express this property in terms of the socalled restitution force. Consider the motion in the area
}.
(xp (~-),T)
t
{(~, ~) • G'x(~,(~), ~)[~, - ~,(~)] + G,(~,(~), ~)~ > 0}
Then, if # --~ oc,
along the paths of the "limit" system (7). The term
(y~(s), y"~ (s)) -. (~(s), ~(s))
uniformly on
[0, s* + el,
t
z(~) - G'~ (x~(~), ~)[~, - ~,(~)] + G,(~(~), ~)~ and .for all sufficiently large # there exists s , - inf{s > 0"
characterizes the constraint violation, and usually the restitution force can be expressed in terms of Z, as d2 F(s) - - ~ s Z ( S ) . Suppose that one can ascertain that
(9)
a(x~(~ + ~-~/~), ~ + ~ - ~ / ~ ) - 0,
'1
Gt (x~(~-),~-)+ G'x ](xp(~-),~-) ~ ( ~ + " - ~ / ~ )
this restitution force is a visco-elastic one and that
l(xp(t),x.,,(t),u(t),t)
< a;(x~, t) F ~(~, x~, ~(t), t) >l
'
}/
(xp(t),x.v(t),u(t),t) }"
z{t- a(~,(t), t) - 0}. (14)
where 5 Conclusions • function •
is defined as the shift operator The new approach to unilaterally constrained mechanical system based on the ideas of dynamical systems with controllable singularities is developed. A rigorous representation of the limit behavior of this class in terms of differential equations with measure is derived. This representation is shown to be connected to
t~(xp, x,, w(.), 7, s) along the paths of the general solution
• p(Xp x , w ( . ) T , s ) - - { ,
,
,
e2p(xp'x"'w(')'T's)
k
\ ~ ( X p , x~, w(.), T, s) /
3696
[11] N. H. McClamroch, "Singular Systems of Differential Equations as Dynamic Model for Constrained Robot Systems," Proc. of IEEE Int. Conf. on Robotics and Automation, San Francisco, Apr. 8-10, pp. 21-28.
the detailed multi-scale system description via a spacetime transformation and a limit procedure. Applying the approach presented to control problems for systems with controllable singularities one can reduce the optimal control problems for unilaterally constrained mechanical systems to a class of standard problems.
[12] N. H. McClamroch, "A Singular Perturbation Approach to Modelling and Control of Manipulators Constrained by a Stiff Environment," in Proceedings of 1989 IEEE Conference on Decision and Control, Tampa, FL.
References
[13] B. M. Miller, "The generalized solutions of nonlinear optimization problems with impulse control," SIAM J. Contr. Optimiz. vol. 34, no. 4, 1996.
[1] J. Bentsman and B. M. Miller, "Control of dynamic systems with unilateral constraints and differential equations with measures," Preprints of the ~th IFAC Nonlinear Control Systems Design Symposium 1998 (NOL COS '98). University of Twente, Enschede, The Netherlands, July 1-3, 1998. Eds. H. J. C. Huijberts, H. Nijmeijer, A. J. van der Schaft and J. M. A. Scherpen. University of Twente, vol. 2, 411-416, 1998. [2] J. Bentsman and tems with Controllable Limit Representations 2001 IEEE Conference volume.
[14] B. M. Miller, "The Generalized Solutions of Ordinary Differential Equations in the Impulse Control Problems," J.Math. Syst., Est. Control, vol. 6, no. 4, 1996. [15] B.M. Miller and J. Bentsman, "A new approach to control of dynamic systems with unilateral constraints," in Proceedings of l~th IFA C World Congress, Beijing, China, v. C, pp. 199-203, 1999.
B. M. Miller, "Dynamical SysSingularities: Multi-Scale and and Optimal Control," Proc. on Decision and Control, this
[16] B.M.Miller and J. Bentsman, "Generalized solutions in dynamic systems with unilateral constraints," in Proceedings of 5th IFA C Symposium "Nonlinear Control Systems" (NOLCOS'01), Saint-Petersburg, Russia, July 4-6, 2001, Preprints, pp. 625-628.
[3] A. Bressan and F. Rampazzo, "On differential systems with quadratic impulses and their applications to Lagrangian mechanics," SIAM J. Control Optim., vol. 31, pp. 1206-1220, 1993.
[17] J. K. Mills and C. Nguyen, "Robotic manipulator collision: Modeling and simulation." Trans. A SME J. Dyn. Sys. Meas. and Contr., vo1.114, no. 4, 1993.
[4] A. Bressan and M. Motta, "A class of mechanical systems with some coordinates as controls. A reduction of certain optimization problems for them. Solution methods," Atti. Acc. Lincei Rend., (9), II, pp. 7-30, 1993.
[18] M. D. P. Monteiro-Marques, Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction. Boston, MA: Birkhauser, 1993.
[5] B. Brogliato and A. Zavala Rio, "On the control of complementary-slackness juggling mechanical systems," IEEE Trans. Automat. Contr., vol. 45, no. 2, 2000.
[19] J. J. Moreau, "Bounded variation in time," in: Topics in Nonsmooth Mechanics, Eds: J.J.Moreau, P.D.Panagiotopolous, G.Strang. Birkhauser Verlag, Basel, Boston, Berlin, pp. 1-74, 1988.
[6] B. Brogliato, Nonsmooth Impact Mechanics. Models, Dynamics and Control, Springer Communications and Control Engineering Series: London, 2000.
[20] J. J. Moreau, "Unilateral contacts and dry friction in finite freedom dynamics", in Nonsmooth Mechanics and Applications, CIMS Course and Lectures, No 302, Springer-Verlag, Wien. New York, pp. 1-82, 1988.
[7] F. H. Clarke, Optimization and Nonsmooth Analysis. New York: John Wiley & Sons, 1983.
[21] Yu. V. Orlov, Theory of optimal systems with generalized controls [in Russian], Nauka, Moscow, 1988.
[8] F . H . Clarke and R. B. Vinter, "Optimal multiprocesses," S I A M J. Contr. and Optim., vol. 27, no. 5, pp. 1072-1091,1989.
[22] A. Ph. Phillipov, Differential Equations with Discontinuous Right-Hand Side. Dortrecht, The Netherlands: Kluwer, 1988.
[9] R. Gabasov, F. M. Kirillova, N. V. Balashevich, and J. Bentsman, "Stabilization with the Help of Bounded Impulse Feedbacks," Preprints of the 2nd International Federation of Automatic Control Symposium on Robust Control Design, Budapest, Hungary, pp. 143-148, June 25-27, 1997.
[23] F. Pfeifer and C. Glocker, Multi-Body Dynamics with Unilateral Constraints, New-York: Wiley, 1996. [24] J. Warga, "Variational problems with unbounded controls," J. SIAM. Ser. A, Control, vol. 3, no. 2, 1965.
[10] M. Jean and J. J. Moreau, Dynamics of elastic or rigid bodies with .frictional contact: numerical methods, Publications of Laboratory of Mechanics and Acoustics, Marseille, April, No 124, 1991.
[25] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.
3697
