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On the transition phase in robotics- Part 1: Impact models and dynamics
Preprints of the Fourth IFAC Symposium on Robot Control September 19-21, 1994, Capri, Italy
On the transition phase in robotics- Part 1: impact models-and dynamics B.Brogliato, P. Orhant, R. Lozano t Laboratoire d'Automatique de Grenoble, URA C.N .R.S. 228 B.P. 46, Saint Martin d 'Heres, France Heudiasyc , U.T.C., URA C.N.R.S . 817 rue P. de Roberval, B.P. 649 , 60206 Compiegne cedex e-mail: [email protected] t
Abstract
we point out the difference between such problems and exogeneous impulsive forces. This is in particular cruThe aim of this note is to study impact dynamics in view cial for the variational approach, that is necessary if one of control of manipulators subject to unilateral kinematic wants to write properly Lagrange equations. The note is constraints during a complete robotic task with contact organized as follows: in section 2, we recall some of the and non-contact phases. In particular, the relationships results in [5] and we point out the difference between exbetween continuous-dynamics model that approximate ogeneous impacts and unilateral constraints. In section the rigid case and the variational approach (Lagrange 3, we study approximating physical models composed of equations) are discussed. Along the whole paper, the springs and dampers at the contact point. In section 4 analysis is developed using simple examples, and we try we study tha variational approach to impact dynamics to systematically review the existing literature related to and the Lagrange equations. the problems we deal with.
1
Introduction
Impact phenomenons between perfectly rigid bodies have attracted the interest of scientists since a long time, and is still a topic of deep investigations. It is worth noting that the problems related to impact dynamics have attracted the interest of physicians (the impact physical laws were in particular studied initialy by researchers like Newton, Huygens or Poisson and the well-known Newton's restitution coefficient is still well alive as a basic model for rigid bodies collisions), and during the last fifteen years as a benchmark example to illustrate chaotic dynamics, mathematicians [7] [12] [13] [15], researchers from mechanical engineering [3] [9] [18] [19] , from robotics [21] [11], and control theory [14] [6] . Since our goal is mainly to deal with stabilization of robot manipulators with unilateral constraints during tasks that involve contact and non-contact phases, we study the mathemat~cal nature of impacts that we shall need to write and study properly the equations of the system. Simply speaking, we show that the corresponding equations are differential equations with distributions in coefficients. The study is done for general mechanical systems such as rigid robot manipulators. It then seems important to understand clearly the dynamics of such systems when there are unilateral constraints, and
2
Distributional model of impacts
Starting from the physical nature of an impact between two rigid bodies that is currently used in the literature, we have studied in [5] the impact forces and dynamics using Schwartz disributions. It has then been proved that an impulsive force yields a discontinuity in the velocity at the time of impact, through a very simple example and carefully using the distributions theory. Namely we have the following: eexample le Assume that a mass m moving on a line, with gravity center coordinate x, is submitted to an impact of magnitude Pi at the instant te; the dynamic equation is given by: (1)
and the following is true [5] e claim 1. Assume the mass is submitted to an impulsive pecussion at t = te. Then there is a discontinuity <7 x in the velocity at the time te while the position x remains continuous. Conversely if the velocity is discontinuous and the popsition continuous, then there is an impulsive percussion.
Relationships with dynamical problems with unilateml constmints
565
The above analysis is valid each time there is an imfunction of bounded variation, a useful fact since pulsive force that acts on the system, and the discontiin certain cases the impacts are likely to occur innuity in the velocity is from a logical point of view to finitely often in a finite time interval. be seen as a consequence of the impulse. Conversely, if 2. In the literature, the analysis generally starts by a the mass collides a rigid wall at t = t e , then x must be priori assuming that during the impact, the change discontinuous at te except if x(t;;) = 0, since it must the bodies'position can be neglected (see e.g. in happen that sgn(x(t;;)) = -sgn(x(tt)). Then the peror that rigid body collisions can be considered [21]), cussion is rather a consequence of the discontinuity. Now as a time-dependent process where changes in mowhy should the position remain continuous at the impact menta occur without changes in configuration since instant? If x has a discontinuity at te,then the impact the total collision period is very small [18] [19]: The force has the form PI btc + P28tc· From a mathematical formulation given here provides a rigorous explanapoint of view, nothing a priori hampers the interaction of this assumption. Moreover the use of Dirac tion force to be like this. Then it means that the dynamimeasures to model percussions makes the (vague) cal "rigid" problem P (i.e. the differential equation with notion of "infinitely large force" acting during a distribution in coefficients together with its initial data) short time [2] [13] disappear, as it clearly alvery can be approximated by a sequence of "compliant" or lows one to separate the magnitude of the impact " continuous-dynamics" problems P n where the interacforce Pi and its distribution on the time axis bt r . tion force has the value Plb n + P28 n [8] chp.l §2, where This is quite natural, since it is not difficult to realbn is a sequence of functions with continuous first derivathat an impact force need not at all be large, but ize tives that determines the Dirac measure and 8n detercan be as small as desired, and in particular can be mines the distribution 8tc [1] §2 lemma 2.2.5. However it much smaller than the other "regular" forces actis really not clear which kind of physical model should coring on the system. This problem is also studied in respond to such approximating problems. In the sequel [12], where the author uses the fact that a function we shall prove that for very simple continuous-dynamics of bounded variation on a closed interval can be models P2 == o. Moreover we shall also use energetical aruniquely decomposed into the sum of an absolutely guments (namely the work done by the impulsive forces continuous, a continuous function and a so-called at impacts) to prove that the position cannot be discon"jump-function", or singular part. It is proved in tinuous. Another point of view is to give the approximat[12] that if the external action is expressed as dg ing problems an a priori physical meaning by considering where g is of bounded variation, then the solution state-dependent forces 1/Jn(x) that possess certain propand g possess the same jump-function. Thus both erties; one has to prove that the limit (with respect to approaches in [12J and in this note are quite similar, a certain notion of convergence) problem is a dynamical although the ways used to get the result are differproblem with unilateral constraints [7] [15]. The available ent (Note also that more complicated mechanical studies are however restricted to penalizing [7] functions systems do not fit within the analysis proposed in 1/Jn which are velocity independent, i.e. the impacts are [12], and also that we prove continuity of the posienergy-Iossless. tion and discontinuity of the velocity). An open issue is to prove that we can associate to any rigid problem P with any energetical behaviour at im3 pacts a compliant problem Pn , and that any passive or This analysis can be applied to more complex systems strictly passive I compliant environment is a possible can- such as rigid manipulators [5J. Now using straightfordidate for approximation. We shall discuss more on this ward manipulations we can write the dynamics of a rigid in the next sections. manipulator subject to an impact in Cartesain space as follows: Remark. (2) 1. Following [16] the solution of (1) (hence x) is necessarily continuous from the right if the Dirac measure is taken equal to it where h is the Heavyside where a(·,·,·) and b(·) have obvious definitions and defunction rightcontinuous . (We could have also de- pend on the dynamical terms, h is the Heavyside funcfined h as a leftcontinuous function since Stieltjes tion, and XI = q (the vector of generalized coordinates). measures can indifferentely be defined from func- Now take Y2 = X2 - b(xdh, then (2) becomes: tions left as well as rightcontinuous [10] p.133). It = Y2 + b(xdh is then possible to initialize the system on the con(3) straint surface, by setting x(O-) or X(O+) depending = a(xI' X2, t) -It.{xd (Y2 + b(xdh) h on whether h is left or rightcontinous respectively [8] §2 chp.1. This is true if h is replaced by any The equation in (3) satisfies the Caratheodory conditions 1 In the control theory sense on existence and uniqueness of solutions for initial data
566
section when we study continuous-dynamics models as approximating models for the perfect rigid case. In relation with the discussion on problems with unilateral constraints and discontinuous position in the foregoing section, let us note that if the position has a discontinuity, then the integrand of the impulsive work contains OteOte which, as already noticed, has no meaning, but would represent as the limit of approximating sequences an "object" (not a function, not a distribution) with mass +00 concentrated at te [17] p.1l7: Clearly physical passive pr strictly passive compliant models cannot yield such results, hence a continuous position.
outside the discontinuities of h [8) (that form a zeromeasure set as long as h is of bounded variation). Thus there exists a maximal solution z(t) to the system in (3) and this solution is a continuous time function [8) (chp.I) . Therefore assuming the control input u has been suitably designed, there is no finite escape time in the system, Xl and Y2 are continuous, and X2 jumps at the instant of the percussion. Thus clearly such mechanical systems excited by impulses belong to the class of systems with singular distributions in coefficients that can be reduced to a Caratheodory (ordinary) system by a change in the unknown function (see [8) for other examples of such manipulations). This fact will prove to be very useful when we deal with stabilization of manipulators during complete tasks with switching controllers.
2. Using the above technique allows to get quite simply the relationship between Pi (the impact magnitude in Cartesian space) and a q (the jump in the generalized velocity). However this is not sufficient to determine which components of q are discontinuous. A way to proceed is to consider Newton's restitution law applied to the last link with position Cartesian vector X = X(q), where X = J(q)q wil have a discontinuity. As long as the Jacobian is full-rank, this allows to express q(tt) as a function of q(t e ), q(t~) and physical data.
Remark. 1. Consider the differential equation :i; = f(x) + g( X )t5te , X E JR., f, 9 smooth functions of x. Then
applying the same technique as in [5) on separation of regular and singular distributions we should get {:i;} +ax t5 te = f(x) + g(x)Ot e, so that i) {:i;} = f(x), and ii) a x = g(x). But the second equality is meaningless: Indeed the term g(x)Ot e represents in fact a distribution, i.e. for any function cp(t) with support K", containing te and continuous at t e, < g(x)t5 te , cp >= g(x)t5te cp(t)dt = g(x(te))cp(t e)· Thus we should write ii) a x = g(x(t e)). But x(t e) is not well-defined and in general g(x(t e)) is neither (Of course if g(x) is replaced by a function of time g(t) then the technique can be employed provided g(t) is continuous at t e, see [8) p.40-41). Such problems are treated e.g. in [8]: Such differential equations do not have in general a unique solution (independently of the choice of the initial data), and the obtained solution strongly depends on the sequence of problems considered to approximate the equation [8] th.4, chp.l, §3. [4] studies existence and uniqueness of solutions of equations like:i; = f(x) + g(x)u, x E JR.n, U scalar of bounded variation. [4] does not choose the distributional a~ proach and overcomes the above problem by considering properties of the input/state map iI>(u). It is seen on an example that the approach has the effect of "smoothing" the solution, i.e. the jumps at isolated points may be ignored. Although the differential equations we shall deal with will not fall into this category, one should note that the work of the impulsive force at the impact time is given by J~l :i;(t)t5te dt, with te E [to, td. Since:i; is discontinuous at t e , the integrand is meaningless, and this is exactly the same kind of problems as in the above differential equations. This reveals a limitation of the theory of distributions in the impact dynamics modelling. We shall discuss about this in the next
JK",
3
3
Continuous-dynamics of impacts
models
In this section we study the relationships between impulsive and physical models of percussions in a simple case, i.e. we show that the interaction force converges towards a Dirac measure when the stiffness and/or the damping coefficient grow unbounded.
567
• From damped to plastic impact (e = 0) Consider the same system as in example 1 but with a compliant "purely dissipative" environment with damping coefficient f. The dynamic equations are:
o ~ t ~ te
mx
=
O,x(O)
=
xo,:i;(O)
=:i;o
(4)
mx+f:i; =0 We obtain after the impact :i; :i;oe-,f,t, x = -~(1 - e-,f,t). One sees that if f -+ +00, then x(t) -+ x(O) = 0 and :i; -+ 0 for all t ~ O. For any increasing positive sequence of values of dam~ ing coefficient Un}, let us denote Pn(r) = fn:i;(r) for 0 ~ r ~ Pn(r) == 0 elsewhere. Then
j"i,
Jo~ Pn(r)dr + £/n
=
Jt
X )
fn:i;dr, with
£/n
=
m±oe-~. We get Pn(-) -+ m±ot5o and £/n -+ 0 as n -+ +00. In the limit, the equation describing the system becomes mx = (p - ffia±)oo, with a± = :i;(0-) = :i;o.
tem. As an illustration, consider the classical bouncing • The generol case (0 < e ~ 1) Consider now that the environment is represented ball problem, that corresponds to adding a constant force by a spring k and a damper I. During the contact (gravity) to the mass: Then it can be shown that the imwe get mx + Ix + kx = o. Assume now that !:l. = pulse function acting on the ball for 0 < e < 1 has the 12 -4km < O. Then we obtain x(t) = ~ertsin(wt), form P = Pk6tk ' where t+oo < +00 is an accumulax(t) = xoe rt [;Ssin(wt) + cos(wt)] , with Xo = x(O), tion point of the sequence {td. A fundamental property r = ~ w = Now for tl = ~, x(tt} = 0 and of this sequence is that the step function h( t) ~ L~=o Pk x(tt} :~: -xoe':;- ~ Let us choose 0 {3 ~ 1, and let on [tk. tk+d , n.2 0 is of bounded variati.on on [to, t+oo] . ~ Hence from [17] p.25 and th .2, p.53, P = h can be considus see what happens if 1 = 2Iln({3)1 (".2+~~\(t3l) 2, ered as a Schwartz's distribution (Note that this would when k -. +00 (Such an 1 guarantees !:l. < 0 for not be the case if Pk = +00, since in this case P o < {3 ~ 1) : We get tl -. 0 and e~ -. {3. Thus is not defined (it takes infinitely large values) as a funcx(td -. -{3xo as k -. +00 (if {3 = 1 then 1 == 0 and tion from V in JR.). Then from density of V in V· [17] we retrieve the above case, and if 0 < {3 < 1, then th.I5, chp.3, we can approximate P and h by sequences of 1 -. 00 as k -. 00) . Simple calculations show that smooth functions {Pn} and {h n }. From [1] lemma 2.2.5 , fo:: Pn(r)dr ~ fo:: Inx(r)+knx(r)dr = mxo({3+ 1) , itn = Pn since the functions hn are continuous. Are there where {In} and {k n } are sequences of damping and sequences of damping and stiffness coefficients such that stiffness coefficients defined as previously. Thus the corresponding compliant model is an approximating once again the sequence of force functions Pn(-) sequence for this problem? The physical intuition would during the collision time converges towards a Dirac yield a positive answer. A rigorous mathematical proof distribution. Note that to show this, we have con- is needed. sidered a sequence of damping coefficients that depends on the mass m. This is at first sight surpriz- Remark. It is worth noting that there is a fundamental ing, as one can expect the nature of the collision difference between the cases of purely dissipative shocks to be independent of the mass of the bodies that (e = 0) and elastic shocks (0 < e ~ 1): Indeed in the first collide. However what really matters is not how case the impact corresponds to an instantaneous dissipathe sequences {In} and {k n } are defined but rather tion of the whole energy of the system, but the compliant that they do exist, i.e. we are able to associate a model does not tend towards an infinitely rigid model; sequence of compliant models to the rigid limiting on the contrary when a spring is added in the compliant model. 3 model then the limit is an infinitely rigid surface. The work performed by the contact forces during the impact is given this time by WIO,td = f~' x(t)(Jnx(t) + knx(t))dt which is found after 4 Dynamics lenghty but straightforward calculations to be equal of percussions-Lagrange equa(r.l2 - 1) < 0 t o 2".ln·(P)+8".3+211n3JPlIH"'·lln(plImx 8".(".2+1 n (13)) 0 /J _. tions Thus the approximating model allows us to give a value to the impulsive work, that is consistent with In most of the textbooks on analytical mechanics conthe energetical behaviour of the impact (here a loss taining a chapter on impacts it is shown that one can deof energy as long as {3 < 1). rive extensions to basic tools of classical mechanics, e.g .. the lundamental principle 01 dynamics and the Lagronge We thus have the following equations in the case of impacts phenomenons. Following • claim 2. Consider the equation in (1) that repre- the philosophy we have presented above, it can be shown sents the dynamics of a rigid mass colliding a rigid envi- that the theorems of the linear and angular momentum ronment, without any external action. Then for anyener- can be easily retrieved. We now focus our attention on getical behaviour of the materials at the impacts (namely the variational formulation of impact dynamics. In this for any restitution coefficient 0 ~ e ~ 1) we can associate section, we first treat this problem as if there were no an approximating sequence of compliant models such that difficulties in applying variational techniques to systems the approximating solutions converge towards the solu- subject to impacts, i.e. we simply take the Lagrange tion of (1). equations and suppose that a generalized external impulWe have been able to prove the above because the sive force acts on the system (We have already noted a considered problem is integrable and the exit times can significant difference between this problem and the probbe calculated. As pointed out in the previous section, lem with unilateral constraints in the foregoing section). in the general case the problem is much more involved. Then we discuss about the variational approach to sysA future work is to find out arguments proving that se- tems with unilateral constraints. quences {In} and {k n } exist that yield the same results The Lagrange equations of the system submitted to when for instance an external force u(t) acts on the sys-
Lt:;;
fA.
<
Lt:;;
568
a generalized force F + TJbt c ' where F represents all the generalized forces without taking into account the impulsive ones, are given by we get:
ft
(~)
-
~
= F
+ TJbt
c '
Thus
(5)
°
from which we deduce:
(6)
and
In case the kinetic energy is a quadratic form of the velocity, ~ is the positive definite inertia matrix and we obtain: 17q
=
°
and
~17q = ~(tt) - ~(t;)
(7)
so that finally the Lagrange equations become in case of impact: 8 2 T{oo}
8Q2" q
impact dynamics something else than merely curves, and this is quite consistent with the analysis we have made above starting from the physical nature of the interaction force between two rigid bodies colliding. It is worth noting that in this limit case of zig-zag curve with e = 1, the corresponding impact function is not a Schwartz's distribution since h is not of bounded variation on any interval of strictly positive measure as £ -+ 0, since the percussion magnitude is 2ml±01 > at each impact and the "flight"-time is b. k ~ ~. In the sequel we shall always deal with systems such that the impact function is of bounded variation on compact time intervals. Let us now formulate the variational problem. First notice that in order for this problem to make sense, the integral action to be minimized must "contain" the impacts 2. Indeed consider the bouncing ball example, and assume that one wants to search for the extremals of a classical variational problem I(x) = ft~' L(x, ±)dt, with fixed end points and the unilateral condition x ~ 0: Then for any to, t1> x(to) > 0, x(td ~ 3 the minimization process will always lead to a smooth solution curve, because the end point conditions uniquely determine the initial data; roughly speaking, the solution will always be such that it takes the whole interval [to , td to reach the constraint x = 0. As a consequence the impacts will always be absent of such a formulation! It is therefore necessary for the solution curve to contain impacts that the end point conditions be modified by fixing for instance to, t1, x(to) and ±(to). Then the constraint x = 0 will in general be reached at t2 < t1 and with a nonzero velocity ±(t2)' It remains now to examine how the integral action whose extremals are the solution of the impact problem can be written. As suggested by the study in the foregoing sections, the perfectly rigid case should be regarded as the limit of sequences of continuous-dynamics problems Pn . A formulation in this direction is the one in [7] [15]: Roughly, the authors in [7] [15] consider approximating variational problems Pn whith Lagrangian L(qn, qn) = T(qn, tin) - U(qn) - Un(qn), where the last term accounts for the potential elastic energy when there is contact. The limiting or bounce problem P has La.grangian L(q,q) = T(q,ti) - U(q) - ?P(q)p" where the unilateral constraints are ?P(q) ~ 0, and p, is a bounded positive measure that represents the impact forces. The study is restricted to the case when the impacts are lossless (e = 1) . The cornerstone of the analysis is to show that, simply speaking, the set of Lipschitz solution curves to the problem P is dense in the set of solutions ofP, and that solutions of P n are suitable sequences. Convergence
+ 8q8q 82T . _ q
8(T-U) = 8q
F (8)
°
Classically (8) is obtained by integrating the Lagrange equations over infinitely short time intervals of shocks durations [13] [21] [2]. We now turn our attention on the variational formulation of impact dynamics. First notice that there are basically two situations: Either the system is submitted to an external force which is impulsive (this is the foregoing analysis) or the system is submitted to unilateral kinematic constraints: The variational approach for impact dynamics is concerned with the latter. More precisely, we look for a extended Lagrangian function Le such that the minimization of the corresponding integral action provides the Lagrange equations for impacts. Classicaly, two problems arise [20]: i) Existence of the extremal curves (Le. in which space do the extremals have to be defined, or what is there nature?), ii) Necessary and sufficient conditions to be verified by the extremals. In order to answer to the first point, consider the following one-dof simple example, where the dynamics can be integrated at hand: Let a disc with radius d move without friction on a horizontal plane between two parallel rigid "walls", situated at a distance d+2£ one from each other. The restitution coefficient is taken to be e = 1, and the initial conditions on position and velocity are such that the problem is well-pbsed. Then the graph of the position 2For instance, if the integrand of [(x) is equal to (1+x 2)(1+[:i: 2 of the disc center with respect to time is a "saw-tooth" or IJ2) with endpoint conditions 0, I , x(O) = x(I) = 0, then the mini"zig-zag" diagram. By letting £ approach zero, this curve mizing curve is "naturally" an infinitesimal zig-zag [20J p .I59. We tends towards an infinitesimal zig-zag curve, that is not a may say that the integral action "contains" the irregularities. This curve, but a generalized curve [20] chp.6, Le. an element is not the case for a mechanical problem if a classical Lagrangian as the bouncing ball example shows. of the dual space of continuous functions (Young's gen- is considered JObviously if the end point conditions do not satisfy the coneralized curves [20]) or smooth functions (Schwartz's dis- straints the problem possesses no solution. tributions). It is therefore clear that one really needs for
569
is understood in the sense of the so-called r -convergence. To the best of our knowledge a variational formulation of the bouncing ball problem when e < 1 has not yet been proposed. Difficulties are mainly that the approximating problems must be dissipative and that the limit" problem possess an accumulation point of discontinuities.
5.1
[11] Y. Hurmuzlu, D.B. Marghitu , "Rigid body collisions
Schwartz's distributions
• definition 1 V is the subspace of smooth C , with bounded support.
[9] W. Goldsmith, Impact: The Theory and physical Behaviour of Colliding Solids, London: Edward Arnold Publishers, 1960.
[10] A. Gramain, Integration, Hermann, Paris, 1988.
Appendix
5
[8] A.F. Filipov, Differential equations with discontinuous right hand sides, Kluwer academic publishers, Dordrecht, NL, 1988.
4
functions cp : JR.n
of planar kinematic chains with multiple contact points", Int. J . of Robotics Research, vo1.l3, no 1, pp .82-92, february 1994.
--+
[12] M. Laghdir, Solutions des equations regissant le mouvement des particules en contact avec frottement sec et recevant des impulsions Ph. D. thesis, USTL, Mathematiques fondamentales et appliquees, France,Montpellier, 1987.
• definition 2 A distribution D is a continuous linear form defined on the vector space V .
This means that to any cp E V, D associates a complex number D(cp), noted < D, cp >. The space of distributions on V is the dual space of V and is noted V*.
[13] J .J . Moreau, "Standard inelastic shocks and the dynamics of unilateral constraints" , CISM courses and lectures, no 288, Springer-VerJag, pp.173-221, 1985.
[14] J.K . Mills and C .V. Nguyen, "Robotic manipulator collisions: modeling and simulation", AS ME J . Dyn. Syst. Meas. and Contr., vol. 114, pp.650-658, decem[1] P. Antosik, J. Mikusinski, R. Sikorski, Theory of ber 1992. distributions- the sequential approach, Elsevier Scientific publishing company, Amsterdam; PWN- Pol- [15] D. Percivale, "Uniqueness in the elastic bounce probish Scientific publishers, Warszawa, 1973. lem", Journal of differential equations, 56, 206-215,
References
[2] P. Appel, Traite de mecanique rationnelle, 3eme edition, 2d tome, dynamique des systemes-mecanique analytique, Paris, Gauthiers-Vi liars 1911 . [3] R.M. Brach, Mechanical impact dynamics, John Wiley, New York, 1991. [4] A. Bressan, "On differential systems with impulsive controls", Rend. Sem. Mat. Univ. Padova, vo1.78, pp.227-236, 1987. [5] B. Brogliato, P. Orhant, "On the transition phase in robotics: impact models, dynamics and control" , IEEE Conf. on Robotics and Automation, San Diego, may 1994. [6] M. Buhler, D.E. Koditshek, P.J. Kindlmann, "A family of robot control strategies for intermittent dynamical environments" , IEEE Control Systems magazine, p.I6-22, f~bruary 1990. [7] G. Buttazzo, D. Percivale, " On the approximation of the elastic bounce problem on Riemannian manifolds", Journal of Differential equations, 47, 227-275 , 1983. 4 i.e.
1985. [16] W.W. Schmaedeke, " Optimal control theory for nonlinear vector differential equations containing me&sures, SIAM J. Control, ser.A, vol.3, no 2, pp.231280, 1965. [17] L. Schwartz, TMorie des distributions, Hermann, publications de l'institut de mathematique de l'universire de Strasbourg, Paris, 1966. [18] R. Souchet, " Analytical dynamics of rigid body impulsive motions", Int. J. Engng. ScL, vol.31, no 1, pp.85-92, 1993. [19] W.J. Stronge, "Rigid body collision with friction", Proc. R. Soc. Lond. A, vol.431, pp.169-181, 1990. [20] L.C. Young, Lectures on the calculus of variations and optimal control theory, Chelsea publishing company, NY, 1980. [21] Y.F. Zheng and H. Hemami, "Mathematical modeling of a robot collision with its environment", J. of Robotic Systems, 2(3), pp.289-307, 1985.
indefinitely differentiable
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On the transition phase in robotics- Part 1: impact models-and dynamics B.Brogliato, P. Orhant, R. Lozano t Laboratoire d'Automatique de Grenoble, URA C.N .R.S. 228 B.P. 46, Saint Martin d 'Heres, France Heudiasyc , U.T.C., URA C.N.R.S . 817 rue P. de Roberval, B.P. 649 , 60206 Compiegne cedex e-mail: [email protected] t
Abstract
we point out the difference between such problems and exogeneous impulsive forces. This is in particular cruThe aim of this note is to study impact dynamics in view cial for the variational approach, that is necessary if one of control of manipulators subject to unilateral kinematic wants to write properly Lagrange equations. The note is constraints during a complete robotic task with contact organized as follows: in section 2, we recall some of the and non-contact phases. In particular, the relationships results in [5] and we point out the difference between exbetween continuous-dynamics model that approximate ogeneous impacts and unilateral constraints. In section the rigid case and the variational approach (Lagrange 3, we study approximating physical models composed of equations) are discussed. Along the whole paper, the springs and dampers at the contact point. In section 4 analysis is developed using simple examples, and we try we study tha variational approach to impact dynamics to systematically review the existing literature related to and the Lagrange equations. the problems we deal with.
1
Introduction
Impact phenomenons between perfectly rigid bodies have attracted the interest of scientists since a long time, and is still a topic of deep investigations. It is worth noting that the problems related to impact dynamics have attracted the interest of physicians (the impact physical laws were in particular studied initialy by researchers like Newton, Huygens or Poisson and the well-known Newton's restitution coefficient is still well alive as a basic model for rigid bodies collisions), and during the last fifteen years as a benchmark example to illustrate chaotic dynamics, mathematicians [7] [12] [13] [15], researchers from mechanical engineering [3] [9] [18] [19] , from robotics [21] [11], and control theory [14] [6] . Since our goal is mainly to deal with stabilization of robot manipulators with unilateral constraints during tasks that involve contact and non-contact phases, we study the mathemat~cal nature of impacts that we shall need to write and study properly the equations of the system. Simply speaking, we show that the corresponding equations are differential equations with distributions in coefficients. The study is done for general mechanical systems such as rigid robot manipulators. It then seems important to understand clearly the dynamics of such systems when there are unilateral constraints, and
2
Distributional model of impacts
Starting from the physical nature of an impact between two rigid bodies that is currently used in the literature, we have studied in [5] the impact forces and dynamics using Schwartz disributions. It has then been proved that an impulsive force yields a discontinuity in the velocity at the time of impact, through a very simple example and carefully using the distributions theory. Namely we have the following: eexample le Assume that a mass m moving on a line, with gravity center coordinate x, is submitted to an impact of magnitude Pi at the instant te; the dynamic equation is given by: (1)
and the following is true [5] e claim 1. Assume the mass is submitted to an impulsive pecussion at t = te. Then there is a discontinuity <7 x in the velocity at the time te while the position x remains continuous. Conversely if the velocity is discontinuous and the popsition continuous, then there is an impulsive percussion.
Relationships with dynamical problems with unilateml constmints
565
The above analysis is valid each time there is an imfunction of bounded variation, a useful fact since pulsive force that acts on the system, and the discontiin certain cases the impacts are likely to occur innuity in the velocity is from a logical point of view to finitely often in a finite time interval. be seen as a consequence of the impulse. Conversely, if 2. In the literature, the analysis generally starts by a the mass collides a rigid wall at t = t e , then x must be priori assuming that during the impact, the change discontinuous at te except if x(t;;) = 0, since it must the bodies'position can be neglected (see e.g. in happen that sgn(x(t;;)) = -sgn(x(tt)). Then the peror that rigid body collisions can be considered [21]), cussion is rather a consequence of the discontinuity. Now as a time-dependent process where changes in mowhy should the position remain continuous at the impact menta occur without changes in configuration since instant? If x has a discontinuity at te,then the impact the total collision period is very small [18] [19]: The force has the form PI btc + P28tc· From a mathematical formulation given here provides a rigorous explanapoint of view, nothing a priori hampers the interaction of this assumption. Moreover the use of Dirac tion force to be like this. Then it means that the dynamimeasures to model percussions makes the (vague) cal "rigid" problem P (i.e. the differential equation with notion of "infinitely large force" acting during a distribution in coefficients together with its initial data) short time [2] [13] disappear, as it clearly alvery can be approximated by a sequence of "compliant" or lows one to separate the magnitude of the impact " continuous-dynamics" problems P n where the interacforce Pi and its distribution on the time axis bt r . tion force has the value Plb n + P28 n [8] chp.l §2, where This is quite natural, since it is not difficult to realbn is a sequence of functions with continuous first derivathat an impact force need not at all be large, but ize tives that determines the Dirac measure and 8n detercan be as small as desired, and in particular can be mines the distribution 8tc [1] §2 lemma 2.2.5. However it much smaller than the other "regular" forces actis really not clear which kind of physical model should coring on the system. This problem is also studied in respond to such approximating problems. In the sequel [12], where the author uses the fact that a function we shall prove that for very simple continuous-dynamics of bounded variation on a closed interval can be models P2 == o. Moreover we shall also use energetical aruniquely decomposed into the sum of an absolutely guments (namely the work done by the impulsive forces continuous, a continuous function and a so-called at impacts) to prove that the position cannot be discon"jump-function", or singular part. It is proved in tinuous. Another point of view is to give the approximat[12] that if the external action is expressed as dg ing problems an a priori physical meaning by considering where g is of bounded variation, then the solution state-dependent forces 1/Jn(x) that possess certain propand g possess the same jump-function. Thus both erties; one has to prove that the limit (with respect to approaches in [12J and in this note are quite similar, a certain notion of convergence) problem is a dynamical although the ways used to get the result are differproblem with unilateral constraints [7] [15]. The available ent (Note also that more complicated mechanical studies are however restricted to penalizing [7] functions systems do not fit within the analysis proposed in 1/Jn which are velocity independent, i.e. the impacts are [12], and also that we prove continuity of the posienergy-Iossless. tion and discontinuity of the velocity). An open issue is to prove that we can associate to any rigid problem P with any energetical behaviour at im3 pacts a compliant problem Pn , and that any passive or This analysis can be applied to more complex systems strictly passive I compliant environment is a possible can- such as rigid manipulators [5J. Now using straightfordidate for approximation. We shall discuss more on this ward manipulations we can write the dynamics of a rigid in the next sections. manipulator subject to an impact in Cartesain space as follows: Remark. (2) 1. Following [16] the solution of (1) (hence x) is necessarily continuous from the right if the Dirac measure is taken equal to it where h is the Heavyside where a(·,·,·) and b(·) have obvious definitions and defunction rightcontinuous . (We could have also de- pend on the dynamical terms, h is the Heavyside funcfined h as a leftcontinuous function since Stieltjes tion, and XI = q (the vector of generalized coordinates). measures can indifferentely be defined from func- Now take Y2 = X2 - b(xdh, then (2) becomes: tions left as well as rightcontinuous [10] p.133). It = Y2 + b(xdh is then possible to initialize the system on the con(3) straint surface, by setting x(O-) or X(O+) depending = a(xI' X2, t) -It.{xd (Y2 + b(xdh) h on whether h is left or rightcontinous respectively [8] §2 chp.1. This is true if h is replaced by any The equation in (3) satisfies the Caratheodory conditions 1 In the control theory sense on existence and uniqueness of solutions for initial data
566
section when we study continuous-dynamics models as approximating models for the perfect rigid case. In relation with the discussion on problems with unilateral constraints and discontinuous position in the foregoing section, let us note that if the position has a discontinuity, then the integrand of the impulsive work contains OteOte which, as already noticed, has no meaning, but would represent as the limit of approximating sequences an "object" (not a function, not a distribution) with mass +00 concentrated at te [17] p.1l7: Clearly physical passive pr strictly passive compliant models cannot yield such results, hence a continuous position.
outside the discontinuities of h [8) (that form a zeromeasure set as long as h is of bounded variation). Thus there exists a maximal solution z(t) to the system in (3) and this solution is a continuous time function [8) (chp.I) . Therefore assuming the control input u has been suitably designed, there is no finite escape time in the system, Xl and Y2 are continuous, and X2 jumps at the instant of the percussion. Thus clearly such mechanical systems excited by impulses belong to the class of systems with singular distributions in coefficients that can be reduced to a Caratheodory (ordinary) system by a change in the unknown function (see [8) for other examples of such manipulations). This fact will prove to be very useful when we deal with stabilization of manipulators during complete tasks with switching controllers.
2. Using the above technique allows to get quite simply the relationship between Pi (the impact magnitude in Cartesian space) and a q (the jump in the generalized velocity). However this is not sufficient to determine which components of q are discontinuous. A way to proceed is to consider Newton's restitution law applied to the last link with position Cartesian vector X = X(q), where X = J(q)q wil have a discontinuity. As long as the Jacobian is full-rank, this allows to express q(tt) as a function of q(t e ), q(t~) and physical data.
Remark. 1. Consider the differential equation :i; = f(x) + g( X )t5te , X E JR., f, 9 smooth functions of x. Then
applying the same technique as in [5) on separation of regular and singular distributions we should get {:i;} +ax t5 te = f(x) + g(x)Ot e, so that i) {:i;} = f(x), and ii) a x = g(x). But the second equality is meaningless: Indeed the term g(x)Ot e represents in fact a distribution, i.e. for any function cp(t) with support K", containing te and continuous at t e, < g(x)t5 te , cp >= g(x)t5te cp(t)dt = g(x(te))cp(t e)· Thus we should write ii) a x = g(x(t e)). But x(t e) is not well-defined and in general g(x(t e)) is neither (Of course if g(x) is replaced by a function of time g(t) then the technique can be employed provided g(t) is continuous at t e, see [8) p.40-41). Such problems are treated e.g. in [8]: Such differential equations do not have in general a unique solution (independently of the choice of the initial data), and the obtained solution strongly depends on the sequence of problems considered to approximate the equation [8] th.4, chp.l, §3. [4] studies existence and uniqueness of solutions of equations like:i; = f(x) + g(x)u, x E JR.n, U scalar of bounded variation. [4] does not choose the distributional a~ proach and overcomes the above problem by considering properties of the input/state map iI>(u). It is seen on an example that the approach has the effect of "smoothing" the solution, i.e. the jumps at isolated points may be ignored. Although the differential equations we shall deal with will not fall into this category, one should note that the work of the impulsive force at the impact time is given by J~l :i;(t)t5te dt, with te E [to, td. Since:i; is discontinuous at t e , the integrand is meaningless, and this is exactly the same kind of problems as in the above differential equations. This reveals a limitation of the theory of distributions in the impact dynamics modelling. We shall discuss about this in the next
JK",
3
3
Continuous-dynamics of impacts
models
In this section we study the relationships between impulsive and physical models of percussions in a simple case, i.e. we show that the interaction force converges towards a Dirac measure when the stiffness and/or the damping coefficient grow unbounded.
567
• From damped to plastic impact (e = 0) Consider the same system as in example 1 but with a compliant "purely dissipative" environment with damping coefficient f. The dynamic equations are:
o ~ t ~ te
mx
=
O,x(O)
=
xo,:i;(O)
=:i;o
(4)
mx+f:i; =0 We obtain after the impact :i; :i;oe-,f,t, x = -~(1 - e-,f,t). One sees that if f -+ +00, then x(t) -+ x(O) = 0 and :i; -+ 0 for all t ~ O. For any increasing positive sequence of values of dam~ ing coefficient Un}, let us denote Pn(r) = fn:i;(r) for 0 ~ r ~ Pn(r) == 0 elsewhere. Then
j"i,
Jo~ Pn(r)dr + £/n
=
Jt
X )
fn:i;dr, with
£/n
=
m±oe-~. We get Pn(-) -+ m±ot5o and £/n -+ 0 as n -+ +00. In the limit, the equation describing the system becomes mx = (p - ffia±)oo, with a± = :i;(0-) = :i;o.
tem. As an illustration, consider the classical bouncing • The generol case (0 < e ~ 1) Consider now that the environment is represented ball problem, that corresponds to adding a constant force by a spring k and a damper I. During the contact (gravity) to the mass: Then it can be shown that the imwe get mx + Ix + kx = o. Assume now that !:l. = pulse function acting on the ball for 0 < e < 1 has the 12 -4km < O. Then we obtain x(t) = ~ertsin(wt), form P = Pk6tk ' where t+oo < +00 is an accumulax(t) = xoe rt [;Ssin(wt) + cos(wt)] , with Xo = x(O), tion point of the sequence {td. A fundamental property r = ~ w = Now for tl = ~, x(tt} = 0 and of this sequence is that the step function h( t) ~ L~=o Pk x(tt} :~: -xoe':;- ~ Let us choose 0 {3 ~ 1, and let on [tk. tk+d , n.2 0 is of bounded variati.on on [to, t+oo] . ~ Hence from [17] p.25 and th .2, p.53, P = h can be considus see what happens if 1 = 2Iln({3)1 (".2+~~\(t3l) 2, ered as a Schwartz's distribution (Note that this would when k -. +00 (Such an 1 guarantees !:l. < 0 for not be the case if Pk = +00, since in this case P o < {3 ~ 1) : We get tl -. 0 and e~ -. {3. Thus is not defined (it takes infinitely large values) as a funcx(td -. -{3xo as k -. +00 (if {3 = 1 then 1 == 0 and tion from V in JR.). Then from density of V in V· [17] we retrieve the above case, and if 0 < {3 < 1, then th.I5, chp.3, we can approximate P and h by sequences of 1 -. 00 as k -. 00) . Simple calculations show that smooth functions {Pn} and {h n }. From [1] lemma 2.2.5 , fo:: Pn(r)dr ~ fo:: Inx(r)+knx(r)dr = mxo({3+ 1) , itn = Pn since the functions hn are continuous. Are there where {In} and {k n } are sequences of damping and sequences of damping and stiffness coefficients such that stiffness coefficients defined as previously. Thus the corresponding compliant model is an approximating once again the sequence of force functions Pn(-) sequence for this problem? The physical intuition would during the collision time converges towards a Dirac yield a positive answer. A rigorous mathematical proof distribution. Note that to show this, we have con- is needed. sidered a sequence of damping coefficients that depends on the mass m. This is at first sight surpriz- Remark. It is worth noting that there is a fundamental ing, as one can expect the nature of the collision difference between the cases of purely dissipative shocks to be independent of the mass of the bodies that (e = 0) and elastic shocks (0 < e ~ 1): Indeed in the first collide. However what really matters is not how case the impact corresponds to an instantaneous dissipathe sequences {In} and {k n } are defined but rather tion of the whole energy of the system, but the compliant that they do exist, i.e. we are able to associate a model does not tend towards an infinitely rigid model; sequence of compliant models to the rigid limiting on the contrary when a spring is added in the compliant model. 3 model then the limit is an infinitely rigid surface. The work performed by the contact forces during the impact is given this time by WIO,td = f~' x(t)(Jnx(t) + knx(t))dt which is found after 4 Dynamics lenghty but straightforward calculations to be equal of percussions-Lagrange equa(r.l2 - 1) < 0 t o 2".ln·(P)+8".3+211n3JPlIH"'·lln(plImx 8".(".2+1 n (13)) 0 /J _. tions Thus the approximating model allows us to give a value to the impulsive work, that is consistent with In most of the textbooks on analytical mechanics conthe energetical behaviour of the impact (here a loss taining a chapter on impacts it is shown that one can deof energy as long as {3 < 1). rive extensions to basic tools of classical mechanics, e.g .. the lundamental principle 01 dynamics and the Lagronge We thus have the following equations in the case of impacts phenomenons. Following • claim 2. Consider the equation in (1) that repre- the philosophy we have presented above, it can be shown sents the dynamics of a rigid mass colliding a rigid envi- that the theorems of the linear and angular momentum ronment, without any external action. Then for anyener- can be easily retrieved. We now focus our attention on getical behaviour of the materials at the impacts (namely the variational formulation of impact dynamics. In this for any restitution coefficient 0 ~ e ~ 1) we can associate section, we first treat this problem as if there were no an approximating sequence of compliant models such that difficulties in applying variational techniques to systems the approximating solutions converge towards the solu- subject to impacts, i.e. we simply take the Lagrange tion of (1). equations and suppose that a generalized external impulWe have been able to prove the above because the sive force acts on the system (We have already noted a considered problem is integrable and the exit times can significant difference between this problem and the probbe calculated. As pointed out in the previous section, lem with unilateral constraints in the foregoing section). in the general case the problem is much more involved. Then we discuss about the variational approach to sysA future work is to find out arguments proving that se- tems with unilateral constraints. quences {In} and {k n } exist that yield the same results The Lagrange equations of the system submitted to when for instance an external force u(t) acts on the sys-
Lt:;;
fA.
<
Lt:;;
568
a generalized force F + TJbt c ' where F represents all the generalized forces without taking into account the impulsive ones, are given by we get:
ft
(~)
-
~
= F
+ TJbt
c '
Thus
(5)
°
from which we deduce:
(6)
and
In case the kinetic energy is a quadratic form of the velocity, ~ is the positive definite inertia matrix and we obtain: 17q
=
°
and
~17q = ~(tt) - ~(t;)
(7)
so that finally the Lagrange equations become in case of impact: 8 2 T{oo}
8Q2" q
impact dynamics something else than merely curves, and this is quite consistent with the analysis we have made above starting from the physical nature of the interaction force between two rigid bodies colliding. It is worth noting that in this limit case of zig-zag curve with e = 1, the corresponding impact function is not a Schwartz's distribution since h is not of bounded variation on any interval of strictly positive measure as £ -+ 0, since the percussion magnitude is 2ml±01 > at each impact and the "flight"-time is b. k ~ ~. In the sequel we shall always deal with systems such that the impact function is of bounded variation on compact time intervals. Let us now formulate the variational problem. First notice that in order for this problem to make sense, the integral action to be minimized must "contain" the impacts 2. Indeed consider the bouncing ball example, and assume that one wants to search for the extremals of a classical variational problem I(x) = ft~' L(x, ±)dt, with fixed end points and the unilateral condition x ~ 0: Then for any to, t1> x(to) > 0, x(td ~ 3 the minimization process will always lead to a smooth solution curve, because the end point conditions uniquely determine the initial data; roughly speaking, the solution will always be such that it takes the whole interval [to , td to reach the constraint x = 0. As a consequence the impacts will always be absent of such a formulation! It is therefore necessary for the solution curve to contain impacts that the end point conditions be modified by fixing for instance to, t1, x(to) and ±(to). Then the constraint x = 0 will in general be reached at t2 < t1 and with a nonzero velocity ±(t2)' It remains now to examine how the integral action whose extremals are the solution of the impact problem can be written. As suggested by the study in the foregoing sections, the perfectly rigid case should be regarded as the limit of sequences of continuous-dynamics problems Pn . A formulation in this direction is the one in [7] [15]: Roughly, the authors in [7] [15] consider approximating variational problems Pn whith Lagrangian L(qn, qn) = T(qn, tin) - U(qn) - Un(qn), where the last term accounts for the potential elastic energy when there is contact. The limiting or bounce problem P has La.grangian L(q,q) = T(q,ti) - U(q) - ?P(q)p" where the unilateral constraints are ?P(q) ~ 0, and p, is a bounded positive measure that represents the impact forces. The study is restricted to the case when the impacts are lossless (e = 1) . The cornerstone of the analysis is to show that, simply speaking, the set of Lipschitz solution curves to the problem P is dense in the set of solutions ofP, and that solutions of P n are suitable sequences. Convergence
+ 8q8q 82T . _ q
8(T-U) = 8q
F (8)
°
Classically (8) is obtained by integrating the Lagrange equations over infinitely short time intervals of shocks durations [13] [21] [2]. We now turn our attention on the variational formulation of impact dynamics. First notice that there are basically two situations: Either the system is submitted to an external force which is impulsive (this is the foregoing analysis) or the system is submitted to unilateral kinematic constraints: The variational approach for impact dynamics is concerned with the latter. More precisely, we look for a extended Lagrangian function Le such that the minimization of the corresponding integral action provides the Lagrange equations for impacts. Classicaly, two problems arise [20]: i) Existence of the extremal curves (Le. in which space do the extremals have to be defined, or what is there nature?), ii) Necessary and sufficient conditions to be verified by the extremals. In order to answer to the first point, consider the following one-dof simple example, where the dynamics can be integrated at hand: Let a disc with radius d move without friction on a horizontal plane between two parallel rigid "walls", situated at a distance d+2£ one from each other. The restitution coefficient is taken to be e = 1, and the initial conditions on position and velocity are such that the problem is well-pbsed. Then the graph of the position 2For instance, if the integrand of [(x) is equal to (1+x 2)(1+[:i: 2 of the disc center with respect to time is a "saw-tooth" or IJ2) with endpoint conditions 0, I , x(O) = x(I) = 0, then the mini"zig-zag" diagram. By letting £ approach zero, this curve mizing curve is "naturally" an infinitesimal zig-zag [20J p .I59. We tends towards an infinitesimal zig-zag curve, that is not a may say that the integral action "contains" the irregularities. This curve, but a generalized curve [20] chp.6, Le. an element is not the case for a mechanical problem if a classical Lagrangian as the bouncing ball example shows. of the dual space of continuous functions (Young's gen- is considered JObviously if the end point conditions do not satisfy the coneralized curves [20]) or smooth functions (Schwartz's dis- straints the problem possesses no solution. tributions). It is therefore clear that one really needs for
569
is understood in the sense of the so-called r -convergence. To the best of our knowledge a variational formulation of the bouncing ball problem when e < 1 has not yet been proposed. Difficulties are mainly that the approximating problems must be dissipative and that the limit" problem possess an accumulation point of discontinuities.
5.1
[11] Y. Hurmuzlu, D.B. Marghitu , "Rigid body collisions
Schwartz's distributions
• definition 1 V is the subspace of smooth C , with bounded support.
[9] W. Goldsmith, Impact: The Theory and physical Behaviour of Colliding Solids, London: Edward Arnold Publishers, 1960.
[10] A. Gramain, Integration, Hermann, Paris, 1988.
Appendix
5
[8] A.F. Filipov, Differential equations with discontinuous right hand sides, Kluwer academic publishers, Dordrecht, NL, 1988.
4
functions cp : JR.n
of planar kinematic chains with multiple contact points", Int. J . of Robotics Research, vo1.l3, no 1, pp .82-92, february 1994.
--+
[12] M. Laghdir, Solutions des equations regissant le mouvement des particules en contact avec frottement sec et recevant des impulsions Ph. D. thesis, USTL, Mathematiques fondamentales et appliquees, France,Montpellier, 1987.
• definition 2 A distribution D is a continuous linear form defined on the vector space V .
This means that to any cp E V, D associates a complex number D(cp), noted < D, cp >. The space of distributions on V is the dual space of V and is noted V*.
[13] J .J . Moreau, "Standard inelastic shocks and the dynamics of unilateral constraints" , CISM courses and lectures, no 288, Springer-VerJag, pp.173-221, 1985.
[14] J.K . Mills and C .V. Nguyen, "Robotic manipulator collisions: modeling and simulation", AS ME J . Dyn. Syst. Meas. and Contr., vol. 114, pp.650-658, decem[1] P. Antosik, J. Mikusinski, R. Sikorski, Theory of ber 1992. distributions- the sequential approach, Elsevier Scientific publishing company, Amsterdam; PWN- Pol- [15] D. Percivale, "Uniqueness in the elastic bounce probish Scientific publishers, Warszawa, 1973. lem", Journal of differential equations, 56, 206-215,
References
[2] P. Appel, Traite de mecanique rationnelle, 3eme edition, 2d tome, dynamique des systemes-mecanique analytique, Paris, Gauthiers-Vi liars 1911 . [3] R.M. Brach, Mechanical impact dynamics, John Wiley, New York, 1991. [4] A. Bressan, "On differential systems with impulsive controls", Rend. Sem. Mat. Univ. Padova, vo1.78, pp.227-236, 1987. [5] B. Brogliato, P. Orhant, "On the transition phase in robotics: impact models, dynamics and control" , IEEE Conf. on Robotics and Automation, San Diego, may 1994. [6] M. Buhler, D.E. Koditshek, P.J. Kindlmann, "A family of robot control strategies for intermittent dynamical environments" , IEEE Control Systems magazine, p.I6-22, f~bruary 1990. [7] G. Buttazzo, D. Percivale, " On the approximation of the elastic bounce problem on Riemannian manifolds", Journal of Differential equations, 47, 227-275 , 1983. 4 i.e.
1985. [16] W.W. Schmaedeke, " Optimal control theory for nonlinear vector differential equations containing me&sures, SIAM J. Control, ser.A, vol.3, no 2, pp.231280, 1965. [17] L. Schwartz, TMorie des distributions, Hermann, publications de l'institut de mathematique de l'universire de Strasbourg, Paris, 1966. [18] R. Souchet, " Analytical dynamics of rigid body impulsive motions", Int. J. Engng. ScL, vol.31, no 1, pp.85-92, 1993. [19] W.J. Stronge, "Rigid body collision with friction", Proc. R. Soc. Lond. A, vol.431, pp.169-181, 1990. [20] L.C. Young, Lectures on the calculus of variations and optimal control theory, Chelsea publishing company, NY, 1980. [21] Y.F. Zheng and H. Hemami, "Mathematical modeling of a robot collision with its environment", J. of Robotic Systems, 2(3), pp.289-307, 1985.
indefinitely differentiable
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