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A Global Study of the Stability of Nonholonomic Dissipative Systems1
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A GLOBAL STUDY OF THE STABILITY OF NONHOLONOMIC DISSIPATIVE SYSTEMS I Miguel C. Muiioz-Lecanda' F. Javier Yaniz-Ferwindez'
• Departamento de Matematica Aplicada IV. Universidad Politecnica de CatalurIa. Edificio C-8. CIlordi Girona 1. E-08084 Barcelona. Spain (matmcml@mat. upc. es. yaniz@rnat. upc. es)
Abstract:Dissipative nonholonomic mechanical systems are studied from a geometric point of view. Stability and global asymptotic stability of the equilibrium points submanifold are established using the generalized La Salle theorem. The control case is formulated and passivity based controls are designed in order to stabilize the system. Cop;''right ~ 2003 JFA C Keywords: Differential geometric methods, l'onlinear control systems, 11echanical systems, Stability, Asymptotic stability, Passive elements.
1. I:\"TRODUCTIO\,
submanifold, and in Section 4 these are applied to dissipative nonholonomic mechanical s~·stems. In Section 5. we stud~' the passivity-based control for nonholonomic mechanical control s~·stems .
Although the specific properties of the equilibrium points of dissipative s~'stems have been well known for years. the s~'stematic study of such s~'stems is fairl~' recent. The results and the usual methods. however. are only local. They use coordinate expressions and conceal the geometric properties of the state spaces and the vector field defining the systems.
Thoughout this paper we suppose that the manifolds and maps are differentiable. The manifolds are Hausdorff. As a reference for differential geometry. notation emplo~'ed , and concepts. see (Abraham and :-'Iarsden. 1978) and (Abraham et al. , 1988).
In this paper we give a geometric intrinsic formulation of some aspects of nonholonomic mechanical dissipative s~·stems. and the use of these properties to design controls by passi\'it~·. A local point of \'iew of some of these problems can be found in (Bloch et al.. 1992) and (Ortega et al .. 1998) .
2. :\"O:\"HOLO:\"O!>.IIC :-'IECHA:\"ICAL SYSTE:-'IS . Let (Q. g) be a Riemannian manifold. dim Q = n. and V the Le\'i-Ci\'ita connection associated to the Riemannian metric g. Let TQ be the tangent bundle of Q with TQ : TQ -> Q the natural projection. Consider D C TQ a non integrable distribution over Q with rang(D) = n - m. such that TQ(D) = Q. Throughout this paper we IIse the same notation for considering D as a distribution in Q or as a submanifold of TQ .
The paper is organized as follows. Section 2 is de\'oted to stating the s~'stems to be used . In Section 3. stability and asymptotical stabilit~· properties are established in the case of equilibrium points I
Partiall~'
supported b~' the Spanish CICYT PB98 - 0920
project
I11
m
The context clarifies which of the two situations we are dealing with. :\'otice that the dimension of D as a submanifold of TQ is 2n - m. Let [(D) be the set of sections of D: that is to say, [(D) = {X E X(Q) I X(p) E Dp. \:Jp E Q}.
Y
=F
0 -,
where FV and ZJ are the vertical lifting of F and Zj from Q to TQ. Xg E X(TQ) is the geodesic vector field corresponding to the connection \i' and V's are such that Y(v q ) E TvqD. \:IVq E D. :\otice that Y satisfies the second order condition, that is. its integral curves in TQ are canonical liftings to TQ of curves in Q. It is known that if L = (Q.g. U. D) is a simple nonholonomic mechanical system. then the energy E = T + TQU E ex (TQ) is constant along the solutions.
+ Z 0 '1
-:,(t) E D
(3)
j=!
A differentiable curve -; : I -. Q is a solution of the nonholonomic system. given by (Q , g) and D. associated to the exterior force vector field F E X (Q), if r satisfies t he next differential equation with constraints
\i' "r~
= Xg + P ' + L).j Zj
(1 )
where Z E X (Q, TQ) is the constraint force. This force is unknown , and consequently up to this point the equations are indetermined . \Ve need to introduce a new condition: d 'Alembert Principle.
2.1 Equilibrium points sub manifold.
Consider the nonholonomic mechanical system given by L = (Q , g, F,D). The equlibrium points of the system are the points of the phase space such that the vector field Y E X(TQ) associated to L is zero. Since Y satisfies the second order condition. the equilibrium points are of the form Oq E T qQ. We denote the set of equilibrium points of the system L by EP E .
D' Alembert Principle: Given the previous nonholonomic system with D as the constraint submanifold, the constraint force Z E X(Q. TQ) satisfies that Z( up) E D;. for every up E D. Observe that if D is an integrable distribution, then the constraint force is perpendicular to the integral submanifolds of D.
Proposition 1. Given the nonholonomic system L = (Q , g. F , D) , the set of equilibrium points of the system satisfies the following condition
We denote by L = (Q.g , F,D) a nonholonomic mechanical system. It is called simple if there exists a function U E eOC(Q), the potential function, such that F = - grad U.
EPE
Assume that the distribution D C TQ is given by the zeros of a finite family of differential I-forms. that is, there exist w 1 •... • w m E n1 (Q) linearly independent at every point such that Vq E D if and only if wi(v q ) = O. i = 1. .... m. Then the dynamic equation (1) can be written as
= {Oq
E TQ
I 7r D F(q) = O}.
If EPE is a submanifold. its dimension is greater than or equal to n - m.
\,Ve denote by 7rD and 7iD ~ the projections over D and D.L, and by ).~p : T pQ -. T Up (TQ) the vertical lift.
m
\i' ·/r = F
0 ")
+ 2:).j
0 -:; )
Zj
PROOF. Let Y E X(TQ) be the vector field associated to the nonholonomic s~'stem L Since }I D E X(D) and Oq E D , then Y(Oq) E TOqD. \Ve haw that:
0 ")
j= 1
~
(t) E D
(2)
where Zj E X(Q) is such that i(Zj)g = ...;j. because D .L is generated b~' ZI ... · , Zm' The coefficients V E ex (TQ) are called Lagrange multipliers. It is known that condition -:f (t) E D allows us to determine their values on D C TQ . For more details on nonholonomic mechanical system see (:\eimark et al.. 1972).
A well known fact about the
m
Y(Oq)
= Xg(Oq) + P
'(Oq) + L
)"(Oq)Z~(Oq)
;= 1 m
= FV(Oq) + L).' (Oq)Z,t'{Oq)
E V·Oq(TQ).
,=1 where Fu p (TQ) = ker T Up TQ. From the previous expression we obtain that
d~'namical
equations of mechanics is that the~' are second order differential equations in the configuration manifold Q. Thus. there is a corresponding first order equation in the tangent bundle TQ. Hence we have a vector Y E X(TQ) associated to the equation (2). This vector field is of the form
m
Y(Oq) = (F(q)
+L
).i(Oq)Z,(q))" E T6~D
;= 1
and since T~~,D = Vup(TQ) n TupD = ).~P(Dp). we have that F(q) + 2::;:1 ).i(Oq)Zi(q) E D q.
112
a a 1 a 1 a + tan()- + - - - - + - - - a;1' ay cos() aVl cos() aV2 a a a X 2 = -----+-a() at 'l aV2
Furthermore, from d'Alembert principle, Z,(q) E
Xl = -
Di. In consequence. the Lagrange multipliers are the only functions such that - 7rD - F(q).
2::7=1)"i(Oq)Z,(q) =
Therefore Y(Oq) = 7rDF(q), and the equilibrium points are those which satisfy the following equation
Since g(grad U. X 2 ) = O. it follows that 1iD(grad U) = g (grad u..:,)X '. After some calculations we ob9(-""',·.'\') tain that:
0= Y(Oq) = 7r D F(q)
o
y(X l . XIl7rD(grad U) m2
+
3. STABILITY OF EQUlLIBRIU:-'l POI:\TS SUB~IA:\IFOLD I:" CO~IPLETE RIE:-'IA:\"="IA:\ ~lA:\IFOLDS. Let {-'I. g) be a Riemannian manifold. It is known that from the metric 9 we can define in 1'1 a distance d in such a way that the induced metric topology from d coincides with the original one in l\I. Then (l\I.g) is a complete manifold if the induced distance is complete. Throughout this section we assume that (1H. g) is a complete Riemannian manifold. For equivalent conditions on the completeness of the Riemannian manifold. see (do Carmo, 1992). Let X E X(l\I) and consider the associated dynamical system X 0 - , = -=;. \\. here -; : I ~ ;\1 is a differentiable curve. We denote by -, (t: x) the solution with initial condition -, (0: x) = x E JI. We say that a solution ') (t: x) is bounded if its image is bounded as a subset of the metric space .H.
.
+ iJ sin () - "2 (VI + l'2) = 0 - i sin () + iJ cos () = 0 . - t'2)
a a l'2
It is well known that in a nonholonornic dissipative mechanical system, the energy E = T + TQU decreases along the solutions.
Assume that the s~'stem rolls without siiping, then the nonholonomic constraints are:
1·
mJI1 , cos a sin () cos ()
+ - - - =2- - - -
If R E X(Q. TQ) satisfies gq(7rDR(vq). vq) ::::; -oYq(Vq , Vq) for every Vq E D , with 0 E JR , a> O. then R is called strictly dissipative.
with potential function U = mgy cosa. where m is the total mass of the s~·stem. J is the moment of inertia of the bar and J11.' the moment of the \\'heels.
.
mJu ' cos a sin () a -cos 2 () al'l
A nonholonomic mechanical system ~ is dissipative if F = -grad U + Rand R E X(Q. TQ) is dissipative over D, that is , gp(1iDR(vq), vq) ::::; 0 for every Vq E D.
The configuration space is locally coordinated by (x , y , (). 'L '1.l '2) and the kinetic energy of the system is given by
() - "2 (VI
a
2.2 Nonholonomic dissipative mechanical systems.
Example 2. Consider the movement of a uniform bar with two wheels at its end points rolling on an inclined plane with angle a. Let ~'l and 7l'2 be the rotation angles of the wheels with respect to an arbitrar~' initial state. (x. y) the position of the center of mass and () the orientation with respect to the horizontal line of the plane.
1·
m 2 cos a sin 2 ()
Therefore, the equilibrium points submanifold of the s~'stell1 is EP~ = JR:2 X {O. 7r} X SI X 51.
For mechanical systems without constraints, the set of equilibrium points is {Oq I grad U (q) = O}. which is generically a submanifold of dimension zero. Thus, the equilibrium points are isolated . This does not occur in the case of nonholonomic mechanical systems, since the set of equilibrium points are generically a submanifold of dimension greater than zero. Consequently, in these cases there is no asymptotically stable equilibrium points.
cos ()
a
----::--- - + ------,,---::--cos () ax cos 2 () ay
If F = -grad U. then EP~ = {Oq E TQ I 7rD(grad U)(q) = O}. ;\"otice that when the system has no exterior forces , all the points in Q are equilibrium points. Examples of this situation can be found in (Lewis, 2000).
j;
cos a sin ()
=
=0
where the diameter of the wheels and the bar are of length two. The constraint distribution is generated b~' the follO\\'ing orthogonal vector fields:
Assume that the equilibrium points of X haw the structure of a submanifold of l\I. This is denoted by EP. The goal of this section is to
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find conditions for the stability of EP when the Riemannian manifold (M , g) is complete and the vector field is also complete.
proposltlOn hold , and consequently EP is weak stable. ~loreover. we have that: Proposition 4. Let (M, g) be a complete Riemannian manifold. X E '1:(111) and ~7 E ex. (111) a Lyapunov function of X at EP. If EP is compact, then it is stable.
Recall that the equilibrium points submanifold EP c M is stable if for every f. > 0 there exists b > osuch that if D(x. EP) ::; b, then D({(t: x), EP) ::; f.. where D is the point-set distance. Vie also call it weak stable if for every f. > 0 there exists an open set [,'. such that if x E U, then D({(t:x),EP)::; f.. If EP is compact and weak stable, then it is also stable. Assume that X E '1:(111) is a complete vector field, we say that EP is asymptotically stable if it is stable and there exists r E IR. r > O. such that if x E 0 = {x E M I D(x,EP) ::; r}. then lim/_ oc D({(t: x). EP) = O. If 0 = M then EP is globally asymptotically stable.
PROOF. From Proposition 3 we have that given f. > O. there exist Ko C B, such that if x E Ko, then ;(t: x) E B, for every t . Since EP is compact, there exists y E EP with m
where Ko is the complementary set of Ko. It satisfies that m > O. since if m was equal to zero, it follows that y E Ko. But this contradicts the fact that y E EP. If we choose J < m. then Bi5 C Ko and therefore if x E Bo, then 1(t: x) E B, for every
3.1 Lyapunol' functions.
t.
If EP c M is the equilibrium points submanifold of X E '1:(M), we say that V E eOC(lI1) is a Lyapunov function of X at EP if • V(x) ::" 0, for every x E M and V(x)
Let X E '1:(111) be a complete vector field on the complete Riemannian manifold (M, g) . For x EM, consider the set l-V (x) of the points y E M such that there exist a sequence (tk) c JR, tk > 0 and tk ---of oc such that limk_x. 1(tk ; x) = y. This is the set of limit points of the integral curve 1(t: x). The proof ofthe following two propositions follows directly from Propositions 4.3 and 4.4 in (Munoz-Lecanda et al., 2002).
::; 0, Vx E M.
If the previous conditions hold in a open neighborhood U :J EP , we can study the stability in the Riemannian manifold (U, gl U ). Proposition 3. Let (AI, g) be a Riemannian manifold, X a vector field on M and EP the equilibrium points submanifold. If there exists A E JR+ and a Lyapunov function, V E eOC(M) of X at EP such that takes a minimum on the sets aBT = {x E M I D(x , EP) = r}, r < A, then EP is weak stable.
PROOF. Consider
f.
D
3.2 La Salle's theorem .
= 0 if
:r E EP.
• .c x V
= minD(EP.Ko ) = minD(y, Ko)
Proposition 5. (La Salle's theorem). Let X be a complete vector field on a complete Riemannian manifold (M , g). Let EP be the equilibrium points submanifold of X and V E ex. (1\1) a Lyapunov function of X at EP. Assume that x E AI is such that ", (t: x) is bounded. Then
> 0 and BT = {x E M I
D(x.EP) ::; r}, r ::; min(f..A). Let 3 E JR be
(1) W(x) ~ {y E M IC .. V(y) = O}. (2) If K ~ M is the maximal invariant set under the flow of X such that h" ~ {y E M I.cx V(y) = A}, then
the minimum value of the function V on aB T , a E (0.3) and A"o = {x E BT I V(x) < a}. It sufficies to prove that Ko is invariant by the flow of X. Take x E Ko and consider ;(t:x). Since Fh(t:x)) is a decreassing function , it follows that V(":(t:x)) ::; ~"h(O:x)) < o. As a consequence, ":(t: x) E A"o C BT for e\'er~' t ::" O. Then Ko is an im'ariant open set and therefore EP is weak stable. D
lim D(;(t:x).K) = O. t-x. :'\otice that EP is in the maximal invariant set K. Corollary 6. Let (M. g) be a complete Riemannian manifold and X E '1:(M) a complete vector field. If the integral curves of X are bounded and there exists a Lyapunov function \l of X at EP such that .cxV(x) < 0 for every x ~ EP, then limt_x D(; (t: x). K) = 0 for every x E J1.
It is well known that if (M. g) is a complete Riemannian manifold. then the Hopf-Rinow theorem (do Carmo. 1992) assures that a subset of ~U is compact if and only if it is closed and bounded. Thus if EP is a compact set, then aBT = {x E M I D(x. EP) = r} is also compact since it is closed and bounded. Hence. if there exists a L~'apunov function the conditions of the previous
Corollary 7. Let (.H. g) be a complete Riemann manifold and X E '1:(J1) a complete vector field such that EP is a compact set.
114
If it satisfies that R(Oq) = O. 'v'q E Q. then the equilibrium points submallifold is given by EP E = {Oq I 7rD(grad U)(q) = O} according with Proposition 1. Since the points in EP E have the form Oq, we can consider EPE as a subset of the zero section of TQ, alld then we can assume that EP E <;;; Q.
(1) If there exists 10 > 0 and V Liapounov function of X at EP in the neighborhood B" such that Lx V(x) < 0 for x fj EP, x E B" then EP is asymptotically stable. (2) If the integral curves of X are bounded and there exists a Lyapunov function ~T of X at EP such that LxV(x) < 0 for x fj EP. then EP is globally asymptotically stable.
Proposition 9. Let (Q.g) be a complete Riemannian manifold . Consider the Ilonholonomic mechanical system L = (Q. g. F = - grad U + R. D) with R E :£(Q. TQ) dissipative such that R(Oq) = 0 for every q E Q. Suppose that U E Coc (Q) satisfies that U (q) > 0 for every q E Q \ EPE and U(q) = 0 if q E EPE. We have that:
Remark 8. In (1) it is not necessary for the curves to be bounded. Since EP is compact, we can find a neighborhood such that all the integral curves with the initial condition in this neighborhood are bounded.
PROOF.
(1) If there exists A E 1R+ such that the energy E E COC (TQ) takes a minimum 011 the sets aBr = {v q E TQ I D(vq.EP E ) = r}. 'v'r::; A. then EP E is weak stable. (Notice that this situation includes the case R == 0) (2) If EP E is stable. R is strictly dissipative and the solutions are bounded , then EP E is globally asymptotically stable .
• First. let us show (2). Since V is a Lyapunov function it follows that EP is stable. ~lore over, it satisfies that EP <;;; K <;;; {y E !If I Lx v'(y) = O} = EP. From the previous proposition we have that for every x E AI, limt_ex: D(r(t:x),EP) = O. • For (1), let 0 < r < 10 , since EP is stable, there exists cS > 0 such that if x E Bfl, then ,(t: x) E Br for every t. Then , in a way similar to that in the previous paragraph, we get that
W(x) <;;; K <;;; {y E Br
PROOF. (1) From Proposition 3. it is enough to show that E = T + U is a Lyapunov function for Y. That is, E(vq) > 0 for ever:.' Vq E TQ \ EPE and Dy-E(v q ) ::; O. The first condition follmvs from the assumptions about the function U. For the second one. let "'t(t: v q) be a solution of the system. Since R is dissipative, we have that LyE(vq) =g(Ro"'t(t:vq)."'t(t:Vq»lt ~ o = g(R(vq), l'q) ::; O. (2) For the second statement, we use La Salle's theorem. Consider a solution of the system ~, (t: v q ), then
I Lx V(y) = O} = EP.
Therefore, we obtain that for every x E Bfl , !imt_ex: D(r(t: x). EP) = o. 0
4. STABILITY OF NONHOLONOJ\IIC J\IECHA:\,IC AL SYSTEJ\IS. In accordance with the above sections, to nonholonomic systems correspond vector fields on TQ which are tangent to the submanifold D. Then. D C TQ must be provided with a complete Riemannian metric in order to determine the conditions for the equilibrium points submanifold to be globally aS~'mptotically stable.
LyE(l'q) =
-Qgh(t: l'q) . ~i (t:
K <;;; {y
+ RV +
L Ai Zl
-
Q
g(vq.l'q) ::; 0
E
TQ ! LyE(y) = O} <;;; {Oq
I q E Q}. I T.D(grad U)(q) =
is not an equilibrium point, we have that
Y(Oqv) = -(7:'D(grad U))V(qO). and in consequence ~ (t: Oqo) et. {Oq I q E Q} for every t. Therefore K would not be an invariant set. From all the abm·e. we obtain that K = {Oq I T.D(grad U)(q) = O} = EPE. Using La Salle's theorem we obtain that
k
(grad U)t'
=
~Ioreover, we have that {Oq O} <;;; K since these are equilibrium points. Suppose that {Oq I T.D(grad U)(q) = O} 1}:. Thus. there exist qO E Q such that 7:' D (grad U) (qO) =1= 0 and Oqn E 11:. Since it
Consider the nonholonomic mechanical system L = (Q.g.F = -graM/ + R.D) with R E :£(Q. TQ) dissipative. The d~'namical vector field associated to L is
= Xg -
l'q» l t ~(I
\\'ith Q > o. Hence the maximal invariant set K. defined in Proposition 5. satisfies that
This is always possible if we take as a metric on TQ the Sasaki metric gT. It is known that the topolog~' induced b~' gT coincides with the natural one in TQ. ~Ioreover. gT is Riemannian. and if 9 is complete then gT is also complete. Therefore. since D is closed. (D. gm) is a complete Riemannian submanifold. and \\'e can apply the results of the previous section. See (Abraham et al .. 1988) for results on these concepts.
Y
g(R O ~, (t:l'q) . ')(t:l'q» l t ~n ::;
E :£(TQ).
l=!
115
So EP E is globally asymptotically stable.
• EPE is also the equilibrlum points submanifold of the system L = (Q, g. F = -grad U + R. D) where the vector field R(vq) = -8 2:7=1 C,~Vq)gq(C;(Vq), vq), 8 > o and it is stable for L. • If rang (C 1 .··· . C k ) = rang D in a neighborhood 0 of EPE , then EPE is asymptotically' stable for ~. • If rang ( ~l "" , C k ) = rangD and the solutions of L are bounded, then EP E is globally asymptotically stable.
0
Proposition 10. Under the same conditions as in the previous proposition, consider the nonholonomic mechanical system L = (Q, g. F = -gradU + R.D) with R E X(Q.TQ) dissipative. Assume that EPE is a compact set, then: (1) EP E is stable. (:\otice that the case R == 0 is also included) (2) If R is strictly dissipative in a neighborhood o of EP~:;, then it is asymptoticall:-: stable. (3) If R is strictly dissipative and the solutions are bounded , it satisfies that EP E is globall:-' asymptotically stable.
Remark 12. The system L is obtained from L by the feedback u;(v q) = -pgq(C,(Vq). v q). 8 E R 3> O.
PROOF. The statements (1) and (3) follow diPROOF. Statement (1) is a corollary of the first
rectly from Propositions 4 and 9. In order to prove (2), we proceed as in Proposition 4. Let f > 0 such that B, C O. From (1) \ve have that EP E is stable, then there exists J > 0 such that for every Vq E B/i , we have that 1(t: v q) E Bc However Bc is a compact set and consequently 1 (t; v q ) is bounded. Then, Proposition 9 gives limt _ex. D('Y(t: v q), EPE) = 0, 't/Vq E B/i. 0
part of Proposition 10. Statements (2) and (3) follm\" from Propositions 9 and 10. 0
Remark 13. We have obtained that in these conditions , by a suitable choice of the inputs u', we can stabilize the submanifold EP E . For simplicity, in the previous proposition we have assumed that the equilibrium points submanifold is a compact set. A similar proposition can be stated using the conditions of Proposition 3.
5. NONHOLONOMIC l\IECHANICAL SYSTEl\IS WITH COXTROL.
A nonholonomic mechanical system with control is a mechanical system with control such that the solutions must belong to a distribution D t;;; TQ. The dynamic equations take the following form:
\7-) 1
= -(grad U) +
m
k
)=1
;=1
REFERENCES
R. Abraham and J.E. Marsden (1978). Foundations of Mechaniccs. Addison-Wesley. R. Abraham, J.E. l\Iarsden and T . Ratiu (1988). :\Ianifolds, Tensor Analysis, and Applications. Springer-Verlag. A.:\1. Bloch, 1\1. Reyhanoglu and l\'.H. l\IcClamroch (1992). Control and Stabilization of :\'onholonomic Dynamic Systems. IEEE Transactions on Automatic Control 37. 17461757. :\I.P. do Carmo (1992). Riemannian geometry. Birkhuser. Boston. A.D. Lewis (2000). Simple :\Iechanical Control Systems with Constaints. IEEE Transactions on .4 utomatic Control 45. 1420- 1436. Ju.1. :\eimark and :\.A. Fufaev (1972). Dynamics of :\onholonomic Systems. American :\lathematical Society. Providence. Rhobe Island. :\I.C. :\Iuiioz-Lecanda and F.J. Yaniz-Fermindez (2002). Dissipatiw Control of :\lechanical Sytems: a Geometric Approach. SIAM 1. Control and Optimitzation 40, 1505- 1516. R. Ortega. A. Loria. P .J . :\icklasson and H. Sira-Ramfrez (1998). Passivity-base Control of Euler-Lagrange Systems. SpringerVerlag.London.
L ,v Zj + L u;C;
where Cl ,'" . C k are the control forces which usually depend on positions and velocities. and u, E Cex. (TQ) are the imputs or controls. We can assume. without loss of generality, that the condition C;(vq ) E D q. 't/l'q E Dq is satisfied. The natural outputs of a nonholonomic mechanical s~'stem with control are the functions h; (v q ) = g(C,(v q). vq ). From now on. we denote the simple nonholonomic mechanical s:-'stems with control b:-' L = (Q. g , U. D. Cl . ... . C k ). Let (Q. g) be a complete Riemannian manifold and let EP E be the equilibrium points submanifold of the nonholonomic s~'stem L = (Q. g, U. D) . Consider the nonholonomic mechanical system with control (Q.g.U.D.C 1 ... ·.Ck). We haw that:
Proposition 11. If EP E is compact and U(q) > O. for ever~' q ~ EPE and U(q) = 0 if q E EP E . then:
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Copyright tQ IFAC Lagrangian and Hamiltonian Methods for Nonlinear Control, Seville, Spain, 2003
c:
0
l>
Publications www.elsevier.com/locate ·ifac
A GLOBAL STUDY OF THE STABILITY OF NONHOLONOMIC DISSIPATIVE SYSTEMS I Miguel C. Muiioz-Lecanda' F. Javier Yaniz-Ferwindez'
• Departamento de Matematica Aplicada IV. Universidad Politecnica de CatalurIa. Edificio C-8. CIlordi Girona 1. E-08084 Barcelona. Spain (matmcml@mat. upc. es. yaniz@rnat. upc. es)
Abstract:Dissipative nonholonomic mechanical systems are studied from a geometric point of view. Stability and global asymptotic stability of the equilibrium points submanifold are established using the generalized La Salle theorem. The control case is formulated and passivity based controls are designed in order to stabilize the system. Cop;''right ~ 2003 JFA C Keywords: Differential geometric methods, l'onlinear control systems, 11echanical systems, Stability, Asymptotic stability, Passive elements.
1. I:\"TRODUCTIO\,
submanifold, and in Section 4 these are applied to dissipative nonholonomic mechanical s~·stems. In Section 5. we stud~' the passivity-based control for nonholonomic mechanical control s~·stems .
Although the specific properties of the equilibrium points of dissipative s~'stems have been well known for years. the s~'stematic study of such s~'stems is fairl~' recent. The results and the usual methods. however. are only local. They use coordinate expressions and conceal the geometric properties of the state spaces and the vector field defining the systems.
Thoughout this paper we suppose that the manifolds and maps are differentiable. The manifolds are Hausdorff. As a reference for differential geometry. notation emplo~'ed , and concepts. see (Abraham and :-'Iarsden. 1978) and (Abraham et al. , 1988).
In this paper we give a geometric intrinsic formulation of some aspects of nonholonomic mechanical dissipative s~·stems. and the use of these properties to design controls by passi\'it~·. A local point of \'iew of some of these problems can be found in (Bloch et al.. 1992) and (Ortega et al .. 1998) .
2. :\"O:\"HOLO:\"O!>.IIC :-'IECHA:\"ICAL SYSTE:-'IS . Let (Q. g) be a Riemannian manifold. dim Q = n. and V the Le\'i-Ci\'ita connection associated to the Riemannian metric g. Let TQ be the tangent bundle of Q with TQ : TQ -> Q the natural projection. Consider D C TQ a non integrable distribution over Q with rang(D) = n - m. such that TQ(D) = Q. Throughout this paper we IIse the same notation for considering D as a distribution in Q or as a submanifold of TQ .
The paper is organized as follows. Section 2 is de\'oted to stating the s~'stems to be used . In Section 3. stability and asymptotical stabilit~· properties are established in the case of equilibrium points I
Partiall~'
supported b~' the Spanish CICYT PB98 - 0920
project
I11
m
The context clarifies which of the two situations we are dealing with. :\'otice that the dimension of D as a submanifold of TQ is 2n - m. Let [(D) be the set of sections of D: that is to say, [(D) = {X E X(Q) I X(p) E Dp. \:Jp E Q}.
Y
=F
0 -,
where FV and ZJ are the vertical lifting of F and Zj from Q to TQ. Xg E X(TQ) is the geodesic vector field corresponding to the connection \i' and V's are such that Y(v q ) E TvqD. \:IVq E D. :\otice that Y satisfies the second order condition, that is. its integral curves in TQ are canonical liftings to TQ of curves in Q. It is known that if L = (Q.g. U. D) is a simple nonholonomic mechanical system. then the energy E = T + TQU E ex (TQ) is constant along the solutions.
+ Z 0 '1
-:,(t) E D
(3)
j=!
A differentiable curve -; : I -. Q is a solution of the nonholonomic system. given by (Q , g) and D. associated to the exterior force vector field F E X (Q), if r satisfies t he next differential equation with constraints
\i' "r~
= Xg + P ' + L).j Zj
(1 )
where Z E X (Q, TQ) is the constraint force. This force is unknown , and consequently up to this point the equations are indetermined . \Ve need to introduce a new condition: d 'Alembert Principle.
2.1 Equilibrium points sub manifold.
Consider the nonholonomic mechanical system given by L = (Q , g, F,D). The equlibrium points of the system are the points of the phase space such that the vector field Y E X(TQ) associated to L is zero. Since Y satisfies the second order condition. the equilibrium points are of the form Oq E T qQ. We denote the set of equilibrium points of the system L by EP E .
D' Alembert Principle: Given the previous nonholonomic system with D as the constraint submanifold, the constraint force Z E X(Q. TQ) satisfies that Z( up) E D;. for every up E D. Observe that if D is an integrable distribution, then the constraint force is perpendicular to the integral submanifolds of D.
Proposition 1. Given the nonholonomic system L = (Q , g. F , D) , the set of equilibrium points of the system satisfies the following condition
We denote by L = (Q.g , F,D) a nonholonomic mechanical system. It is called simple if there exists a function U E eOC(Q), the potential function, such that F = - grad U.
EPE
Assume that the distribution D C TQ is given by the zeros of a finite family of differential I-forms. that is, there exist w 1 •... • w m E n1 (Q) linearly independent at every point such that Vq E D if and only if wi(v q ) = O. i = 1. .... m. Then the dynamic equation (1) can be written as
= {Oq
E TQ
I 7r D F(q) = O}.
If EPE is a submanifold. its dimension is greater than or equal to n - m.
\,Ve denote by 7rD and 7iD ~ the projections over D and D.L, and by ).~p : T pQ -. T Up (TQ) the vertical lift.
m
\i' ·/r = F
0 ")
+ 2:).j
0 -:; )
Zj
PROOF. Let Y E X(TQ) be the vector field associated to the nonholonomic s~'stem L Since }I D E X(D) and Oq E D , then Y(Oq) E TOqD. \Ve haw that:
0 ")
j= 1
~
(t) E D
(2)
where Zj E X(Q) is such that i(Zj)g = ...;j. because D .L is generated b~' ZI ... · , Zm' The coefficients V E ex (TQ) are called Lagrange multipliers. It is known that condition -:f (t) E D allows us to determine their values on D C TQ . For more details on nonholonomic mechanical system see (:\eimark et al.. 1972).
A well known fact about the
m
Y(Oq)
= Xg(Oq) + P
'(Oq) + L
)"(Oq)Z~(Oq)
;= 1 m
= FV(Oq) + L).' (Oq)Z,t'{Oq)
E V·Oq(TQ).
,=1 where Fu p (TQ) = ker T Up TQ. From the previous expression we obtain that
d~'namical
equations of mechanics is that the~' are second order differential equations in the configuration manifold Q. Thus. there is a corresponding first order equation in the tangent bundle TQ. Hence we have a vector Y E X(TQ) associated to the equation (2). This vector field is of the form
m
Y(Oq) = (F(q)
+L
).i(Oq)Z,(q))" E T6~D
;= 1
and since T~~,D = Vup(TQ) n TupD = ).~P(Dp). we have that F(q) + 2::;:1 ).i(Oq)Zi(q) E D q.
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a a 1 a 1 a + tan()- + - - - - + - - - a;1' ay cos() aVl cos() aV2 a a a X 2 = -----+-a() at 'l aV2
Furthermore, from d'Alembert principle, Z,(q) E
Xl = -
Di. In consequence. the Lagrange multipliers are the only functions such that - 7rD - F(q).
2::7=1)"i(Oq)Z,(q) =
Therefore Y(Oq) = 7rDF(q), and the equilibrium points are those which satisfy the following equation
Since g(grad U. X 2 ) = O. it follows that 1iD(grad U) = g (grad u..:,)X '. After some calculations we ob9(-""',·.'\') tain that:
0= Y(Oq) = 7r D F(q)
o
y(X l . XIl7rD(grad U) m2
+
3. STABILITY OF EQUlLIBRIU:-'l POI:\TS SUB~IA:\IFOLD I:" CO~IPLETE RIE:-'IA:\"="IA:\ ~lA:\IFOLDS. Let {-'I. g) be a Riemannian manifold. It is known that from the metric 9 we can define in 1'1 a distance d in such a way that the induced metric topology from d coincides with the original one in l\I. Then (l\I.g) is a complete manifold if the induced distance is complete. Throughout this section we assume that (1H. g) is a complete Riemannian manifold. For equivalent conditions on the completeness of the Riemannian manifold. see (do Carmo, 1992). Let X E X(l\I) and consider the associated dynamical system X 0 - , = -=;. \\. here -; : I ~ ;\1 is a differentiable curve. We denote by -, (t: x) the solution with initial condition -, (0: x) = x E JI. We say that a solution ') (t: x) is bounded if its image is bounded as a subset of the metric space .H.
.
+ iJ sin () - "2 (VI + l'2) = 0 - i sin () + iJ cos () = 0 . - t'2)
a a l'2
It is well known that in a nonholonornic dissipative mechanical system, the energy E = T + TQU decreases along the solutions.
Assume that the s~'stem rolls without siiping, then the nonholonomic constraints are:
1·
mJI1 , cos a sin () cos ()
+ - - - =2- - - -
If R E X(Q. TQ) satisfies gq(7rDR(vq). vq) ::::; -oYq(Vq , Vq) for every Vq E D , with 0 E JR , a> O. then R is called strictly dissipative.
with potential function U = mgy cosa. where m is the total mass of the s~·stem. J is the moment of inertia of the bar and J11.' the moment of the \\'heels.
.
mJu ' cos a sin () a -cos 2 () al'l
A nonholonomic mechanical system ~ is dissipative if F = -grad U + Rand R E X(Q. TQ) is dissipative over D, that is , gp(1iDR(vq), vq) ::::; 0 for every Vq E D.
The configuration space is locally coordinated by (x , y , (). 'L '1.l '2) and the kinetic energy of the system is given by
() - "2 (VI
a
2.2 Nonholonomic dissipative mechanical systems.
Example 2. Consider the movement of a uniform bar with two wheels at its end points rolling on an inclined plane with angle a. Let ~'l and 7l'2 be the rotation angles of the wheels with respect to an arbitrar~' initial state. (x. y) the position of the center of mass and () the orientation with respect to the horizontal line of the plane.
1·
m 2 cos a sin 2 ()
Therefore, the equilibrium points submanifold of the s~'stell1 is EP~ = JR:2 X {O. 7r} X SI X 51.
For mechanical systems without constraints, the set of equilibrium points is {Oq I grad U (q) = O}. which is generically a submanifold of dimension zero. Thus, the equilibrium points are isolated . This does not occur in the case of nonholonomic mechanical systems, since the set of equilibrium points are generically a submanifold of dimension greater than zero. Consequently, in these cases there is no asymptotically stable equilibrium points.
cos ()
a
----::--- - + ------,,---::--cos () ax cos 2 () ay
If F = -grad U. then EP~ = {Oq E TQ I 7rD(grad U)(q) = O}. ;\"otice that when the system has no exterior forces , all the points in Q are equilibrium points. Examples of this situation can be found in (Lewis, 2000).
j;
cos a sin ()
=
=0
where the diameter of the wheels and the bar are of length two. The constraint distribution is generated b~' the follO\\'ing orthogonal vector fields:
Assume that the equilibrium points of X haw the structure of a submanifold of l\I. This is denoted by EP. The goal of this section is to
113
find conditions for the stability of EP when the Riemannian manifold (M , g) is complete and the vector field is also complete.
proposltlOn hold , and consequently EP is weak stable. ~loreover. we have that: Proposition 4. Let (M, g) be a complete Riemannian manifold. X E '1:(111) and ~7 E ex. (111) a Lyapunov function of X at EP. If EP is compact, then it is stable.
Recall that the equilibrium points submanifold EP c M is stable if for every f. > 0 there exists b > osuch that if D(x. EP) ::; b, then D({(t: x), EP) ::; f.. where D is the point-set distance. Vie also call it weak stable if for every f. > 0 there exists an open set [,'. such that if x E U, then D({(t:x),EP)::; f.. If EP is compact and weak stable, then it is also stable. Assume that X E '1:(111) is a complete vector field, we say that EP is asymptotically stable if it is stable and there exists r E IR. r > O. such that if x E 0 = {x E M I D(x,EP) ::; r}. then lim/_ oc D({(t: x). EP) = O. If 0 = M then EP is globally asymptotically stable.
PROOF. From Proposition 3 we have that given f. > O. there exist Ko C B, such that if x E Ko, then ;(t: x) E B, for every t . Since EP is compact, there exists y E EP with m
where Ko is the complementary set of Ko. It satisfies that m > O. since if m was equal to zero, it follows that y E Ko. But this contradicts the fact that y E EP. If we choose J < m. then Bi5 C Ko and therefore if x E Bo, then 1(t: x) E B, for every
3.1 Lyapunol' functions.
t.
If EP c M is the equilibrium points submanifold of X E '1:(M), we say that V E eOC(lI1) is a Lyapunov function of X at EP if • V(x) ::" 0, for every x E M and V(x)
Let X E '1:(111) be a complete vector field on the complete Riemannian manifold (M, g) . For x EM, consider the set l-V (x) of the points y E M such that there exist a sequence (tk) c JR, tk > 0 and tk ---of oc such that limk_x. 1(tk ; x) = y. This is the set of limit points of the integral curve 1(t: x). The proof ofthe following two propositions follows directly from Propositions 4.3 and 4.4 in (Munoz-Lecanda et al., 2002).
::; 0, Vx E M.
If the previous conditions hold in a open neighborhood U :J EP , we can study the stability in the Riemannian manifold (U, gl U ). Proposition 3. Let (AI, g) be a Riemannian manifold, X a vector field on M and EP the equilibrium points submanifold. If there exists A E JR+ and a Lyapunov function, V E eOC(M) of X at EP such that takes a minimum on the sets aBT = {x E M I D(x , EP) = r}, r < A, then EP is weak stable.
PROOF. Consider
f.
D
3.2 La Salle's theorem .
= 0 if
:r E EP.
• .c x V
= minD(EP.Ko ) = minD(y, Ko)
Proposition 5. (La Salle's theorem). Let X be a complete vector field on a complete Riemannian manifold (M , g). Let EP be the equilibrium points submanifold of X and V E ex. (1\1) a Lyapunov function of X at EP. Assume that x E AI is such that ", (t: x) is bounded. Then
> 0 and BT = {x E M I
D(x.EP) ::; r}, r ::; min(f..A). Let 3 E JR be
(1) W(x) ~ {y E M IC .. V(y) = O}. (2) If K ~ M is the maximal invariant set under the flow of X such that h" ~ {y E M I.cx V(y) = A}, then
the minimum value of the function V on aB T , a E (0.3) and A"o = {x E BT I V(x) < a}. It sufficies to prove that Ko is invariant by the flow of X. Take x E Ko and consider ;(t:x). Since Fh(t:x)) is a decreassing function , it follows that V(":(t:x)) ::; ~"h(O:x)) < o. As a consequence, ":(t: x) E A"o C BT for e\'er~' t ::" O. Then Ko is an im'ariant open set and therefore EP is weak stable. D
lim D(;(t:x).K) = O. t-x. :'\otice that EP is in the maximal invariant set K. Corollary 6. Let (M. g) be a complete Riemannian manifold and X E '1:(M) a complete vector field. If the integral curves of X are bounded and there exists a Lyapunov function \l of X at EP such that .cxV(x) < 0 for every x ~ EP, then limt_x D(; (t: x). K) = 0 for every x E J1.
It is well known that if (M. g) is a complete Riemannian manifold. then the Hopf-Rinow theorem (do Carmo. 1992) assures that a subset of ~U is compact if and only if it is closed and bounded. Thus if EP is a compact set, then aBT = {x E M I D(x. EP) = r} is also compact since it is closed and bounded. Hence. if there exists a L~'apunov function the conditions of the previous
Corollary 7. Let (.H. g) be a complete Riemann manifold and X E '1:(J1) a complete vector field such that EP is a compact set.
114
If it satisfies that R(Oq) = O. 'v'q E Q. then the equilibrium points submallifold is given by EP E = {Oq I 7rD(grad U)(q) = O} according with Proposition 1. Since the points in EP E have the form Oq, we can consider EPE as a subset of the zero section of TQ, alld then we can assume that EP E <;;; Q.
(1) If there exists 10 > 0 and V Liapounov function of X at EP in the neighborhood B" such that Lx V(x) < 0 for x fj EP, x E B" then EP is asymptotically stable. (2) If the integral curves of X are bounded and there exists a Lyapunov function ~T of X at EP such that LxV(x) < 0 for x fj EP. then EP is globally asymptotically stable.
Proposition 9. Let (Q.g) be a complete Riemannian manifold . Consider the Ilonholonomic mechanical system L = (Q. g. F = - grad U + R. D) with R E :£(Q. TQ) dissipative such that R(Oq) = 0 for every q E Q. Suppose that U E Coc (Q) satisfies that U (q) > 0 for every q E Q \ EPE and U(q) = 0 if q E EPE. We have that:
Remark 8. In (1) it is not necessary for the curves to be bounded. Since EP is compact, we can find a neighborhood such that all the integral curves with the initial condition in this neighborhood are bounded.
PROOF.
(1) If there exists A E 1R+ such that the energy E E COC (TQ) takes a minimum 011 the sets aBr = {v q E TQ I D(vq.EP E ) = r}. 'v'r::; A. then EP E is weak stable. (Notice that this situation includes the case R == 0) (2) If EP E is stable. R is strictly dissipative and the solutions are bounded , then EP E is globally asymptotically stable .
• First. let us show (2). Since V is a Lyapunov function it follows that EP is stable. ~lore over, it satisfies that EP <;;; K <;;; {y E !If I Lx v'(y) = O} = EP. From the previous proposition we have that for every x E AI, limt_ex: D(r(t:x),EP) = O. • For (1), let 0 < r < 10 , since EP is stable, there exists cS > 0 such that if x E Bfl, then ,(t: x) E Br for every t. Then , in a way similar to that in the previous paragraph, we get that
W(x) <;;; K <;;; {y E Br
PROOF. (1) From Proposition 3. it is enough to show that E = T + U is a Lyapunov function for Y. That is, E(vq) > 0 for ever:.' Vq E TQ \ EPE and Dy-E(v q ) ::; O. The first condition follmvs from the assumptions about the function U. For the second one. let "'t(t: v q) be a solution of the system. Since R is dissipative, we have that LyE(vq) =g(Ro"'t(t:vq)."'t(t:Vq»lt ~ o = g(R(vq), l'q) ::; O. (2) For the second statement, we use La Salle's theorem. Consider a solution of the system ~, (t: v q ), then
I Lx V(y) = O} = EP.
Therefore, we obtain that for every x E Bfl , !imt_ex: D(r(t: x). EP) = o. 0
4. STABILITY OF NONHOLONOJ\IIC J\IECHA:\,IC AL SYSTEJ\IS. In accordance with the above sections, to nonholonomic systems correspond vector fields on TQ which are tangent to the submanifold D. Then. D C TQ must be provided with a complete Riemannian metric in order to determine the conditions for the equilibrium points submanifold to be globally aS~'mptotically stable.
LyE(l'q) =
-Qgh(t: l'q) . ~i (t:
K <;;; {y
+ RV +
L Ai Zl
-
Q
g(vq.l'q) ::; 0
E
TQ ! LyE(y) = O} <;;; {Oq
I q E Q}. I T.D(grad U)(q) =
is not an equilibrium point, we have that
Y(Oqv) = -(7:'D(grad U))V(qO). and in consequence ~ (t: Oqo) et. {Oq I q E Q} for every t. Therefore K would not be an invariant set. From all the abm·e. we obtain that K = {Oq I T.D(grad U)(q) = O} = EPE. Using La Salle's theorem we obtain that
k
(grad U)t'
=
~Ioreover, we have that {Oq O} <;;; K since these are equilibrium points. Suppose that {Oq I T.D(grad U)(q) = O} 1}:. Thus. there exist qO E Q such that 7:' D (grad U) (qO) =1= 0 and Oqn E 11:. Since it
Consider the nonholonomic mechanical system L = (Q.g.F = -graM/ + R.D) with R E :£(Q. TQ) dissipative. The d~'namical vector field associated to L is
= Xg -
l'q» l t ~(I
\\'ith Q > o. Hence the maximal invariant set K. defined in Proposition 5. satisfies that
This is always possible if we take as a metric on TQ the Sasaki metric gT. It is known that the topolog~' induced b~' gT coincides with the natural one in TQ. ~Ioreover. gT is Riemannian. and if 9 is complete then gT is also complete. Therefore. since D is closed. (D. gm) is a complete Riemannian submanifold. and \\'e can apply the results of the previous section. See (Abraham et al .. 1988) for results on these concepts.
Y
g(R O ~, (t:l'q) . ')(t:l'q» l t ~n ::;
E :£(TQ).
l=!
115
So EP E is globally asymptotically stable.
• EPE is also the equilibrlum points submanifold of the system L = (Q, g. F = -grad U + R. D) where the vector field R(vq) = -8 2:7=1 C,~Vq)gq(C;(Vq), vq), 8 > o and it is stable for L. • If rang (C 1 .··· . C k ) = rang D in a neighborhood 0 of EPE , then EPE is asymptotically' stable for ~. • If rang ( ~l "" , C k ) = rangD and the solutions of L are bounded, then EP E is globally asymptotically stable.
0
Proposition 10. Under the same conditions as in the previous proposition, consider the nonholonomic mechanical system L = (Q, g. F = -gradU + R.D) with R E X(Q.TQ) dissipative. Assume that EPE is a compact set, then: (1) EP E is stable. (:\otice that the case R == 0 is also included) (2) If R is strictly dissipative in a neighborhood o of EP~:;, then it is asymptoticall:-: stable. (3) If R is strictly dissipative and the solutions are bounded , it satisfies that EP E is globall:-' asymptotically stable.
Remark 12. The system L is obtained from L by the feedback u;(v q) = -pgq(C,(Vq). v q). 8 E R 3> O.
PROOF. The statements (1) and (3) follow diPROOF. Statement (1) is a corollary of the first
rectly from Propositions 4 and 9. In order to prove (2), we proceed as in Proposition 4. Let f > 0 such that B, C O. From (1) \ve have that EP E is stable, then there exists J > 0 such that for every Vq E B/i , we have that 1(t: v q) E Bc However Bc is a compact set and consequently 1 (t; v q ) is bounded. Then, Proposition 9 gives limt _ex. D('Y(t: v q), EPE) = 0, 't/Vq E B/i. 0
part of Proposition 10. Statements (2) and (3) follm\" from Propositions 9 and 10. 0
Remark 13. We have obtained that in these conditions , by a suitable choice of the inputs u', we can stabilize the submanifold EP E . For simplicity, in the previous proposition we have assumed that the equilibrium points submanifold is a compact set. A similar proposition can be stated using the conditions of Proposition 3.
5. NONHOLONOMIC l\IECHANICAL SYSTEl\IS WITH COXTROL.
A nonholonomic mechanical system with control is a mechanical system with control such that the solutions must belong to a distribution D t;;; TQ. The dynamic equations take the following form:
\7-) 1
= -(grad U) +
m
k
)=1
;=1
REFERENCES
R. Abraham and J.E. Marsden (1978). Foundations of Mechaniccs. Addison-Wesley. R. Abraham, J.E. l\Iarsden and T . Ratiu (1988). :\Ianifolds, Tensor Analysis, and Applications. Springer-Verlag. A.:\1. Bloch, 1\1. Reyhanoglu and l\'.H. l\IcClamroch (1992). Control and Stabilization of :\'onholonomic Dynamic Systems. IEEE Transactions on Automatic Control 37. 17461757. :\I.P. do Carmo (1992). Riemannian geometry. Birkhuser. Boston. A.D. Lewis (2000). Simple :\Iechanical Control Systems with Constaints. IEEE Transactions on .4 utomatic Control 45. 1420- 1436. Ju.1. :\eimark and :\.A. Fufaev (1972). Dynamics of :\onholonomic Systems. American :\lathematical Society. Providence. Rhobe Island. :\I.C. :\Iuiioz-Lecanda and F.J. Yaniz-Fermindez (2002). Dissipatiw Control of :\lechanical Sytems: a Geometric Approach. SIAM 1. Control and Optimitzation 40, 1505- 1516. R. Ortega. A. Loria. P .J . :\icklasson and H. Sira-Ramfrez (1998). Passivity-base Control of Euler-Lagrange Systems. SpringerVerlag.London.
L ,v Zj + L u;C;
where Cl ,'" . C k are the control forces which usually depend on positions and velocities. and u, E Cex. (TQ) are the imputs or controls. We can assume. without loss of generality, that the condition C;(vq ) E D q. 't/l'q E Dq is satisfied. The natural outputs of a nonholonomic mechanical s~'stem with control are the functions h; (v q ) = g(C,(v q). vq ). From now on. we denote the simple nonholonomic mechanical s:-'stems with control b:-' L = (Q. g , U. D. Cl . ... . C k ). Let (Q. g) be a complete Riemannian manifold and let EP E be the equilibrium points submanifold of the nonholonomic s~'stem L = (Q. g, U. D) . Consider the nonholonomic mechanical system with control (Q.g.U.D.C 1 ... ·.Ck). We haw that:
Proposition 11. If EP E is compact and U(q) > O. for ever~' q ~ EPE and U(q) = 0 if q E EP E . then:
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