Force/Motion Control of Constrained ManipulatorsWithout Velocity Measurements�Antonio Loria Elena PanteleyCenter for Control Engineering and Computation (CCEC)Department of Electrical and Computer EngineeringUniversity of CaliforniaSanta Barbara, CA 93106-9560, USAaloria@hamilton.ece.ucsb.eduelena@ruapehu.ece.ucsb.eduAbstractIn this note we deal with the problem of force/tracking control of constrained manipulatorswithout velocity measurements interacting with an in�nitely sti� environment. We use a reducedorder model for which the coordinates space is restricted to a subset where the constraintsJacobian is guaranteed to be non singular. In contrast to other similar approaches, we do notassume that the generalized trajectories belong to the subset all the time thus, our contributionis to prove that if the trajectories start within the subset , they remain in it. Furthermore, weprove uniform asymptotic stability considering only position and force measurements.Keywords: Output feedback, tracking control, holonomic constraints.1 IntroductionStarting with [22, 13, 10] the interaction with an in�nitely sti� environment is modeled by holonomic(algebraic) constraints imposed to the manipulator's motion. This kind of motion is often referredto in the literature as constrained robot motion. Unfortunately, the constraints equation is singular,hence in order to cope with this di�culty, various techniques for deriving so-called reduced ordermodels have been proposed in the literature based on the projection of the dynamic robot equationsonto a submanifold described by the algebraic equation of constraints.Concerning the force/position control, several schemes have been proposed for stabilization ofconstrained manipulators [13, 5, 15]; the position/force regulation problem was considered in [20],where the Lyapunov's direct method was used to guarantee local stability.Nevertheless, the above cited solutions for constrained manipulators present two major disad-vantages: the �rst is the need of velocity measurements which are often contaminated with noise [3].The second is that it is assumed that the constraints equation is globally solvable. It is supposedeither that the constraint Jacobian is non singular in the whole state space ([13, 4, 21, 5, 8]) or thatthe actual trajectory is contained in the neighbourhood of the desired set-point ([20, 1]). However,it should be noted that these assumptions are rather restrictive, for instance the �rst may be notful�lled even in very simple cases as that of a planar two-link revolute-joint manipulator whoseendpoint is constrained to a plane.To the best of our knowledge the force/position control problem without velocity measurementswas �rst treated by [9] and [16]; Huang and Tseng studied the (open loop) observer design for�This work was supported in part by the INTAS France-Russia collaboration project and by CONACyT, Mexico.1

constrained robots. Later in [16] the authors proposed a nonlinear observer and proved for the�rst time, asymptotic stability of the closed loop system considering an in�nitely sti� environment.Some other results in this direction are [12, 8, 6]. In [12] we proposed the �rst force/positioncontroller without velocity measurements and with uncertain gravity knowledge considering anelastic environment. Under the assumption that the constraints Jacobian never degenerates, theauthors of [8] proposed a \semi-global" adaptive force/motion controller for manipulators underholonomic constraints.In this paper we consider that the end-tool is interacting with an in�nitely sti� environment.We use the reduced order model introduced in [15] for which the coordinates space is restricted to asubset where the constraints Jacobian is guaranteed to be non singular. In contrast to the othersimilar approaches mentioned above, we do not assume that the generalized trajectories belong tothe subset all the time thus, our contribution is to prove that if the trajectories start within thesubset , they remain in it. Furthermore, we prove uniform asymptotic stability considering onlyposition and force measurements using standard Lyapunov stability theory.Notation. In this paper we use k � k for the Euclidean norm of vectors and the induced L2norm of matrices. The largest and smallest eigenvalues of a matrix K are denoted by kM and kmrespectively, where K and its eigenvalues may have a subindex.2 Dynamic model2.1 PreliminariesWe consider in this note the standard model of a rigid revolute joint robot manipulatorD(q)�q + C(q; _q) _q + g(q) = u+ f (1)where D(q) = D>(q) > 0 is the robot inertia matrix, g(q) is the gravitational forces vector, C(q; _q) _qrepresents the Coriolis and centrifugal forces, u 2 IRn are the applied torques and f 2 IRn is thereaction forces vector. In order to introduce our notation, we recall that there exist some positiveconstants kc, dm and dM such that 8 q 2 IRn, dmI � D(q) � dMI and kC(q; _q)k � kck _qk.We assume that the manipulator's end-e�ector interacts with an in�nitely sti� environmenthence, that its motion is constrained to a smooth (n�m){dimensional submanifold �, de�ned by�(q) = 0 (2)where the function � : IRn ! IRm is at least twice continuously di�erentiable, and m is the numberof holonomic constraints. AssumptionA1 below, concerns the solvability of the constraints equation(2) and will be used in the sequel in order to derive a reduced order model of constrained motion.A1 We assume that there exists an operating region � IRn de�ned as 4= 1 �2, where 1 isa convex subset of IRn�m, 2 is an open subset of IRm. We also assume the existence of a functionk : 1 ! IRm twice continuously di�erentiable, such that �(q1; k(q1)) = 0 for all q1 2 1. Underthese conditions, the vector q2 can be uniquely de�ned by the vector q1 such that q2 = k(q1) forall q1 2 1.Remark 1 (Jacobian structure) Notice that under this assumption the matrix J(q) =@�(q)=@q can be partitioned as J(q) = [J1(q); J2(q)], where J1(q) 4= @�(q)=@q1, J2(q) 4= @�(q)=@q2and the Jacobian matrix J2(q) 2 IRm�m never degenerates in the set . Necessary and su�cientconditions for global solvability have been given in [16] laying on results of Sandberg [19].2

Thus, without loss of generality we can assume that for all q 2 there exist positive constants�1; �2; �3 such that, for all i = 1; 2; : : : ; n,0 < �1 � kJ2(q)k � kJ(q)k � �2 (3) @J(q)@qi < �3:Remark 2 1In words, Assumption A1 guarantees the solvability of (2) only in the set , whichis equivalent to assume that the Jacobian J2(q) is not singular only if q 2 . This is the maindi�erence with other similar reduced order models used in the literature where it is supposed thatJ2(q) is full rank in the entire state space.Considering that the end-e�ector motion is constrained to the submanifold �, the above de�-nition of the set allows a parameterization of the generalized coordinates vector in which onlyn�m independent coordinates need to be controlled [13, 15, 14]. Thus, without loss of generalitywe can choose q1 and q2 as independent and dependent coordinates respectively and we can expressthe generalized velocities vector _q in terms of the independent velocities as_q = H(q) _q1 (4)where H(q) = � In�m� J�12 (q)J1(q) � : (5)Similarly to the parameterization of the vector _q, the generalized reaction forces f which do notdeliver power on admissible velocities i.e., _q>f = 0, can be parameterized by the vector of Lagrangemultipliers � 2 IRm as f = J>(q)�: (6)2.2 Reduced order model and its propertiesIn this section we present the reduced order model introduced in [15] and we stress some propertieswhich are fundamental for further analysis. First, using equations (4) { (6) in (1) we obtainD�(q)�q1 + C�(q; _q) _q1 + g�(q) = H>(q)u� = Z(q) �C�(q; _q) _q1 + g(q) � u�where we have de�ned D�(q) 4= H>(q)D(q)H(q) (7)C�(q; _q) 4= D(q) _H(q) + C(q; _q)H(q)C�(q; _q) 4= H>(q)C�(q; _q)g�(q) 4= H>(q)g(q)Z(q) 4= �J(q)D�1(q)J>(q)��1 J(q)D�1(q)Next, similarly to [13] we introduce a decoupled control scheme which allows to control gener-alized positions and constraint forces separately, thus consider the control input u of the formu = H+>(q)ua � J>(q)ub1Notice that the inequality kJ2(q)k � kJ(q)k in (3) is not obvious. Interested readers are referred to [7], ch. X,sec. 6,7. 3

where ua 2 IRn�m; ub 2 IRm and H+(q) = �H>(q)H(q)��1H>(q) which under Assumption A1,exists and is bounded for all q 2 . Then the decoupled reduced order dynamic model for robotmanipulators under holonomic constraints can be rewritten in the form ([15])D�(q)�q1 + C�(q; _q) _q1 + g�(q) = ua (8)� = Z(q)�C�(q; _q) _q1 + g(q) �H+>(q)ua�+ ub (9)Equation (8) de�nes the dynamics of the system in the position direction i.e., the dynamics of theindependent coordinates, which are to be controlled. Equation (9) de�nes the dynamics in theforce direction i.e., of the force exerted by the manipulator's end-tool when interacting with itsenvironment. Due to the decoupled nature of this reduced order model we are able to design twoseparate position and force control laws ua and ub respectively.2.2.1 PropertiesP1 For all q 2 IRn, the matrix D�(q) = D>� (q) � dm�I > 0. Moreover, for all q 2 there exists0 < dM� <1 such that D�(q) � dM�I. Moreover for a suitable choice of C�(q; _q) we have that forall q 2 IRn the matrix _D�(q)� 2C�(q; _q) is skew-symmetric.It is worth remarking the existence of dM� < 1 only for all q 2 . This stems from theassumption that the Jacobian J2(q) is nonsingular only in this subset. See (5) and (7).P2 For all q 2 the norm of the matrix H(q) satis�es the inequality1 � kH(q)k � 1 + �2=�1moreover there exist a constant �4 such thatkH+(q)k � �4P3 For all x 2 IRn�m the matrix C�(q; _q1)x satis�es the followingC�(q;H(q) _q1)x = C�(q;H(q)x) _q1moreover for all q 2 there exists a kc� > 0 such thatkC�(q;H(q) _q1)k � kc�k _q1kFor a proof of Property P1 see [17]; for a proof of P2 and P3 see [11].3 Problem statement and its solution3.1 MotivationIn the previous section we have shown that the dynamic model in the position direction, that is, eq.(8) possesses similar properties than the model of unconstrained manipulators provided AssumptionA1 is satis�ed. Hence, algorithms used to control unconstrained manipulators can also be appliedin presence of holonomic constraints, provided one can ensure that the trajectories never leave theset where contact is possible and the reduced order model is well de�ned. In order to overcomethis di�culty, most of reduced order models considered in the literature rely on the (implicit orexplicit) assumption that, either = IRn or that the manipulator remains in a neighbourhood ofthe desired trajectory during the whole transient. However, it must be noted that assuming that = IRn is restrictive and unrealistic. In order to illustrate this idea, and the type of result we seekfor, consider the simple example of the planar 2-link manipulator shown in Figure 1.4

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Figure 1: Two-link planar constrained manipulatorFor this manipulator the constraints equation is given byl1 sin(q1)� l2 sin(q2 � q1) = 0: (10)Let us assume that l2 < l1 and that the manipulator cannot go into the surface, in this case it iseasy to see that contact of the end-tool with the surface is only possible if either�q1 2 IR : 0 < q1 < arcsin� l2l1�� (11)or �q1 2 IR : � � arcsin� l2l1� < q1 < �� : (12)This clearly shows that the set cannot coincide with IR2. On the other hand, in order to satisfyAssumptionA1, must be de�ned in a way such that the dependent coordinate be uniquely de�nedas a twice continuously di�erentiable function of the independent coordinate. For this example letus choose q1 to be the independent coordinate; then from (10) q2 4= k(q1) = q1+arcsin� l1l2 sin(q1)�.Notice that k(q1) is C2 only if (11) or (12) hold thus, for the reduced order model (8) { (9) to bewell de�ned, 1 must be chosen according to (11) or (12) depending on the initial condition q1(t0).To strengthen the justi�cation of this choice notice further that for this example, (with q1 asindependent coordinate) H(q) = 1l1 cos(q1)cos(q2�q1) + 1 ! ;then from the de�nition of k(q1) it follows that _q = H(q) _q1 grows unboundedly if q1 = ��arcsin� l2l1�or q1 = arcsin� l2l1�, hence the reduced order model (8) { (9) is not valid at these points.For these reasons, before exploring the stability properties of the closed loop system of theconstrained manipulator (1) { (2), with a control algorithm commonly used for unconstrainedmanipulators, one should be concerned by the permanence of the solutions in the set . Thissimple example illustrates our motivation to consider the following important problem.3.2 Problem statementDe�ne I 4= [0;1), assume that only position and force measurements are available, then design asmooth control law u = u(t; q; f) such that� q(t) 2 8 t � t0; to 2 I� limt!1 ~q(t) 4= limt!1[q(t)� qd(t)] = 0;� limt!1 ~f(t) 4= limt!1[f(t)� fd(t)] = 0;where fd(t) and qd(t) stand for the desired values of force and trajectory respectively, which aresupposed to satisfy the following restrictions: 5

A2 The desired trajectory qd(t) 2 C2 and maxt2I fkqd(t)k; k _qd(t)k; k�qd(t)kg � Bd.A3 The desired trajectory qd(t) 2 for all t � 0. Furthermore, it belongs to the interior of (i.e.�) with some distance from the boundary of the set (i.e. @)infq1b2@1 kq1d(t)� q1bk > �; (13)where � is a positive constant whose choice may depend on the initial conditions and the robotparameters.A4 For all desired trajectories qd(t) 2 the constraint equation �(qd(t)) = 0 is satis�ed. Further-more, the desired contact force satis�es fd(t) = J>(qd(t))�d(t) for all t � 0.3.3 Main resultProposition 3 Consider the model (8) { (9) in closed loop with the control lawua = D�(q)�q1d + C�(q; _qd) _q1d + g�(q)� kp~q1 �Kd# (14)ub = Z(q)H+>(q)D�(q)�q1d + �d � kf ~� (15)� _qc = �A(qc +B~q1)# = (qc +B~q1) (16)where A, B, and Kd are diagonal positive de�nite, kp; kf > 0 and ~� 4= � � �d. De�ne x(t) 4=col[~q1(t); _~q1(t); #(t)], then there always exist some constants 0 < �5 < �4 < � with � as in A3,such that for any initial condition (t0; x0) 2 (I � x) wherex 4= n~q1; _~q1; # 2 IRn�m : k~q1k; k _~q1k; k#k < �5o (17)the independent coordinates q1(t) remain in the set 1 for all t � t0. Moreover there always existmatrices A and B su�ciently large to ensure that the closed loop system (8) { (9), (14){ (16) isuniformly asymptotically stable. �It is important to remark that it is not necessary to assume that the end-tool is always in contactwith the surface. Putting aside the problem of impact, we can consider the problem of not loosingcontact under the assumption that initially, the manipulator interacted with its environment. Thiscan be easily proven as done in [21] by showing that the manipulator always exerts some forceagainst its environment hence, assuming that constraints are unilateral and that the manipulatorcan only push the surface, we need to prove that fi > 0 for all i = 1; : : : ;m. From equation (33)and the boundedness of q and _q, there exists kZ > 0 such thatk ~fk = kf � fdk � kZ + kfdkkf + 1 :Now, let kld be a constant such that fdi > kld for all i � m. Notice that one can always choose kfsuch that k ~fk < kld. Then we obtain that fi � �k ~fk+ fdi > �kld + kld that is, fi > 0 as required.After the original submission of this paper, we have noticed an increasing interest on the problemof control of manipulators during their complete task: free-motion, impact, constrained motion.Even though it is beyond the scope of this paper to deal in full detail with this problem, it is worthpointing out the importance of guaranteeing that the trajectories does not escape the domain wherethe Jacobian's rank does not degenerate and henceforth where the reduced order model is valid.6

To be more precise, let us recall the stability criterion established in [2] for the problem of therobot control problem during its complete task. In that reference the authors divide the time spanof the complete task into successive intervals corresponding to the constrained, unconstrained andimpact phases: IR+ = �0 [ I0 [ �1 [ �2 [ � � �where the �'s are transition phases and the I's, impact phases.Based on this assumption the authors introduce the concept of weak stability which roughlyspeaking, says that a system is weakly stable if it is stable for all t � 0, t 2 �. In that referencenew tools to prove weak stability are given based on Lyapunov like techniques. In this paper weshow using simple Lyapunov stability theorems how one can guarantee that the trajectories do notescape a set (where the Jacobian does not degenerate) based on the basic assumption that theinitial conditions are contained by and that the desired reference does not lay \too close" to itsboundaries.Even though in [2] the authors consider intervals of time while we use subsets of coordinates,it is our belief that there is a strong relation between these concepts. Thus, our methodology ofproof may contribute to open a new way for proving weak stability in the sense of [2]. Besides thefact that, looking at the constrained motion problem as part of the complete task problem, it iscrucial to ensure that the model remains valid for all time in some interval (or equivalently for allq1(t) 2 1).4 Proof of main resultIt is worth recalling that under Assumption A1 it holds that the Jacobian of the constraintsequation is full rank only if q1(t) 2 1 8 t � t0. However we do not suppose that this happens tohold for all t > t0, then we must prove that under the conditions of Proposition 1, AssumptionsA1 { A4 and for all initial conditions x0(t0) 4= col[~q1(t0); _~q1(t0); #(t0)] 2 x, the solution x(t) isalways bounded, moreover we prove that q1(t) 2 1 8 t � t0. This constitutes the beginning andmost important part of the proof while the second one concerns the uniform asymptotic stabilityof the error system.4.1 Boundedness of solutionsA fundamental tool to prove our main result is the following lemma borrowed from [18]Lemma 4 Let P1, P2 be compact sets in B� � IRn such thatf0g � �P 1 � P1 � �P 2 � P2 � �B� � IRnwhere �P denotes the interior of P , and �B� is the closure of B�. Let V : B� � I 7! IR for all(x; t) 7! V (x; t) be a function of class C1 and let a be a constant such that(a) (8 t 2 I) and (8 x 2 @P2), V (x; t) � a(b) (8 t 2 I) and (8 x 2 @P1), V (x; t) < a(c) (8 t 2 I) and (8 x 2 �P2 �P1), _V (x; t) � 0.If t0 2 I and x0 2 P1, then the solution x(t; t0; x0) of the di�erential equation _x = f(t; x), wheref(t; x) is continuous and locally Lipschitzian, is de�ned and belongs to P2 for all t � t0. �7

In order to apply the Lemma 4 we �rst de�neB� 4= fx 2 IR3(n�m) : k~q1k < �; _q1; # 2 IRn�mgnext, consider the functionV (x; t) = 12 _~q1>D� _~q1 + kp2 ~q1 2 + 12#>KdB�1#+ "~q1> _~q1 � "#> _~q1 (18)where " is a small positive constant to be de�ned. Using Property P1 and Assumption A3 it iseasy to see that V (x; t) is a smooth function in (B� � I). Now, let us de�neP1 4= n~q1; _~q1; # 2 IRn�m : k~q1k; k _~q1k; k#k � �5oP2 4= n~q1; _~q1; # 2 IRn�m : k~q1k; k _~q1k; k#k � �4o (19)where �5 and �4 as in Proposition 1. Next, we prove that that the function (18) meets all conditionsof Lemma 4.Condition (a). Consider the auxiliary functionW1(x; t) 4= 14 _~q1>D� _~q1 + kp4 ~q1 2 + 14#>KdB�1#;then it can be easily shown that W2(x; t) 4= V (x; t)�W1(x; t) is positive de�nite if" < 1p8 min((kpdm�)1=2; �kdmdm�bM �1=2) (20)then we have that V (@P2 � I) �W1(@P2 � I), which in its turn implies thatV (@P2 � I) � supt 2 I;x 2 @P2 W1(x; t) = 14(dm� + kp + kdm=bM )�24then the �rst condition holds with a � 14 (dm� + kp + kdm=bM )�24.Condition (b). First notice that, since x 2 @P1 then k~q1k � �5 and since q1d(t) 2 1 byhypothesis, it follows that q1(t) is bounded. Now, let B�5;q1d be the ball centered at q1d(t) and radius�5, if infq1b2@1 kq1d(t)� q1bk > �5 (21)then, B�5;q1d � 1. Since � > �5, the latter happens to hold observing A3. Moreover, sincekq1(t)� q1d(t)k � �5 it follows that q1(t) 2 B�5;q1d , hence q1(t) 2 1.This implies that for all x 2 @P1, there exists a dM� such that 1 > dM� > kD�k in accordancewith Property P1. Thus we can write for all (x; t) 2 (@P1 � I)V (x; t) � 12 ["+ kp] k~q1k2 + �"+ 12dM�� k _~q1k2 + 12 �"+ kdMbm � k#k2 (22)hence V (@P1 � I) � 12 �dM� + kp + kdMbm + 4"��25 (23)8

it then follows that both conditions (a) and (b) are met if�4 > �5s2(dM� + kp + kdM =bm + 4")dm� + kp + kdm=bM (24)Condition (c). Consider the �rst error equation, i.e. (8) and (14), using Property P3 we areable to write D�(q)�~q1 + [C�(q; _q) + C�(q; _qd)] _~q1 + kp~q1 +Kd# = 0_# = �A#+B _~q1 (25)Now, using Properties P1 { P3 and Assumptions A2 { A4, we will prove that the timederivative of (18) along the trajectories of (25) is negative de�nite for all (x; t) 2 (P2 � I). Firstnotice that under Assumption A3, if x 2 P2 then ~q1 � �4, and hence q1 2 1. Thus, aftersome straightforward calculations one obtains that the time derivative of (18), is bounded for all(x; t) 2 (P2 � I), as_V (x; t) � �12 � k~q1kk#k �T Q1z }| {� "kp=dM� �"(kp + kdM )=dm��"(kp + kdM )=dm� 2kdmam=3bM � � k~q1kk#k ��"" k~q1kk _~q1k #T Q2z }| {� kp=2dM� �kc�Bd=dm��kc�Bd=dm� bm=3 �" k~q1kk _~q1k #�" k _~q1kk#k #T Q3z }| {� "bm=3 �"(kc�Bd=dm� + aM=2)�"(kc�Bd=dm� + aM=2) kdmam=3bM �" k _~q1kk#k #� 1z }| {�kdmam3bM � "kdMdm� � k#k2 � 2z }| {h"3bm � kc�Bd � 2"�4kc�=dm� � "i k _~q1k2: (26)Now Q1, Q3 and 1 are positive de�nite if" < min� 2d2m�kpkdmam3dM�bM (kp + kdM )2 ; 4d2m�kdmambm9bM (2kcBd + aMdm�)2 ; kdmamdm�3kdM bM � (27)while Q2 is positive de�nite if kpbm > 6k2c�B2ddM�d2m� (28)and 2 > 0 if bm > 6 and " > kc�Bdbm6 � 2kc��4dm� : (29)Conditions (20) and (27) hold for a su�ciently small " > 0, condition (28) holds for su�cientlylarge bm while the last condition imposes a lower bound on " which can be made arbitrarily smallby increasing bm. As a matter of fact, it can be proven [11] that for su�ciently large A = aI andB = bI satisfying b 4= 12�1�4kc�dm� (30)with �1 > max�kc�B2ddM��4kpdm� ; dm��4kc� ; 1� (31)9

_V (x; t) is negative de�nite for all (x; t) 2 (P2 � I). It follows that condition (c) of Lemma 4 holdsas well, then k~q1(t)k � �4 for all t � t0. Since �4 < �, under Assumption A3 we conclude thatq1(t) 2 1 for all t � t0, moreover from A1 it follows that q(t) 2 for all t � t0.4.2 Asymptotic stabilityThe proof of asymptotic stability follows directly from the results of previous section using thefollowing theorem of [18]Theorem 5 Let V : P2 � I 7! IR be a function of class C1 and a Lyapunov function on �P2 � I. If(a) V (x; t)! 0 when x! 0 uniformly for t 2 I(b) _V (x; t) is negative de�nite on P2 � I(c) V (x; t) � c for all (x; t) 2 P2 � I, and c 2 IRthen all solutions x(t; t0; x0) such that x(t; t0; x0) 2 P2 for every t � t0 tend to 0 as t!1 uniformlyin t0, x0. �We proceed to prove that V (x; t) de�ned by (18) meets all the conditions of Theorem 5. The�rst condition follows straightforward under Assumptions A2 { A4 about the desired trajectory,while the second condition was already shown in the previous section. The third condition followsobserving that, since q1(t) 2 1 for all t � t0 we are able to write using (22)V (P2 � I) � V (@P2 � I) � 12 �dM� + kp + kdMbm + 4"��24 (32)In the previous section we proved that the solutions x(t; t0; x0) 2 P2 for all t � t0 thus, usingTheorem 5, we conclude that the error system in the position direction, i.e. (25), is uniformlyasymptotically stable. Now, looking at the second error equation i.e., in the force direction, wehave using P3 that(kf + 1)~� = Z(q)n[C�(q; _q) + C�(q; _qd)] _~q1 +H+>[kp~q1 +Kd#]o : (33)The proof is completed by noticing that since q1 2 1, then Z(q) is bounded. Moreover the righthand side terms of (33) tend uniformly asymptotically to zero, then it follows that ~�(t)! 0, hence~f(t)! 0 for all t � t0 and for all to 2 I. �5 Simulation resultsFor the sake of illustration, we have performed some brief simulations on a two-link planar robotlike that of Figure 1. We have considered that l1 = 2m, l2 = 1m. The masses of the links arem1 = 5, m2 = 2:5, while the momenta of inertia about their centres of mass are I1 = 6:33, andI2 = 0:83. For both links, it was considered that the centres of mass are at the middle of the links.We have considered that the initial con�guration of the manipulator is similar to that shownin Figure 1, hence 1 := fq1 2 IR : 0 < q1 < �=6 � 0:01g and we have chosen q1 to be theindependent coordinate. The control gains were set to kp = 50000, Kd = diagf[20000; 20000]g,A = diagf[3000; 3000]g, and B = diagf[1000; 1000]g. The desired trajectory is q1d = �18(1 +cos(6�t)), while the desired force was set to 2N and f(t0) = 2:7N. In the �rst part of Figure 2we show the resulting trajectory q1(t) and its reference, q1d(t). Then, we show the resulting forcetracking error, ~f . Notice that the trajectory is bounded between -0.1 and �=6 � 0:01, hence thatq1(t) 2 1, as it was expected. 10

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Figure 2: Actual and reference independent trajectories q1(t) and q1d(t) and force values.6 Concluding remarksWe have studied the problem of output feedback tracking control of manipulators under holonomicconstraints. We used a reduced order model whose particular feature, and advantage in relation toother models available in the literature, is that the generalized coordinates space is constrained toa smooth manifold where it can be guaranteed that the constraints Jacobian is regular.We proposed a controller which uses the output of a linear �lter instead of the velocities vector,hence only position measurements are needed. We proved that the proposed control law keeps theindependent trajectories bounded in the set where the solvability of the constraints equation isguaranteed. Furthermore, we proved local uniform asymptotic stability in the set in both, theforce and the tracking directions, using the direct Lyapunov's method.We believe that our results and proof methodology may be extended and contribute to themore general problem of robot control during the complete task: free motion, impact, constrainedmotion.References[1] S. Arimoto, Y.H. Liu, and T. Naniwa. Principle of orthogonalization for hybrid control ofrobot arms. In Proc. 12th. IFAC World Congress, Sydney, Australia, 1993.[2] B. Brogliato and S. Niculescu and M. Monteiro-Marques. On tracking control of rigid manip-ulators with unilateral constraints. In Proc. 4th. European Contr. Conf., Brussels, Belgium,1997. Paper no. 872.[3] P. B�elanger. Estimation of angular velocity and acceleration from shaft encoder measurements.In Proc. IEEE Conf. Robotics Automat., volume 1, pages 585{592, Nice, France, 1992.[4] A. Caiti and G. Cannata. Hybrid position/force control of constrained manipulators in presenceof uncertainties. In Proc. 4th. Symposium on Robot Control, pages 647{652, Capri, Italy, 1994.[5] R. Carelli and R. Kelly. An adaptive impedance/force controller for robot manipulators. IEEETrans. Automat. Contr., 36:967{971, 1991.[6] M. S. de Queiroz, J. Hu, D. Dawson, T. Burg, and S. Donepudi. Adaptive position/forcecontrol of robot manipulators without velocity measurements: Theory and experimentation.IEEE Trans. Syst. Man and Cybernetics, 1996. (to appear).[7] F. R. Gantmacher. The theory of matrices, vols. I and II. Chelsea publishing Co., New York,1960. 11

[8] J. Hu, M. de Queiroz, T. Burg, and D. Dawson. Adaptive position/force of robot manipulatorswithout velocity measurements. In Proc. IEEE Conf. Robotics Automat., pages 887{892, 1995.[9] H.P. Huang and W.L. Tseng. Asymptotic observer design for constrained robot systems. IEEProceedings-D, 138:211{216, 1991.[10] H.N. Koivo and R.K. Kankaanrantaes. Dynamics and simulation of compliant motion of amanipulator. In Proc. IEEE Conf. Robotics Automat., volume 2, pages 163{173, 1988.[11] A. Loria. On output feedback control of Euler-Lagrange systems. PhD thesis, Univ.of Technology of Compiegne, Compi�egne, France, October 1996. Available fromhttp://www-ccec.ece.ucsb.edu/people/aloria/index.html.[12] A. Loria and R. Ortega. Force/position regulation for robot manipulators with unmeasurablevelocities and uncertain gravity. Automatica, 32(6):939{943, 1996. See also: Proc. IFACSymposium on Motion Control, Muncih, 1995.[13] H. McClamroch and D. Wang. Feedback stabilization and tracking of constrained robots.IEEE Trans. Automat. Contr., 33:419{426, 1988.[14] J. K. Mills and A. A. Goldenberg. Force and position control of manipulators during con-strained motion tasks. IEEE Trans. Robotics Automat., 5(1):30{46, 1989.[15] E. Panteley and A. Stotsky. Adaptive trajectory/force control scheme for constrained robotmanipulators. Int. J. Adapt. Control Signal Process., 7(6):489{496, 1993.[16] E. Panteley and A. Stotsky. Asymptotic stability of constrained robot motion observer basedcontrol schemes. In Proc. 2nd. European Contr. Conf., pages 1255{1260, Groningen, TheNetherlands, 1993.[17] E. Panteley and A. Stotsky. Adaptive trajectory/force control scheme for constrained robotmaqnipulators. Technical Report no. 112, Inst. for Problems of Mech. Engg., Academy of Sc.of Russia, June 1995.[18] N. Rouche and J. Mawhin. Ordinary di�erential equations II: Stability and periodical solutions.Pitman publishing Ltd., London, 1980.[19] I. Sandberg. Global implicit function theorems. IEEE Trans. Circuits and Systems, 28:145{149, 1981.[20] D. Wang and H. McClamroch. Position force control for constrained manipulator motion:Lyapunov's direct method. IEEE Trans. Robotics Automat., 9:308{313, 1993.[21] B. Yao, S. P. Chan, and D. Wang. Robust motion and force control of robot manipulators inthe presence of environmental constraint uncertainties. In Proc. 31st. IEEE Conf. DecisionContr., pages 1875{1880, Tucson, Arizona, 1992.[22] T. Yoshikawa. Dynamics hybrid position/force control of robot manipulators{description ofhand constraints and calculation of joints driving forces. In Proc. IEEE Conf. Robotics Au-tomat., pages 1393{1398, San Francisco, CA, 1986.12