STABILITY OF HYBRID POSITION/FORCE CONTROL APPLIED TO MANIPULATORS WITH FLEXIBLE JOINTS

Transcription

1 International Journal of Robotics and Automation, Vol. 14, No. 4, 1999 STABILITY OF HYBRID POSITION/FORCE CONTROL APPLIED TO MANIPULATORS WITH FLEXIBLE JOINTS P.B. Goldsmith, B.A. Francis, and A.A. Goldenberg Abstract Hybrid position/force control enables a robot to apply specified forces to a constraint while moving along a prescribed path. Flexibility of the motor transmissions (joint flexibility) is a major concern in the design of hybrid controls, particularly with regard to system stability. While much work on hybrid control has been toward the development of advanced schemes specifically for flexible-joint robots, little is known about how joint flexibility affects the stability of simple well-established hybrid controls. This paper examines the stability of the most basic hybrid control, which requires no robot dynamic model. For both rigid and compliant constraints, we prove that the closed-loop system is always asymptotically stable. Yet we also show that joint flexibility can destabilize a well-known model-based control. We then propose a control that achieves asymptotic tracking when applied to rigid-joint manipulators and is asymptotically stable when applied to flexible-joint manipulators. Experiments on a constrained flexible-joint robot support the key theoretical results. Key Words Manipulator control, constrained robots, hybrid position/force control, flexible joints, stability, elastic constraints 1. Introduction The goal of hybrid position/force control is to control simultaneously the position of a robot s tool and the force it applies to a constraint. For this purpose, a variety of control algorithms, called hybrid controls, have been proposed in the literature. A major issue in the design of hybrid controls is the flexibility of robot motor transmissions (joint flexibility), which is a potential source of instability. To compensate for joint flexibility, many sophisticated control schemes have been proposed. However, it is not known whether joint flexibility destabilizes simple hybrid controls designed for rigid robots. Such an analysis can help assess the need for more complex controls. Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Canada, T2N 1N4; pgold@enme.ucalgary.ca Department of Electrical and Computer Engineering, The University of Toronto, Toronto, Canada, M5S 1A4; francis@control.utoronto.ca Department of Mechanical and Industrial Engineering, The University of Toronto, Toronto, Canada, M5S 1A4; golden@mie.utoronto.ca (paper no ) 146 Approaches for controlling constrained manipulators can be divided into hybrid controls [1] and impedance controls [2]. Hybrid controls regulate force in constrained directions and position in unconstrained directions, whereas impedance controls regulate position in all directions and achieve a desired impedance via choice of gains. Impedance controls can be used for explicit force control by specifying a suitable trajectory inside the constraint surface (and optionally adding a force feedback loop). However, because there is position feedback in constrained directions, errors in the position or stiffness of the constraint produce force errors [3]. For this reason, hybrid control is preferred for explicit control of force. In [4] hybrid controls were categorized as either modelbased controls [5 8], which contain a model of the manipulator inertia, or non-model-based controls, which do not contain such a model [1]. Model-based controls usually feedback linearize the constrained robot dynamics and achieve asymptotic tracking of position and force trajectories when applied to rigid manipulators in contact with rigid constraints [3, 6, 8]. Non-model-based controls do not achieve asymptotic tracking but may achieve asymptotic stability [9]. Joint flexibility is often the dominant source of compliance in the constrained system and should be included in stability analyses. For example, in relative terms, a harmonic drive transmission is much more flexible than a typical workpiece. Analyses of single-link robots under force feedback have revealed that flexibility between the actuator and the force sensor can destabilize the system [10, 11]. This problem can be avoided by inserting a lowpass filter in the force feedback loop [12], or by disabling the force feedback and using only feed-forward to control the contact force [13]. Unlike force feedback, position feedback is essential and must be designed to ensure stability. Much of the research on the effect of position feedback on the stability of flexible-joint robots focuses on unconstrained manipulators. An extensive survey may be found in [14], while more recent work in this area may be found in [15] and [16]. The stability of a constrained flexible-joint robot was investigated in [17] for a control in which the desired configuration is specified in link angles. Hybrid controls for constrained flexible-joint robots have been developed using a variety of approaches. Integral manifold control [18] and composite control [19] are based on a singular perturbation model of the system dynamics.

2 Exact feedback linearization of the flexible-joint system was investigated in [13, 20, 21]. The stability of adaptive controls was studied in [22], and sliding mode control was considered in [23]. In [24], a parameter adaptive law was applied within a composite control strategy. A similar approach was taken in [19], except that an adaptive neural network control scheme was used. In spite of the research trend toward complex hybrid controls that compensate for joint flexibility, this paper investigates the stability of a simple hybrid control originally designed for rigid robots. Previous results on the stability of such rigid hybrid controls model the joint flexibility as a singular perturbation [25], which assumes that joint stiffness is large compared to other stiffness parameters in the system, such as constraint stiffness and control stiffness (gains). However, as this is not true for many practical systems, the present analysis considers the joint stiffness to be finite and arbitrary. The control investigated in this paper is the hybrid version of an impedance control called stiffness control [26], which is obtained from the impedance version by setting the position gains to zero in constrained directions. In [4] this hybrid version was called stiffness control as well, but we will call it basic hybrid control to distinguish it from the impedance version of [26]. The term basic distinguishes this control from model-based hybrid controls and from the original hybrid control of Raibert and Craig [1], which used the Jacobian inverse instead of the transpose and had well-documented stability problems [4]. Basic hybrid control does not require a dynamic model of the flexible-joint robot (or even a rigid one) or feedback of higher order states introduced by the joint flexibility. The stability of basic hybrid control was proven in [9] for the case of compliant constraints and in [3] for the case of rigid constraints. The present paper proves that this control is always asymptotically stable when applied to a flexible-joint manipulator in contact with either a rigid or compliant constraint. This result holds for any values of the system s dynamic parameters, including the (unknown) joint and constraint stiffness. In contrast, we show that an established model-based approach can be destabilized by joint flexibility. We then propose a modification to basic hybrid control that yields asymptotic tracking when applied to rigid manipulators without sacrificing stability in the presence of joint flexibility. Finally, we present the results of experiments conducted on a constrained flexiblejoint robot. 2. Manipulator Dynamics Neglecting gravitational effects, we write the dynamic equations of a flexible-joint manipulator as in [14] as: D lq (q) q + v q (q, q) + B q ( q q m ) + K q (q q m ) + F q = 0 (1) M q q m B q ( q q m ) K q (q q m ) = τ (2) (q, q m, q, q m, τ, F q ) R n R n R n R n R n R n where: 147 q R n is the vector of joint coordinates q m R n is the vector of actuator coordinates τ R n is the vector of actuator forces/torques M q R n n is a symmetric positive definite matrix of motor inertias D lq (q) R n n is a symmetric positive definite matrix of link inertias K q R n n is a diagonal positive definite matrix of joint stiffnesses B q R n n is a diagonal positive definite matrix of damping coefficients v q (q, q) R n is a vector of centrifugal and Coriolis terms, quadratic in q F q R n is the vector of generalized forces applied by the end effector, expressed with respect to joint coordinates In (1) and (2), n is the number of manipulator joints, assumed equal to the number of degrees of freedom of the end effector when unconstrained. In particular, n = 3 if the end effector is a point in R 3, and n = 6 if the end effector is a rigid body in R 3. The constraint acting on the end effector may be modelled as either rigid or compliant. If the constraint is rigid, it is modelled as in [6] as a holonomic constraint, expressed as: Φ q (q) = 0 (3) where Φ q : R n R n is a smooth function such that rank Φ q q (q) = m throughout the region of interest. As shown in [3, 6], there exists a smooth diffeomorphism x = X(q), q = X 1 (x), defined for all x in an open region Θ x, a vector partition: x = x 1, x 1 R n m, x 2 R m (4) x 2 and an open region Θ x1 R n m, such that for each x 1 Θ x1, Φ q (q) = 0 implies x 2 = 0. We refer to generalized coordinates x chosen in this manner as task coordinates for the constraint. Note that the generalized constraint force F q in (1) is expressed with respect to the joint coordinates q. This force may be expressed with respect to the task coordinates x using the relation: F x = J(q) T F q (5) where J(q) = X(q)/ q. Let F x1 and F x2 be the components of F x associated with x 1 and x 2, respectively. Then F x2 represents constraint forces and F x1 represents frictional forces applied by the end effector to the constraint. We model the constraint in task coordinates as: F x1 = B f x 1 (6) x 2 = 0 (7) (x 1, x 1 ) Θ x1 R n m, where B f R (n m) (n m) is a diagonal matrix of nonnegative dynamic friction coefficients.

3 We model a compliant constraint in task coordinates as: F x1 = B f x 1 (8) F x2 = B e x 2 + K e x 2 (9) (x, x) Θ x R n, where K e R m m and B e R m m are diagonal positive definite stiffness and damping matrices and Θ x is an open subset of R n. The damping term B e x 2 is included to account for the fact that no real material can store elastic energy without some dissipation. To combine the manipulator dynamic equations (1) and (2) with the constraint equations, we first express the former with respect to task coordinates. Differentiating q = X 1 (x) twice with respect to time gives: q = J(x) 1 x (10) q = J(x) 1 x + J(x) 1 ẍ (11) (x, x, ẍ) Θ x R n R n, where, by abuse of notation, J(x) J(X 1 (x)). Let x m be the measured end effector position, which we assume is obtained from the actuator displacements q m via the transformation x m = X(q m ). We premultiply (1) by J(x) T and (2) by J(x m ) T, and substitute (5), (10), and (11) into the result to obtain: D l (x)ẍ + v l (x, x) + J(x) T B q [J(x) 1 x J(x m ) 1 x m ] +J(x) T K q [X 1 (x) X 1 (x m )] + F x = 0 (12) M(x m )ẍ m + v m (x m, x m ) J(x m ) T B q [J(x) 1 x J(x m ) 1 x m ] J(x m ) T K q [X 1 (x) X 1 (x m )] = J(x m ) T τ (13) (x, x m, x, x m, τ, F x ) Θ x Θ x R n R n R n R n where: D l (x) = J(x) T D lq (x)j(x) 1 (14) M(x m ) = J(x m ) T M q J(x m ) 1 (15) v l (x, x) = J(x) T v q (x, x) + J(x) T D lq (x) J(x) 1 x (16) v m (x m, x m ) = J(x m ) T M q (x) J(x m ) 1 x m (17) Note that v l (x, x) is quadratic in x, and that v m (x m, x m ) is quadratic in x m. It is convenient to define: [ ] E 1 = I n m 0 R (n m) n (18) [ ] E 2 = 0 I m R m n (19) Then the equations of the rigid constraint, (6) and (7), may be written as F x = E T 1 B f x 1 + E T 2 F x2 and x = E T 1 x 1. Substitution of these expressions into (12) and (13) gives: D l (x 1 )E T 1 ẍ 1 + v l (x 1, x 1 ) 148 +J(x 1 ) T B q [J(x 1 ) 1 E T 1 x 1 J(x m ) 1 x m ] +J(x 1 ) T K q [X 1 (E T 1 x 1 ) X 1 (x m )] +E T 1 B f x 1 + E T 2 F x2 = 0 (20) M(x m )ẍ m + v m (x m, x m ) J(x m ) T B q [J(x 1 ) 1 E T 1 x 1 J(x m ) 1 x m ] J(x m ) T K q [X 1 (E T 1 x 1 ) X 1 (x m )] = J(x m ) T τ (21) where, by abuse of notation: Let: D l (x 1 ) D l (E T 1 x 1 ) (22) v l (x 1, x 1 ) v l (E T 1 x 1, E T 1 x 1 ) (23) D l11 (x 1 ) = E 1 D l (x 1 )E T 1 (24) G l (x 1 ) = E 2 E 2 D l (x 1 )E T 1 D l11 (x 1 ) 1 E 1 (25) We may decompose (20) into state and output equations by multiplying through by E 1 and G l (x 1 ) in turn to give: D l11 (x 1 )ẍ 1 + E 1 v l (x 1, x 1 ) +E 1 J(x 1 ) T B q [J(x 1 ) 1 E T 1 x 1 J(x m ) 1 x m ] +E 1 J(x 1 ) T K q [X 1 (E T 1 x 1 ) X 1 (x m )]+B f x 1 = 0 (26) F x2 = G l (x 1 ) { J(x 1 ) T B q [J(x 1 ) 1 E T 1 x 1 J(x m ) 1 x m ] +J(x 1 ) T K q [X 1 (E T 1 x 1 ) X 1 (x m )] + v l (x 1, x 1 ) + E T 1 B f x 1 } (x 1, x m, x 1, x m, τ) Θ x1 Θ x R n m R n R n. (27) The state equations of the system are (21) and (26), while the output equation is (27). In the case of a compliant constraint, the system is obtained from (8), (9), (12), and (13) and as: D l (x)ẍ + v l (x, x) + J(x) T B q [J(x) 1 x J(x m ) 1 x m ] +J(x) T K q [X 1 (x) X 1 (x m )] +E T 1 B f x 1 + E T 2 [B e x 2 + K e x 2 ] = 0 (28) M(x m )ẍ m + v m (x m, x m ) J(x m ) T B q [J(x) 1 x J(x m ) 1 x m ] J(x m ) T K q [X 1 (x) X 1 (x m )] = J(x m ) T τ (29) F x2 = B e x 2 + K e x 2 (30) (x, x m, x, x m, τ) Θ x Θ x R n R n R n Fig. 1 clarifies the meaning of the dynamic parameters introduced in this section. Shown is a model of a planar Cartesian robot arm with compliant joints in contact with a compliant constraint. The dynamic parameters of this robot are M q = diag(m 1, m 2 ), D lq = diag(d 1, d 2 ), K = diag(k 1, k 2 ), B = diag(b 1, b 2 ), K e = k e, and B e = b e.

4 F x2 = G l (x 1 ) { J(x 1 ) T B q [J(x 1 ) 1 E T 1 x 1 J(x m ) 1 x m ] +J(x 1 ) T K q [X 1 (E T 1 x 1 ) X 1 (x m )] + v l (x 1, x 1 ) + E T 1 B f x 1 } (34) (x 1, x m, x 1, x m, x d, F d ) Θ x1 Θ x R n m R n R n m R m If x d is fixed in Θ x1 and F d = 0, then the system (32), (33), (34) has an equilibrium point at: x 1 = x d (35) x m = E T 1 x d (36) x 1 = 0 (37) x m = 0 (38) We translate this equilibrium point to the origin by defining: Figure 1. Planar flexible joint robot and compliant constraint. 3. Stability of Basic Hybrid Control In this section, we investigate the stability of basic hybrid control applied to the constrained flexible-joint manipulators derived in the preceding section. We call this property robust stability, as all dynamic parameters in the system are assumed to be completely unknown. The stability of this control applied to a rigid-joint manipulator was analysed in [4] and [9]. This control may be written as: τ = J(x m ) T E T 1 [K p (x d x m1 ) K v x m1 ] + J(x m ) T E T 2 F d (31) where K p R (n m) (n m) and K v R (n m) (n m) are positive definite gain matrices. Note that τ is computed using the measured end effector position x m, obtained from measured actuator displacements q m via the transformation x m = X(q m ). Force feedback is not used since it was shown to destabilize single-dof manipulators in [10] and [11]. 3.1 Rigid Constraint Substitution of the control (31) into the equations of a flexible-joint manipulator in contact with a rigid constraint (21), (26), (27) gives: D l11 (x 1 )ẍ 1 + E 1 v l (x 1, x 1 ) +E 1 J(x 1 ) T B q [J(x 1 ) 1 E T 1 x 1 J(x m ) 1 x m ] +E 1 J(x 1 ) T K q [X 1 (E T 1 x 1 ) X 1 (x m )]+B f x 1 = 0 (32) M(x m )ẍ m + v m (x m, x m ) J(x m ) T B q [J(x 1 ) 1 E T 1 x 1 J(x m ) 1 x m ] J(x m ) T K q [X 1 (E T 1 x 1 ) X 1 (x m )] +E T 1 [K v x m1 + K p (x m1 x d )] E T 2 F d = 0 (33) 149 w 1 = x 1 x d (39) w 2 = x m E1 T x d (40) z 1 = w 1 (41) w 2 Θ z1 = {z 1 R 2n m z 1 < c} (42) where c > 0 is chosen small enough that: x d x d Θ x1, z 1 Θ z1 = + z 1 Θ x1 Θ x (43) E1 T x d Substitution of (39) and (40) into (32), (33), and (34) gives: D l11 (x d + w 1 )ẅ 1 + E 1 v l (x d + w 1, w 1 ) (44) +E 1 J(x d + w 1 ) T B q [J(x d + w 1 ) 1 E T 1 w 1 J(E T 1 x d + w 2 ) 1 w 2 ] +E 1 J(x d + w 1 ) T K q {X 1 [E T 1 (x d + w 1 )] X 1 (E T 1 x d + w 2 )} + B f w 1 = 0 M(E T 1 x d + w 2 )ẅ 2 + v m (E T 1 x d + w 2, w 2 ) (45) J(E T 1 x d + w 2 ) T B q [J(x d + w 1 ) 1 E T 1 w 1 J(E T 1 x d + w 2 ) 1 w 2 ] J(E T 1 x d + w 2 ) T K q {X 1 [E T 1 (x d + w 1 )] X 1 (E T 1 x d + w 2 )} +E T 1 [K v E 1 w 2 + K p E 1 w 2 ] = 0 F x2 = G l (x d + w 1 ) (46) { J(xd + w 1 ) T B q [J(x d + w 1 ) 1 E T 1 w 1 J(E T 1 x d + w 2 ) 1 w 2 ] +J(x d +w 1 ) T K q {X 1 [E T 1 (x d +w 1 )] X 1 (E T 1 x d +w 2 )} + v l (x d + w 1, w 1 ) + E T 1 B f w 1 } (z 1, z 1, x d ) Θ z1 R 2n m Θ x1.

5 Let z 2 = z 1 and z = z 1. Then for each x d in Θ x1, z 2 z = 0 is an equilibrium point of (44) and (45). Note that at equilibrium, F x2 = 0 by (46). To prove the stability of the system (44), (45), (46), we will make use of the following lemma. Lemma 1: Let H and T be j j matrices, and suppose that T is invertible. If T T HT is positive definite, then so is H. (Reference: [27], page 399) Theorem 1: Consider the system (44), (45), (46). For every fixed x d Θ x1, the equilibrium point z = 0 is asymptotically stable. Proof: Linearization of (44) and (45) about z = 0 gives: where: D l11 (x d )ẅ 1 + E 1 B(E T 1 w 1 w 2 ) +E 1 K(E T 1 w 1 w 2 ) + B f w 1 = 0 (47) M(E T 1 x d )ẅ 2 B(E T 1 w 1 w 2 ) K(E T 1 w 1 w 2 ) +E T 1 K v E 1 w 2 + E T 1 K p E 1 w 2 = 0 (48) B = J(E T 1 x d ) T B q J(E T 1 x d ) 1 (49) K = J(E T 1 x d ) T K q J(E T 1 x d ) 1 (50) Here, we have used the fact that: X 1 (x) = J(x) 1 (51) x We may write (47) and (48) as: where: H 2 z 1 + H 1 z 1 + H 0 z 1 = 0 (52) H 2 = D l11(x d ) 0 (53) 0 M(E1 T x d ) H 1 = E 1BE1 T + B f E 1 B (54) BE1 T B + E1 T K v E 1 H 0 = E 1KE1 T E 1 K (55) KE1 T K + E1 T K p E 1 Since D l11 (x d ) and M(E1 T x d ) are positive definite, so is H 2. It remains to show that H 1 and H 0 are positive definite. Let: T = whose inverse is given by: E 1 I n m E2 T E 2 E1 T T 1 = ET 1 (56) I n (57) 0 E Now: T T H 1 T = B 0 0 K v + ET 1 B f E 1 E1 T B f B f E 1 B f (58) which is positive definite since B and K v are positive definite and the second matrix in the sum is positive semidefinite. Similarly: T T H 0 T = K 0 (59) 0 K p which is positive definite. Therefore, H 1 and H 0 are positive definite by Lemma 1. Since H 0, H 1, and H 2 are positive definite, all eigenvalues of (52) lie strictly in the left half-plane [28]. Therefore, by Lyapunov s Linearization Theorem ([29], page 179), the nonlinear system (44), (45), (46) is asymptotically stable. QED Remark 1: It follows from Theorem 1 and continuity of (46) that x 1 converges asymptotically to x d and F x2 converges asymptotically to zero [3]. Remark 2: Since Theorem 1 holds for each set point x d in the region Θ x1 where the system equations are defined, the stability of the system is independent of manipulator configuration. Moreover, this stability result holds for arbitrary inertial parameters, joint stiffness and damping, constraint friction coefficients, and control gains. Remark 3: The special case of a workless holonomic constraint is obtained by setting the matrix of friction coefficients B f to zero. In this case, inspection of (58) reveals that T T H 1 T remains positive definite and thus the system remains asymptotically stable. It may similarly be shown (using a different choice for T ) that H 1 remains positive definite when K v = 0 and B f is positive definite, which indicates that either velocity feedback or constraint friction is sufficient to ensure stability. Remark 4: Since no joint friction was assumed in the dynamic model, Theorem 1 indicates that joint friction is not required for stability provided that joint damping B (associated with joint flexibility) is present. Remark 5: The assumption of linear joint stiffness is not necessary. For example, consider a nonlinear elastic torque at joint i of µ i (q i q mi ) 3 +k qi (q i q mi ). Linearization of this term about q i = q mi = 0 (which corresponds to z = 0 in Theorem 1) gives k qi (q i q mi ), which is identical to the linear stiffness terms assumed in equations (1) and (2). Therefore, when the closed-loop system (44), (45) is linearized about z = 0, the same result (52) is obtained as before, and Theorem 1 follows. In general, stability is guaranteed for any nonlinear stiffness function whose linearization results in a positive definite stiffness matrix K q. The same is true for the damping terms. 3.2 Compliant Constraint The equations of a flexible joint manipulator in contact with a compliant constraint are given by (28), (29), (30). Substitution of basic hybrid control (31) into these equations gives:

6 D l (x)ẍ + v l (x, x) + J(x) T B q [J(x) 1 x J(x m ) 1 x m ] +J(x) T K q [X 1 (x) X 1 (x m )] + E T 1 B f x 1 +E T 2 [B e x 2 + K e x 2 ] = 0 (60) M(x m )ẍ m + v m (x m, x m ) J(x m ) T B q [J(x) 1 x 1 J(x m ) 1 x m ] J(x m ) T K q [X 1 (x) X 1 (x m )] +E T 1 [K v x m1 + K p (x m1 x d )] E T 2 F d = 0 (61) F x2 = B e x 2 + K e x 2 (62) (x, x m, x, x m, x d, F d ) Θ x Θ x R n R n R n m R m If x d is fixed in Θ x1, where now Θ x1 {x 1 x Θ x, x 2 = 0}, and F d = 0, then the system (60), (61), (62) has an equilibrium point at: x = E T 1 x d (63) x m = E T 1 x d (64) x 1 = 0 (65) x m = 0 (66) We translate this equilibrium point to the origin by (re)defining: w 1 = x E T 1 x d (67) w 2 = x m E1 T x d (68) z 1 = w 1 (69) w 2 Θ z1 = {z 1 R 2n z 1 < c} (70) where c > 0 is chosen small enough that: x d Θ x1, z 1 Θ z1 = ET 1 x d + z 1 Θ x Θ x (71) E1 T x d Substitution of (67) and (68) into (60), (61), and (62) gives: D l (E T 1 x d + w 1 )ẅ 1 + v l (E T 1 x d + w 1, w 1 ) +J(E T 1 x d +w 1 ) T B q [J(E T 1 x d +w 1 ) 1 x J(E T 1 x d +w 2 ) 1 w 2 ] +J(E T 1 x d +w 1 ) T K q [X 1 (E T 1 x d +w 1 ) X 1 (E T 1 x d +w 2 )] +E T 1 B f x 1 + E T 2 [B e E 2 w 1 + K e E 2 w 1 ] = 0 (72) M(E T 1 x d + w 2 )ẅ 2 + v m (E T 1 x d + w 2, w 2 ) J(E T 1 x d + w 2 ) T B q [J(E T 1 x d + w 1 ) 1 w 1 J(E T 1 x d + w 2 ) 1 w 2 ] J(E T 1 x d +w 2 ) T K q [X 1 (E T 1 x d +w 1 ) X 1 (E T 1 x d +w 2 )] +E T 1 (K v E 1 w 2 + K p E 1 w 2 ) = 0 (73) F x2 = B e E 2 w 1 + K e E 2 w 1 (74) (z 1, z 1, x d ) Θ z1 R 2n Θ x1 151 Let z 2 = z 1 and z = z 1. Then for each x d in Θ x1, z 2 z = 0 is an equilibrium point of (44) and (45). Theorem 2: Consider the system (72), (73), (74). For every fixed x d Θ x1, the equilibrium point z = 0 is asymptotically stable. Proof: Linearization of (72)and (73) about z = 0 gives: D l (E T 1 x d )ẅ 1 + B( w 1 w 2 ) + K(w 1 w 2 ) +E T 1 B f E 1 w 1 + E T 2 [B e E 2 w 1 + K e E 2 w 1 ] = 0 (75) M(E T 1 x d )ẅ 2 B( w 1 w 2 ) K(w 1 w 2 ) +E T 1 K v E 1 w 2 + E T 1 K p E 1 w 2 = 0 (76) where B and K are defined by (49) and (50). We write (75) and (76) as: H 2 z 1 + H 1 z 1 + H 0 z 1 = 0 (77) where: H 2 = D l(e1 T x d ) 0 (78) 0 M(E1 T x d ) H 1 = B + ET 1 B f E 1 + E2 T B e E 2 B (79) B B + E1 T K v E 1 H 0 = K + ET 2 K e E 2 K (80) K K + E1 T K p E 1 Since D l (E1 T x d ) and M(E1 T x d ) are positive definite, so is H 2. It remains to show that H 1 and H 0 are positive definite. Let: I n T = ET 1 E 1 (81) E2 T E 2 I n = whose inverse is given by: Now (81) and (79) give: ET 1 E 1 E T 1 E T 2 E T 2 E 2 E T 1 E T 2 I n I n T 1 = 0 E 1 E 2 0 (82) (83) B 0 T T H 1 T = 0 K v 0 + ET 1 B f E 1 E1 T B f E 1 E1 T B f E 1 E1 T B f E 1 0 B e (84)

7 which is positive definite since B, K v, and B e are positive definite and the second matrix in the sum is positive semidefinite. Similarly: K 0 T T H 0 T = 0 K p 0 0 K e (85) which is positive definite. Thus, by Lemma 1, H 1 and H 0 are positive definite as well. Therefore, all eigenvalues of (77) lie strictly in the left half plane, and the nonlinear system (72), (73), (74) is thus asymptotically stable by Lyapunov s Linearization Theorem. QED Remark 1: Remarks 1-4 for the case of a rigid constraint similarly apply to this case of a compliant constraint. Moreover, the stability of the system is independent of the constraint stiffness and damping. Remark 2: Remark 5 of Theorem 1 (which assumes a rigid constraint) explained that the assumption of linear joint stiffness and damping is unnecessary. A similar argument can be applied to the constraint stiffness and damping. In particular, it follows from the above proof that stability is guaranteed for any nonlinear stiffness and damping terms whose linearization results in positive definite stiffness and damping matrices K e and B e in the linearized system (75), (76). For example, this is true when the constraint force along each constrained task coordinate x 2i is given by the nonlinear function µ i x 3 2i + k eix 2i + ν i sgn( x 2i ) x 2 2i + b ei x 2i. Remark 3: With constraint friction B f set to zero in (84), T T H 1 T remains positive definite and hence the system remains asymptotically stable. Thus, constraint friction is not required to stabilize the system as long as velocity feedback K v x m1 is applied in unconstrained directions. By a similar argument, the constraint damping B e x 2 is not required as long as a velocity feedback K vc x m2 is applied in constrained directions. 4. Tracking In the previous section, it was shown that basic hybrid control is robustly stable about a constant set point x d = x d0, F d = 0. In this section, we consider the inputs to be arbitrary trajectories x d (t) and F d (t), and examine the problem of limiting the position error x 1 x d and the force error F x2 F d. It was shown in [30] that a system which is asymptotically stable about every set-point in a region (such as in Theorems 1 and 2) will track (with arbitrarily small error) trajectories that vary sufficiently slowly in the region. However, this result does not provide a quantitative relationship between the tracking error and the trajectory speed, dynamic and kinematic parameters, and so on. Although input-output stability theorems can provide quantitative bounds on the tracking error, for such bounds to be meaningful, the input trajectory must be eliminated in favour of its derivative (since the tracking error really depends on the speed of the trajectory, not its size [30]). Finding such a transformation for our constrained flexible system is not trivial, if one even exists. 152 In light of the above discussion, we propose a useful yet mathematically tractable design goal: design a control that simultaneously achieves asymptotic tracking when applied to a known rigid system and asymptotic stability when applied to an unknown flexible system. Here, asymptotic tracking means that the position error x 1 x d and the force error F x2 F d approach zero asymptotically for all smooth time-varying reference inputs x d and F d. We will say that a control achieves nominal tracking if it achieves asymptotic tracking when applied to a perfectly rigid robot/constraint with precisely known dynamics. Hence, our design goal is to simultaneously achieve nominal tracking and robust stability. Although Theorems 1 and 2 guarantee the robust stability of basic hybrid control (31), this control does not achieve nominal tracking. Controls designed for nominal tracking are called model-based controls, because they contain an internal model of the system dynamics. In Section 4.1 we will show that the standard model-based approach does not achieve stability robustness. Then in Section 4.2, we propose a modification to basic hybrid control that achieves both nominal tracking and stability robustness. 4.1 Resolved Acceleration Control It was shown in [6] that nominal tracking can be achieved by a hybrid control proposed in that paper. In [3], that control was shown to be identical to the resolved acceleration control proposed earlier in [5]. It was further shown in [12] that the operational space approach proposed in [7] is also identical to resolved acceleration control. We shall therefore refer to the controls of [5, 6, 29] as resolved acceleration control. A generalization of this control to redundant manipulators was presented in [8]. Resolved acceleration control may be written as: τ = D q (x m )J(x m ) 1 E T 1 [ẍ d + K p (x d x m1 ) + K v ( x d x m1 )] +J(x m ) T v(x m, x m ) + J(x m ) T E T 2 F d (86) where D q (x m ) = D lq (x m ) + M q is the manipulator inertia matrix. Note that in (86), position measurements x m are taken from the motor side of the joints. In the special case of rigid joints, we have x m = x. We wish to examine the stability of the control (86) applied to a manipulator with flexible joints in contact with a rigid constraint or a compliant constraint. For comparison, we shall also consider the case of a rigidjoint manipulator. Our analysis is based on a planar manipulator in point contact with a flat wall, shown in Fig. 2. We vary the set point x d and determine the stability at a number of manipulator positions, with F d set to zero. For each value of x d, we linearize the closed-loop system about the associated equilibrium point and plot the system poles. The symbols used to plot the poles indicate the approximate arm position in accordance with Fig. 2. For example, poles corresponding to xd between -0.5 m and -0.3 m are denoted by the + symbol. Increments in xd of 0.05 m are used from -0.5 m to 0.5 m while increments

8 of 0.02 m are used from 0.5 m to 0.51 m, where the o symbol is used to plot poles. According to Lyapunov s Linearization Theorem, poles appearing in the open right half plane indicate that the original nonlinear system is unstable at the corresponding configuration Nm/rad and 20 Nms/rad, respectively. It can be seen that the system becomes unstable at certain manipulator positions. Similarly, Fig. 3d shows instability for the case of a flexible-joint manipulator in contact with a compliant constraint. Thus, resolved acceleration control can be destabilized by joint compliance. Note from the plot symbols that this instability is not confined to configurations that are near signularities. 4.2 Dynamic Hybrid Control In this section, we investigate the stability of a model-based version of basic hybrid control applied to a flexible-joint manipulator. This control, referred to as dynamic hybrid control, was proposed in [3] as: Figure 2. Planar manipulator used for root loci. The dynamic parameters assumed for the stability analyses are typical of a commercial robot The link lengths are l 1 = l 2 = 0.3 m, and the motor and link inertia matrices are respectively: M = (kg m 2 ) (87) and: D l (q) = + cos q 1 (kg m 2 ) (88) The gains for resolved acceleration control (86) are assumed as K p = 1000 N/m and K v = 44 Ns/m. Fig. 3a shows root loci for the nominal case of a rigidjoint manipulator in contact with a rigid constaint. The plot indicates that the (second-order) system is stable and the closed-loop poles are the same for all configurations considered. It was shown in [3, 6] that this is the case for all rigid manipulators and rigid constraints, and that the general closed-loop system has the form: ë + K v e + K p e = 0 (89) F x2 = F d (90) where e = x 1 x d. Thus, the control (86) is a feedback linearizing control for the rigid-joint, rigid-constraint system. Fig. 3b shows root loci for the case of a compliant constraint. The stiffness, damping, and friction of the wall are assumed to be 10 4 N/m, 20 Ns/m, and 0 Ns/m, respectively. It can be seen that two of the four poles vary with manipulator configuration, but the system is always stable. It was proven in [3] that this control is in fact stable for any rigid-joint manipulator in contact with any compliant constraint. Fig. 3c shows root loci for the case of a flexible-joint manipulator in contact with a rigid constraint. The joint stiffness and damping at each joint are assumed to be 153 τ = J(x m ) T E T 1 [K p (x m )(x d x m1 ) K v (x m ) x m1 ] +J(x m ) T E T 2 F d +D q (x m )J(x m ) 1 E T 1 ẍ d + J(x m ) T v(x m, x m ) (91) where K p (x m ) = k p E 1 D(x m )E T 1 and K v (x m ) = k v E 1 D(x m ) E T 1, for positive scalars k p and k v. It was shown in [3] that (91) linearizes the equations of a rigid-joint manipulator in contact with a rigid constraint to give the closed-loop system: ë + k v e + k p e = 0 (92) F x2 = F d E 2 D(x 1 )E T 1 (k v e + k p e) (93) Note that in (92), the position error e = x 1 x d converges to zero asymptotically since k v and k p are positive. Similarly, as e approaches zero, F x2 approaches F d by (93). Thus, dynamic hybrid control achieves nominal tracking. Note that this control results in a force error asymptotically approaching zero, whereas resolved acceleration control yields zero force error at all times. We will now examine the stability of dynamic hybrid control applied to a manipulator with flexible joints. If x d is constant, the control (91) simplifies to: τ = J(x m ) T E T 1 [K p (x m )(x d x m1 ) K v (x m ) x m1 ] +J(x m ) T E T 2 F d + J(x m ) T v(x m, x m ) (94) Note that (94) is identical to (91), except that now K p and K v are functions of x m, and the term J(x m ) T v(x m, x m ) has been added. However, these changes do not affect the stability proofs of Theorems 1 and 2. When (94) is substituted into the constrained robot dynamics and the resulting closed-loop system is linearized about the equilibrium point corresponding to x d, the functions K p (x m ) and K v (x m ) become constant positive definite matrices K p (x d ) and K v (x d ), respectively. Also, since the term J(x m ) T v(x m, x m ) is quadratic in x m, it vanishes upon linearization. It is also evident that (94) achieves asymptotic stability even when the estimate of the manipulator inertia D(x) is inaccurate. The only restriction on the estimate of D(x) is that it be positive definite so that K p (x m ) and K v (x m ) are positive definite.

9 Figure 3. Root loci for resolved acceleration control varying configuration. (a) Rigid contact and rigid joints. (b) Compliant contact and rigid joints. (c) Rigid contact and compliant joints. (d) Compliant contact and compliant joints. 5. Experiments Stability and tracking experiments were performed on the Robotics and Automation Laboratory s Robotwin arm, which has highly flexible joints. The first set of experiments test the validity of Theorems 1 and 2 (or more precisely, the validity of their assumptions, since the theorems themselves have been proven). A second set of experiments test the tracking capabilities of basic hybrid control during a simple hybrid task, namely drawing a circle on a rigid surface. The Robotwin arm consists of modular links and joints, which may be assembled in numerous configurations. For our experiments, the robot is configured as a three-axis SCARA robot, whose joint and link parameters are identified in Fig. 4. This robot s joint flexibility results from its harmonic drives and from the light construction of connecting pins in the joints. Two different end effectors are employed in the experiments: a sphere and a pencil. The end effector is constrained by a steel wall whose position is 154 Figure 4. Three-DOF manipulator constrained by a wall. Each joint of the arm is actuated by a DC motor equipped with an optical encoder to sense the motor angle. The torque applied at each joint is limited by software to ±20 Nm. Feedback control is performed by a personal computer at 1000 Hz. The length of link one (l 1 in Fig. 4) is m and the length of link two depends on which of the two end effectors is used. The inertial parameters of the arm are

10 Figure 5. Initial condition response of basic hybrid control rigid constraint. (a) Initial condition response: p1(0) = -18 cm. (b) Initial condition response: p1(0) = 9 cm. unknown. The motor transmissions are harmonic drives with gear ratios of 100 at joint 1 and 50 at joints 2 and 3. According to the manufacturer s data, the nominal stiffness of the harmonic drives is 10 4 Nm/rad. However, the joint stiffness was observed to be much lower for small deflections. In particular, negligible torque was required to manually deflect joint 1 about one degree, while the other two joints exhibited smaller but detectible regions of nearzero stiffness. These regions were not dead zones, as the output shafts would return to their original positions when unloaded. The joints also exhibited static and dynamic friction moments of 2 to 5 Nm. 5.1 Stability In these tests, the desired position and force are constant, and basic hybrid control is required to respond to large initial errors. The set-up is as shown in Fig. 4, where the end effector is a m diameter smooth aluminum sphere. The sphere makes sliding point contact with 155 the wall, which has been covered with a greased wooden surface to minimize contact friction. This set-up represents the case of a flexible joint robot in contact with a rigid constraint. Joint 2 is fixed at q 2 = 0 so that arm motion is restricted to the p 1 p 2 plane. The length l 2 of the second link is m to the center of the sphere. The end effector is considered to be the center of the sphere, with task coordinates x = p 1. The wall constrains the end p 2 effector to the line p 2 = 42 cm. The desired task vector is x d = p 1d = 0, F d = 5 N, and the control gains are K p = 500 N/m and K v = 0. Velocity feedback is not used because the motors provide dynamic friction, which plays the same role in the closedloop equations. For each test, the sphere starts in contact with the wall at some distance from x d, and then the controller is activated. The initial condition response of the system is shown in Figs. 5a and 5b for two different values of initial position.

11 For each test, the first two plots give the measured values of the end effector coordinates p 1 and p 2, denoted by p m1 and p m2, respectively. The primary source of error in these measurements is the joint flexibility, as the encoders are mounted on the motor side of the harmonic drives. The second two plots give the torque produced by the two motors used in this experiment. In Fig. 5a, the sphere is initially set to the left of the target at p 1 (0) = 18 cm. After the controller is activated at 0.03 seconds, the arm moves to the desired equilibrium position while maintaining contact with the wall. The apparent wall deflection indicated by p m2 is actually a result of joint flexibility and is 0.3 cm in steady state due to the 5 N contact force. The 2 cm increase in p m2 at the start of the test represents a large transient contact force caused by the saturation of the joint 1 torque at - 20 Nm; the initial torque at joint 1 calculated by the control algorithm was actually -40 Nm. In Fig. 5b, the initial position error is halved to p 10 = 9 cm so that the joint torques do not saturate, and the sphere now starts on the right hand side of the target. The plot of p m2 indicates that the sphere loses contact with the wall before reaching its desired equilibrium position. (The sharp reversal of p m2 at seconds corresponds to the second joint striking its hard limit.) The sphere loses contact with the wall because the inertial interaction force is larger than the 5 N bias force due to F d. Figure 6. Effect of position gain on tracking performance. 156

12 These tests verify that basic hybrid control is not destabilized by joint compliance, as predicted by Theorem 1, Section 3.1b. Note that the system achieved its desired equilibrium in spite of very large initial position errors. Much smaller initial errors would be expected in practice. 5.2 Tracking Theorems 1 and 2 pertain to the stability of basic hybrid control with fixed position and force inputs applied. A system which is stable in this sense will not necessarily follow a time-varying trajectory, unless the trajectory varies sufficiently slowly [30]. In this section, we investigate experimentally the performance of the system when executing a 3-DOF hybrid tracking task. Specifically, basic hybrid control is applied to draw a circle on a sheet of paper. The experimental set-up is again as shown in Fig. 4, and a sheet of paper has been affixed to the wall, which is now steel. The end effector is an HB pencil mounted on the second link using a special fixture. With the pencil in place, the length of the second link is m. We idealize the end effector as a single point at the tip of the pencil having task coordinates x = (p 1 p 2 p 3 ) T. The task consists of four phases. The pencil tip starts at an arbitrary position away from the wall. During the first four seconds, the tip moves along a straight line to a specified position near the surface of the wall. Basic hybrid control (31) is used in this phase, but with position feedback applied to all three degrees of freedom (m = 0). Then control of p 2 is switched from position to force control with F d = 7 N for 0.2 seconds. This is to overcome joint stiction and drive the pencil tip toward the paper. The robot then pauses at this point for 1.8 seconds with F d reduced to 0.4 N. Finally, with F d = 0.4 N, the robot draws a 10 cm diameter circle on the paper. Fig. 6 shows the effect of position gain on tracking performance; velocity feedback was not used. Here, I 2 is the 2x2 identity matrix. The circles that appear in Fig. 6 are scaled down reproductions of the actual 10 cm diameter circles drawn by the robot. The wall is fixed at p 2 = 50 cm and the center of each circle is at p 1 = 29 cm, p 3 = 0 cm. Each circle is drawn clockwise at an angular velocity of 2 rad/s. It can be seen that the circle quality improves with gain. When the position gains were increased to 5000 N/m, control chattering occurred at the transition from pure position control to hybrid control. This chattering was also observed when the constraint was removed and the desired force was set to zero. The tracking accuracy in Fig. 6 is fundamentally limited by the joint flexibility. Even if the controller achieves perfect tracking based on motor encoder readings, there are errors in the actual trajectory due to the joint deflections. In particular, the one degree of play (near-zero stiffness) in joint one corresponds to about a five percent radial error in the circle (based on our wall position of p 2 = 50 cm, which corresponds to a q 1 of about 45 degrees). This accuracy limit is approached in Fig. 6d for a control stiffness of 4000 N/m. Thus, there would be little benefit in stabilizing the system for higher gains. Fig. 7 shows the effect of varying the velocity of the reference trajectory. The center of the circle is at p 1 = 29 cm, p 3 = 0 cm. Also, K p = 3000I 2 (N/m), F d = 0.2 N. Fig. 7a shows that when the angular velocity of the circle is doubled to 4 rad/s, the resulting circle does not change much from that of Fig. 6c. However, when the angular velocity is increased to 6 rad/s, Fig. 7b indicates that the pencil loses contact with the paper and bounces as it moves around the circle. Figure 7. High-speed tracking performance. 157

13 Figure 8. Robustness to uncertainty in wall position. Finally, Fig. 8 demonstrates the fundamental advantage of hybrid control over impedance control (which specifies position along constrained directions). In impedance control, a significant change or uncertainty in wall position will cause the pencil to fall short of the wall or break. In contrast, hybrid control is robust to uncertainty in wall position because only force (and not position) is controlled in the direction normal to the constraint. In Fig. 8a, the wall is fixed at p 2 = 52 cm and the center of the circle is at p 1 = 28 cm, p 3 = 0 cm, and K p = 3000I 2 (N/m), F d = 0.5 N. The wall is then moved to p 2 = 50 cm and the same program (with no modifications) is executed again. Fig. 8b shows that the circle is drawn again in spite of the 2 cm change in the wall position. 6. Summary and Conclusion Basic hybrid control was proved to be asymptotically stable when applied to a flexible-joint robot in contact with a rigid or compliant constraint. This result holds for all values of dynamic parameters in the system model and requires no knowledge about the stiffness and damping of the robot joints or constraint, the motor and link inertias, and so on. It applies to all nonredundant manipulator geometries and all holonomic constraints. A modified version of this control achieves nominal tracking without sacrificing stability robustness, whereas an established hybrid control that achieves nominal tracking can be destabilized by joint flexibility. Experiments on a 3-DOF flexible-joint robot constrained by a wall verified the stability of basic hybrid control in the presence of significant joint compliance. Set point regulation was achieved in spite of large initial position errors. The robot was also able to draw circles at 158 speeds of up to 20 cm/s. Instability was observed for sufficiently high position gains. By Theorem 1, this instability cannot be caused by joint flexibility alone and is thus due to unmodelled effects (such as joint stiction and motor dynamics). The results presented here support the use of basic hybrid control for controlling hybrid tasks. In addition to being robust, it is easy to implement since it does not require a dynamic model of the robot. If such a model is available, the dynamic hybrid control proposed in this paper is preferable to resolved acceleration control, as it is robust with respect to joint flexibility. References [1] M.H. Raibert & J.J. Craig, Hybrid position/force control of manipulators, ASME J. Dynamic Meas. Control, 102, 1981, [2] N. Hogan, Impedance control: An approach to manipulation, part I theory, part II implementation, part III applications, ASME J. Dynamic Systems Measurement Control 7(1), 1985, [3] P.B. Goldsmith, Stability of hybrid position/force control applied to robots with flexible joints, Ph.D. thesis, University of Toronto, [4] C.H. An & J.M. Hollerbach, The role of dynamic models in Cartesian force control of manipulators, Int. J. Robotics Research, 8(4), 1989, [5] K.G. Shin & C.P. Lee, Compliant control of robotic manipulators with resolved acceleration, Proc. 24th IEEE Conf. Decision and Control, 1985, Ft. Lauderdale, [6] N.H. McClamroch & D. Wang, Feedback stabilization and tracking of constrained robots, IEEE Trans. Automatic Control, 33(5), , [7] O. Khatib, A unified approach for motion and force control of robot manipulators: The operational space formulation, IEEE J. Robotics Automation, 3(1), 1987, [8] T. Yoshikawa, Dynamic hybrid position/force control of robot manipulators-description of hand constraints and calculation

14 of joint driving force, IEEE J. Robotics Automation, 3(5), 1987, [9] T. Yabuta, A.J. Chona, & G. Beni, On the asymptotic stability of the hybrid position/force control scheme for robot manipulators, Proc. IEEE Int. Conf. Robotics Automation, Philadelphia, PA, 1988, [10] C.H. An, C.G. Atkeson, & J.M. Hollerbach, Dynamic stability issues in force control of manipulators, Proc. IEEE Int. Conf. Robotics Automation, Raleigh, NC, 1987, [11] D. Eppinger, & W. Seering, Three dynamic problems in robot force control, Proc. IEEE Int. Conf. Robotics Automation, Scottsdale, AZ, 1989, [12] C.H. An & J.M. Hollerbach, Model-based control of a robot manipulator (Cambridge, MA: MIT Press, 1988). [13] S. Ahmad, Constrained motion (force/position) control of flexible joint robots, IEEE Trans. System Man Cybernetics, 23(2), 1983, [14] M.W. Spong, Control of flexible joint robots: A survey, Technical Report (University of Illinois at Urbana-Champaign, 1990). [15] M. 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Zhang, Kinematic stability of robot manipulators under force control, Proc. IEEE Int. Conf. Robotics Automation, 1989, Scottsdale, Arizona, [29] H.K. Khalil, Nonlinear systems (New York: Macmillan, 1992). [30] M.A. Kelemen, Stability property, IEEE Trans. Automatic Control, 31(8), 1986, Biographies Peter Goldsmith received his B.Sc. and M.Eng. Degrees from the University of Calgary in 1987 and 1989, respectively, and his Ph.D. from the University of Toronto in 1995, all in mechanical engineering. He has held research positions with Nova, DRES, DREV, and DCIEM. Since 1995, he has been an Assistant Professor of Mechanical and Manufacturing Engineering at the University of Calgary. His current interests are in kinematics, dynamics, and control of robots, biomedical robots, robots for sports shoe testing, and learning control theory. Bruce A. Francis received his B.A.Sc. and M.Eng. degrees in mechanical engineering and his Ph.D. degree in electrical engineering from the University of Toronto in 1969, 1970, and 1975, respectively. He has held research and teaching positions at the University of California at Berkeley, the University of Cambridge, Concordia University, McGill University, Yale University, the University of Waterloo, Caltech, and the University of Minnesota. He is presently a Professor of Electrical and Computer Engineering at the University of Toronto. Professor Francis is a Fellow of the IEEE. He has co-authored three books, including Optimal sampled-data control systems with T. Chen, published in His current research is in digital control and signal processing. A. Goldenberg received his B.A.Sc. and M.A.Sc. degrees from Technion-Israel Institute of Technology, Haifa, Israel, in 1969 and 1972, respectively, and his Ph.D. degree from the University of Toronto, Toronto, Ontario in 1976, all in electrical engineering. From 1975 to 1981, he was employed by Spar Aerospace Ltd., Toronto, where he worked mainly on control, analysis, and design of the space shuttle remote manipulator system, and satellite controls. During he was an Assistant Professor of Electrical Engineering and from 1982 to 1987 he was an Associate Professor of Mechanical Engineering at the University of Toronto. Since 1987 he has been a Professor of Mechanical Engineering at the University of Toronto. He holds cross appointments in the Department of Electrical Engineering and the Institute of Biomedical Engineering. He founded the Robotics and Automation Laboratory in