New Tuning Conditions for a Class of Nonlinear PID Global Regulators of Robot Manipulators.

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1 Memorias del Congreso Nacional de Control Automático Cd. del Carmen, Campeche, México, al 9 de Octubre de New Tuning Conditions for a Class of Nonlinear PID Global Regulators of Robot Manipulators. J. Orrante-Saanassi a, V. Santibañez a and V. Hernandez-Guzman b a Instituto Tecnologico de La Laguna, b Facultad de Ingenieria, Universidad Autonoma de Queretaro. a AP 9-, Torreon, Coah., CP, Mx. b AP -, Queretaro, Qro., CP 65, Mx. aos@ieee.org, santibanez@ieee.org, vmhg@uaq.mx Abstract The motivation of this wor relies on the fact that, until now, the tuning conditions presented in the literature for assuring global asymptotic stability are conservative in the sense that controller gains for oints with a little or nothing of gravitational torques are tuned in the same form that the oints which support large gravitational torques. In this paper we extend the more recent tuning conditions presented in the literature for a class of nonlinear PID global regulators of robot manipulators. Such an extension has permitted to carry out by the first time experimental essays for a class of nonlinear PID global regulators, which are presented in this paper using a two degrees of freedom robot manipulator. Keywords: robot manipulators, nonlinear PID, global stability, tuning conditions. I. INTRODUCTION The study of PID controllers in robot manipulators has been subect of extensive researches in many years over the past. Several wors have been presented proving that PID controllers guarantee semiglobal asymptotically stability of the closed-loop equilibrium point in the case of setpoint control (Arimoto, 98), (Arimoto and Suzui, 99), (Kelly, 995), (Ortega et al., 995), (Alvarez-Ramirez et al., ), (Meza et al., ), (Hernandez et al., 8). Due to the classical linear PID controllers has only been proved to be semiglobally asymptotically stable, several nonlinear PID global controllers have been proposed in some wors (Arimoto, 995), (Kelly, 998), (Santibañez and Kelly, 998). These nonlinear PID controllers, unlie the linear PID controller, yield global asymptotical stability of the closed-loop equilibrium point. In (Santibañez and Kelly, 998) it was proposed a class of nonlinear PID global regulators for robot manipulators, which encompasses the particular cases of (Arimoto, 995) and (Kelly, 998). However, an important drawbac exists: the tuning conditions for assuring global asymptotic stability for the aforementioned nonlinear PID controllers are far from being acceptable in practical applications. The latter in the sense that the tuning conditions previously reported are conservative and overestimated because such conditions are established as expressions of norms of gain parameter vectors and matrices. This mae that all the oint gains be restricted by the same rule, in such a way that the conditions for larger and smaller oints is the same. This paper, inspired in (Hernandez-Guzman et al., 8) and (Hernandez-Guzman and Silva-Ortigoza, ), addresses a tuning procedure which allows to obtain conditions less restrictive for oints in order to get better performances. Such tuning procedure has permitted to carry out experimental essays which are reported in this wor using a two degrees of freedom robot manipulator. Finally, some remars on notation. Given some vector x R n and some matrix A(x) R n n, the Euclidean norm of x is defined as x = x and the spectral norm of A(x) is defined as A(x) = λ max (A T A) where λ max (A T A) is the largest eigenvalue of the symmetric matrix A T A. In the case where A(x) R n n is a symmetric matrix, then A = max i λ i (A), where λ i (A) and are eigenvalues of A(x) and the absolute value function, respectively. λ min stands for the smallest eigenvalue of A for all x R n. We use symbol y i to represent the i th component if y is a vector or the i th diagonal entry if Y is a diagonal matrix. A. Robot Dynamics II. PRELIMINARIES Consider the dynamic model for a serial n lin rigid robot, given by M(q) q + C(q, q) q + g(q) + F v q = τ () where q, q, q R n are the vectors of oint accelerations, oint velocities and oint positions, respectively, τ R n is the vector of applied torques, M(q) R n n is the symmetric positive definite manipulator inertia matrix, C(q, q) R n n is the matrix of centripetal and Coriolis torques, g(q) is the vector of gravitational torques obtained as the gradient of the robot potential energy, i. e. g(q) = U(q) q, and F v R n n is a constant diagonal definite positive matrix which represent the viscous friction coefficient at each oint. We assume that all oints of the robot are of the revolute type. B. Properties of the robot dynamics In the following two properties of the dynamics () are presented. D.R. AMCA Octubre de

2 Property. Matrices M(q) and C(q, q) satisfy: «q T M(q) C(q, q) q = () M(q) = C(q, q) + C T (q, q). () Furthermore, there exists a positive constant c such that, for all x, y, z R n : C(x, y)z c y z. () Property. The gravitational torque vector g(q) is bounded for all q R n. This means that there exist finite constants i such that: sup g i (q) q R n i i =,, n, (5) where g i (q) stands for the elements of g(q). Equivalently, there exists a constant such that g(q) for all q R n. Furthermore there exists a positive constant gi such that = max q g i (q) q g, i for all q R n and i =,,..., n. Finally, we present some useful results. Theorem : (Kelly et al., 5) Let f : R n R be a continuously differentiable function with continuous partial derivatives up to at least second order. Assume that: f() = R (6) f x () = Rn () If the Hessian matrix / x[ f(x)/ x] is positive definite for all x R n, then f(x) is a globally positive definite and radially unbounded function. Theorem : (Kelly et al., 5) (Mean value theorem for vectorial functions). Consider the continuous vectorial function f : R n R m. If f i (z, z,..., z n ) has continuous partial derivatives for i =,..., m, then for each pair of vectors x, y R n and each w R m there exists ξ R n such that: [f(x) f(y)] T w = w T f(z) (z) (x y) (8) z=ξ where ξ is a vector on the line segment that oins the vector x and y. Definition : Let A be a n n matrix with a i representing its element at row i and column. The matrix A is said to be strictly diagonally dominant if: a ii > =, i a i, i =,..., n. (9) Definition : If A is a n n symetric and strictly diagonally dominant matrix and if a ii > for all i =,..., n, then A is positive definite. Definition : F( a, ε, x) with a >, ε >, and x R n denotes the set of all continuous differentiable increasing functions sat(x) = [sat(x ) sat(x )... sat(x n )] T such that sat( x) = sat(x) sgn(x) = sgn(sat(x)) x sat(x) a x, x R : x < ε ε sat(x) a ε, x R : x ε (d/dx)sat(x) { a x, if x < ε sat(x) a ε, if x ε Definition : Given positive constants l and m, with l < m, a function Sat(x; l, m) : R R : x Sat(x; l, m) is said to be a strictly increasing linear saturation function for (l, m) if it is locally Lipschitz, strictly increasing, C differentiable and satisfies: ) Sat(x; l, m) = x when x l ) Sat(x; l, m) < m for all x R. Lemma : Let A R n be a matrix with a i representing the element in row i and column, y x, sat(x) two vectors such that: x = 6 x x. x n 5, sat(x) = 6 sat(x ) sat(x ). sat(x n) 5 () where sat( ) is a function lie that described in Definition. If A is symmetric and strictly diagonally dominant, namely n a ii > a i, i =,,..., n, () =, i and a ii > for all i =,,..., n, then sat(x) T Ax >, x R n, with x R n ; i.e., sat(x) T Ax is a positive definite continuous function. Outline of the proof: The proof is based in the Sylvester Theorem proof by using the following nice property of the saturation function sat(x) F( a, ε, x): (x i + x )(sat(x i ) + sat(x )), (x i x )(sat(x i ) sat(x )), x i, x R III. MAIN RESULT A. A class of nonlinear PID controllers Consider a class of nonlinear PID controllers lie that presented in (Santibañez and Kelly, 998), which can be written as: τ = q U a( q) K v q + K i w w(t) = [αsat( q(σ)) q(σ)]dσ + w() () where K v and K i are diagonal positive definite n n matrices, q = q d q denotes the position error vector, α > is a constant scalar, U a ( q) is a ind of C artificial potential energy induced by a part of the proportional term of the controller whose properties were established in (Santibañez and Kelly, 998). Two examples of this ind of nonlinear PID regulators are: Controller presented in (Kelly, 998) τ = K p q K v q + K i w () w(t) = [αsat( q(σ)) q(σ)]dσ + w() D.R. AMCA Octubre de

3 This controller has associated an artificial potential energy U a ( q) given by U a( q) = qt K p q () Controller presented in (Santibañez and Kelly, 998) τ = K psat( q) K v q + K i Z w t (5) w(t) = [αsat( q(σ)) q(σ)]dσ + w() This controller has associated an artificial potential energy U a ( q) given by U a( q) = K psat(r)dr := pi sat(r i )dr i (6) A particular case of this control law was presented in (Arimoto, 995), with 8 < sin(x) for x π sat(x) = Sin(x) = for x > π : for x < π. The stability conditions for the latter nonlinear PID controllers presented in (Santibañez and Kelly, 998), as well as in (Kelly, 998) for the first nonlinear PID controller and (Arimoto, 995) for the second one, are very conservative in the sense that are far from being acceptable in practical applications, specially when they are used in robot manipulators with many degrees of freedom. In the following a new tuning procedure which allows to obtain stability conditions less restrictive to assure global asymptotic stability is proposed. B. Stability analysis (New tuning conditions) Inspired in (Hernandez-Guzman et al., 8) and (Hernandez-Guzman and Silva-Ortigoza, ), in this section we present new tuning conditions for the stability analysis of the class of nonlinear PID controllers introduced in (Santibañez and Kelly, 998). B.. Consider the control law (): τ = K p q K v q + K i w w = [αsat( q(σ)) q(σ)]dσ + w(), which was proposed in (Kelly, 998) and reanalyzed in (Hernandez-Guzman and Silva-Ortigoza, ). This control law has been rewritten to express it lie the equation (). K p, K v and K i are n n diagonal positive definite matrices, q = q d q denotes the position error vector and α > is a small constant suitably selected. By using the control law () into the dynamics (), we obtain the following closed-loop system q q d dt 6 q z = 6 5 M(q) [K p q K v q + K i z C(q, q) q F v q g(q) + g(q d )] αsat( q) q where z is defined as z = [αsat( q(σ)) q(σ)]dσ + w() K i g(q d ) {z } z() 5 () (8) so that () is an autonomous differential equation whose unique equilibrium is: [ q T q T z T ] T = R n. (9) Proposition. There always exist a positive scalar constant α and positive definite matrices K p, K v, K i R n n such that the equilibrium point (9) from () is globally asymptotically stable. Proof. Based on the Lyapunov theory, we propose the following Lyapunov function candidate where V ( q, q, z) = V ( q, q,z) + V ( q) + V ( q) () V = [ q αsat( q)]t M(q)[ q αsat( q)] + zt K i z + αf vi sat(r i )dr i + α vi sat(r i )dr i V = qt K p q α sat( q)t M(q)sat( q) V = U(q d q) U(q d ) + qt K p q + g(q d ) T q, where K p = K p + Kp. Thus, () will be a radially unbounded positive definite function provided that V and V be also a radially unbounded positive definite function. To this end, we provide lower bounds on the following terms: qt K p q = α satt ( q)m(q)sat( q) H( q i ) = pi q i () α λmax{m(q)}sat ( q i ) α λmax{m(q)}h( q i) () 8 < q i, q i ε : ε, q i > ε From () and () we can conclude that there always exist large enough positive constants pi, i =,..., n, such that: and hence p i q i > α λmax{m(q)}h( q i) q R n () V = qt K p q α sat( q)t M(q)sat( q) > q R n On the other hand, according to Theorem, and Definition and, V is definite positive if (Hernandez-Guzman et al., 8) p i > gi () Finally, we can conclude that () is definite positive and radially unbounded if () and () are satisfied. It is possible to verify that, using Property, the time derivative of () along the traectories of the closed-loop system () is given by: V = q T F v q α sat( q)m(q) q αsat( q) T C(q, q) T q αsat( q) T [K p q + g(q d ) g(q)] q T K v q. (5) D.R. AMCA Octubre de

4 Negative semidefiniteness may be proved by upper bounding each element of (5): q T F v q λ min {F v} q q T K v q λ min {K v} q αsat( q) T C(q, q) T q αε c q α sat( q) T M(q) q αλ max{m} q and using Theorem, we have αsat( q) T [g(q d ) g(q)] = αsat( q) T g(z) z q, (6) z=ξ for some ξ belonging to the line that oins q d and q. Finally the time derivative V can be upper bounded by: V [λ min {F v} + λ min {K v} α(ε c + λ max{m})] q " αsat( q) T K p + g(z) # q () z z=ξ By using Definition, Definition [ and Lemma, we can ] conclude that αsat( q) T K p + g(z) z q is definite [ z=ξ ] positive if the matrix K p + g(z) z fulfills z=ξ pi > gi ; (8) and by choosing a small enough value for α > such that [λ min {F v} + λ min {K v} α(ε c + λ max{m})] >, (9) we have that (5) is negative semidefinite. Therefore, we can use the LaSalle invariance principle to assure global asymptotic stability provided that (), () and (9) are satisfied. This completes the proof of Proposition. B.. Consider a control law similar to that presented in (Arimoto, 995) and (Santibañez and Kelly, 998): τ = K psat(b q;l, m) K v q + K i w () w = [αsat(b q(σ); l, m) q(σ)]dσ + w(), where now, Sat( ) is defined in Definition and we have used a new gain B = diag{β, β,..., β n } multiplying q inside the saturation function. B, K p, K v and K i are n n diagonal positive definite matrices, l and m are the saturation limits and α > is a small constant suitably selected. By using the control law () into the dynamics (), we obtain the following closed-loop system q q d dt 6 q z = 6 5 M(q) [K psat(b q;l, m) K v q + K i z C(q, q) q F v q g(q) + g(q d )] αsat(b q; l, m) q 5 () where z is defined as z = [αsat(b q(σ);l, m) q(σ)]dσ + w() K i g(q d ) () {z } z() so that () is an autonomous differential equation whose unique equilibrium is: [ q T q T z T ] T = R n. () From here, the saturation limits l and m are omitted for reasons of space. Proposition. There always exist a positive scalar constant α and positive definite matrices B, K p, K v, K i R n n such that the equilibrium point () from () is globally asymptotically stable. Proof. In base to Lyapunov theory, we propose the following Lyapunov function candidate where V = V = V = V ( q, q, z) = V ( q, q,z) + V ( q) + V ( q) () [ q αsat(b q)]t M(q)[ q αsat(b q)] + zt K i z + αf vi Sat(β i r i )dr i + α vi Sat(β i r i )dr i K psat(br)dr α Sat(B q)t M(q)Sat(B q) [ g(q d r) + KpSat(Br; l, m) + g(q d )] T dr, (5) and K p = K p +K p. Thus, () will be a radially unbounded positive definite function provided that V and V be also a radially unbounded positive definite function. Consider first V. Note that according to definition and by direct integration we find that: Also note that: Sat T (Br; l, m)k pdr = G i( q i), 8 p i βi q i, βi qi l >< p G i( q i) = i l β + pi l( q i l i β ), β i i q i > l >: p i l β pi l( q i + l i β ), β i i q i < l α α SatT (B q)m(q)sat(b q) λmax{m(q)}sat (β i q; l, m) α 8 H( q i) = < βi q i, βi qi m : ε, β i q i > m () From (6) and () we can conclude that there always exist large enough positive constants pi, i =,..., n, such that: G i ( q i ) > α λmax{m(q)}h( q i), q i R, i =,...,n (8) and hence: (6) Sat T (Br)K pdr α SatT (B q)m(q)sat(b q) > q R n (9) Now, we show that q [ g(q d r) +Kp Sat(Br; l, m) + g(q d )] T dr is a positive definite function. First assume that q i l/β i, for i =,..., n. Hence, according to Definition we can write: q [ g(q d r) + KpSat(Br; l i, m i ) + g(q d )] T dr = q [ g(q d r) + KpBr + g(q d )] T dr. This mean that: q " q Z # q [ g(q d r) + Kp Br + g(q d )]T dr = g(q) q + K p B, q i l/β i, i =,...,n () D.R. AMCA Octubre de 5

5 which, according to (Hernandez-Guzman and Silva- Ortigoza, ) and (Hernandez-Guzman et al., 8), is a positive definite if: p i β i > gi > = max q g i (q) q, i =,...,n. () and, in base to Theorem, this proves that q [ g(q d r)+k p Br+g(q d)] T dr is positive definite as long as q i l/β i, for i =,..., n. Now suppose that and represent all integers from to n such that q < l/β and q < l/β. According to (Zavala-Rio and Santibañez, ), Definition and Property, we can write: where [ g(q d r) + KpSat(Br; l, m) + g(q d )] T dr = [ g i (r i ) + g i (q d ) + p i Sat(β i r i ; l,m)]dr i = P + P () g (r ) = g (q d r, q d,...,q dn ) g (r ) = g (q,q d r, q d,...,q dn ). g n (r n) = g n(q,q,...,q n, q dn r n) and P = [ g i (r i ) + g i (q d ) + p i β i r i ]dr i,i + X Z l/β [ g (r ) + g (q d ) + p β r ]dr + X Z l/β [ g (r ) + g (q d ) + p β r ]dr P = X [ g (r ) + g (q d ) + p Sat(β r ;l, m)]dr l/β + X [ g (r ) + g (q d ) + p Sat(β r ; l,m)]dr l/β X [p l ]dr + X Z l/β [p l ]dr l/β q = X q p l l «+ X p β l q l «β () Note that P is always positive if: p l > i, () because q > l and q < l. Finally, since we have proven that P is positive we conclude that q [ g(q d r) + KpSat(Br; l, m) + g(q d )] T dr is a positive definite and radially unbounded function of q. The latter prove that () is positive definite and radially unbounded. It is possible to verify that, using Property, the time derivative of () along the traectories of the closed-loop system () is given by: V = q T F v q α Sat(B q)m(q) q αsat(b q) T C(q, q) T q αsat(b q) T [g(q d ) g(q)] αsat(b q) T K psat(b q) q T K v q(5) Negative semidefiniteness may be proved by upper bounding each element of (5): q T F v q λ min {F v} q q T K v q λ min {K v} q αsat(b q) T C(q, q) T q α c m n q αsat(b q) M(q) q αλ max{b}λ max{m} q αsat(b q) T K psat(b q) α pi Sat(β i q i ) αsat(b q) T [g(q d ) g(q)] αh T (B q)eh(b q) (6) where β i, i =,,..., n, h(b q) = [ Sat(β q )... Sat(β n q n ) ] T, and E is a symmetric matrix with all of its entries bounded (Hernandez-Guzman and Silva-Ortigoza, ). Finally the time derivative V can be upper bounded by: V [λ min{f v} + λ min{k v} α(m n c + λ max{b}λ max{m})] q α [λ min{k p E}] Sat(B q i) () By choosing K p such that λ min (K p E) > (8) and a small enough value for α > such that [λ min {F v} + λ min {K v} α(m n c + λ max{b}λ max{m})] >, (9) we have that (5) is negative semidefinite. Therefore, we can use the LaSalle invariance principle to ensure global asymptotic stability provided that (), (), (8) and (9) are satisfied. This complete the proof of Proposition. IV. EXPERIMENTAL RESULTS In this section is presented a experimental essays on the CICESE robot, using the controllers () and (). The CICESE is a -dof robot manipulator with revolute oints whose dynamic model is presented in (Kelly and Santibañez, ). The numerical values of the parameters for the CICESE robot are shown in Table I. λ max {M} and c were calculated following the procedure proposed in (Kelly and Santibañez, ). TABLE I NUMERICAL VALUES OF THE PARAMETERS FOR THE CICESE ROBOT. Parameter Joint Joint Units g i [N m/rad] i.98.8 [N m] τmax 5 5 [N m] λmax{m}.5 [g m ] c.6 [g m ] In the following, the experimental results for each control law presented above are shown. The responses obtained in the experimental results is the best obtained for their respective controllers.. Consider the control law () τ = K p q K v q + K i w w = [αsat( q(σ)) q(σ)]dσ + w() D.R. AMCA Octubre de 6

6 which we have used to perform a real-time experimental essay on the CICESE robot. The controller gains used are presented in Table II. The oint errors and torques obtained in the experiment essay are shown in Fig.. [deg] 8 6 q q 6 8 t[seg] TABLE II CONTROLLER GAINS. gains Joint Joint Units Kp [N m/rad] Kv 6 [N m s/rad] K i 5. [N m / rad] α 5 [/s] L, M.,. [rad] [Nm] 5 5 τ τ t[seg] Figura. Position errors and torque obtained using control law ().. Consider the control law () τ = K psat(b q;l, m) K v q + K i w w = [αsat(b q(σ); l, m) q(σ)]dσ + w(), which we have used to perform a real-time experimental essay on the CICESE robot. The controller gains used are presented in Table III. The oint errors and torques obtained in the experiment essay are shown in Fig.. [deg] 8 6 q q 5 t[s] TABLE III CONTROLLER GAINS. gains Joint Joint Units Kp 5.5 [N m/rad] Kv 5 [N m s/rad] K i [N m / rad] B [N m / rad] α. [/s] L, M.,. [rad] [Nm] t[seg] Figura. Position errors and torque obtained using control law (). The controller parameters for both controllers were selected in such a way that the torque of each actuator remains τ τ under the maximum permitted torque ( τ i < τ maxi ). Due to the control law () has a saturation function in the proportional part, it is possible choose larger proportional gains than those of the control law (). This fact allows to achieve the desired position faster for the controller () than the controller () (see Fig. and ). It is easy to prove that controller parameters fulfill the tuning conditions for assuring global asymptotic stability. V. CONCLUSIONS In this paper new tuning conditions for a class of nonlinear PID global regulators were found. By using Lyapunov s stability theory new tuning conditions were established, which are less restrictive than those shown in (Santibañez and Kelly, 998), (Arimoto, 995) and (Kelly, 998). For validation purposes, experimental results were reported using the CICESE robot manipulator. VI. ACKNOWLEDGMENTS This research was partially supported by DGEST and CONACyT (grant 5). REFERENCES Alvarez-Ramirez, J., I. Cervantes and R. Kelly, PID regulation of robot manipulators: stability and performance, System and Control Letters, vol., pp. -8,. Arimoto, S., Stability and robustness of PID feedbac control for robot manipulators of sensory capability, Robotics Researches: First International Symposium, MIT Press, Cambridge, pp. 8 99, 98. Arimoto, S., Fundamental problems of robot control: Part I, Innovations in the realm of robot servo-loops, Robotica, vol., pp. 9, 995. Arimoto, S. and H. Suzui, Asymptotic stability and robustness of PID local feedbac for position control of robot manipulators, Proc. International Conference on Automation Robotics and Computer Vision, Singapur, pp8-86, 99. Hernandez-Guzman, V. M., V. Santibañez and R. Silva-Ortigoza, A New Tuning Procedure for PID Control of Rigid Robots, Advanced Robotics, vol.,, 8. Hernandez-Guzman, V. M., V. Santibañez and A. Zavala-Río, A Saturated PD controller for Robots Equipped with Brushless DC-motors, Robotica, vol. 8, 5-,. Hernandez-Guzman, V. M. and R. Silva-Ortigoza, Improving Performance of Saturation-Based PID controllers for Rigid Robots, Asian Journal of Control., vol., no., December. Kelly, R., A tuning procedure for stable PID control of robot manipulators, Robotica, vol., pp. -8, 995. Kelly, R., Global Positioning of Robot Manipulators via PD Control Plus a Class of Nonlinear Integral Actions, IEEE Transaction on Automatic Control, vol., no., July 998. Kelly, R. and V. Santibañez, Control de Movimiento de Robots Manipuladores, Prentice Hall, Madrid,. Kelly, R., V. Santibañez and A. Loria, Control of robot manipulators in oint space, Springer-Verlag, Berlin, 5. Khalil, H., Nonlinear Systems, Prentice Hall, New Jersey,. Meza, J. L., V. Santibañez and R. Campa, An Estimate of the Domain of Attraction for the PID Regulator of Manipulators, International Journal of Robotics and Automation, vol., pp. 8-95,. Ortega, R., A. Loria and R. Kelly, A semiglobally stable output feedbac PI D regulator for robot manipulators, IEEE Transaction on Automatic Control, vol., pp. 6, 995. Santibañez, V. and R. Kelly, A class of nonlinear PID global regulators of robot manipulators, Proc. of the IEEE International Conference on Robotics and Automation, pp. 6-68, 998. Zavala-Rio, A., and V. Santibañez, A natural saturating extension of the PD-with-desired-gravity-compensation control law for robot manipulators with bounded inputs. IEEE Trans. on Robotics, vol., 86-9,. D.R. AMCA Octubre de