A Cascaded-Based Hybrid Position-Force Control for Robot Manipulators with Nonnegligible Dynamics

Transcription

1 21 American Conference Marriott Waterfront, Baltimore, MD, USA June 3-July 2, 21 FrA16.4 A Cascaded-Based Hybrid Position-Force for Robot Manipulators with Nonnegligible Dynamics Antonio C. Leite, Fernando Lizarralde and Liu Hsu Abstract This workaddresses the hybridposition-force control problem for robot manipulators performing interaction tasks on constraint surfaces with regular curvature. A novel hybrid control law, based on an orientationdependent term, is proposed to circumvent the performance degradation owing to modeling uncertainty, particularly with respect to the geometry of the constraint surface. As in our previous work, the effect of the robot dynamics can be included by using a cascade control strategy. In contrast, instead of only position, the presented approach provides complete robot posture control. Simulation results illustrate the performance and feasibility of the proposed control scheme. I. Introduction For many years, the academic community has actively investigated the simultaneous application of position and force control to increase the degree of autonomy and flexibility in robotic systems, in order to perform interaction tasks. The feedback provided by position and force sensing has been used to develop several stable control strategies based on the hybrid control approach. The hybrid control strategy combines force and torque information with end-effector position and orientation data, according to the concept of complementary orthogonal subspaces in force and motion formalized by Mason [1]. In this context, the efficiency of the hybrid control method was first experimentally verified on a Scheinman-Stanford arm by Raibert and Craig [2]. On the other hand, some works have proposed a correction in the original hybrid position-force control scheme in order to avoid the problem of kinematic instability [3],[4],[5]. Other works have been developed to establish sufficient stability conditions for kinematic and dynamic hybrid position-force control approaches [6],[7]. Furthermore, the presence of uncertainties in the system parameters have also been of concern. Several approaches were proposed to estimate the geometry of the constraint surface considering the robot kinematics and dynamics fully known [8], [9], [1]. Some exceptions can be found in recently published papers [11], [12]. However, to the best of our knowledge, the orientation control problem was not rigorously taken into account in the control design, particularly when the robot motion is constrained on surfaces with nonplanar geometry. In this paper, the position and force control problem for nonredundant robot manipulators using a force sensor mounted on the robot wrist is considered. A hybrid control method is used to combine position control and direct force control in the presence of constraint surfaces with regular curvature. A novel hybrid control law, based on an orientationdependentterm, is proposed to solve the interaction problem This work was partially supported by CAPES, CNPq and FAPERJ. The authors are with the Department of Electrical Engineering - COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil. [toni, fernando, liu]@coep.ufrj.br on smooth surfaces with unknown geometry. The orientation control uses the unit quaternion formulation which is free of singularities and computationally efficient [13]. The stability analysis of the overall closed-loop system is developed based on the Lyapunov approach. A posture control method for robots with nonnegligible dynamics is presented using a cascade control strategy. Simulation results illustrate the performance and feasibility of the proposed control scheme. II. Nomenclature The following notation is adopted in what follows: Ē a = [ x a y a z a ] denotes the orthonormal frame a and x a, y a, z a denote the unit vectors of the frame axes. For a given vector ν R n, its elements are denoted by ν i for i=1 n, that is, ν =[ ν 1 ν 2 ν n ] T. Q( ) denotes the skew-symmetric operator, such that, for any vector ν R 3 implies that Q(ν) = ν 3 ν 2 ν 3 ν 1 ν 2 ν 1. III. Robot Kinematic Consider the kinematic control problem for a robot manipulator. In this framework, the robot motion can be simply described by: θ i = u i, i = 1,, n, (1) where θ i and θ i are the angular position and the angular velocity of the i-th joint respectively, and u i is the velocity control signal applied to the i-th joint motor drive. This can be applied to most commercial robots with high gear ratios and/or when the task speed is slow. Let p R m be the end-effector position with respect to the robot base, expressed in the tool frame Ēe and R be be the rotation matrix of the tool frame Ē e with respect to the base frame Ēb. Now, let q=[ q s qv T ] T be the unit quaternion representation for R be, where q s R and q v R 3 are the scalar and vectorial part of the quaternion respectively, subject to the constraint q T q = 1 [13]. In this context, the end-effector configuration x R m is given by the forward kinematics map, represented by a m-dimensional vector function [ ] p x = = h(θ), (2) q where θ R n is the vector of manipulator joint angles. In general, h( ) : R n R m is a nonlinear transformation describing the relation between the joint space Q and the operational space O. Note that, considering a robot arm with six-degree-of-freedom (6-DOF), m=7 and n= /1/$ AACC 526

2 The end-effector velocity v = [ ṗ ω ] T, composed by the linear velocity ṗ and the angular velocity ω, both expressed in the tool frame Ēe, is related to the joint velocity θ by v = J(θ) θ, (3) where J(θ) R n n is the manipulator Jacobian. Thus, from (3) and considering θ i as the control input u i (i=1,..., n) one obtains the following control system: v = J(θ) u. (4) A cartesian control law v k (t) can be transformed to joint control signals by using the inverse kinematics algorithm [ ] u = J 1 vh (θ) v k, v k =, (5) provided that matrix J(θ) is nonsingular. Therefore, from (4) and (5) one has that [ ] [ ] ṗ vh =, (6) ω and naturally, v h and v o are designed to control the endeffector position and orientation respectively. Here, the following two assumptions are considered: (A1) the robot kinematics is known; (A2) the control law v k (t) does not drive the robot to singular configurations. The failure of any these assumptions is an important and challenging issue in robotics area, that deserves future investigations. v o v o IV. Hybrid Scheme In the hybrid control approach, different directions of the task space T are simultaneously controlled using position and force sensing, based on the concept of complementary orthogonal subspaces [1]. Some issues about the validity of the orthogonal complements concept in the hybrid control theory were raised by Duffy [14], particularly with respect to the dependence on the choice of units and dimensional inconsistency. However, the hybrid control scheme proposed in this section is free from such problems, since: (a) the control actions can be split and treated separately in translational and rotational motions; (b) the formulation of the control scheme only requires the end-effector position and the interaction force between the end-effector and environment, respectively. Hence, the position and force constraints can be separately considered and the controllers are not affected by mutual interferences. These constraints are specified in an appropriate coordinate system for the task execution named the constraint frame and denoted by Ēs. From the selection matrices S R 3 3 and I S, which determine what degrees of freedom must be controlled by force and position, the control signals are decoupled and the control laws for each subspace can be independently designed in order to achieve simultaneously different force and position requirements for a given interaction task. Thus, the hybrid control law can be given by v h = v hp + v hf, (7) where v hp and v hf are the decoupled control signals acting respectively in the position and force subspaces, such that, v hp =R es (I S)R T es v p, v hf =R es S R T es v f, (8) where v p is the position control signal, v f is the force control signal and R es is the rotation matrix of the constraint frame Ē s with respect to the tool frame Ēe. Remark 1: Note that, from (8) the decoupling of control variables can be performed in the constraint space C, where the task is naturally described and the selection matrices have a diagonal form with and 1 elements. A. Constraint Geometry Now, considering that the constraint surface in the task space T can be described by ϕ(p)=, where ϕ(p) : R n R is a smooth mapping, the constrained motion of the end-effector satisfies D(p)ṗ=, where D(p) = ϕ(p) denotes the normal vector of the constraint surface. Then, when the constraint geometry is known p the constraint frame Ē s = [ x s y s z s ] can be conveniently chosen with unit normal vector z s(p)= D(p) and arbitrary D(p) orthonormal vectors x s, y s. Thus, the rotation matrix of the constraint frame Ēs with respect to the base frame Ēb can be derived by R bs =[( x s) b ( y s) b ( z s) b ], and the desired orientation of the end-effector on the surface can be obtained by R d = R bs (R es) T d, (9) where (R es) d denotes the desired rotation matrix of the constraint frame Ē s with respect to the tool frame Ē e. Remark 2: Without loss of generality, the reorientation of the end-effector on the constraint surface considers the alignment problem of the tool frame Ēe with respect to the constraint frame Ēs, such that (Res) d =I and thus R d =R bs. On the other hand, considering that the robot end-effector interacts with a constraint surface with unknown geometry, the normal vector z s at the contact point (and consequently x s, y s) could be estimated from online measurements of the contact force f and the infinitesimal displacement p on the surface [15], that is, z s = ( f f t) / f f t, ft=( f t) t, where t= p/ p and x s = t 1/ t 1, t 1 = (I z s z s ) x e, and finally, y s=q( z s) x s. Hence, the estimated orientation of the constraint frame Ēs with respect to the tool frame Ēe is given by ˆR es = [ ( x s) e ( y s) e ( z s) e ], (1) and thus, the desired end-effector orientation on the constraint surface can be obtained as R d =R be ˆRes. B. Position Consider the position control problem for a 3-DOF kinematic manipulator. Here, one assumes that the control objective is to track the desired time-varying trajectory p d (t) from the current end-effector position p, that is, p p d (t), e p = p d (t) p, (11) 5261

3 where e p is the position error. Considering v h =v p and from (6), one has that ṗ = v p. Thus, using a feedforward plus proportional control law v p = ṗ d + K p e p, (12) where K p = k p I, the position error dynamics is governed by ė p + K p e p =. Hence, by a proper choice of k p as a positive constant, implies that lim t e p(t)=. C. Force Consider the force control problem for a 3-DOF kinematic manipulator endowed with a wrist-force sensor 1. Here, one assumes that the control objective is to regulate the measured contact force f to a constant desired force f d along the constraint surface, that is, f f d, e f = f f d, (13) where e f is the force error. Similar to Hooke s law, the generalized force at the end-effector can be modeled by f e = K s (p p s), (14) where p is the position of the contact point, K s =k s I is the elasticity matrix, k s > is the spring constant assumed to be known and p s is the free length of spring. Note that, the vector of contact force points in the opposite direction than the vector of end-effector force, that is, f e= f. Then, from (13) and (14), the force error equation is given by ė f = K s ṗ. Considering v h = v f and from (6), one has that ṗ=v f. Thus, using a proportional control law v f = K f e f, (15) where K f = k f I, the force error dynamics is governed by ė f +k s k f e f =. Hence, for a proper choice of k f as a positive constant implies that lim t e f (t)=. D. Orientation Consider the orientation control problem for a kinematic manipulator. Here, one assumes that the control objective is to drive the current orientation matrix R SO(3) to a desired time-varying attitude R d (t), that is, R R d (t), R q =R T R d (t) I, (16) where R q SO(3) is the orientation error matrix expressed in the tool frame Ēe. Note that, taking R=R be and from (9) one has that R q =R es (R es) T d. Let e q = [ e qs e T qv ] T be the unit quaternion representation for R q such that e q =q 1 q d (t), where q d is the unit quaternion representation for R d and denotes the quaternion product operator. Note that, e q = [1 T ] T if and only if R and R d are aligned. Thus, from (6) one has that ω = v o and using a feedforward plus proportional control law v o = ω d + K o e qv, (17) where ω d is the desired angular velocity and K o is a positive definite matrix, the equilibrium point (e qs, e qv) = (±1, ) is almost globally 2 asymptotically stable. 1 In order to avoid hard impacts that could damage the parts in contact during the interaction task, a compliance behavior of the arm tip could be achieved by means of a linear spring mounted to the force sensor plate, aligned with the end-effector approach axis. Proof: for a proof see [15]. E. Kinematic Hybrid Consider the hybrid position-force control problem for a kinematic robot interacting on a constraint surface with unknown nonplanar geometry. The proposed hybrid control law includes a new term in the decoupled position and force control signals (8), which depends on the end-effector orientation: v hp = ˆR es (I S) ˆR T es v p ˆR es (I S)Q(ω es) ˆR T es e p, (18) v hf = ˆR es S ˆR T es v f ˆR es S Q(ω es) ˆR T es K 1 s e f, (19) where ω es R 3 is the angular velocity of the constraint frame Ēs with respect to the tool frame Ēe, assumed to be measurable. Now, let ξ p R 2 be decoupled position error and ξ f R be the decoupled force error, both expressed in the constraint frame Ēs, after selecting the position and force control directions, such that, ξ T p ξ p = ē T p ē p, ē p = (I S) ˆR T es e p, (2) ξ T f ξ f = ē T f ē f, ē f = S ˆR T es e f, (21) where ē p R 3 e ē f R 3. Then, the following theorem can be stated: Theorem 1: Consider the closed-loop system described by (4) and (5) with the hybrid control law given by (7) and (18)-(19), the position controller (12), the force controller (15) and the orientation controller (17). Assume that the reference signal p d (t) is piecewise continuous and uniformly bounded in norm, f d is constant and q d (t) is the unit quaternion representation for R d (t) SO(3). Under the assumptions (A1) and (A2), the following properties hold: (i) all signals of the closed-loop system are uniformly bounded; (ii) lim t ξ p(t)=, lim t ξ f (t)= and lim t e qv(t)=, lim t e qs(t) = ±1. Thus, the overall closed-loop system is almost globally asymptotically stable. Proof: The closed-loop stability analysis uses the following Lyapunov function candidate: 2V = ξ T p Λ p ξ p + ξ T f Λ f ξ f + 2 (e qs 1) e T qv e qv, (22) where Λ p =Kp 1 e Λ f = (K s K f ) 1. The time-derivative of V along the system trajectories assumes the form V = ē T p Λ p ē p + ē T f Λ f ē f + 2(e qs 1)ė qs + 2 e T qv ė qv, (23) and resorting to ē p = (I S)( ˆRT es e p + ˆR T es ė p ), ė p =ṗ d v h, ē f = S ( ˆR T es e f + ˆR T es ė f ), ė f = K s v h, and based on the quaternion error propagation equation [16] ė qs = 1 2 et qv ω, ė qv = 1 2 ( eqs I + Q(eqv) ) ω, ω=ω d v o, one obtains V = ē T p Λ p (I S) ˆRT es e p + ē T p Λ p (I S) ˆR T es (ṗ d v h ) + ē T f Λ f S ˆRT es e f ē T f Λ f S ˆR T es K s v h + e T qv (ω d v o). 2 In this work, one uses the term almost globally to indicate that the domain of attraction is the entire state space, except for a set of measure zero. 5262

4 Reminding that ˆRT es = Q(ω es) ˆR T es and from (7), (17), (18) and (19), one has that V = ē T p Λ p (I S)Q(ω es) ˆR T es e p + ē T p Λ p (I S) ˆR T es ṗ d ē T f Λ f S Q(ω es) ˆR T es e f ē T p Λ p (I S) (I S) ˆR T es v p +ē T p Λ p (I S)(I S)Q(ω es) ˆR es T e p +ē T p Λ p (I S)S Q(ω es) ˆR es T Ks 1 e f ē T p Λ p (I S)S ˆR T es v f ē T f Λ f S (I S)K s ˆRT es v p + ē T f Λ f S (I S)K s Q(ω es) ˆR T es e p ē T f Λ f S S K s ˆRT es v f + ē T f Λ f S S K s Q(ω es) ˆR T es K 1 s e f e T qv K o e qv. Using the concept of complementary orthogonal subspaces [1], that is, (I S)S=S (I S)=, (I S) (I S) k =(I S), S S k =S, where k Z one obtains V = ē T p Λ p (I S) ˆR T es ṗ d ē T p Λ p (I S) ˆR T es v p ē T f Λ f S K s ˆR T es v f e T qv K o e qv. Finally, resorting to (12) and (15) yields V = ξ T p ξ p ξ T f ξ f e T qv K o e qv. This implies that V (t) V () and, therefore, that ξ p, ξ f, e qs and e qv are uniformly bounded. The time-derivative of V is given by V = 2(ξp T ξ p + ξf T ξ f + e T qv K o ė qv ) and one can show that ξ p, ξf and ė qv are also uniformly bounded. Thus, V is bounded, and, hence, V is uniformly continuous. Since V is radially unbounded and V over the entire state space, applying the usual argument based on Barbalat s lemma [17] one has that lim t V (t) = and consequently that lim t ξ p(t)=, lim t ξ f (t)=, lim t e qv(t)= and lim t e qs(t)=±1 which proves the almost global stability of the closed-loop system. The above Lyapunov analysis allow us to establish passivity properties [17] for the hybrid control scheme. In fact, considering that the hybrid control system as being driven by a fictitious external input signal ν =[ ν p ν o ] T, the error system can be rewritten as ė p = ṗ d (v h + ν p), (24) ė f = K s (v h + ν p), (25) ė q = 1 2 JT q (e q)(ω d v o ν o). (26) Thus, one can state the following corollary. Corollary 1: (Passivity) Consider the error system (24)-(26) with the hybrid control law (18)-(19) and the orientation control law (17). Then, the maps ν p ˆR es [(I S)Kp 1 ē p+ S K 1 f ē f ] and ν o e qv are output strictly passive [2] with positive definite storage function (22). Proof: Considering (23) and (24)-(26) one has that: V ξp T ξ p ξf T ξ f [ ē T p Kp 1 (I S)+ē T f K 1 f S ] ˆR es T ν p e T qv ν o. V. Dynamic Robot Now, considering the hybrid control problem for a robot manipulator with nonnegligible dynamics (e.g., direct-drive manipulators), an extension of the proposed controller to include the robot dynamics is presented. The nonlinear dynamic model of the robot manipulator in contact with the environment can be expressed in the generalized coordinates θ R n by [13] M(θ) θ + C(θ, θ) θ + g(θ) = τ J T (θ)f, (27) where τ R n is the vector of actuator torques and F R n is the vector of generalized forces (force and moment) exerted by the end-effector on the environment. It is worth mentioning that, M(θ) R n n represents the manipulator inertia matrix, C(θ, θ) θ R n gives the Coriolis and centrifugal forces terms and g(θ) R n is the vector of gravitational forces. Other forces acting at the joints (e.g., viscous friction) could be also considered in the dynamic model (27). In this section, the key idea is to introduce a cascade control strategy (see Fig. 1) to solve the hybrid control problem for a robot manipulator with nonnegligible dynamics, analogous to the case that only the position control problem using the visual servoing approach was considered [18], [19]. To achieve this, one can assume that there exists a control law τ = Γ(θ, θ, θ m, θ m, θ m) J T (θ) F, (28) which guarantees the control goal defined by θ θ m(t), e = θ θ m(t), (29) where θ m denotes the desired time-varying trajectory expressed in the joint space Q. Now, one supposes that it is possible to define the desired trajectory θ m and its derivatives θ m, θ m in terms of a cartesian control signal v k such that one has (6) except for a vanishing term, that is, v = v k + J(θ) L(s)e, (3) where L( ) denotes a linear operator with s being the differential operator. Then, one can conclude that the cartesian control law v k (t) can be applied to (3). Moreover, one can obtain some intuition if the parameters of the robot dynamic model (27) are assumed to be exactly known. A passivity-based control algorithm in the generalized coordinates could be used to solve the tracking problem: τ = M(θ) ż + C(θ, θ) z K d σ + g(θ) + J T (θ) F + ν r, (31) where K d is a positive definite gain matrix, ν r is a fictitious external input which drives the closed-loop system and z = θ m Λ e, σ = θ z = ė + Λ e, (32) with Λ = λ I and λ >. The closed-loop system (27), (31) and (32) can be written in the form M(θ) σ + (C(θ, θ) + K d ) σ = ν r, (33) and the stability analysis can be carried out by using a passivity approach [17], driving a stable closed-loop system. Indeed, (33) defines an output strictly passive map ν r σ with positive definite storage function 2V r = σ T M(θ)σ and using the Barbalat s lemma one can show that all signals are bounded and lim t σ(t) =, which implies that lim t ė(t) = e lim t e(t) =. Then, based on the cascade control strategy one can define θ m=j 1 (θ) v k + λ e, and from (29) one has that v = v k + J(θ) σ, (34) where σ satisfies a stable closed-loop dynamics. Since this approach only differs from the kinematic control approach by a 5263

5 θ m, θ m, θ m Dynamic θ τ Robot + Environment θ f Forward Kinematics q p Cascade Strategy v k Σ v h v o Hybrid Orientation v f v p q d, ω d Force Position f d p d, ṗ d Fig. 1. Block diagram of the cascade structure. vanishing term σ(t), using the general result for interconnected passive systems [17] one can demonstrate that the control signals v h (t) and v o(t), computed for the kinematic control case, can be applied to the case of dynamic robot control and the closed-loop stability can be proved. Note that, in the case of parametric uncertainty on the robot system (3) and (27), adaptive or robust control strategies [2] can also be used for the dynamic robot control [18], [19]. Furthermore, the proposed hybrid position-force control scheme has passivity properties which make it possible to guarantee stability when cascaded to another control system with similar passivity properties, as presented in [19]. VI. Simulation Results In this section, simulations results are presented to illustrate the performance of the proposed kinematic hybrid control scheme. In the simulations, a 6-DOF robot manipulator similar to Puma 56 arm has to perform the tracking of a reference trajectory on a cylindrical surface with unknown geometry. The links lengths are l 1 = mm and l 2 = mm for link 1 and 2 respectively. The reference trajectory is specified in YZ plane and described by p d = [ r n sin(ω nt) r n cos(ω nt) ] T, where r n = 1 mm and ω n = π 5 rads 1 are the radius and the angular velocity of the trajectory respectively. The control parameters are: k p = 2 mms 1, k f = 4 mms 1 N 1 e K o = 1 I rads 1. Other parameters are: f d =1 N, k s = N mm 1. The simulations are performed in the presence of friction force and measurement noise. The model used to describe the friction is based on the Coulomb friction model [13], where f t µ f n and µ is the friction coefficient of material. During the interaction task, the static and kinetic coefficient of friction were µ s =.2 and µ k =.15, which represent the approximate coefficients of friction between wood and metal [13]. The noise levels used are typically found in the HEDS-55 optical incremental encoder (Agilent Tech.) and JR3 force sensor (JR3 Inc.). In the simulation, the end-effector orientation was kept constant until the contact point has been reached as t=5 s. Figures 2(a) and 2(b) describe the time history of the position and force errors. The maximum position and force errors in the steady-state were around 1.5 mm and.5 N, respectively. The time history of decoupled position and force control signals are presented in Figures 3(a) and 3(b). Figures 2(c) and 3(c) describe the time history of the orientation error and orientation control signal respectively. The norm of orientation error in the steady-state was around The tracking of the reference trajectory is shown in Figure 4, where it can be observedthata remarkable performance was achieved, during the interaction task. In Figure 5 is illustrated the endeffector trajectory on the cylindrical surface located in the workspace. [mm] 1 1 (a) position error e p,x e p,y [N] 1 5 (c) orientation error (b) force error e f,z Fig. 2. Position, force and orientation errors. VII. Conclusion and Future Works This work considers the hybrid position-force control problem for a robot manipulator using a wrist force sensor. A novel hybrid control law is proposed to solve the interaction problem on constraint surfaces with unknown geometry. The stability and convergence analysis of the overall closed-loop control system is developed based on the Lyapunov approach. A posture control method, which takes into account the robot dynamics, is presented by using a cascade control strategy. Simulation results illustrate the performance and the feasibility of the proposed control scheme. Some topics for future research using the ideas discussed here are relaxing the complete knowledge of the robot kinematics, to consider the stiffness coefficient of the constraint surface uncertain and to include the nonlinear uncertain robot dynamics in the control design. e q,x e q,y e q,z 5264

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