Distributed Force/Position Consensus Tracking of Networked Robotic Manipulators

Transcription

1 180 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL 2014 Distribute Force/Position Consensus Tracking of Networke Robotic Manipulators Lijiao Wang Bin Meng Abstract In this paper, we aress the tracking problem of istribute force/position for networke robotic manipulators in the presence of ynamic uncertainties. The en-effectors of the manipulators are in contact with flat compliant environment with uncertain stiffness an istance. The control objective is that the robotic followers track the convex hull spanne by the leaers uner irecte graphs. We propose a istribute aaptive force control scheme with an aaptive force observer to achieve the asymptotic force synchronization in constraine space, which also maintains a cascae close-loop structure separating the system into kinematic moule an ynamic moule. A ecentralize stiffness upating law is also propose to eal with the environment uncertainties. The convergence of tracking errors of force an position is prove using Lyapunov stability theory an input-output stability analysis tool. Finally, simulations are performe to show effectiveness of the theoretical approach. Inex Terms Networke robotic manipulators, istribute aaptive force control, force/position consensus tracking, compliant environment, aaptive force observer. I. INTRODUCTION DISTRIBUTED control of networke robotic manipulators has gaine significant attention in recent years. The motivation of the research is that a multi-robot approach offers several avantages, such as parallel execution of tasks, robustness by aing reunancy, an elimination of the single point of failure that is present in single robot systems 1, etc. The major purpose of istribute control is to achieve a global group behavior with only local interaction. Distribute control scheme performe on multi-robot networks can provie great benefits with lower cost, higher versatility an easier maintenance 2. Consensus, as a funamental problem in istribute control, means that a group of vehicles reach an agreement on a common value by interacting with their local neighbors. The robotic agent in this paper is moele as the fully actuate Euler-Lagrange system. The stuies on istribute control of multiple Euler-Lagrange systems mainly focus on the position coorination problems 30. The consensus tracking problem with a single or multiple ynamic leaers is consiere in 6 8, 10. In 6, a unifie extene Slotine an Manuscript receive June 19, 2013; accepte October 16, This work was supporte by National Basic Research Program (973) of China (2013CB733100), an National Natural Science Founation of China ( , , ). Recommene by Associate Eitor Jie Chen Citation: Lijiao Wang, Bin Meng. Distribute force/position consensus tracking of networke robotic manipulators. IEEE/CAA Journal of Automatica Sinica, 2014, 1(2): Lijiao Wang is with Beijing Institute of Control Engineering, Beijing , China ( lijiaowang@126.com). Bin Meng is with the Science an Technology on Space Intelligent Control Laboratory, Beijing Institute of Control Engineering, Beijing , China ( mengb@amss.ac.cn). Li controller is propose for both leaerless consensus an consensus tracking in the presence of ynamic uncertainties on irecte graphs with a spanning tree. Cascae framework is introuce which provies a convenient approach to hanle the consensus problem. Yet, for the tracking case, the esire trajectory is assume to be available to all the agents. To remove this assumption, a istribute sliing moe estimator which aims at estimating the esire trajectory in a istribute manner is constructe in 7 for multiple linear agents, an then extene to multiple Euler-Lagrange systems with ynamic uncertainties in 8 for irecte graphs containing a spanning tree. The work of 9 evelops the cascae framework in 6, an solves task-space consensus problem on strongly connecte graphs. The work of 10 further proposes an aaptive task-space position observer such that task-space velocity-free synchronization on irecte graphs containing a spanning tree can be achieve. Nevertheless, the above literature concentrates on the istribute position control in free space without consiering the case when the robots interact with the environment. In practical applications, such as inspection, assembly, polishing, grining as well as scribing, the en-effectors of the robot manipulators are usually require to be in contact with the environment, which involves the interactive force between the manipulators an the environment. In orer to realize safe an elaborate operation in the presence of uncertainties, various aaptive force/position control laws are evelope for single robotic agent 115, aiming at controlling both the position of the en-effector in the unconstraine irection an its contact force with the environment in the constraine irection. Aaptive force control is propose in 11 to estimate unknown environment stiffness with exact environment position when the robot inertia matrix is constant. In 12, aaptive control is also use to eal with stiffness uncertainties after fully linearizing an ecoupling system ynamics. The work of 13 proposes a passivity-base aaptive law to cope with stiffness uncertainties of the environment by use of a scalar factor, while the robot moel an surface position are require to be exactly known. The work of 14 takes both stiffness uncertainties an surface kinematic uncertainties into account, an evises a new aaptive force/position tracking controller for a soft robot finger in compliant contact with a rigi flat surface when the ynamic parameters of the robot are exactly known. When it comes to large-scale assembly tasks in moern manufactures an space applications, multiple robot manipulators may perform tasks in a common environment an the research of istribute force control becomes as significant as istribute motion control in these cases. Another example is that when multi-robots push or carry a common flat object, they must coorinate both position an force to realize movement of the object without rotating or tumbling.

2 WANG AND MENG: DISTRIBUTED FORCE/POSITION CONSENSUS TRACKING OF NETWORKED ROBOTIC MANIPULATORS 181 Therefore, it is necessary to consier the force synchronization problem in many applications ue to the requirement of a balance movement. The work of 15 consiers multiple mobile manipulators grasping a common object in contact with a rigi surface, an proposes an aaptive robust control law in a cooperative manner, but not in a istribute manner. In aition, when the environment is compliant, the methos in rigi conition are no longer applicable, since the bouncing effect cause by the contact force can eventually lea to instability of the manipulators 16. To our best knowlege, no work has focuse on istribute compliant force control of networke robotic manipulators. When the information topology is irecte with a spanning tree an uncertain environment an robotic ynamics are consiere, ue to the asymmetric Laplacian matrix, the nonlinearities an uncertainties of the robot-environment moel, the istribute force control scheme is not a straightforwar extension of the single agent case. In this paper, we consier the force synchronization problem in constraine space, where multiple robotic manipulators on irecte graphs interact with uncertain compliant environment. We propose a istribute aaptive hybri controller such that force an free-space position asymptotically synchronize to the convex hull constitute by the leaers uner irecte graphs containing a spanning tree. A novel aaptive force observer is constructe to achieve force synchronization in the presence of environment uncertainties, which further removes the requirement to measure force erivatives. In aition, by efining an appropriate force reference signal, our strategy eliminates the reliance on the estimation of surface istance, thus avoiing overparamterization in the kinematic loop esign. The propose scheme has a quite simple an irect form in a cascae framework. Notation. In the subsequent sections, if there are no special illustrations, (ˆ ) enotes the estimation of ( ), ( ) = (ˆ ) ( ), an ( ) = ( ) ( ) with ( ) being the esire value of ( ). In aition, 1 m an 0 m enote m 1 column vector containing all ones an all zeros respectively, an I m enotes the m m ientity matrix. II. PRELIMINARIES AND PROBLEM DESCRIPTION A. Robotic Dynamics an Kinematics We consier m robotic followers labele as agents 1 to m, interacting with flat compliant environment. The single-agent moel is illustrate in Fig. 1. The ynamic moel of the ith follower in joint space ignoring friction forces can be written as 14 M i (q i ) q i + C i (q i, q i ) q i + g i (q i ) + J i (q i ) T n si f i = τ i, (1) where q i R p is the joint variable, M i (q i ) R p p is the inertia matrix, C i (q i, q i ) R p p is the couple centripetal an Coriolis matrix, g i (q i ) R p is the gravitational force, τ i R p is the exerte joint control torque, J i (q i ) is the Jacobian matrix which will be efine later, f i R is the normal contact force, an n si R 3 is the surface normal vector. Let x bi -y bi -z bi an x ei -y ei -z ei be the base frame, an the surface frame, respectively, an R i R 3 3 be the rotation matrix from the base frame to the surface frame. Let x si R 3 an x i R 3 enote the surface/environment position an the generalize en-effector position, respectively. The aforementione variables are illustrate in Fig. 1, where x i is shown as x mi. Clearly, we have R T i n si = T. Accoring to 17, x i can be expresse as Fig. 1. A robot manipulator interacting with a flat compliant surface. x i = h i (q i ), where h i ( ) : R p R 3 is a nonlinear mapping from joint space to task space. Then, the en-effector velocity ẋx i is relate to the joint-space velocity as 18 ẋx i = J i (q i ) q i, where J i (q i ) R 3 p is referre to as the Jacobian matrix. For simplicity of analysis, we make the following assumption. Assumption 1. The robot manipulators are operate in finite task space, an are non-reunant, i.e., J i (q i ), i = 1,, m, are nonsingular an square matrices. We procee to get the eformation of the compliant environment as 14 δx fi = x T i n si x T sin si, which is assume to be linearly elastic with environment stiffness k si, i.e., δx fi = θ 0i f i, where θ 0i = 1 k si. The istance from the base center to the surface is presente as θ 1i = x T si n si. Let x i be the task-space position vector in the surface frame. Then, we have x i = Ri T θ1i x i, (2) an the task-space velocity x i = J i (q i ) q i, (3) where J i (q i ) = Ri TJ i(q i ) is the Jacobian matrix in the surface frame. Therefore, x i is partitione into two parts, i.e., xfi x i =, x pi where x fi = δx fi is the environment eformation in the constraine space utilize for force tracking, an x pi is the position in the unconstraine space utilize for motion tracking. As a result, (1) can be reformulate as M i (q i ) q i + C i (q i, q i ) q i + g i (q i ) + J i T fi = τ 0 i. (4) 2 It is well known that the ynamics (4) have the following properties 18. Property 1. There exist positive constants k m1, k m2, k C an k g such that k m1 I p M i (q i ) k m2 I p, C i (y 1, y 2 )y 3

3 182 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL 2014 k C y 2 y 3 for all vectors y 1, y 2 an y 3 R p, an g i (q i ) k g. Property 2. The ynamics (4) is linearly parametric with respect to a group of unknown ynamic parameters a i R r, i.e., M i (q i )ξ 1 + C i (q i, q i )ξ 2 + g i (q i ) = Y i (q i, q i, ξ 2, ξ 1 )a i, where Y i R p r is the ynamic regressor matrix, ξ 1 R p, an ξ 2 R p. Property 3. Ṁ i (q i ) 2C i (q i, q i ) is skew symmetric if C i (q i, q i ) is appropriately selecte. B. Graph Theory We assume there are N agents with m robotic followers an N m virtual leaers. Let the vertex i enote the ith agent an the vertex set V = {1,, N}. The ege (i, j) enotes the information flow between agents i an j. Let E enote the ege set. An ege (j, i) E means that agent i can obtain information from agent j. The vertex sets for the followers an leaers are enote by V F = {1,, m} an V L = {m + 1,, N}, respectively. The following assumption is mae for the information topology. Assumption 2. For each follower, there exists at least one leaer that has a irecte path to the follower. The ajacency matrix W = w ij R N N associate with the interaction graph E is efine as w ij > 0 if (j, i) E, an w ij = 0, otherwise. We assume the self eges o not exist, i.e., w ii = 0. Then we efine the Laplacian matrix L = l ij R N N as N w ik, i = j, l ij = k=1 w ij, i j. Here, L has the form of 8 L1 L L = 2, (5) 0 (N m) m 0 (N m) (N m) where L 1 R m m an L 2 R m (N m) satisfy the following lemma. Lemma 1 8. The matrix L 1 is non-singular, an all eigenvalues of L 1 have positive real parts, if an only if Assumption 2 hols. Moreover, each entry of L1 L 2 is nonnegative, an all row sums of L 1 L 2 equals to one, if Assumption 2 hols. In this paper, for the robotic followers represente by (3) an (4) with the uncertainties of ynamic parameter a i, environment stiffness k si an istance θ 1i, our control objective is to rive both f i an x pi to their esire values f i an x pi by esigning τ i in a istribute manner, where f i an x pi are etermine by the convex hull spanne by the leaers, i.e., f := T f 1 f m = (L 1 L 2)f L, (6) x p := x T p1 x T T pm = (L 1 L 2 I 2 ) x pl, (7) where f L an x pl are the column stack vectors of f i an x pi (i V L ), respectively. To guarantee the convergence of sliing moe estimators, we introuce the following assumption 7. Assumption 3. f L an x pl as well as their first-orer, secon-orer an thir-orer erivatives are all boune. III. THE DESIGN OF FORCE/POSITION CONSENSUS TRACKING CONTROLLER In this section, we give the cascae structure esign scheme, which ivies the overall system into kinematic moule an ynamic moule. The kinematic loop is first constructe to achieve force/position synchronization an to get a reference joint-space velocity for the controller, while the ynamic loop is to pose the control law such that the conitions require for the kinematic loop esign are satisfie. A. Kinematic Loop Design In this part, for the kinematic equation (3) in the presence of uncertain environment stiffness k si an istance θ 1i, we aim at esigning an appropriate joint-space reference velocity such that f i f i an x pi x pi as t, i V F, an giving the esign conitions, which will be satisfie by the ynamic loop. We first construct the sliing moe estimators to estimate the esire trajectory in a istribute manner 7, i.e., ˆv fi = β f1 sgn w ij (ˆv fi ˆv fj ) + w ij (ˆv fi f j ), (8) ˆv pi = β p1 sgn w ij (ˆv pi ˆv pj ) + w ij (ˆv pi x pj ), (9) â fi = β f2 sgn w ij (â fi â fj ) + w ij (â fi f j ), (10) âa pi = β p2 sgn w ij (âa pi âa pj ) + w ij (âa pi x pj ), (11) where ˆv fi an ˆv pi are estimates of f i an ẋx pi, respectively, â fi an âa pi are estimates of fi an x pi, respectively, β f1, β p1, β f2 an β p2 are ajustable positive constant gains esigne to satisfy the following conition. Conition 1. β f1 > f, β p1 > x p, βf2 >... f an β p2 >... x p. Uner Conition 1, the sliing moe estimators are ensure to converge in finite time T 7 0. Then, for i V F, we esign the estimate joint-space reference velocity base on the estimate esire trajectory where ˆ q ri = J i ˆ x ri, (12) ˆ x ri = ˆv xoi ˆθ 0i w ij (f oi f oj ) w ij (f oi f j ) j V Λ i F j V L, w ij ( x pi x pj ) (13)

4 WANG AND MENG: DISTRIBUTED FORCE/POSITION CONSENSUS TRACKING OF NETWORKED ROBOTIC MANIPULATORS 183 ˆv xoi = ˆθ0iˆv fi ˆv T pi T, (14) ˆθ 0i = Γ 0i ( f oi + Λ oi (f oi f i ))(f oi f i ), (15) f oi = x fi ˆθ 0i Λ oi (f oi f i ). (16) Here, f oi is the observe force, an ˆ x ri is the estimate taskspace reference velocity. Equation (15) is the environment stiffness aaptation law. The estimate stiffness parameter ˆθ 0i is assume to be nonsingular in the whole upating process. Equation (16) is the ecentralize force observer to obtain the filtere f i. Λ i = iag{λ fi, Λ pi }, where Λ fi an Λ pi are ajustable positive efinite iagonal matrices. Γ 0i is the ajustable positive efinite upating gain, an Λ oi is the ajustable positive observer gain. x fi in (16) is the projection of x i in the force irection, an x i can be erive from (3). Now, we procee to analyze quality of the kinematic loop. The following lemmas will be aopte. Lemma Let y = G(p 1 )u, where G(p 1 ) is an m 1 n 1 strictly proper an exponentially stable transfer matrix with the Laplace variable p 1. Then u(t) L n1 2 implies y(t) L m1 2 L m1, ẏy(t) L m1 2, y(t) is continuous an y(t) 0 as t. Lemma If y L 2 is uniformly continuous, then y 0 as t. Denote the estimate joint-space sliing moe vector an the estimate task-space sliing vector respectively as an ŝs i = q i ˆ q ri (17) ŝs xi = x i ˆ x ri. (18) Obviously, from (3), (12), (17) an (18), we have ŝs i = J i ŝs xi. (19) Here, ˆ q ri an ŝs i will be use later in Section III-B for the ynamic loop esign. Substituting (13) an (16) into (15), we get ŝs xi = ˆθ0i ( f oi f ri ) x pi x pri + Λ oi ˆθ0i (f oi f i ), (20) where f ri = ˆv fi Λ fi w ij (f oi f oj ) w ij (f oi f j ), x pri = ˆv pi Λ pi w ij ( x pi x pj ). Let f oi = f oi f i. Then, (20) can be rewritten as ˆθ0i f ˆθ 0i w ij (f oi f oj ) oi j V = Λ x i F pi w ij ( x pi x pj ) ˆθ0i w ij (f oi f j ) Λ i j V L + σ i, (21) where σ i = ˆθ0i (ˆv fi f i ) ˆv pi x pi Λ oi ˆθ0i (f oi f i ) + ŝs xi. Quality of the kinematic loop is given in the following theorem. Theorem 1. For the system constitute by (15), (16), (8), (9) an (21), if Assumptions 2 an 3, Conition 1 an the following Conition 2 hol, then f i 0 an x pi 0 as t. Conition 2. For t T 0, ŝs xi L ; for t > T 0, ŝs xi L 2 L. Proof. We first concentrate on the proof of convergence in the force irection. Let us begin with analysis of the observer ynamic loop constitute by (15) an (16). Denote x foi = θ 0i f oi an x foi = x foi x fi. Then, multiplying ˆθ 0i on both sies of (16) an by simple transformation, we can rewrite (16) as x foi = Λ oi x foi θ 0i ( f oi + Λ oi (f oi f i )). (22) The Lyapunov function V fi is then constructe for the observer ynamics as V fi = 1 2 x T foi x foi + θ 0i 2 θ T 0iΓ 0i θ 0i. (23) Differentiating V fi with respect to time an substituting (15) an (22) into it, we obtain V fi = x T foiλ oi x foi 0. (24) This inicates that x foi L 2 L an θ 0i L. Therefore, we have f oi f i L 2 L an ˆθ 0i L. Now, we continue to prove the bouneness of f i in finite time for boune initial values f i (0), ˆθ 0i (0) an ˆv fi (0). If Conition 1 hols, (8) guarantees that ˆv fi f i 0 in finite time T 8 0. Meanwhile, ˆv fi L can be ensure, provie that Assumption 3 hols. For boune f i, we have boune f i an then boune f oi. Thus, f ri is boune. Since we have ŝs xi L from Conition 2 an f oi f i L from the above observer loop analysis, f oi L is guarantee base on (20). From (22), we can further obtain that f oi f i L. Then, f i L. Therefore, we can conclue that f i is boune in finite time T 0. When t > T 0, Conition 2 gives ŝs x L 2, an thus σ i L 2. From (21), we get f o + Λ f L 1 f o L 2, where f o is the stack vector of f oi, an Λ f = iag{λ f1,, Λ fm }. Since L 1 is Hurwitz accoring to Assumption 2 an Lemma 1, f oi L 2 L an f oi L 2 L are erive from the input-output stability analysis in Lemma 2. Then we have f i = f oi (f oi f i ) L 2 L an f i L. From Lemma 3, the uniform continuity of f i an the result that f i L 2 etermine that f i 0 as t. Similar to the stability analysis in the force irection, we can get that x pi 0 as t. Remark 1. Our construction of the ecentralize force observer (16) not only removes the requirement for the measurements of force erivatives, but also provies a simple an irect solution to the force synchronization problem with environment uncertainties uner irecte graphs containing a spanning tree. In fact, if we o not introuce (16), the reference

5 184 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL 2014 task-space velocity is still constructe as (13) with f oi replace by f i, i.e., ˆθ 0i w ij (f i f j ) j V ˆ x ri = ˆv xoi Λ i F V L, (25) w ij ( x pi x pj ) where ˆθ 0i shoul be upate online by a new aaptation law erive from Lyapunov stability tool or input-output stability analysis. Substituting (25) into (18), we get the close-loop kinematic equation x = Λ(L 1 I 3 ) x + ŝs x + θ 0 Y s + (v x o x ), (26) where x, ŝs x, v xo an Y s are the column stack vectors of x i, ŝs xi, v xoi an Y si, respectively, Y si = ˆv fi + Λ fi w ij (f i f j ), 0 T 2 T, θ 0 = iag{ θ 01 I 3,, θ 0m I 3 } an Λ = iag{λ 1,, Λ m }. Since the matrix L 1 is asymmetric in (26), we cannot obtain the asymptotic convergence of tracking error x via the traitional Lyapunov tool. If we turn to input-output stability analysis, θ 0 Y s L 2 must be guarantee to get an asymptotically convergent x. However, it is ifficult for us to construct an inirect stiffness upating law satisfying θ 0 Y s L 2. Therefore, the straightforwar extension of traitional tools for consensus tracking problem is not applicable to solve the force synchronization problem with the asymmetric L 1 an the term θ 0 Y s introuce by uncertain stiffness k si. Then, the force observer (16) is propose in our paper to eal with the afore-mentione problem, where the stiffness uncertainty is first cope with in the observer loop (22) via Lyapunov tool, an then, synchronization is achieve in the kinematic loop (21) via input-output stability analysis. B. Dynamic Loop Design In this part, we will evise the control law τ i for (3) such that not only Conition 2 for the kinematic loop esign is satisfie, but the velocity tracking errors of all the robotic followers are also riven to zero, i.e., { L, t T ŝs xi 0, L 2 L, t > T 0, f i 0 an x pi 0 as t 0, i V F. Now, it is time for us to esign the aaptive control law. On the basis of (12) an (13), we efine the estimate task-space reference acceleration as ˆθ0i â ˆ x ri = fi Λ i âa pi + ˆθ0i f ri ˆθ 0i w ij ( f oi f oj ) w ij ( f oi f j ) j V F j V L w ij ( x pi x pj ) the estimate joint-space reference acceleration as ˆ q ri =, (27) J i (ˆ x ri Jiˆ q ri ), (28) an the ynamic regressor matrix Y i as ˆM i (q i )ˆ q ri + Ĉi(q i, q i )ˆ q ri + ĝg i (q i ) = Y i (q i, q i, ˆ q ri, ˆ q ri )âa i, where ˆ x ri an f ri are given in (13) an (20), respectively. Then, we propose the following aaptive control law τ i = J i T K i ŝs xi + J i T fi + Y 0 i (q i, q i, ˆ q ri, ˆ q ri )âa i, 2 (29) âa i = Γ i Y T i (q i, q i, ˆ q ri, ˆ q ri )ŝs i, (30) where K i an Γ i are the ajustable positive efinite matrices. Now, for the overall system constitute by kinematic loop an ynamic loop, we present the following theorem. Theorem 2. Consier the multiple robotic manipulator systems escribe by (3) (5) with uncertain ynamic parameter a i, environment stiffness k si an istance θ 1i. Suppose Assumptions 1 3 an Conition 1 are satisfie. Then, the controller (29), the observer (16), an the upating laws (15) an (30) give rise to f i 0, f i 0, x pi 0, an x pi 0 as t, i V F. Proof. We first substitute (29) into (4) to get the close loop ynamics M iˆṡs i + C i ŝs i = J T i K i ŝs xi + Y i (q i, q i, ˆ q ri, ˆ q ri )ãa i, (31) where ˆṡs i = q i ˆ q ri. Since Conition 1 gives rise to ˆv fi f i 0 an ˆv pi x pi 0, âfi f i 0 an âapi x pi 0 in finite time 7 T0, we also provie our proof in two parts separate by T 0 as the proof in Theorem 1. In the first part where t 0, T 0, we shoul prove that for boune initial values q i (0), q i (0), âa i (0), ˆθ 0i (0), ˆv fi (0), ˆv pi (0), â fi (0) an âa pi (0), q i an q i will remain boune. From (8) (11), the bouneness of ˆv fi, ˆv pi, â fi an âa pi are ensure. For boune q i an q i, we can get boune x i, x i an Ji, implying the bouneness of f i, f i, x pi an x pi. In Section III-A, we have prove f oi f i L 2 L an ˆθ 0i L. Therefore, ˆ x ri L from (13), further inicating the bouneness of ˆ q ri in (12). Then, ŝs i L an ŝs xi L are obtaine. On the basis of (16), we get f oi L an ˆθ 0i L. Accoring to (27), we erive ˆ x ri L, further implying that ˆ q ri L base on (28). Then, from (30), âa i is boune, which implies the bouneness of âa i. Therefore, we can get from (31) that ˆṡs i L, resulting in the bouneness of q i. Now we conclue that for boune q i (0), q i (0), âa i (0), ˆθ 0 (0), ˆv fi (0), ˆv pi (0), â fi (0) an âa pi (0), q i, q i an ŝs xi remain boune in finite time T 0. Now, we procee our proof for the secon part where t (T 0, ). Since the sliing moe estimators (8) (11) converge in this perio, by simple computation, we have t ˆ x ri = ˆ x ri, an then, tˆ q ri = ˆ q ri an tŝs i = ˆṡs i can be erive. Therefore, for t > T 0, (31) can be rewritten as (q i, q i, ˆ q ri, tˆ q ) ri ãa i. M i tŝs i + C i ŝs i = J T i K i ŝs xi + Y i Let us take the Lyapunov-like function as V i = 1 2ŝs T i M i ŝs i + 1 2ãa T iγ i ãa i. (32)

6 WANG AND MENG: DISTRIBUTED FORCE/POSITION CONSENSUS TRACKING OF NETWORKED ROBOTIC MANIPULATORS 185 We ifferentiate V i with respect to time an substitute (30) into it. Due to the skew symmetry property of Ṁ i 2C i, we have V i = ŝs T i ( ) M i tŝs i + C i ŝs i + ãa T iγ âa i i = ŝs T xik i ŝs xi 0, which implies that ŝs xi L 2, ŝs i L an ãa i L, an thus ŝs xi L. Then, Conition 2 in Theorem 1 hols. By Assumptions 2 an 3, Conitions 1 an 2, we can erive that x i L 2 L, ˆθ 0i L, ˆθ 0i L, f oi L, x pi L, an x i 0, as t from Theorem 1, yieling ˆ x ri L from (27). Meanwhile, we have ˆ q ri L an q i = ŝs i +ˆ q ri L, leaing to the bouneness of Ji. Then, base on (28), we get tˆ q ri L. Therefore, tŝs i L can be erive from (32), which implies the bouneness of q i, an then the bouneness of x i. Because of the fact that x i L 2 L an the uniform continuity of x i, we get that x i 0 as t from Lemma 3. Combining with Theorem 1, we obtain that f i 0, f i 0, x pi 0 an x pi 0 as t. Remark 2. In Theorem 2, the uncertain istance θ 1i is compensate via the stiffness aaptation (15) an the force observer (16). In fact, in the efinition of estimate task-space reference velocity (13), the construction of the force observer enables us to utilize ˆθ 0i f oi as the estimate eformation in the surface normal irection. From (2), we can get the expression of the estimate istance θ 1i as x T i n si ˆθ 0i f oi. Yet, the variable ˆθ 1i is not utilize in our esign. Remark 3. The overall system can be ecompose as a cascae of two subsystems, i.e., (21) an (31). The former is T the kinematic loop with states ˆθ0i f oi x T pi an the latter is the ynamic loop with states (ŝs i, ãa i ). The ynamic loop is an extension of the Slotine an Li controller 20. Fig. 2. Interaction graph of the agents. For simplicity, the gravitational force is neglecte an the transformation matrix R i is assume to be ientity matrix in our simulation. In aition, the stiffness of the environment an the istance from the base to the surface are assume to be k si = 1000 an θ 1i = 0.55, respectively. The initial joint position an initial velocity of the manipulator are set as q i (0) = (i + 1)π 36 (i + 1)π T an q i (0) = T, 0.05i i respectively. The observer gains are selecte as Λ oi = 10, β f1 = 10, β p1 = 10, β f2 = 10 an β p2 = 10. The gains for the controller an upating laws are chosen as Λ i = iag{2, 10}, K i = 100 iag{2, 1}, Γ 0i = 5e 8 an Γ i = 0.001I 3. The sampling perio is selecte as T s = s. Fig. 3 illustrates the transients of force tracking errors. The en-effectors of the robots are not in contact with the environment in the initial perio, so the contact force remains zero uring the early perio until the en-effector touches the surface. Fig. 4 shows the transients of position tracking errors. The force an position of all robotic agents asymptotically converge to the esire values, i.e., the convex hull forme by the leaers. Fig. 5 shows converging process of the force observation errors. The aaptation of ˆθ 0i begins when the ith robotic agent interacts with the environment, as illustrate in Fig. 6. IV. SIMULATION In this section, we use six two-link planar robotic manipulators to valiate our control algorithm. For the ith manipulator, physical parameters of the two links are selecte as m 1 i = i, l 1 Ci = i, r 1 i = i, m 2 i = i, l 2 Ci = i, r 2 i = i, where mīi is the mass of the īth link, līci is the position of the mass center, līci + rīi is the length of the īth link, an the moment of inertia relative to the mass center can be calculate as Iīi = 1 12 m1 i (līci + rīi )2, ī {1, 2}, i {1,, 6}. We assume that there are two leaers. The information graph associate with the leaers an the followers is shown in Fig. 2, where L1 an L2 enote the two virtual leaers an R1,, R6 enote the six robotic followers. We choose w ij = 1 if (j, i) E, an w ij = 0 otherwise, i, j {1,, 8}. The leaers force an position trajectories are f Lk = ( cos(0.1t))k an x plk = 0.1k sin (0.1t), k {1, 2}, respectively. Fig. 3. Force tracking errors of the agents. V. CONCLUSION In this paper, we consier force/position consensus tracking of multiple robotic manipulators uner irecte graphs, where the en-effectors of the robots are in contact with flat compliant surfaces with uncertain stiffness an istance. We propose a istribute aaptive hybri controller to realize both force an task-space position tracking to the convex hull spanne by the leaers. A novel ecentralize force observer is constructe to achieve consensus tracking in the force irection with environment uncertainties, which also helps to get ri of the reliance on the measurements of force erivatives. A ecentralize stiffness estimator is constructe to estimate environment parameters online. Our scheme removes the requirement for

7 186 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 1, NO. 2, APRIL 2014 the estimation of surface istance, an offers a simple an irect solution to force synchronization problem in a cascae framework. Simulations are also performe to examine effectiveness of the propose strategy. Our future work will focus on investigating the feasibility of extening our strategy to the case when the surface-orientation uncertainties as well as the robotic kinematic an ynamic uncertainties are involve. Fig. 4. Fig. 5. Position tracking errors of the agents. Observation errors of the force observers. 4 Chopra N, Spong M W, Lozano R. Synchronization of bilateral teleoperators with time elay. Automatica, 2008, 44(8): Sun D, Shao X Y, Feng G. A moel-free cross-couple control for position synchronization of multi-axis motions: theory an experiments. IEEE Transactions on Control Systems an Technology, 2007, 15(2): Nuno E, Ortega R, Basanez L, Hill D. Synchronization of networks of nonientical Euler-Lagrange systems with uncertain parameters an communication elays. IEEE Transactions on Automatic Control, 2011, 56(4): Cao Y, Ren W, Meng Z. Decentralize finite-time sliing moe estimators an their applications in ecentralize finite-time formation tracking. Systems & Control Letters, 2010, 59(9): Mei J, Ren W, Ma G. Distribute containment control for Lagrangian networks with parametric uncertainties uner a irecte graph. Automatica, 2012, 48(4): Wang H. Task-space synchronization of networke robotic systems with uncertain kinematics an ynamics. IEEE Transactions on Automatic Control, to be publishe 10 Wang L J, Meng B, Wang H T. Aaptive task-space synchronisation of networke robotic agents without task-space velocity measurements. International Journal of Control, DOI: / Careli R, Kelly R, Ortega R. Aaptive force control of robot manipulators. International Journal of Control, 1990, 52(1): Chiaverini S, Siciliano B, Villani L. Force an position tracking: parallel control with stiffness aaptation. IEEE Control Systems Magazine, 1998, 18(1): Villani L, e Wit C C, Brogliato B. An exponentially stable aaptive control for force an position tracking of robot manipulators. IEEE Transactions on Automatic Control, 1999, 44(4): Doulgeri Z, Karayianniis Y. Force/position tracking of a robot in compliant contact with unknown stiffness an surface kinematics. In: Proceeings of IEEE International Conference on Robotics an Automation. Roma, Italy: IEEE, Li Z, Li J, Kang Y. Aaptive robust coorinate control of multiple mobile manipulators interacting with rigi environment. Automatica, 2010, 46(12): Cortesao R, Coutinho F. Environment stiffness estimation with multiple observers. In: Proceeings of IEEE 35th Annual Conference on Inustrial Electronics. Porto, Portugal: IEEE, Craig J J. Introuction to Robotics: Mechanics an Control. Upper Sale River, NJ: Prentice Hall, Kelly R, Santibanez V, Loria A. Control of Robot Manipulators in Joint Space. Lonon: Springer, Lozano R, Brogliato B, Egelan O, Maschke B. Dissipative Systems Analysis an Control. Lonon: Springer-Verlag, Slotine J J E, Li W. Applie Nonlinear Control. Englewoo Cliffs, NJ: Prentice Hall, Lijiao Wang Ph. D. caniate at Beijing Institute of Control Engineering, China Acaemy of Space Technology. Her research interest covers aaptive control an istribute control of Euler-Lagrange systems. Fig. 6. The upating process of ˆθ 0i. REFERENCES 1 Jones C V, Mataric M J. Behavior-base coorination in multi-robot systems. Autonomous Mobile Robots: Sensing, Control, Decision-Making an Applications. Unite States: Marcel Dekker, Inc., Bullo F, Cortes J, Martinez S. Distribute Control of Robotic Networks. Princeton: Princeton University Press, Ren W. Distribute leaerless consensus algorithms for networke Euler-Lagrange systems. International Journal of Control, 2009, 82(11): Bin Meng Senior engineer at Beijing Institute of Control Engineering, China Acaemy of Space Technology. Her research interest covers hypersonic flight vehicle control, characteristic moeling, aaptive control an intelligent control. Corresponing author of this paper.