Robust Controlled Synchronization of Interconnected Robotic Systems

Transcription

1 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July, WeB. Robust Controlled Synchronization of Interconnected Robotic Systems Yen-Chen Liu and ikhil Chopra Abstract In this paper we study controlled synchronization of interconnected robotic systems with dynamic uncertainty. It is first demonstrated that a previously developed algorithm for robust tracking [] renders the closed loop system semi-passive. For the special case of two identical agents and in the absence of communication delays, the proposed control law ensures state synchronization for a large coupling gain. In the general case of heterogeneous agents communicating on balanced graphs, the control law guarantees ultimate boundedness of the synchronization and the tracking errors both in the delayed and the delay free case. umerical examples are presented to verify the efficacy of the proposed control algorithms. I. ITRODUCTIO The problem of cooperative control of multi-agent systems, studied in [], [3], [], [], is important in several applications. Synchronization is an important mechanism of cooperative control. To get a broad perspective on the various notions of synchronization, the reader is referred to []. The concept of controlled synchronization was proposed in [7]. In controlled synchronization, the goal is to state synchronize the agents while tracking a desired trajectory. Since the scheme of [7] required all-to-all coupling and acceleration measurements, the scheme did not scale well with the number of robots. These results were extended in [] using contraction theory where different time-scales for tracking and synchronization were exploited to guarantee synchronization and tracking. The passivity and dissipativity property of dynamical systems has been exploited to study synchronization between interconnected systems in [9], [], []. In [9], it was demonstrated that it is possible to output synchronize nonlinear passive systems, provided the storage function is positive definite and the interagent communication graph is balanced. The results of [9] have been successfully applied to controlled synchronization of mechanical systems in []. However, the control laws developed in [], using passivity property, or [], using contraction theory, primarily rely on the adaptive trajectory tracking algorithm of [3]. As discussed in [] and [], due to unmodeled dynamics the adaptive tracking control algorithm of [3] may not necessarily lead to desirable tracking performance. Therefore, an alternative robust control approach to address the tracking problem was proposed in []. This work was supported by the ational Science Foundation under grant 93 Y. C. Liu is with the Department of Mechanical Engineering, University of Maryland, College Park, 7 MD, USA yliu9@umd.edu. Chopra is with the Department of Mechanical Engineering and The Institute for Systems Research, University of Maryland, College Park, 7 MD, USA nchopra@umd.edu In this paper we exploit the semi-passive closed loop behavior of the control law developed in [] to study synchronization and ultimate boundedness for interconnected robotic systems. It is to be noted that the synchronization results in [9] were developed with the passivity assumption on the system dynamics; however the problem synchronization of semi-passive systems and controlled synchronization were not addressed. Synchronization of semi-passive system has been studied in [] and we use their main result for demonstrating synchronization of two identical robotic systems under an appropriate control law. However, communication delays and the case of heterogeneous agents were not discussed in that paper. The objective of this paper is to study the feasibility of controlled synchronization using the robust tracking control of []. Additionally, the effect of network delays on synchronization is also studied. In the case of two identical agents, the proposed formulation allows the application of the synchronization results of []. For the case of heterogeneous agents governed by Lagrangian dynamics, a control law is developed to guarantee ultimate boundedness of the synchronization and tracking errors. Moreover, numerical simulation results are also provided to illustrate the performance of the proposed controller. This paper is organized as follows. A brief background on semi-passivity, graph theory and properties of Lagrangian equations is discussed in Section II. Subsequently, the main results on robust controlled synchronization are presented in Section III, which is followed by a numerical example in Section IV. Finally, Section V concludes the results of this paper and summarizes future research directions. II. PRELIMIARIES In the sequel we utilize the semi-passivity property [] of dynamical systems. Definition [] The nonlinear system Σ is said to be semipassive if there exists a C storage function V : R n R + and a function S : R n R such that for any initial conditions x and any admissible input u, the following inequality Vx u T y Sx holds for all t, where the function Sx is nonnegative outside some ball in the sense that β > s.t. x β Sx The agents in this paper are modeled as robotic systems. Following [], in the absence of friction, the dynamics of //$. AACC 3

2 a n-link robotic with revolute joints can be described as Mq q+cq, q q+gqu 3 where q R n is the vector of generalized configuration coordinates, u R n is the vector of generalized forces acting on the system, Mq R n n is a symmetric, positive definite matrix, Cq, q R n is the vector of Coriolis/Centrifugal forces and gq H q Rn is the gradient of the potential function Hq. The above nonlinear equations of motion exhibit certain fundamental properties due to their Lagrangian dynamic structure []. Property.: The matrix Mq is symmetric positive definite and there exists positive constant λ m and λ M such that λ m I Mq λ M I Property.: Under an appropriate definition of the matrix C, the matrix Ṁ -C is skew symmetric. Property.3: The Lagrangian dynamics are linearly parameterizable which gives that Mq q+cq, q q+gqyq, q, qθ where Θ is a constant p-dimensional vector and Y is an n p matrix of known functions of the generalized coordinates and their higher derivatives. Communication topology and information exchange between agents can be represented as a graph. Some basic terminology and definitions from graph theory [], which are sufficient to follow the subsequent development, are outlined below Definition By a graph G we mean a finite set V G {v i,...,v }, whose elements are called nodes or vertices, together with set EG V V, whose elements are called edges, which is an ordered pair of distinct vertices. An edge v i,v j is said to be incoming with respect to v j and outgoing with respect to v i and can be represented as an arrow with vertex v i as its tail and vertex v j as its head. The in-degree of a vertex v G is the number of edges that have this vertex as a head. Similarly, the out-degree of a vertex v G is the number of edges that have this vertex as the tail. If the in-degree equals the out-degree for all vertices v V G, then the graph is said to be balanced. If, for all v i,v j EG, the edge v j,v i EG then the graph is said to be undirected. Otherwise, it is called a directed graph. A path of length r in a directed graph is a sequence v,...,v r of r+ distinct vertices such that for every i {,...,r }, v i,v i+ is an edge. A weak path is a sequence v,...,v r of r+ distinct vertices such that for each i,...,r either v i,v i+ v i+,v i is an edge. A directed graph is strongly connected if any two vertices can be joined by a path and is weakly connected if any two vertices can be joined by a weak path. In the sequel we assume that communication topology of the interconnected robotic system is balanced and strongly connected. III. ROBUST COTROLLED SYCHROIZATIO Consider heterogeneous agents whose uncertain dynamics are given by 3. The objective of this paper is to develop a control scheme for studying state synchronization of the agents while tracking a time-varying trajectory q d t. While it is possible to use adaptive tracking [3] or robust control [] algorithms for the individual agents to ensure that they track a feef trajectory and consequently synchronize, additional coupling between the agents [7], [] improves the transient tracking performance. Furthermore, when using the robust control methods, for example [], the tracking error does not approach the origin and hence mutual coupling between the agents can improve synchronization between the agents. An interesting comparison between the adaptive and robust control methods for tracking has been provided in [], and in this paper we study the algorithm of [] when there is mutual coupling between the agents. The desired trajectory q d t is assumed to be twice differentiable and hence the signals q d t, q d t are well defined and are assumed to be bounded. We first characterize an interesting passivity property associated with the robust tracking algorithm of []. A. Semi-Passivity of the Robust Control Algorithm The Euler-Lagrange equations of motion for an n-link robot 3 are linearly parameterizable Property.3. In this paper, it is assumed that the parameter vector Θ is unknown and there exists Θ R p and ρ R +, such that Θ : Θ Θ ρ By using Property.3, the Lagrangian dynamics for Θ can be written as M q q+c q, q q+g qyq, q, qθ where M, C, and g are the matrices corresponding to Θ. Following [], define a nominal control vector u as u t M qat+c q, qvt+g q K t st Yq, q,v,aθ K t st 7 where vt, at, and st are given as qtqt q d t, vt q d t Λ qt at q d t Λ qt, st qt+λ qt where qt is also denoted as the tracking error. The matrices K t and Λ are both n n positive definite diagonal matrices. Let the control input be given as ut u t+yq, q,v,aδθt+τt 9 Yq, q,v,aθ + δθt K t st+τt where Θ is the fixed nominal parameter vector and δθt is an additional control input defined as { ρ Y T s δθ Y T s, if Y T s >ε ρ ε Y T s, if Y T s ε 3

3 Let the estimate of parameter vector ˆΘt : Θ + δθt, thus ˆΘt ΘΘ + δθt Θ Θ+δΘt where Θ Θ Θ. Then the closed loop system can be written as, Mqṡ+Cq, qs+k t s Yq, q,v,a Θ+δΘ+τ q Λ q+s If we define Xt[st qt] T as the state for this system, according to [7], the state Xt is related to the vector Xt[ qt qt] T by a linear diffeomorphism, Xt T Xt, where T R n n is a nonsingular positive definite matrix, which is defined by [ ] T In n Λ Ø n n I n n where I n n is an n n identity matrix, and Ø n n is an n n zero matrix. Lemma 3.: Given ε > and < γ <, the dynamical system defined by and is semi-passive with τ,s as the input-output pair. Proof: Consider a positive definite storage function for the system as VX st Mqs+ q T Λ T K t q Differentiating VX along the trajectories and using Property., VXs T Mṡ+ st Ṁs+ q T ΛK t q s T Y Θ+δΘ Cs K t s+τ+ st Ṁs+ q T ΛK t q q T K t q q T Λ T K t Λ q+s T Y Θ+δΘ+s T τ 3 Following [], from, if Y T s >ε, then Y T s T Θ+δΘ Y T s T Θ ρ Y T s Y T s Y T s Θ ρ < The above equation is derived from the Cauchy-Schwartz inequality and the assumption of Θ. On the other hand, if Y T s ε, then Y T s T Θ+δΘ Y T s T ρ Y T s Y T s + δθ Y T s T ρ Y T s Y T s ρ ε Y T s ρ Y T s ρ ε Y T s The maximum value of the above equation is ερ/ when Y T s ε/. Therefore, the derivative of storage function 3 can be written as VX q T K t q q T Λ T K t Λ q+ερ/+s T τ α 3 X + ερ/+s T τ where the constant α 3 λ min Q denotes the minimum eigenvalue of Q R n n, which is defined by [ ] K Q t Ø n n Λ T 7 K t Λ Ø n n Recalling that XtT Xt, the inequality can be rewritten as VX α 3 λ min T T T X + ερ/+s T τ Define SX : α 3 λ min T T T X ερ/ see. Then, SX γα 3 λ min T T T X ερ/ +γα 3 λ min T T T X γα 3 λ min T T T X Xt β 9 where β : ερ and < γ <. Therefore, the γα 3 λ min T T T system is semi-passive in the sense of and. B. Robust Trajectory Tracking with Interagent Coupling Assuming that the dynamics for the individual agents are described by 3 and using the preliminary control law 9, the closed loop dynamics for the individual system can be written as M i q i ṡ i +C i q i, q i s i + K ti s i Y i q i, q i,v i,a i Θ i + δθ i +τ i q i Λ q i + s i i,..., Let the coupling control law for the individual agents be given as τ i tk s s j t s i t i,..., where, for simplicity, the synchronizing gain matrix K s is taken as a positive constant. Moreover, the suffix j i denotes the neighbor communicating to the i th agent, and we denote n i as the number of neighbors connecting to the i th agent. The agents are said to output synchronize if lim s it s j t i, t It is to be noted that s j s i q j + Λ q j q i + Λ q i q j + Λq j q i + Λq i ė ij + Λe ij 3 where e ij tq j t q i t denotes the synchronization error between any two agents. Equation 3 represents an exponentially stable linear system with the input s j t s i t. As shown, for example, in [], it follows that if s j t s i t is a signal that converges asymptotically to zero and e ij t is bounded, then lim e ijt i, t Hence, if the agents output synchronize in the sense of, then they state synchronize as well. If we assume two identical agent whose dynamics are described by, then according to Lemma 3. the control law renders the agents semi-passive. It is easy to verify 3

4 that the agent dynamics satisfy the conditions of Theorem in [] and hence with coupling control, where K s is sufficiently large see [] for details, the agents output synchronize in the sense of. Following 3, the two agents additionally state synchronize. However, we are interested in the more general case of heterogeneous dynamic agents and this class of systems is considered in the rest of the paper. The equation 3 demonstrates that the coupling between the agent s outputs results in coupling between their states. Define E i t{e ij j i } as the synchronization state of the i th agent. Let Z i t[s i t q i t E i t] T denote the state of the i th agent, and Z i t[ qt q i t E i t] T. The state for the interconnected mechanical systems is defined by Zt[Z t... Z t] T. Following [7], the agent state Zt is related to the vector Zt[ Z t... Z t] T by a linear diffeomorphism, ZtT Zt, where T is a nonsingular positive definite matrix. The above discussion leads us to the main result of the paper Theorem 3.: Given ε >, < γ <, consider the dynamical system described by,, and. If the interagent communication graph is balanced and strongly connected, then the synchronization errors and all solutions of the coupled dynamical system are uniformly ultimately bounded. Proof: Consider a positive definite storage function for the agent system as where V i Z i VZV Z + +V Z i V i Z i s T i M i q i s i + q T i Λ T K ti q i + K s e T ijλe ij oting Property., there exists K functions α a,α b such that α a Z VZ α b Z Taking the derivative of the storage function along trajectories, and substituting the control law yields V i V i i q T i K ti q i q T i Λ T K ti Λ q i +s T i Y i Θ i + δθ i +s T i τ i + K s i e T ijλė ij i q T i K ti q i q T i Λ T K ti Λ q i + s T i Y i Θ i + δθ i +K s i s j s i T s i + K s i e T ijλė ij 7 As the information exchange graph is assumed to be balanced, i s T i s i i s T i s i + i s T j s j From the above equation and using 3, the derivative of storage function can be rewritten as V i K s i + q T i K ti q i q T i Λ T K ti Λ q i + s T i Y i Θ i + δθ i i i i ė ij + Λe ij T ė ij + Λe ij e T ijλė ij q T i K ti q i q T i Λ T K ti Λ q i K s e T ijλ T Λe ij s T i Y i Θ i + δθ i K s α i Z i + i i ė T ijė ij ε i ρ i / 9 where the second term is due to the control input see Proof of Lemma 3., and α i : λ min Q i with K ti Ø n n Ø n n Λ T K ti Λ Ø n n Q i K. Ø s n n ΛT Λ Ø n n 3... Ø.. n n where Q i R +n in +n i n with n i denoted as the number of neighbors connecting to the i th agent. It is to be noted that strong connectivity of the communication graph is required to achieve the first term 9 in the derivative of the storage function. Let K α : minα i, i,...,, and Γ : i ε iρ i /. Hence, the inequality 9 can be rewritten as VZ K α Z + Γ 3 Recalling that ZtT Zt and define λ T,min λ min T T T VZ K α λ T,min Z + Γ γk α λ T,min Z γk α λ T,min Z + Γ γk α λ T,min Z Zt β s 3 where β s : Γ γk α λ T,min.AsK α is bounded away from zero, therefore VZ<, Zt β s. From Property. and inequality, we can obtain that VZ VZ i i ZT i P i Z i ZT i P i Z i i i λ min P i Z i K min P Z λ Max P i Z i K Max P Z where P i can be defined according to V i Z i, and KP min : minλ min P i, KP Max : Maxλ Max P i, i,...,. The class K functions can be taken as α a Z KP min Z and α b Z KP Max Z Thus, the ultimate bound for the system is given by b αa KP α b β s Max βs ΓKP Max. K min P γk α λ T,minK min P 37

5 Hence, using Theorem. [9], if the mechanical systems are connected with balanced graph, the tracking and synchronization errors of this system are uniformly ultimately bounded. C. Communication Delay In the presence of communication delays between the agents, let the coupling controls be given as τ i tk s s j t T ji s i t i,..., 33 The state for this system is given as Z t Zϕ;ϕ [t T,t] where T maxt ij i, j. It is possible to demonstrate that the trajectories are ultimately bounded in this case as well, and this is outlined in the next result. Theorem 3.3: Given ε >, consider the dynamical system described by,, and 33. If the interagent communication graph is balanced and strongly connected, then the synchronization and tracking errors for the interconnected system are bounded. Proof: Consider a positive definite storage functional for this system as VZ t V Z + +V Z + K s i t t T ji s T jws j wdw where the individual storage functions V i Z i are given by. Differentiating along trajectories of the system yields VZ t i q T i K ti q i q T i Λ T K ti Λ q i + s T i Y i Θ i + δθ i +s T i τ i + K s s T j s j s j t T ji T s j t T ji 3 Exploiting the balanced graph assumption and [9] and the control input see Proof of Lemma 3., the derivative of the storage function can be rewritten as VZ t i q T i K ti q i q T i Λ T K ti Λ q i + ε i ρ i / K s s j t T ji s i T s j t T ji s i 3 It is evident from the above equation that β o s.t for Z t β o, VZ t. As VZ t is positive definite, this implies that the state vector is ultimately bounded. IV. UMERICAL EXAMPLE umerical simulations are presented in this section to demonstrate the efficacy of the proposed scheme. In the simulations, four agents modeled as nonlinear DOF planar robots, are interconnected based on the topology shown in Figure. The communication topology in the example is balanced and strongly connected. The reader is referred to Chapter 7 in [] for the dynamic model used in this simulation. During the simulation, the agents track a timevarying trajectory, q d [.7sint+.sint sin.t.sint] T. The agents parameters are listed in Table I. 3 Fig.. Balanced topology for the interconnected mechanical system. TABLE I SIMULATIO PARAMETERS FOR DOF PLAAR ROBOT Agent Mass LinkLength Θ st.,..,..7,.99,.,.9,.3 nd.,.,.,.,.,.,. 3 rd.9,..3,..9,.7,.,.,. th,.,. 7.,.,.,.,. TABLE II COTROL PARAMETERS FOR SYCHROIZATIO SIMULATIO Agent Θ ρ st.,.,.,.9,.. nd.,.7,.,., rd.7,.97,.3,.3,.7.3 th 7.,.,.7,.,..9 The nominal parameters Θ and ρ are listed in Table II, where ρ Θ Θ. Even though a lower ε leads to a smaller ultimate bound [], in practice the value of ε is chosen keeping in mind possible vibrations and chattering in the control signal []. The performance improvement obtained by the proposed controlled synchronization algorithm can be demonstrated by selecting a higher ε and lower tracking gains in the example. The control gains for the following simulation are K t, Λ and ε. In the absence of mutual coupling, as seen in Figure, the robots do not follow the desired trajectory solid black line. The tracking errors in the closed loop system are shown in Figure 3, where both the tracking and synchronization errors are large in the absence of controlled synchronization. When the proposed scheme is implemented with constant communication delays and K s, T.sec, T 3.sec, T 3.sec, T.9sec, and T.sec, the tracking performance improves as shown in Figure. Figure demonstrates that the agents achieve better performance with lower tracking and synchronization errors. V. COCLUSIOS AD FUTURE WORK In this paper controlled synchronization of interconnected mechanical systems using robust tracking control with mutual coupling between the agents was studied. Using a robust control law developed in [], it was demonstrated that the proposed algorithm rendered the closed loop system semipassive []. For the special case of two identical agents, in the absence of communication delays and for a large coupling gain, the proposed control law ensured state synchronization of the interconnected systems. In the general case of heterogeneous agents with dynamic uncertainty, and which are communicating on balanced graphs, the proposed control

6 law lead to ultimate boundedness of the synchronization and tracking errors both in the delayed and the delay free case. Furthermore, numerical examples were presented to validate the proposed algorithm. Simulations indicated that the proposed algorithm can reduce the synchronization errors between the agents when compared to the case of only using the robust control scheme for tracking. Future work entails demonstrating synchronization for the general case as well exploring the mechanism for the effects of delays on the tracking performance. 3 Position rad Time sec Time sec Position rad R EFERECES Fig.. Configuration of the agents using only robust tracking control. Tracking error rad Synchronization error rad Time sec Fig. 3. Time sec Tracking and synchronization errors without synchronization. Position rad Time sec Time sec Position rad Fig.. Configuration of the agents with synchronization and delays. Synchronization error rad Tracking error rad Time sec Fig.. Time sec Agents errors with synchronization and delays. [] M. W. Spong, On the robust control of robot manipulators, IEEE Transactions on Automatic Control, vol. 37, no., pp. 7 7, 99. [] A. Jadbabaie, J. Lin, and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control, vol., no., pp. 9, Jun. 3. [3] W. Ren and R. W. Beard, Consensus seeking in multi-agent systems using dynamically changing interaction topologies, IEEE Transactions on Automatic Control, vol., no., pp., May. [] R. Olfati-Saber and R. Murray, Consensus problems in networks of dynamic agents with switching topology and time-delays, IEEE Transactions on Automatic Control, vol. 9, pp. 33, Sep.. [] D. J. Lee and M. W. Spong, Stable flocking of multiple inertial agents on balanced graphs, IEEE Transactions on Automatic Control, vol., no., pp. 9 7, 7. [] H. ijmeijer and A. Rodriguez-Angeles, Synchronization of mechanical systems. World Scientific, River Edge, J, 3. [7] A. Rodriguez-Angeles and H. ijmeijer, Mutual synchronization of robots via estimated state feedback: a cooperative approach, IEEE Transactions on Control Systems Technology, vol., no., pp., July. [] S.-J. Chung and J.-J. E. Slotine, Cooperative robot control and concurrent synchronization of lagrangian systems, IEEE Transactions on Robotics, vol., no. 3, pp. 7, Jun. 9. [9]. Chopra and M. Spong, Passivity-based control of multi-agent systems, in Advances in Robot Control: From Everyday Physics to Human-Like Movements, S. Kawamura and M. Svinin, Eds. Springer Verlag,, pp [] A. Y. Pogromsky, Passivity based design of synchronizing systems, International Journal of Bifurcation and Chaos, vol., pp. 9 39, 99. [] G.-B. Stan and R. Sepulchre, Analysis of interconnected oscillators by dissipativity theory, IEEE Transactions on Automatic Control, vol., no., pp. 7, 7. []. Chopra and Y. C. Liu, Controlled synchronization of mechanical systems, in ASME Dynamic Systems and Control Conference, Oct.. [3] J.-J. Slotine and L. Weiping, Adaptive manipulator control: A case study, Automatic Control, IEEE Transactions on, vol. 33, no., pp. 99 3, ov 9. [] R. Ortega, A. Loria, P. J. icklasson, and H. Sira-Ramirez, Passivitybased control of Euler-Lagrange Systems:Mechanical, Electrical and Electromechanical Applications, ser. Comunications and Control Engineering Series. Springer Verlag, London, 99. [] M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control. ew York: John Wiley & Sons, Inc.,. [] C. Godsil and G. Royle, Algebraic graph theory, in Springer Graduate Texts in Mathematics o. 7. Springer,. [7] M. W. Spong, R. Ortega, and R. Kelly, Comments on adaptive manipulator control: a case study by J. Slotine and W. Li, IEEE Transactions on Automatic Control, vol. 3, no., pp. 7 7, Jun. 99. [] E. Sontag, A remark on the converging-input converging-state property, IEEE Transactions Automat. Control, vol., no., pp. 33 3, 3. [9] H. K. Khalil, onlinear systems. ew Jersey: Prentice Hall,. [] A. Jaritz and M. W. Spong, An experimental comparison of robust control algorithms on a direct drive manipulator, IEEE Transactions on Control Systems Technology, vol., no., pp. 7, ov