Adaptive position feedback consensus of networked robotic manipulators with uncertain parameters and communication delays
2015; Institution of Engineering and Technology; Volume: 9; Issue: 13 Linguagem: Inglês
10.1049/iet-cta.2014.1125
ISSN1751-8652
Autores Tópico(s)Adaptive Control of Nonlinear Systems
ResumoIET Control Theory & ApplicationsVolume 9, Issue 13 p. 1956-1963 Research ArticlesFree Access Adaptive position feedback consensus of networked robotic manipulators with uncertain parameters and communication delays Lijiao Wang, Lijiao Wang Beijing Institute of Control Engineering, Beijing, 100190 People's Republic of ChinaSearch for more papers by this authorBin Meng, Corresponding Author Bin Meng [email protected] Science and Technology on Space Intelligent Control Laboratory, Beijing Institute of Control Engineering, Beijing, 100190 People's Republic of ChinaSearch for more papers by this author Lijiao Wang, Lijiao Wang Beijing Institute of Control Engineering, Beijing, 100190 People's Republic of ChinaSearch for more papers by this authorBin Meng, Corresponding Author Bin Meng [email protected] Science and Technology on Space Intelligent Control Laboratory, Beijing Institute of Control Engineering, Beijing, 100190 People's Republic of ChinaSearch for more papers by this author First published: 01 August 2015 https://doi.org/10.1049/iet-cta.2014.1125Citations: 13AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract In this study, the authors address the leaderless consensus problem of multiple robotic manipulators with communication delays. Taking dynamic uncertainties of the robotic agents into consideration, they perform the distributed adaptive position feedback control scheme design for strongly connected information topology, where the communication is subject to constant and bounded time delays. The novel distributed sliding observer is first proposed such that the joint velocities can be estimated online. Then, the distributed adaptive controller with dynamic parameter updating is constructed based on the observed information. The convergence of consensus errors is proved with Lyapunov stability analysis tool. Finally, simulations are performed on networked robotic manipulators to validate the effectiveness of the theoretical approach. 1 Introduction Synchronisation and distributed control of multi-agent systems have gained considerable attention in recent years [1–7]. As one important branch for multi-agent control, consensus of multiple Euler–Lagrange systems shows great prospective advantage in many applications, for example, teleoperation, multiple robotic manipulators achieving a cooperative task, formation of multiple spacecraft, attitude alignment of clusters of satellites and so on [8]. The distinct challenge for control of networked Euler–Lagrange agents lies in the non-linearity and the uncertainties, which makes it difficult to apply directly the distributed control strategies developed for linear systems. In this paper, we concentrate on the leaderless consensus problem of multiple robotic manipulators modelled as the full-actuated Euler–Lagrange systems, where the objective is to drive all the robotic agents reach an agreement on certain variables of interest in the absence of any leaders in the network. A number of researchers have devoted to adaptive leaderless consensus control of multiple Euler–lagrange systems [8–13]. The backstepping-based adaptive technique is employed in [9] to solve consensus problem of multiple robotic manipulators interacting on undirected graphs. Ren [10] proposes three distributed leaderless consensus algorithms for networked Euler–Lagrange systems on undirected graphs. The work in [8] introduces the cascade framework and provides a simple and direct solution for consensus problem under directed graphs containing a spanning tree. In real implementations, the communication time delays among the agents may lead to instability of an otherwise stable system, and thus make the consensus problem more challenging. Passivity-based adaptive scheme is utilised in [11, 12] to achieve asymptotic consensus of bilateral teleoperators in the presence of dynamic uncertainties and time delays. The passivity-based strategy is extended to multiple Euler–Lagrange systems in [13] and asymptotic consensus is achieved on balanced digraphs independent of constant time delays. The work in [5] proposes a novel extended Slotine and Li controller to handle time delays and uncertainties for networked Euler–Lagrange systems on strongly connected graphs, where asymptotic convergence of consensus errors is analysed via frequency-domain analysis tools. In practical applications, nearly all commercially available robotic manipulators do not have link velocity sensors. Even for those equipped with velocity sensors, velocity measurements are often contaminated by high levels of noise which constrains the system performance [14]. Then, various velocity-free control strategies are developed for a single robotic manipulator. Sliding observer has been extensively used to estimate the velocities from the partially known state variables [14, 15]. The non-linear sliding observer integrating the full robot dynamics is constructed in [15] to estimate the joint velocity in a tracking task, and further extended to the case when the robot dynamics is uncertain in [14]. A model independent discontinuous output feedback controller is proposed in [16] for a class of uncertain mechanical systems whose dynamics are first-order differentiable, where a novel filter design is developed and semi-global asymptotic tracking is achieved. When it comes to synchronisation of multiple robots, the local interactions between the agents imply the challenges that are not considered in the design of tracking controllers for a single agent. An observer-based leader-following consensus of multiple agents is investigated in [17] based on a low-gain output feedback method to compensate for input saturation, applicable to the case when the robotic dynamics can be simplified to be linear. Position synchronisation of two or more robots under undirected graphs is researched in [18], where coupling errors are introduced to create interconnections that render mutual synchronisation of the robots and a set of model-based non-linear observers are designed such that only position measurements are used in the consensus tracking controller. However, the dynamic parameters are assumed to be exactly known and the desired tracking trajectory is required to be available to all manipulators. Taking uncertain dynamics and disturbances into consideration, an adaptive neural network consensus tracking controller is designed in [19] and a high gain observer is utilised to estimate the joint velocities of the robotic manipulators, where the communication graph of the robots is required to contain a spanning tree with the root robot having access to the desired trajectory. To the best of our knowledge, no efforts have been made on joint-position leaderless consensus of networked robotic manipulators without joint velocity measurements. Especially, the absence of a common reference trajectory, the asymmetric Laplacian matrix as well as the communication delays propose the challenge for our design and make it not a straightforward solution of the existing literature. In this paper, we solve the leaderless consensus problem of multiple robotic manipulators with dynamic uncertainties and communication delays, provided that joint velocities of the agents are not measurable. We give the distributed adaptive position feedback control scheme for strongly connected information topology. A novel distributed sliding observer is first proposed such that the joint velocity can be estimated online. By use of the observed variables, an adaptive controller is then designed to deal with the uncertainties and the communication delays. The convergence of consensus errors is proved with Lyapunov stability analysis tool. Finally, simulations are performed to validate the effectiveness of the theoretical approach. Compared with the existing literature, the major novelty of the our work lies in the proposal of a distributed sliding observer relying on non-linear couplings among the agents, which eliminates velocity measurements in the consensus problem. In contrast to the sliding observer in [14] for the tracking of a single manipulator, our observer design bears a different form and can solve the velocity-free leaderless consensus problem under directed graphs. In contrast to the work in [8] where an extended Slotine and Li distributed controller is proposed, our control protocol removes the reliance on joint velocities via introducing the observed information in the controller design. 2 Preliminaries and problem formulation 2.1 Modelling We consider m robotic agents labelled as agents 1 to m. The dynamic model of the i th agent can be written as the following Euler–Lagrange equation [20, 21] (1)where is the joint variable, is the symmetric and positive definite inertia matrix, is the coupled centripetal and Coriolis matrix, is the gravitational torque, and is the exerted joint torque. The properties of the dynamics can be summarised as follows [14, 20, 21]. Property 1.For some positive constants , , and , we have , where In denotes the n × n identity matrix, for all vectors ξ1, ξ2, and . Property 2.For ∀ξ1, ξ2, ξ3, , Ci(ξ1, ξ2)ξ3 = Ci(ξ1, ξ3)ξ2 and Ci(ξ1, ξ2 + ξ4)ξ3 = Ci(ξ1, ξ2)ξ3 + Ci(ξ1, ξ4)ξ3. Property 3.The dynamics of the agent is linearly parametric with respect to a group of unknown constant dynamic parameters , that is, (2)where is referred to as the dynamic regressor matrix. Property 4.The matrix is skew symmetric with an appropriate choice of . Introduce xi = qi and . Then, (1) can be rewritten in the state-space representation as (3)where ηi = − Hi(xi)−1 (Ci(xi, vi)vi + gi(xi)) + Hi(xi)−1 τi. 2.2 Graph theory A directed graph (digraph) is used in this paper to describe network communication topology among the robotic agents. Here, we first introduce some basic notations on digraph theory [22]. Denote the vertex set as and the edge set . The neighbours of agent i constitute a set . The adjacency matrix associated with the graph is defined as wij > 0 if (j, i) ∈ ℰ and wij = 0 otherwise. Assume the self edges do not exist, that is, wii = 0. Then, define the out-degree of node i as The Laplacian matrix associated with the digraph is defined as A digraph contains a spanning tree if there exists a node such that there exists a directed path from this node to every other node. Furthermore, a digraph is strongly connected if there is a directed path connecting any two arbitrary nodes of the graph. The properties of the digraphs are represented in the following lemmas. Lemma 1 [2].If the communication topology among the m robotic agents is directed with a spanning tree, L has a simple zero-eigenvalue and all other eigenvalues of L have positive real parts, and is the only vector in the kernel of L. Moreover, there exists a non-negative vector , that is, γi ≥ 0, , such that γT L = 0. In addition, if the communication topology is strongly connected, the vector γ is positive, that is, γi > 0, . Lemma 2 [23].For a digraph with Laplacian matrix L, all the eigenvalues of L are located in the disk where D denotes the maximum node out-degree of the digraph, that is, . Lemma 3 [8, 24, 25].For a connected graph, there exists a matrix with the following properties: (P1) SST = Im −1; (P2) ; (P3) S 1m = 0m −1; (P4) (S ⊗ In)y = 0n (m −1) ⇔ y = 1m ⊗ yc for some ; and (P5) −SLST is Hurwitz. Lemma 4 [8].Let where , , p1 is the Laplace variable, Tj is a non-negative constant, β is a positive constant, and is the matrix with all elements equals to zero except one off-diagonal element equals to one of the wij such that holds, where , and denotes the number of the elements in the set . If the digraph is connected, the asymptotic convergence of u gives rise to as t → ∞. 2.3 Problem formulation In this work, for the m robotic manipulators represented by (1) with uncertain dynamic parameters adi and communication delays, our control objective is to design the distributed adaptive position feedback control torque τi such that the positions of all the agents reach consensus, that is, qi − qj → 0 and as t → ∞, provided that the measurements of joint velocity are not available, ∀i, . 3 Main result In this section, we give the distributed controller design and convergence proof. For the sake of clarity of exposition, we first deal with the consensus problem in the absence of time delays. 3.1 Case without time delays In this part, we consider the problem of the adaptive position feedback leaderless consensus of (1) with uncertain parameters for the case without time delays. First, the decentralised sliding observer is proposed to estimate the joint velocities online. Then, the distributed adaptive controller with dynamic parameter updating is constructed based on the observed information. Finally, the convergence of consensus errors is proved. We first design the sliding observer. For the i th agent represented by (3), we give the reference velocity (4)where α > 0 is an adjustable constant. Then, introduce the sliding vector as (5)Differentiating (4) with respect to time, we obtain (6)Since the measurement of vi is not available, we introduce the following estimated variables (7) (8)where , and are the estimates of , si and vi, respectively. Then, we proceed to introduce the auxiliary variables (9)where , , sgn(·) is the sign function, , is the estimate of xi, updated by the sliding observer (12), and is also given in (12). By use of Property 1 and (9), we obtain (10)Let (11)where . Then, it is time for us to propose the following decentralised sliding observer (12)where | · | is defined as , , , is the estimate of Hi (xi), is the estimate of , , , , are adjustable symmetric and positive definite matrices, and D is given in Lemma 2. Now, we introduce the dynamic regressor matrix (13)where is the estimate of adi, and is the estimate of . Note that and can be calculated by and On the basis of the variables defined above, we give the adaptive control law (14)with updated by (15)where is an adjustable symmetric and positive definite matrix. Subtracting (3) from (12), we get the observer error dynamics (16)(17)where . Here, (16) has the following property. Lemma 5 [14].For (16), if , we can get that the set is invariant. The region characterised by , is known as the sliding patch . According to Lemma 5, (16) and Fillippov's solution concept, the following equation holds in the sliding patch [14] (18)Here, Filippov's solution concept indicates that the dynamics on the switching surface is an average of the dynamics on each side of the discontinuity surface. Substituting (18) into (9), we get in the sliding patch (19)Before performing the stability analysis, we introduce the following lemma. Lemma 6 [26].Let , where G (p1) is an m1 × n1 strictly proper, exponentially stable transfer matrix in the Laplace variable p1. Then, implies , , is continuous and as t → ∞. Now, it is time for us to give the following result. Theorem 1.Consider the m robotic agents represented by (1) with uncertain dynamic parameters adi. The observer (12) and the adaptive control law (14) and (15) give rise to the asymptotic consensus of the agents, that is, qi − qj → 0 and as t → ∞, , provided that the agents are interconnected on directed graphs containing a spanning tree and the following conditions hold. (C 1) The initial value of is set to satisfy . (C 2) The initial value of e is chosen to satisfy ∥e (0) ∥ 2 ≤ [{λmin (A)}/{λmax (A)}] ∥ Λ1 ∥ 2, where e is the stack vector of ei, , A = diag[A1, …, Am], and , is chosen as , where is an adjustable positive constant. Proof.Substituting (14) into (1) and then utilising (2), (5)–(7) and (13), we derive the closed-loop dynamics (20)where the property is utilised, and . Reformulate (20) in the matrix form as (21)where ⊗ denotes the Kronecker product, s, and are the column stack vectors of si, and , respectively, ; H = diag[H1 (x1), …, Hm(xm)], , C = diag [C1 (x1, v1), …, Cm(xm, vm)], Yd = diag [Yd 1, …, Ydm], and K = diag[K1, …, Km].If conditions (C1) and (C2) hold, we get from Lemma 5 that the state is initially in the sliding patch . Our proof is then divided into two steps, where the first is to show the stability in the sliding patch and the latter is to prove that the state dose not leave the sliding patch if (C1) and (C2) are satisfied. Define the Lyapunov-like function (22)where Γd = diag[Γd 1, …, Γdm]. Differentiating V with respect to time and utilising (21) and Property 4, we have (23)Substituting (15), (17) and (19) into (23), we obtain where . When , we simply obtain (24)Now, we consider the case . By Lemma 2, (18) and (C2), we have Therefore, we obtain (25)From (22), (24) and (25), we derive (26)Now, we substitute (4) into (5) and rewrite it in the matrix form as (27)Since the agents are interconnected on connected graphs, we can introduce the coordinate transform using the matrix S, the properties of which are given in Lemma 3. Then, (27) can be reformulated as (28)By (P5), (26) and (28), we get z ∈ ℒ2 ∩ ℒ∞, and ∥z ∥ →0 from Lemma 6, further leading to by use of (4). Then, is guaranteed by (5). Moreover, ∥z ∥ →0 means qi − qj converges to zero according to (P4). We further have from (6). Then, is derived by (20), yielding si → 0 as t → ∞.Now, we proceed to prove that all trajectories with initial conditions (C1) and (C2) guarantee that the overall system dynamics remains in the sliding patch for all t > 0. From (22) and (25), we derive (29)By (C2), we have (30)which gives rise to , further implying that the state does not leave the sliding patch for all t > 0 from Lemma 5. □ Remark 1.Since the joint positions can be measured, it is always possible for us to satisfy (C1) via setting . Additionally, Condition (C2) gives the constraint for the initial state estimation error, the bound of which relies on the global information of the agents. However, in practical applications, we can utilise trial and error to ensure this condition without knowing the exact value of the bound. 3.2 Case with time delays In this part, we will take communication delays among the agents into consideration based on the design procedure in Section 3.1. The existence of time delays hampers the use of the compact form (21) and (27), making the strategy in the preceding subsection no longer applicable. A novel delay-robust adaptive control strategy based on a distributed sliding observer is then proposed to conquer the challenge caused by time delays. We introduce the distributed consensus algorithm with time-delayed information in the neighbourhood. The reference velocity, the reference acceleration and the estimated reference velocity become (31) (32) (33)where Tij denotes the constant and bounded time delay from agent j to agent i. The sliding observer for joint velocity vi is modified as (34)On the basis of the observed variables, we modify the distributed controller as (35)with the updating law (36)Then, the convergence of consensus errors for the time-delay case is obtained. Theorem 2.Consider the m leaderless robotic agents represented by (1) interacting on strongly connected graphs. The agents are subject to constant and bounded communication delays Tij as well as uncertain dynamic parameters adi. Suppose condition (C1) and the following conditions hold. (C 3) The initial value of e is chosen to satisfy , where e is the stack vector of ei, , A = diag[(A1), …, (Am)], , and , where γ is defined in Lemma 1. (C 4) The gain for the controller is set as , where is defined in Property 1. Then, observer (34), controller (35) and updating law (36) give rise to the asymptotic consensus of the agents, that is, qi − qj → 0 and as t → ∞. Proof.We first substitute (35) into (1) and utilise (2), (5), (13), (32) and (33) to get the closed-loop dynamics as (37)where and . Subtracting (3) from (8), we get the observer error dynamics (38)(39)If conditions (C1) and (C3) hold, we get from Lemma 5 that the state is initially in the sliding patch , where (18) holds. Furthermore, we can derive (19). Now, we first analyse the stability in the sliding patch. On the basis of (20) and (39), we define the Lyapunov-like function for the overall system (40)where (41) (42)and we have γi > 0 from Lemma 1 with the information topology being strongly connected. Differentiate V1i with respect to time and then substitute (5), (19), (32), (36) and (37) into . Utilising Property 4, we obtain (43)It is easy to obtain from the definition of and Property 1 that (44)Differentiating V2i with respect to time and substituting (18) and (39) into , we have (45)Then, combining (43)–(45), we obtain (46)where γ = [γ1, …, γm]T, F1 = [∥ v1 ∥ 2, …, ∥ vm ∥ 2]T and . According to Lemma 1 and (C4), we further obtain (47)where .Now, we proceed to prove that all trajectories with initial conditions (C1) and (C3) will remain in the sliding patch for all t > 0. From (47) and (40), we derive (48)where vj(t) = 0 and for t < 0 is assumed. By (C3), we have (49)which implies , and thus the state does not leave the sliding patch for all t > 0.From (40) and (47), we get that si ∈ ℒ2 ∩ ℒ∞, , and . Thus, we obtain vi ∈ ℒ∞ from (5) and from (33), which further leading to the boundedness of . Then, the fact that is ensured according to (37), yielding that si → 0 as t → ∞. Now, we reformulate (5) in the matrix form as [8] (50)where and are defined in Lemma 4, and , k ∈ {1, …, n}. According to Lemma 4, we have with , and then . Now, we rewrite (30) as . Therefore, can be guaranteed, further indicating qi − qj → 0 as t → ∞ by virtue of Lemma 1. □ Remark 2.Compared with the sliding observer design for a single robotic manipulator in [14], the distributed sliding observer (34) performs observation by use of non-linear coupling among the agents in the neighbourhood. Together with the distributed adaptive controller (35), our observer-based consensus protocol can deal with constant and bounded time delays and achieve velocity-free asymptotic synchronisation of multiple robotic manipulators. 4 Simulation In this section, we examine our control algorithm with six two-DOF planar robotic manipulators. For the i th manipulator, physical parameters of the two links are selected as , , , , , . where is the mass of the th link, is the position of the centre of mass, is the length of the th link and the moment of inertia relative to the centre of mass can be calculated as , , i ∈ {1, …, 6}. For simplicity, the gravitational force is neglected in our simulation. Then, the dynamic parameters, the inertia matrix and the centripetal and Coriolis matrix can be denoted by , respectively, where denotes the j th element of adi, j ∈ {1, 2, 3}, and denotes the joint position of the j th link for the i th robotic agent, j ∈ {1, 2}. 4.1 Case without communication delays The information graph associated with the robotic agents is shown in Fig. 1, where R1, …, R6 denote the six agents, respectively. The gains for the observer, the controller and parameter updating laws are selected as , , , α = 3, , , , Ki = 8I2 and Γdi = 0.01I3, i ∈ {1, …, 6}. To eliminate the chattering effects caused by the sliding mode observer, the signum function is replaced by (1 − exp(− ay))/(1 + exp(− ay)), a > 0 [27]. Here, we set a = 200 in our simulation. Fig. 1Open in figure viewerPowerPoint Interaction graph for the no-time-delay case Fig. 2 illustrates the response of the joint positions of the six agents. Since agent 5 is the single rooted node in the information topology, the other agents are illustrated to asymptotically synchronise to a common position, that is, position of R5. The control torques of the six agents are given in Fig. 3, from which we can see that the chattering effect is avoided by the approximation of the signum function. Fig. 2Open in figure viewerPowerPoint Transient response of joint positions without time delays Fig. 3Open in figure viewerPowerPoint Control torques without time delays 4.2 Case with communication delays The information topology in this case is restricted to strongly connected graphs, as shown in Fig. 4. The gains for the observer, the controller and parameter updating laws are selected as , , , α = 2, , , , , Ki = 15I2 and Γdi = 0.01I3 for all i ∈ {1, …, 6}. The approximate factor a for the signum function is still set as a = 200. Fig. 4Open in figure viewerPowerPoint Interaction graph of the agents for the time-delay case Figs. 5 and 6, respectively, illustrate the transients of joint positions and control torques with different time delays. The robots are illustrated to asymptotically converge to a common position despite of communication delays. Fig. 5Open in figure viewerPowerPoint Transient response of joint positions with time delays a Tij = 0.02 b Tij = 1 Fig. 6Open in figure viewerPowerPoint Control torques with time delays a Tij = 0.02 b Tij = 1 5 Conclusion In this paper, we solve the consensus problem of multiple robotic manipulators with dynamic uncertainties provided that the joint velocities of the agents are not measurable. Distributed control schemes are designed for both no-time-delay case and constant-time-delay case under directed graphs, where sliding observers are first constructed such that the joint velocity can be estimated online and then, the distributed adaptive controllers with dynamic parameter updating are proposed based on the observed information. Convergence for consensus errors is proved by resorting to Lyapunov stability analysis tools. Finally, simulations are performed on networked robotic manipulators to validate the effectiveness of the theoretical approach. Note that our handling of the time-delay case relies on a strongly connected information topology. In the future, we will work at relaxing the information topology to directed graphs containing a spanning tree. 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