Globally Stable Adaptive Formation Control of Euler-Lagrange Agents via Potential Functions

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1 09 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June -12, 09 ThA18.6 Globally Stable Adaptive Formation Control of Euler-Lagrange Agents via Potential Functions Ademir Rodrigues Pereira, Liu Hsu and Romeo Ortega Abstract This paper proposes a formation control strategy for uncertain Euler-Lagrange mobile agents based on artificial potential functions and robust adaptive control. A desired kinematic model is derived from a given potential function. The system parametric uncertainties and external disturbances are compensated by a robust adaptive control algorithm named binary adaptive control which combines the good transient properties and robustness of Sliding Mode Control with the desirable steady-state properties of parameter adaptive systems. For a given communication graph, conditions for global stability of the multiagent system are established using a conveniently shaped Lyapunov function for centralized and partially decentralized control schemes. Keywords - formation control; multi-agent systems; robust adaptive control; Euler-Lagrange systems I. INTRODUCTION Several formation control strategies can now be found in the literature, e.g., leader-follower ([2], [3] and [13]), behavior-based ([1], [8]), virtual structure ([16]) and artificial potential functions ([9]), ([4]), ([5]), ([]). In this paper, we adopt the potential function approach which is designed using the idea of repulsion and attraction forces in order to generate interaction rules for the group members with each other or with the environment. This approach allows formation control by prescribing a set of steady-state intervehicle distances. In principle, the implementation of the resulting formation control laws requires the position and velocity information of each agent and of its neighbors (centralized case). In practice, only the agent positions may be available and the velocities of the neighbors would have to be estimated by each individual vehicle. In a previous work [12], we have obtained semiglobal stability results for the case all-to-all communication between the agents, using a partially decentralized control where the information required for each agent is its own position and velocity and only the positions of the neighbors. This seems to be of interest in order to reduce the control computational load. In this paper, we further show that the multi-agent system can be made globally stable with a more general information topology both for the centralized and the This work was supported by the CNPq (Brazilian Research Council) and FAPERJ (Rio de Janeiro Research Foundation) A. R. Pereira is with Technological Center of the Brazilian Army (CTEx), Rio de Janeiro, Brazil ademir@ctex.eb.br L. Hsu is with the Department of Electrical Engineering, COPPE/Fed. Univ. of Rio de Janeiro, Brazil liu@coep.ufrj.br R. Ortega is with the Laboratoire de Signaux et Systemes of Supelec, Centre National de la Recherche Scientifique, Gif-sur-Yvette,France Romeo.Ortega@lss.supelec.fr partially decentralized cases. To this end, a helpful tool is an appropriate Lyapunov function shaping 1. In the proposed method, the parametric uncertainties are counteracted by the so called binary adaptive control (BAC) similar to the Binary Model Reference Adaptive Control (B- MRAC) [7]. Basically, the binary adaptive control consists of a high gain gradient adaptive law with parameter projection. The projection warrants that the adaptive parameter vector is kept bounded within some closed finite ball in the parameter space. As the adaptation gain is increased, the binary behaves as a sliding mode control (SMC) as discussed in [7]. However, the binary control is continuous thus avoiding the chattering phenomena of SMC. The paper is organized as follows. In Section II, we present the multiagent mathematical model, the problem statement, the information topology (using some graph-theoretic tools) and briefly describe the potential function approach. In Section III, we develop the stable binary adaptive control schemes both for the centralized and the partially decentralized cases. In Section IV, simulations are presented to illustrate the performance of the proposed strategy. Finally, conclusions are presented in Section V. II. PROBLEM STATEMENT In what follows, the Euclidean norm of a vector v and the corresponding induced norm of a matrix A are denoted by v and A, respectively. Consider a group of N vehicles fully actuated and modeled by the following set of Euler-Lagrange (EL) equations H i (y i )ÿ i + C i (ẏ i, y i )ẏ i = τ i, i = 1,..., N. (1) where H i IR n n corresponds to the inertia matrix, C i IR n n to the matrix of Coriolis and centripetal forces, τ i IR n denotes the control inputs, y i IR n is the position of the i-th agent. The considered class of EL systems can be used to represent mathematical models of ships, satellites and terrestrial and submarine robots and is assumed to have the following properties for all i = 1,..., N: the inertia matrix H i satisfies h i m x 2 x T H i (y i )x h i M x 2, with positive constant h i m and h i M ; the inertia matrix H i is differentiable; the matrix C i is chosen based on the Christoffel symbols so that v T ( Ḣ i 2C i) v = 0 v IR n (2) 1 for simplicity we will use simply the term Lyapunov function even if it would be more appropriate to use Lyapunov-like function /09/$ AACC 2606

2 We allow the matrices H i and C i to be uncertain in the sense that the parameters of model (1) are only known nominally. We consider the particular formation problem which consists in designing control laws so that all N agents converge to a desired configuration pattern defined by given inter-agent distances, ẏ i (t) 0 y i (t) y j (t) d ij for given constants d ij 0, i, j = 1,..., N. A. Information topology The information topology of the group of N agents can be described by a graph G := {V, E}, named information graph, where V := { v 1,..., v N} is a set of nodes, which represent the agents, E V V is the set of edges. The set of indices corresponding to the information neighbors of the agent i is named N i and is defined by N i := { j e ij = (v i, v j ) E } (3) In this work, we assume that the information topologies are represented by strongly connected graphs, i.e., there exists at least one directed path between any two nodes in both directions. From Fig. 1, examples of strongly connected graphs can be seen. For more details in graph theory, see [11] and references therein. In this work, we consider only Fig. 1. (a)cyclic (b) undirected (c)other the case when the information flow is bidirectional for each pair of neighbors, e.g., case (b) in Fig. 1). B. Artificial potential function approach Definition - An inter-agent potential function J ij (y ij ) is a twice (continuously) differentiable, nonnegative, radially unbounded function of the distances y ij = y i y j between agents i and j, such that J ij attains unique minimum when the agents are located at a desired distance d ij. Let us define the vector of interagent relative positions (or interagent position errors) as the vector ỹ formed by stacking the absolute position of one particular vehicle, say y 1, with the interagent relative positions given by of the following set: {y ij i = 1,...,N 1, j N i, j > i} (4) where y ij = y i y j. The need to include the element y 1 in the vector ỹ will become clear shortly. Consider the class of potential functions J(ỹ) defined by J(ỹ) = N 1 i j N i, j>i J ij (y ij ) + J 1b (y 1 ) (5) where J 1b is a radially unbounded function of y 1. This term will guarantee that all agents remain in some compact set of the position frame. The bounding potential function has the form { J 1b = 0 ; if y 1 rb Ψ( y 1 ) ; if y 1 > rb (6) where r b 0 is an arbitrary constant, Ψ( y 1 ) > 0, y i > r b, and y 1 implies J 1b. Having assumed that the information graph is connected, then the vector position error between any two agents can always be given as the sum of a set of interagent position errors between neighboring agents. For example, with the undirected information graph b, Fig. 1, (y 1 y 3 ) = (y 1 y 2 )+(y 2 y 3 ). Hence, it can be concluded that the potential function is radially unbounded w.r.t. the relative distances (y ij ) between any pair of agents as well as to their absolute positions (y i ). The motion of each agent is required to asymptotically obey the following first order kinematic model C. The sliding function ẏ i = y ij(ỹ) = g i (ỹ) (7) Let us define the function s i as s i = ẏ i + g i (ỹ) (8) The control objective is to make s i (t) 0 as t, so that each agent obeys the desired kinematic model (7) asymptotically. The functions s i will be called sliding function to remind the similarity with the switching function used in sliding mode control. However, in order to avoid high frequency control switching (chattering phenomena), no sliding mode using discontinuous control law is intended in the present strategy. We will only seek for the asymptotic convergence of the sliding function to zero with a continuous binary adaptive control law, as explained in what follows. III. BINARY ADAPTIVE CONTROL OF MULTI-AGENT SYSTEMS The next step consists in designing the control signals τ i such that the sliding functions tend to zero in spite of the system uncertainties. The derivative of (8) is given by ṡ i = ÿ i + ġ i (ỹ, ỹ) (9) Premultiplying (9) by H i and considering (1), one gets A. Centralized control H i ṡ i + C i s i = τ i + H i ġ i + C i g i () In this section we assume that the position and velocity of all neighbors of each agent are available for each individual agent control. In this case we can design a centralized control system where each agent has all necessary position and velocity information from the neighbors. We then consider the well known linear parametrization Y i θ i = (H i ġ i +C i g i )[14], where Y i is a regressor matrix 2607

3 composed of known functions of ỹ and ỹ and θ i IR mi is a parameter vector where m i is the number of unknown parameters for the i-th agent. Such vector is assumed uncertain in the sense that it is known only nominally. The nominal parameters θnom i can be used in the initialization of the parameter adaptation law in order to improve the adaptation transient. Now, equation () can be written as H i ṡ i + C i s i = τ i Y i θ i (11) Then, the following control law is proposed τ i = Y i θ i K i D si (12) where K i D is symmetric positive definite and θi = [θ i 1...θ i m i ] T is an adaptive parameter vector. Now, introducing the parameter mismatch θ i = θ i θ i, one can rewrite (11) as H i ṡ i + C i s i = Y i θi K i D si (13) which is a well known form in the adaptive control theory of robot manipulators [14]. The following adaptation law based on binary adaptive control, as introduced in ([6]) to design robust Model Reference Adaptive Control law for linear plants (B-MRAC), is proposed θ i = σθ i Γ i Y it s i (14) The σ-factor, also called projection factor, is defined as: { 0 ; if θ σ = i < Mθ i or σ eq < 0 σ eq ; if θ i M i (15) θ and σ eq 0 where σ eq = θ it Γ i Y it s/ θ i 2 and M i θ (> θ ) is a constant. Let B i θ = {θi : θ i M i θ }. Assuming that θi (0) B i θ, the projection factor acts as follows. If at any time θi (t) is on the sphere θ i = M i θ and the term Γi Y it s i points outwards such sphere, the update vector is projected onto the tangent plane of the sphere; alternatively, if it points inwards, the σ-factor is equal zero and θ i (t) moves to the interior of the sphere. Then, it is straightforward to prove that the closed ball B i θ is invariant ([6]), i.e., t 0, θi (t) B i θ. Hence, the binary adaptation law consists simply of a well known gradient adaptation law with parameter projection, however the idea is to exploit the useful properties of the controller when the adaptation Γ increases. This will be shown in the next section (see also [6]). 1) Stability analysis: Consider the system (13), (14), (15) which has the state vector x = [y T, ẏ T, θ T ] T. Then, assume θ i (0) B i θ with constant Mi θ θ i. Consider the invariant equilibrium set Ω e = {x : s = 0; g(ỹ) = 0; θ B θ } where the vector s concatenates all individual sliding functions and θ concatenates all adaptive parameter mismatches, and B θ = Bθ 1 B2 θ... BN θ. Note that if s = 0, then g(ỹ) = 0 is equivalent to ỹ = 0. For simplicity of analysis and without loss of generality, assume Γ i = γi. To explain the rational behind binary adaptation, consider first the Lyapunov Function V 0 = ( 1 H i s i + 1 ) 2 sit 2γ θ i 2 (16) Using the skew-symmetry property, the time-derivative of (16) is given by V 0 = [ s it KD i si σ ] γ ( θ i + θ i ) T θi (17) Note that the second term within brackets is non positive and that, by parameter projection, the parameter vector is norm bounded by a constant, say, θ M for some positive constant M. These facts, together with the positive definite lower and upper bounds of s it H i s i as stated below (1) it is straightforward to show that V 0 = λv 0 + O(γ 1 ) (18) where λ is a positive constant independent of γ. Then, by using a well known Comparison Lemma, one concludes that V (t) tends exponentially fast to a residual value of order O(γ 1 ). This also implies that s(t) tends exponentially fast to a residual value of order O(γ 1/2 ). This shows that the adaptation transient of the binary adaptation is exponentially fast towards an arbitrarily small residual set, as the adaptation gain is increased. Moreover, it can be argued that the binary controller establishes one bridge from adaptive control to sliding mode control through the adaptation gain γ. Hence such gain can be tuned so as to get the best from both approaches while avoiding their drawbacks, e.g., chattering. We will further pursue the stability analysis and show that s i 0, ẏ i 0 (and g i (ỹ) 0) as t and, thus, the state x(t) converges to Ω e. To this end, consider the following shaped candidate Lyapunov function V = V 0 + αj(ỹ) (19) where α is a nonnegative constant. Now, the time-derivative of (19) is given by where V 0 is given by (18). Note that Then, from (8) 2 Therefore V = V 0 + α d J(ỹ) () dt d J J(ỹ) = dt y ẏ d dt J(ỹ) = st ẏ ẏ 2 (21) V s T K D s + αs T ẏ α ẏ 2 (22) 2 note that (7) is required to hold only asymptotically. 2608

4 where K D = diag { KD 1, K2 D,.. }.,KN D. The above inequality can be rewritten as V [ s T ẏ T] [ K D α 2 I ][ ] ṡ α 2 I αi (23) y For V 0, the Schur complement S [15] should be positive definite, i.e., which holds if S = αi 1 4 α2 K 1 D > 0 (24) α 4 I < K D (25) Then, V is negative semi-definite so that V is uniformly bounded t. With any α > 0, the shaped Lyapunov function allows us to conclude that s and J are both bounded, and thus ỹ is norm bounded due to the radial unboundedness of J. Moreover, since y 1 is in ỹ and the information graph is assumed connected, one concludes that all vehicles remain in a compact set of the position frame. Therefore the whole state remains in some compact set of the state space and thus La Salle s theorem can be applied to conclude that the trajectories tend to the largest invariant set in M = {x : s = 0, g = 0}, which is the equilibrium set Ω e. Thus, the interagent distances tend to constants so that a constant formation is reached asymptotically. The above arguments prove the following Theorem 1: Consider a multi-agent system consisting of N vehicles fully actuated and modeled by the EL dynamics (1) with the centralized control law (12)-(14-15) with regressor Y i defined by Y i θ i = (H i ġ i +C i g i ). Then, K D > 0 the following holds (a) the sliding function s tends exponentially fast to a residual set of order O(γ 1 ); (b) the system trajectories tend asymptotically to the equilibrium set Ω e, i.e., s 0, ẏ 0 as t ; (c) all closed loop signals are uniformly bounded and the multiagent system tends asymptotically to some constant formation corresponding to g(ỹ) = 0. B. Partially decentralized binary adaptive control We now consider a partially decentralized control system scenario where the velocities of the neighbors are not available to each agent. Let us define the new parametrization Y i ϑ i = [ C i g i] and from (), we obtain The control law is H i ṡ i + C i s i = τ i Y i ϑ i + H i ġ i (26) τ i = Y i ϑ i K i Ds i (27) where, K i D is symmetric positive definite and ϑi are the new adaptive parameter (vectors). Let ϑ i = ϑ i ϑ i. Then, substituting the control law in (26), we have H i ṡ i + C i s i = Y i ϑi K i Ds i + H i ġ i (28) The Lyapunov function is chosen as ( 1 V = H i s i + 1 ) 2 sit 2γ ϑ it ϑi + αj(ỹ) (29) The derivative with respect to time of (29) is given by V = ( s it K ids i σγ ) ϑit ϑi + s it H i ġ i + α d dt J(ỹ) Note that [ ġ i 2 J = y 1 y i 2 J... 2 J y 2 y i 2... y i 2 J y N y i and the standard definition of Hessian matrix of J is [ 2 2 ] J J = y i y j i,j=1,...,n () ] ẏ (31) From (21), (31) and (32), we rewrite equation() as (32) V s T K D s + s T [H( 2 J) T + αi]ẏ α ẏ 2 (33) where H = diag { H 1, H 2,...,H N}. Let us define M(α) = H( 2 J) T + αi and we can rewrite the equation (33) as V [ s T ẏ T][ K D M(α) 2 MT (α) 2 αi ] [ s ẏ ] (34) We have already assumed that the inertia matrices H i in (1) are all uniformly norm-bounded. Thus, σ H H for some constant σ H. At this point, we further assume that the Hessian is norm-bounded by a constant σ J, i.e., σ J 2 J. From For V 0, the Schur complement S should be positive definite, i. e., S = αi 1 4 MT (α)k 1 D M(α) > 0 (35) Then, with λ m (.) denoting the smallest eigenvalue of a symmetric positive definite matrix, (35) holds if λ m (K D ) > M(α) 2, (36) 4α which is implied by λ m (K D ) > (α + σ Hσ J ) 2 (37) 4α The minimum value of the right hand side of (36) w.r.t. α occurs at α = σ H σ J. Thus, there exists α > 0 such that V 0 if λ m (K D ) > σ H σ J (38) Then, we can conclude that s i 0 and ẏ i 0 as t and the equilibrium set Ω e is asymptotically reached, as in the centralized case. We can now state Theorem 2: Consider a multi-agent system consisting of N vehicles fully actuated and modeled by the EL dynamics (1) with the partially decentralized control law (27), with regressor Y i defined by Y i ϑ i = C i g i, and with binary 2609

5 adaptation applied to ϑ i. Assume that the Hessian matrix of J is norm-bounded by a constant σ J and that the norm bound of the inertia matrices is σ H. Then, if (38) is valid, the following properties hold (a) the system trajectories tend asymptotically to the equilibrium set Ω e, i.e., s 0, ẏ 0 as t ; (b) all closed loop signals are uniformly bounded and the multiagent system tends asymptotically to some constant formation corresponding to g(ỹ) = 0. Note that the theorems only guarantee that a formation is reached asymptotically. However, the desired formation, may not be reached due to local minima of J which is one drawback of the potential function approach when applied to formation. This can be partially circumvented by heuristic rules, such as renumbering the agents. A. Illustration example IV. SIMULATIONS The simulation results presented in this section illustrate the theoretical results. Only the more difficult problem of partially decentralized case will be discussed due to lack of space. We consider the control of a group of six point masses moving on a plane. The objective is to achieve an ultimate pyramidal pattern as depicted in Fig. 2. The dynamics of each agent is described by M i ÿ i + D i ẏ = τ i ; i = 1,...,6 (39) where y i IR 2 is the position of the i-th vehicles, M i and D i represent scalar mass and damping constants, respectively. We consider that the information topology corresponds to the undirected graph shown in Fig. 2 Substituting (8) and The constants d ij give the desired inter-vehicular distances. For simplicity, the agent parameters are identical and given by, i, M i = and D i = 1. The initial velocities were set equal to zero. For the potential function, the parameters were chosen as i, j, a ij = 0.1, b ij =, d ij = and thus c ij = The adaptive parameter were initialized at the nominal values as θ i (0) =.3 (true values are 1) and the binary control was used with M θ = 2 θ i and adaptation gain Γ i = I 2. In order to verify Theorem 2, two values for the gain matrix K i D = 5ζI 2 were tested. We have evaluated lower bound in (38), i.e., λ m (K D ) = ζ >. For ζ = 5, thus violating this inequality, the agents performed the trajectories shown in Fig.3. The initial positions are indicated by small circles. No instability was verified suggesting that the inequality (38) may just be a sufficient stability condition. Indeed, from Fig. 4, one can see that the agents tend to desired value in steady state, however a rather oscillatory transient is observed. Now, increasing ζ to ζ = 45, the agent coord y coord x Fig. 3. Trajectories of the agents forming a triangle for K D = 5I trajectories are shown in Fig. 6. The transient is faster and much less oscillatory. 55 (9) in (39) we obtain Fig. 2. Information graph M i ṡ i + D i s i = τ i + M i ġ i + D i g i () The control law is given by (27), where Y i = [g i ] and θ i = D i. We have used the potential function based on [4] and it has the form of the equation (5), where J ij is defined by the formula [ J ij (y ij a ij ) = ( y ij 2 + bij c ij y ij 2 )] exp 2 2 c ij where a ij are the attraction constants and b ij are the repulsion constants. The parameters c ij are defined by the following d ij2 c ij = ( ). (41) b log ij a ij y i y j time (sec) Fig. 4. Inter vehicular distances for K D = 5I V. DYNAMIC FORMATION In this section we briefly describe one possible extension of the above developed theory. Suppose that the multiagent system is required to keep a certain dynamic formation, i.e., 26

6 Fig. 5. coord y y i y j coord x Trajectories of the agents forming a triangle for K D = 45I time (sec) Fig. 6. Intervehicular distances for K D = 45I when the group moves in formation ([]). The simplest case is when the formation motion is translational. To this end we define for each agent the modified position as x i = y i f i (t) where f i : IR IR n, f i C 2, which are specified according to the desired dynamic formation. Then, the EL system (1) becomes H i (y i )ẍ i + C i (ẏ i, y i )ẋ i = u i ; (42) u i = τ i + H i (y i ) f i + C i (ẏ i, y i ) f i (43) One simple possibility is to choose a leader, say, agent 1. Then, one specifies a single desired function f(t) and sets f i (t) = f(t); i. To make agent 1 follow f(t), one should make r b = 0 in (6). Since H i and C i are assumed uncertain, the two last terms of (43) have to be obtained adaptively by including such terms in the regressor matrix. Then, one can redesign the formation strategy using the potential function approach. However, the potential function is to be expressed in terms of the new position variables x i. In the original position variables y i, the potential will be time-varying. Adaptive control is essential to allow precise tracking. More details and simulation results on dynamic formation can be found in liu/acc09formation. VI. CONCLUSIONS We have proposed a framework to design formation control for a group of Euler-Lagrange agents. Artificial potential functions were used to generate a desired geometric pattern with prescribed interagent distances. Global stability results were obtained for either centralized as well as for partially decentralized control structures. A shaping term for the Lyapunov function was found instrumental to derive the main results, valid for groups with a strongly connected information graph with bi-directional information flow. Future work includes the extension to the dynamic formation problem, only briefly outlined in the above section, to more general information topologies represented by strongly connected graphs with unidirectional or bidirectional information flow, and to nonholonomic agents. Other issues include responses to typical real-world situations such as range measurement dropouts, communication delays, etc. VII. ACKNOWLEDGMENTS The authors gratefully acknowledge the contribution of the CNPq, FAPERJ and CAPES. REFERENCES [1] T. Balch and R. C. Arkin. Behavior-based formation control for multirobot teams. IEEE Trans. on Robotics and Automation, 14(6): , December [2] A.K. Das, R. Fierro, V. Kumar, J.P. Ostrowski, J. Spletzer, and C.J. Taylor. A vision-based formation control framework. IEEE Trans. on Robotics and Automation, 18(5): , October 02. [3] J. P. Desai, J. Ostrowski, and V. Kumar. Controlling formations of multiple mobile robots. 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