Robust Distance-Based Formation Control of Multiple Rigid Bodies with Orientation Alignment

Transcription

1 Robust Distance-Based Formation Control of Multiple Rigid Bodies with Orientation Alignment Alexandros Nikou Christos K. Verginis Dimos V. Dimarogonas ACCESS Linnaeus Center, School of Electrical Engineering and KTH Center for Autonomous Systems, KTH Royal Institute of Technology, SE- 44, Stockholm, Sweden. {anikou, cverginis, Abstract: This paper addresses the problem of distance- and orientation-based formation control of a class of second-order nonlinear multi-agent systems in 3D space, under static and undirected communication topologies. More specifically, we design a decentralized model-free control protocol in the sense that each agent uses only local information from its neighbors to calculate its own control signal, without incorporating any knowledge of the model nonlinearities and exogenous disturbances. Moreover, the transient and steady state response is solely determined by certain designer-specified performance functions and is fully decoupled by the agents dynamic model, the control gain selection, the underlying graph topology as well as the initial conditions. Additionally, by introducing certain inter-agent distance constraints, we guarantee collision avoidance and connectivity maintenance between neighboring agents. Finally, simulation results verify the performance of the proposed controllers. Keywords: Multi-agent systems, Cooperative systems, Distributed nonlinear control, Nonlinear cooperative control, Robust control.. INTRODUCTION During the last decades, decentralized control of networked multi-agent systems has gained a significant amount of attention due to the great variety of its applications. The main focus of multi-agent systems is the design of distributed control protocols in order to achieve global tasks, such as consensus, network connectivity and collision avoidance. A particular multi-agent problem that has been considered in the literature is the formation control problem, where the agents represent robots that aim to form a prescribed geometrical shape, specified by a certain set of desired relative configurations between the agents. The main categories of formation control that have been studied in the related literature are ([Oh et al., ]) position-based control, displacement-based control, distance-based control and orientation-based control. In distance-based formation control, inter-agent distances are actively controlled to achieve a desired formation, dictated by desired interagent distances. Each agent is assumed to be able to sense the relative positions of its neighboring agents, without the need of orientation alignment of the local coordinate systems. When orientation alignment is considered as a control design goal, the problem is known as orientationbased (or bearing-based) formation control. The desired formation is then defined by relative inter-agent orientations. The literature in distance-based formation control is rich, and is traditionally categorized in single or double inte- This work was supported by the H ERC Starting Grand BU- COPHSYS, the Swedish Research Council (VR), the Knut och Alice Wallenberg Foundation and the European Union s Horizon Research and Innovation Programme under the Grant Agreement No (AEROWORKS). grator agent dynamics and directed or undirected communication topologies (see e.g. [Olfati-Saber and Murray,, Smith et al., 6, Hendrickx et al., 7, Anderson et al., 7, 8, Dimarogonas and Johansson, 8, Krick et al., 9, Dorfler and Francis,, Cao et al.,, Park et al.,, Belabbas et al.,, Oh and Ahn, 4]) Orientation-based formation control has been addressed in [Eren,, Trinh et al., 4, Zhao and Zelazo, 6], whereas the authors in [Trinh et al., 4, Bishop et al.,, Fathian et al., 6] have considered the combination of distance- and orientation-based formation. In most of the aforementioned works in formation control, the two-dimensional case with simple dynamics and point-mass agents has been dominantly considered. In real applications, however, the engineering systems have nonlinear second order dynamics and are usually subject to exogenous disturbances and modeling errors. Another important issue concerns the connectivity maintenance, the collision avoidance between the neighboring agents and the transient and steady state response of the closed loop system, which have not been taken into account in the majority of related woks. Thus, taking all the above into consideration, the design of robust distributed control schemes for the multi-agent formation control problem becomes a challenging task. Motivated by this, we aim to address here the distancebased formation control problem with orientation alignment for a team of rigid bodies operating in 3D space, with unknown second-order nonlinear dynamics and external disturbances. We propose a purely decentralized control protocol that guarantees distance formation, orientation alignment as well as collision avoidance and connectivity maintenance between neighboring agents and in parallel ensures the satisfaction of prescribed transient and

2 steady state performance. The prescribed performance control framework has been incorporated in multi-agent systems in [Karayiannidis et al.,, Bechlioulis and Kyriakopoulos, 4], where first order dynamics have been considered. Furthermore, the first one only addresses the consensus problem, whereas the latter solves the position based formation control problem, instead of the distanceand orientation-based problem treated here. Due to space constraints, a more detailed version of this paper that contains the extended derivations of the proofs and omitted calculations, can be found in [Nikou et al., 6]. The remainder of the paper is structured as follows. In Section notation and preliminary background is given. Section 3 provides the system dynamics and the formal problem statement. Section 4 discusses the technical details of the solution and Section is devoted to a simulation example. Finally, the conclusion and future work directions are discussed in Section 6.. Notation. NOTATION AND PRELIMINARIES The set of positive integers is denoted as N. The real n- coordinate space, with n N, is denoted as R n ; R n and R n > are the sets of real n-vectors with all elements nonnegative and positive, respectively. Given a set S, we denote as S its cardinality. The notation x is used for the Euclidean norm of a vector x R n. Given a symmetric matrix A, λ min (A) = min{ λ : λ σ(a)} denotes the minimum eigenvalue of A, where σ(a) is the set of all the eigenvalues of A and rank(a) is its rank; A B denotes the Kronecker product of matrices A, B R m n, as was introduced in [Horn and Johnson, ]. Define by I n R n n, m n R m n the unit matrix and the m n matrix with all entries zero, respectively; B c (r) = {x R 3 : x c r} is the 3D sphere of radius r and center c R 3. The vector connecting the origins of coordinate frames {A} and {B} expressed in frame {C} coordinates in 3D space is denoted as p C B/A R 3. Given a R 3, S(a) is the skew-symmetric matrix defined according to S(a)b = a b. We further denote as q B/A T 3 the Euler angles representing the orientation of frame {B} with respect to frame {A}, where T 3 is the 3D torus. The angular velocity of frame {B} with respect to {A}, expressed in frame {C} coordinates, is denoted as ω C B/A R 3. We also use the notation M = R 3 T 3. For notational brevity, when a coordinate frame corresponds to an inertial frame of reference {}, we will omit its explicit notation (e.g., p B = p B/, ω B = ω B/ etc.). All vector and matrix differentiations are derived with respect to an inertial frame {}, unless otherwise stated.. Dynamical Systems Consider the initial value problem: ψ = H(t, ψ), ψ() = ψ Ω ψ, () with H : R Ω ψ R n, where Ω ψ R n is a non-empty open set. Definition. ([Sontag, 3]) A solution ψ(t) of the initial value problem () is maximal if it has no proper right extension that is also a solution of (). Theorem. ([Sontag, 3]) Consider the initial value problem (). Assume that H(t, ψ) is: a) locally Lipschitz in ψ for almost all t R, b) piecewise continuous in t for each fixed ψ Ω ψ and c) locally integrable in t for each fixed ψ Ω ψ. Then, there exists a maximal solution ψ(t) of () on the time interval [, τ max ), with τ max R > such that ψ(t) Ω ψ, t [, τ max ). Proposition. ([Sontag, 3]) Assume that the hypotheses of Theorem hold. For a maximal solution ψ(t) on [, τ max ) with τ max < and for any compact set Ω ψ Ω ψ, there exists a t [, τ max ) such that ψ(t ) / Ω ψ..3 Graph Theory An undirected graph G is a pair (V, E), where V is a finite set of nodes, representing a team of agents, and E {{i, j} : i, j V, i j}, with M = E, is the set of edges that model the communication capability between neighboring agents. For each agent, its neighbors set N i is defined as N i = {j,..., j Ni } = {j V s.t. {i, j} E}. If there is an edge {i, j} E, then i, j are called adjacent. A path of length r from vertex i to vertex j is a sequence of r + distinct vertices, starting from i and ending to j, such that consecutive vertices are adjacent. For i = j, the path is called a cycle. If there is a path between any two vertices of the graph G, then G is called connected. A connected graph is called a tree if it contains no cycles. The adjacency matrix A(G) = [a ij ] R N N of a graph G is defined by a ij = a ji =, if {i, j} E, and a ij = otherwise. The degree d(i) of vertex i is defined as the number of its neighboring vertices, i.e. d(i) = N i, i V. Let also (G) = diag{[d(i)] i V } R N N be the degree matrix of the system. Consider an arbitrary orientation of G, which assigns to each edge {i, j} E precisely one of the ordered pairs (i, j) or (j, i). When selecting the pair (i, j), we say that i is the tail and j is the head of the edge {i, j}. By considering a numbering k M = {,..., M} of the graph s edge set, we define the N M incidence matrix D(G) as it was given in [Mesbahi and Egerstedt, ]. The Laplacian matrix L(G) R N N of the graph G is defined as L(G) = (G) A(G) = D(G)D(G) τ. Lemma. [Dimarogonas and Johansson, 8, Section III] Assume that the graph G is a tree. Then, D τ (G)D(G) is positive definite. 3. PROBLEM FORMULATION Consider a set of N rigid bodies, with V = {,,..., N}, N, operating in a workspace W R 3, with coordinate frames {i}, i V, attached to their centers of mass. Each agent occupies a sphere B ri (p i (t)), where p i : R R 3 is the position of the agent s center of mass and r i is the agent s radius. We also denote as q i : R T 3, i V, the Euler angles representing the agents orientation with respect to an inertial frame {}, with q i = [φ i, θ i, ψ i ] τ. By defining x i : R M, v i : R R 6, with x i = [p τ i, qτ i ]τ, v i = [ṗ τ i, ωτ i ]τ, we model each agent s motion with the nd order dynamics: ẋ i (t) = J i (x i )v i (t), (a) M i (x i ) v i (t) + C i (x i, ẋ i )v i (t) + g i (x i ) + w i (x i, ẋ i, t) = u i, (b) where J i : M R 6 6 is a Jacobian matrix given by J i (x i ) = diag{i 3, J q (x i )}, with

3 sin(φ i ) tan(θ i ) cos(φ i ) tan(θ i ) cos(φ i ) sin(φ i ) J q (x i ) = sin(φ i ) cos(φ i ), (3) cos(θ i ) cos(θ i ) for which we make the following assumption: Assumption. The angle θ i satisfies the inequality π < θ i (t) < π, i V, t R. The aforementioned assumption guarantees that J i is always well-defined and invertible, since det(j i ) = cos θ i. Furthermore, M i : M R 6 6 is the positive definite inertia matrix, C i : M R 6 R 6 6 is the Coriolis matrix, g i : M R 6 is the gravity vector, and w i : M R 6 R R 6 is a bounded vector representing model uncertainties and external disturbances. We consider that the aforementioned vector fields are unknown and continuous. Finally, u i R 6 is the control input vector representing the 6D generalized force acting on the agent. The dynamics () can be written in vector form as: ẋ(t) = J(x)v(t), M(x) v(t) + C(x, ẋ)v(t) + ḡ(x) + w(x, ẋ, t) = u, (4a) (4b) where x = [x τ,..., x τ N ]τ : R M N, v = [v τ,..., v τ N ]τ : R R 6N, u = [u τ,..., u τ N ]τ R 6N, and J = diag{[j i ] i V } R 6N 6N, M = diag{[m i ] i V } R 6N 6N, C = diag{[c i ] i V } R 6N 6N, ḡ = [g τ,..., g τ N ]τ R 6N, w = [w τ,..., w τ N ]τ R 6N. It is also further assumed that each agent can measure its own p i, q i, ṗ i, v i, i V, and has a limited sensing range of s i > max{r i + r j : i, j V}. Therefore, by defining the neighboring set N i (t) = {j V : p j (t) B si (p i (t))}, agent i also knows at each time instant t all p i j/i (t), q j/i(t) and, since it knows its own p i (t), q i (t), it can compute all p j (t), q j (t), j N i (t), t R. The topology of the multi-agent network is modeled through the graph G = (V, E), with V = {,..., N} and E = {{i, j} V V s.t. j N i () and i N j ()}. The latter implies that at t = the graph is undirected, i.e., p lk () p mk () < d k,con, {l k, m k } E, () with d k,con = min{s lk, s mk }, l k, m k V, k M. We also consider that G is static in the sense that no edges are added to the graph. We do not exclude, however, edge removal through connectivity loss between initially neighboring agents, which we guarantee to avoid, as presented in the sequel. It is also assumed that at t = the neighboring agents are at a collision-free configuration, i.e., d k,col < p lk () p mk (), {l k, m k } E, with d k,col = r lk + r mk. Hence, we conclude that d k,col < p lk () p mk () < d k,con, {l k, m k } E. (6) Moreover, given the desired formation constants d k,des, q k,des for the edge k M, the formation configuration is called feasible if the set Φ = {x M N : p lk p mk = d k,des, q lk q mk = q k,des, {l k, m k } E}, with l k, m k V, k M, is nonempty. Due to the fact that the agents are not dimensionless and their communication capabilities are limited, the control protocol, except from achieving a desired inter-agent formation, should also guarantee for all t R that (i) the neighboring agents avoid collision with each other and (iii) all the initial edges are maintained, i.e., connectivity maintenance. Therefore, all pairs {l k, m k } V V of agents that initially form an edge must remain within distance greater than d k,col and less than d k,con. We also make the following assumptions that are required on the graph topology: Assumption. The graph G is initially a tree. Formally, the robust formation control problem under the aforementioned constraints is formulated as follows: Problem. Given N agents governed by the dynamics (), under the Assumptions, and given the desired inter-agent distances and angles d k,des, q k,des, with d k,col < d k,des < d k,con, {l k, m k } E, l k, m k V, k M, design decentralized control laws u i R 6, i V such that {l k, m k } E, k M, the following hold: () lim t p lk (t) p mk (t) = d k,des, () lim t [q mk (t) q lk (t) q k,des ] = 3, (3) d k,col < p lk (t) p mk (t) < d k,con, t R. 4. PROBLEM SOLUTION Let p = [p τ,..., p τ N ]τ : R R 3N, q = [q τ,..., q τ N ]τ : R T 3N be the stacked vectors of all the agent positions and Euler angles. We further denote as p, q : R R 3M, with p(t) = [p T l,m (t),..., p T l M,m M (t)] T, q(t) = [q T l,m (t),..., q T l M,m M (t)] T and p lk,m k (t) = p lk (t) p mk (t), q lk,m k (t) = q lk (t) q mk (t), {l k, m k } E, with the edges ordered as in the case of the incidence matrix D(G). Thus, the following holds: p(t) = (D τ (G) I 3 ) p(t), q(t) = (D τ (G) I 3 ) q(t). (7a) (7b) Next, let us introduce the errors e p k : R R, e p k (t) = p lk,m k (t) d k,des and eq k = [eq k, e q k, e q k 3 ] τ : R T 3, e q k (t) = q m k (t) q lk (t) q k,des, for all distinct edges {l k, m k } E, k M, as well as the stack vectors e p (t) = [e p (t),..., ep M (t)]τ R M, e q (t) = [(e q (t))τ,..., (e q M (t))τ ] τ T 3M. By taking the time derivative of e p, e q, we obtain: ė p (t) = F p (x) (D τ (G) I 3 ) ṗ, ė q (t) = (D τ (G) I 3 ) q, where F p : M N R M 3M, with p τ l,m (t)... 3 F p (x) = p τ l M,m M (t) By introducing the stack error vector e(t) = [(e p (t)) τ, (e q (t)) τ ] τ R 4M, (8) can be written as: (8a) (8b) ė(t) = F p (x) D τ (G)[ṗ τ, q τ ] τ, (9) where [ ] Fp (x) F p (x) = M 3M 3M 3M I R 4M 6M, 3M [ ] D(G) I3 D(G) = 3N 3M 3N 3M D(G) I R 6N 6M. 3 Finally, we obtain from (4a): (a) (b)

4 [ṗ q ] = I I J q (x ) } 3 3 {{... J q (x N ) } J(x) ṗ. ṗ N ω. ω N }{{} v(t) = J(x)v(t), () and thus, (9) can be written as: ė(t) = F p (x) D τ (G)J(x)v(t). () The concepts and techniques of prescribed performance control, originally proposed in [Bechlioulis and Rovithakis, 8], are adapted in this work in order to: a) achieve predefined transient and steady state response for the distance and orientation errors e p k, eq k, k M as well as ii) avoid the violation of the collision and connectivity constraints between neighboring agents, as presented in Section 3. As stated in [Bechlioulis and Rovithakis, 8], prescribed performance characterizes the behavior where the aforementioned errors evolve strictly within a predefined region that is bounded by absolutely decaying functions of time, called performance functions. The mathematical expressions of prescribed performance are given by the inequality objectives: C k,col ρ p k (t) < ep k (t) < C k,conρ p k (t), ρq k (t) < eq (t) < ρ q (t), (3) k n k k M, n {,, 3}, where the functions ρ q k (t) = (ρq k, ρ q k, )e lq k t + ρ q k,, ρp k (t) = ( ρp k, c)e lp k t + ρ p k, c, with c = (max{c k,con, C k,col }), are designer-specified, smooth, bounded, and decreasing functions of time, where l p k, lq k, ρp k,, ρq k, R >, k M, incorporate the desired transient and steady state performance specifications respectively, and C k,col, C k,con R >, k M, are associated with the collision and connectivity constraints. In particular, we select C k,col = d k d k,col, C k,con = d k,con d k, (4) k M, which, since the desired formation is compatible with the collision and connectivity constraints (i.e., d k,col < d k,des < d k,con, k M), ensures that C k,col, C k,con R >, k M and, in view of (6) that C k,col ρ p k () < ep k () < ρp k ()C k,con, k M. Moreover, by choosing ρ q k, = ρq k () > max n {,,3} eq k n (), () it is also guaranteed that ρ q k () < eq k n () < ρ q k (). k M, n {,, 3}. Hence, if we guarantee prescribed performance via (3), by employing the decreasing property of ρ p k (t), ρq k (t), k M, we obtain C k,col < e p k (t) < C k,con, ρ q k (t) < eq k n (t) < ρ q k (t), and, consequently, owing to (4) we have d k,col < p lk (t) p mk (t) < d k,con, k M, t R, and, hence, a solution to problem. In the sequel, we propose a decentralized control protocol that does not incorporate any information on the agents dynamic model and guarantees (3) for all t R. Given the errors e p (t), e q (t): Step I-a: Select the corresponding functions ρ p k (t), ρq k (t) and positive parameters C k,con, C k,col, k M, following (3), (), and (4), respectively, in order to incorporate the desired transient and steady state performance specifications as well as the collision and connectivity constraints. In addition, define the normalized errors ξ p k : R R, ξ q k = [ξq k, ξ q k, ξ q k 3 ] τ : R R 3 with ξ p k (t) = (ρ p k (t)) e p k (t), ξq k (t) = (ρq k (t)) e q k (t), k M, as well as the stack vector forms ξ p = [ξ p,..., ξp M ]τ = (ρ p ) e p, ξ q = [(ξ q )τ,..., (ξ q M )τ ] τ = (ρ q ) e q, and ξ = [(ξ p ) τ, (ξ q ) τ ] τ = (ρ) e R 4M, where ρ p = diag{[ρ p k ] k M} R M M, ρ q = diag{[ρ q k I 3] k M } R 3M 3M, ρ = diag{ρ p, ρ q }. Step I-b: Define the transformed errors ε p k : R R, εq k : R 3 R 3 and the signals r p k : R R, rq k : R3 R 3 3 as ( ( ) ( ) ) ε p k (ξp k ) = ln + ξp k ξp k, (6a) C k,col C k,con ( + ξ q ) ( ε [ln q k (ξq k ) = k + ξ q ) ( k + ξ q )] τ ξ q, ln k ξ q, ln 3 ξ q, (6b) k k k 3 r p k (ξp k ) = εp k (ξp k ) ξ p k r q k (ξq k ) = εq k (ξq k ) ξ q k = diag C k,col + C k,con = (C k,col + ξ p k )(C k,con ξ p k ), { [r ] } p = diag (ξ p ) k n k n n {,,3} } { [ (ξ q k n ) ] n {,,3} and design the decentralized reference velocity vector for each agent v i,des = [ṗ τ i,des, ωτ i,des ]τ : R 4M R R 6 as: v i,des (ξ, t) = (ρ p (t)) r p (ξ p )ε p (ξ p )p k ij k ij k ij k ij k i,j (t) ij J i (x i ) j N i () (ρ q (t)) r q (ξ q )ε q (ξ q ) (7) k ij k ij k ij k ij k ij j N i () where k ij M is the edge of agents i, j N i (), i.e., {l kij, m kij } E and l kij = i, m kij = j. We write (7) in vector form v des (ξ, t) = [ṗ τ des (ξp, t), ωdes τ (ξq, t)] τ : v des (ξ, t) = J (x) D(G) F τ p(x)r(ξ)(ρ(t)) ε(ξ), (8) where ṗ des = [ṗ τ,des,..., ṗτ N,des ]τ, ω des = [ω,des τ,..., ωn,des τ ]τ R 3N, ε = [(ε p ) τ, (ε q ) τ ] τ = [ε p,..., εp M, (εq )τ,..., (ε q M )τ ] τ R 4M and J(x), D(G), F p as they were defined in () and (), respectively. Moreover, r = diag{r p, r q } R 4M 4M, r p = diag{[r p k ] k M} R M M and r q = diag{[r q k ] k M} R 3M 3M. It should be noted that J (x) is always well-defined due to Assumption. Step II-a: Define the errors e v : R 4M R R 6N, with e v (ξ, t) = [(e v ) τ (ξ, t),..., (e v N )τ (ξ, t)] τ = v(t) v des (ξ, t), where e v i (ξ, t) = [ev i (ξ, t),..., e v i 6 (ξ, t)] τ = [ṗ τ i (t) ṗ τ i,des (ξp, t), ωi τ (t) ωτ i,des (ξq, t)] τ = v i (t) v i,des (ξ, t), i V, and select the corresponding performance functions ρ v i m : R R >, with ρ v i m (t) = (ρ v i m, ρv i m, )e lv im t + ρ v i m, and ρ v i = m, ρv i m () > e v i m (), li v m, ρ v i R m, >, ρ v i < m, ρ v i m,, i V, m {,..., 6}. Moreover, define the normalized errors ξ v i = [ξv i,..., ξ v i 6 ] τ : R 4M R R 6 with ξ v i (ξ, t) = (ρv i (t)) e v i (ξ, t), ρv i = diag{[ρ v i m ] m {,...,6} } R 6 6, which is written as ξ v (ξ, t) = [(ξ v (ξ, t)) τ,..., (ξ v N (ξ, t))τ ] τ = (ρ v (t)) e v (ξ, t) R 6N, with ρ v (t) = diag { [ρ v i (t)] i V} R 6N 6N.,

5 e p (t)[m] e q (t)[r] - - e q k e q k e q k e p (t)[m] e q (t)[r] e p 3 (t)[m] e q 3 (t)[r] Fig.. The evolution of the distance errors e p k (t) with the performance functions ρ p k (t), k {,, 3}. Step II-b: Define the transformed velocity errors ε v i : R 6 R 6 and the signals ri v : R6 R 6 6 as: ( ) ( )] + ξ v ε v i (ξv i [ln ) = i + ξ v τ i6 ξi v,, ln ξi v, (9a) 6 ri v (ξv i ) = εv i (ξv i ) [ ] ξi v = diag{ ri v m (ξi v m ) } m {,...,6} { [ ] } = diag ( (ξi v m ), (9b) ) m {,...,6} and design the decentralized control protocol for each agent i V as u i : R 6 R R 6 : u i (ξ v i, t) = γ i (ρ v i (t)) r v i (ξ v i )ε v i (ξ v i ), () with γ i R >, i V, which can be written in vector form as: u(ξ v, t) = Γ(ρ v (t)) r v (ξ v )ε v (ξ v ), () where Γ = diag{[γ i I 6 ] i V } R 6N 6N, ε v = [(ε v ) τ,..., (ε v N )τ ] τ R 6N and r v = diag{[r v i ] i V} R 6N 6N. Remark. Notice by (7) and () that the proposed control protocols are distributed in the sense that each agent uses only local information to calculate its own signal. In that respect, regarding every edge k ij, with {l kij, m kij } = {i, j}, the parameters ρ p k ij,, ρq k ij,, lp k ij, l q k ij, as well as the sensing radii s j, j N i (), which are needed for the calculation of the functions ρ p k ij, ρ q k ij, can be transmitted off-line to each agent i V. Moreover, the proposed control law does not incorporate any prior knowledge of the model nonlinearities/disturbances, enhancing thus its robustness. The main results of this work are summarized in the following theorem. Theorem. Consider a system of N agents, subject to (), aiming at establishing a formation described by the distances d k,des and orientation angles q k,des, k M, while satisfying the collision and connectivity constraints between neighboring agents, represented by d k,col and d k,con, respectively, with d k,col < d k,des < d k,con, k M. Then, under Assumptions,, the control protocol (7)-() guarantees: C k,col ρ p k (t) < ep k (t) < C k,conρ p k (t), ρq k (t) < e q k n (t) < ρ q k (t), k M, n {,, 3}, t, as well as the boundedness of all closed loop signals. Proof. The proof can be found in [Nikou et al., 6]. Remark. The transient and steady state performance of the closed loop system is explicitly and solely determined Fig.. The evolution of the orientation errors e q k n (t) with the performance functions ρ q k (t), k, n {,, 3}. by the parameters l p k, lq k, ρp k,, ρq k,, ρp k, and C k,col, C k,con, k M. Nonetheless, fine tuning might be needed in realtime scenarios, to retain the required control input signals within the feasible range that can be implemented by real actuators. Similarly, the input constraints impose an upper bound on the speed of convergence of ρ p k (t) and ρ q k (t), k M, as obtained by the exponentials e lp k t, e lq k t. Remark 3. Regarding Assumption, we stress that, by choosing the initial conditions θ i (), i V, and the desired formation constants θ k,des = q k,des, k M close to zero, the condition π < θ i(t) < π will not be violated, since the agents will be mostly operating near the point θ i =, i V. This is a reasonable assumption for real applications, since θ i represents pitch angles that are desired to be close to zero (consider, e.g., aerial vehicles). Furthermore, notice that the proposed control scheme guarantees collision avoidance only for the initially neighboring agents (at t = ), since that is how the edge set E is defined. Inter-agent collision avoidance with all possible agent pairs is left as future work.. SIMULATION RESULTS To demonstrate the efficiency of the proposed control protocol, we considered a simulation example with N = 4, V = {,, 3, 4} spherical agents of the form (), with r i = m and s i = 4m, i {,..., 4}. We selected the exogenous disturbances as w i = A i sin(ω c,i t)(a i x i a i, ẋ i ), where the parameters A i, ω c,i, a i, a i as well as the dynamic parameters of the agents were randomly chosen in [, ]. The initial conditions were taken as p () = [,, ] T m, p () = [,, ] T m, p 3 () = [, 4, 4] T m, p 4 () = [, 3,.] T m, q () = q () = q 3 () = q 4 () = [,, ] T r, which give the initial edge set E = {{, }, {, 3}, {, 4}}. The desired graph formation was defined by the constants d k,des =.m, q k,des = [ π 4,, π 3 ]T r, k {,, 3}. Invoking (4), we also chose C k,col =.m and C k,con =.7m. Moreover, the parameters of the performance functions were chosen as ρ p k, =., ρq k, = π > max{eq k (), e q k (), e q k 3 ()} = π 3 and l p k = lq k =, k {,, 3}. In addition, we chose ρ v i = m, ev i m () +., li v m = and ρ v i m, =.. Finally, γ i was set to. The simulation results are depicted in Fig. -4. In particular, Fig. and show the evolution of e p k (t) and eq k n (t) along with ρ p k (t) and ρq k (t), respectively, k {,, 3}, n {,, 3}. The velocity errors e v i m (t)

6 e v (t) e v 3(t) e v (t) e v i e v i e v i3 e v i4 e v i e v i e v 4(t) Fig. 3. The evolution of the velocity errors e v i m (t) with the functions ρ v i m (t), i {,..., 4}, m {,..., 6}. Fig. 4. The resulting input signals u i (t), i {,..., 4}. along ρ v i m (t) and the control signals u i are illustrated in Figs. 3 and 4, respectively. As it was predicted by the theoretical analysis, the formation control problem with prescribed transient and steady state performance is solved with bounded closed loop signals, despite the unknown agent dynamics and the presence of external disturbances. 6. CONCLUSIONS AND FUTURE WORK In this work we proposed a robust decentralized control protocol for distance- and orientation-based formation control, collision avoidance and connectivity maintenance of multiple rigid bodies with unknown dynamic models. Future efforts will be devoted towards extending the current results to directed as well as time-varying communication graph topologies. REFERENCES Anderson, B., Yu, C., Dasgupta, S., and Morse, S. (7). Control of a Three-Coleader Formation in the Plane. Systems & Control Letters, 6(9), Anderson, B., Yu, C., Fidan, B., and Hendrickx, J. (8). Rigid Graph Control Architectures for Autonomous Formations. IEEE Control Systems, 8, Bechlioulis, C. and Kyriakopoulos, K. (4). Robust Model-Free Formation Control with Prescribed Performance and Connectivity Maintenance for Nonlinear Multi-Agent Systems. CDC. Bechlioulis, C. and Rovithakis, G. (8). Robust Adaptive Control of Feedback Linearizable MIMO Nonlinear Systems with Prescribed Performance. TAC. Belabbas, A., Mou, S., Morse, S., and Anderson, B. (). Robustness Issues with Undirected Formations. CDC. Bishop, A.N., Deghat, M., Anderson, B., and Hong, Y. (). Distributed Formation Control with Relaxed Motion Requirements. International Journal of Robust and Nonlinear Control, (7), Cao, M., Morse, S., Yu, C., Anderson, B., and Dasgupta, S. (). Maintaining a Directed, Triangular Formation of Mobile Autonomous Agents. Communications in Information and Systems, (),. Dimarogonas, D. and Johansson, K. (8). On the Stability of Distance-Based Formation Control. CDC. Dorfler, F. and Francis, B. (). Geometric Analysis of the Formation Problem for Autonomous Robots. TAC. Eren, T. (). Formation Shape Control Based on Bearing Rigidity. International Journal of Control. Fathian, K., Rachinskii, D., Spong, M., and Gans, N. (6). Globally Asymptotically Stable Distributed Control for Distance and Bearing Based Multi-Agent Formations. IEEE American Control Conference. Hendrickx, J., Anderson, B., Delvenne, J., and Blondel, V. (7). Directed Graphs for the Analysis of Rigidity and Persistence in Autonomous Agent Systems. International Journal of Robust and Nonlinear Control. Horn, R. and Johnson, C. (). Matrix Analysis. Cambridge university press. Karayiannidis, Y., Dimarogonas, D., and Kragic, D. (). Multi-Agent Average Consensus Control with Prescribed Performance Guarantees. CDC. Krick, L., Broucke, M., and Francis, B. (9). Stabilisation of Infinitesimally Rigid Formations of Multi-Robot Networks. International Journal of Control. Mesbahi, M. and Egerstedt, M. (). Graph Theoretic Methods in Multiagent Networks. Princeton University Press. Nikou, A., Verginis, C., and Dimarogonas, D. (6). Robust Distance-Based Formation Control of Multiple Rigid Bodies with Orientation Alignment. Oh, K. and Ahn, H. (4). Distance-Based Undirected Formations of Single-Integrator and Double-Integrator Modeled Agents in n-dimensional Space. International Journal of Robust and Nonlinear Control. Oh, K., Park, M., and Ahn, H. (). A Survey of Multi- Agent Formation Control. Automatica, 3, Olfati-Saber, R. and Murray, R. (). Distributed Cooperative Control of Multiple Vehicle Formations using Structural Potential Functions. IFAC. Park, M., Oh, K., and Ahn, H. (). Modified Gradient Control for Acyclic Minimally Persistent Formations to Escape from Collinear Position. CDC, Smith, S., Broucke, M.E., and Francis, B.A. (6). Stabilizing a Multi-Agent System to an Equilateral Polygon Formation. International Symposium on Mathematical Theory of Networks and Systems, Sontag, E. (3). Mathematical Control Theory: Deterministic Finite Dimensional Systems, volume 6. Springer Science & Business Media. Trinh, M., Oh, K., and Ahn, H. (4). Angle-Based Control of Directed Acyclic Formations with Three- Leaders. ICMC, Zhao, S. and Zelazo, D. (6). Bearing Rigidity and Almost Global Bearing-Only Formation Stabilization. Transactions on Automatic Control.