Leader-follower formation control of underactuated surface vessels

Abstract

Leader-follower formation control of underactuated surface vessels has attracted much attention in past decades. The challenge arises from the second-order nonholonomic constraint on acceleration. To overcome the difficulty, plenty of nonlinear techniques are applied, such as robust adaptive control [1], dynamic surface control [2], and input-output linearization [3]. In addition, many papers study the eventtriggered technique to enhance the management of communication resources between surface vessels, e.g., [4, 5]. Generally, the surface vessel’s states contain both of position and orientation. Most existing results consider these states in a linear space. However, the actual configuration space is a nonlinear manifold [6]. An important class of manifolds which arises naturally in rigid body kinematics is called matrix Lie groups. Regarding surface vessels, the system can be established on the special Euclidean group in two dimensions, i.e., SE(2). Moreover, in many studies, another fact is that the desired formation pattern is defined in the earth-fixed frame. However, the formation cannot rotate in the scenario of curved reference paths but only has the movement of translation. Thus, it will be more practical if the desired pattern is given in the body-fixed frame. Therefore, we investigate the leader-follower formation control of underactuated surface vessels under the framework of SE(2). The desired formation pattern is not predefined in the earth-fixed frame but in the leader’s body-fixed frame. By designing a virtual leader, the formation problem is transformed into the trajectory tracking problem. Then, with relative systems on Lie groups, we convert the trajectory tracking to the stabilization of a relative system. Utilizing a stabilization controller on SE(2), we derive the formation controller eventually. Regarding the advantages, the surface vessel is described on a differentiable manifold without any local coordinates, so that globally effective results can be derived under such a framework. Another advantage is that we consider the desired formation pattern given in the leader’s body-fixed frame, which allows the formation to have rotation as well as translation. Preliminaries. Let g = (R, p) ∈ SE(2) denote the configuration of the surface vessel, where R is the rotation matrix through attitude angle θ, and p = [x y]T is the position vector. Let ξ̂ ∈ se(2) denote the velocity, where se(2) is the Lie algebra associated with SE(2). Then the kinematic equation of the surface vessel is ġ = gξ̂, which is a global description in the sense that it does not rely on local coordinates. Let I represent the inertia tensor; then, the dynamic equation of the surface vessel is given by ̇̂ ξ = I♯(ad ξ̂ I ♭ξ̂+ b̂(ξ̂)+ ∑2 i=1 f̂iui), where f̂i is the control vector field, and ui is the scalar control input. We assume there are only yaw moment and surge force on the surface vessel. For the simplicity, we define a new control input τ̂ = ∑2 i=1 f̂iui, which can be expressed in R3 as τ = [τr τx 0]T. Therefore, we will design the yaw moment τr and surge force τx. Note that the control input τ̂ lies in Lie algebra se(2), while the configuration g is on Lie group SE(2). In order to construct state feedback, g should be mapped into se(2). Thus, we introduce the logarithm map logSE(2) : SE(2) → se(2). Definition 1. For g = (R, p) ∈ SE(2) with trace(g) 6= −1, the logarithm map logSE(2) is defined as X̂ = logSE(2)(g) = [θ̂ A−1(θ)p; 01×2 0], where θ̂ = [0 − θ; θ 0], A−1(θ) = [α(θ) θ/2; − θ/2 α(θ)], α(θ) = θ 2 cot θ 2 . Due to the fact that R3 and se(2) are isomorphic, X̂ can also be mapped into R3 and it is denoted by X = [θ qx qy]T, where q = [qx qy]T = A−1(θ)p. Problem formulation. Consider the formation of two surface vessels, i.e., one leader and one follower. We use subscript “0” and “1” to represent leader and follower respectively, so that their configurations are denoted by g0 and g1. Due to the fact that the desired formation is provided in the leader’s body-fixed frame, we firstly define the relative configuration g01. Referring to errors on Lie groups, the relative configuration of the follower with respect to the leader can be defined as g01 = g −1 0 g1. Then, g01 can be uniquely decided by θ01 and r01 = [rx 01 r y 01] T, where θ01 = θ1 − θ0, rx 01 = (x1 − x0) cos θ0 + (y1 − y0) sin θ0, r 01 = −(x1 − x0) sin θ0 + (y1 − y0) cos θ0. Let θ̄01 denote the desired relative attitude angle and r̄01 = [r̄x 01 r̄ y 01] T denote the desired relative position in the leader’s body-fixed