ural problem arises as how to obtain such feasible motions. We then propose a sys- tematic approach to derive feasible motions in terms of an equivalent control system. In the later part of this chapter, we will also provide typical examples and detailed analysis involving coordination control of constant-speed agents and heterogeneous agents to demonstrate the application of this coordination control framework. 9.1.2 Chapter organization
The rest of this chapter is organized as follows. Section 9.2 introduces preliminary concepts on differential geometry and the problem formulation with a motivating example. Section 9.3 presents the main result on formation feasibility condition by incorporating both heterogeneous kinematic constraint and formation constraint into a unified form. Section 9.4 discusses its application in coordination control of net- worked agents with constant-speed dynamics. In Section 9.5, we discuss the motion generation problem using the formation feasibility analysis, together with a detailed illustrative example on cooperative control of networked heterogeneous agents. Fi- nally, Section 9.6 concludes this chapter.
9.2
Preliminaries, problem formulation and motivating ex-
amples
9.2.1 Preliminaries on differential geometry and system equation
Some standard notions from differential geometry (especially the concepts of distri- bution/codistribution) will be introduced in this subsection. More background can be found in [Isidori, 1995, Chapter 1] and [Murray et al., 1994, Chapter 7].
A distribution ∆(x) on Rn is an assignment of a linear subspace of Rn at each point x. Given a set of k vector fields X1(x),X2(x),· · · ,Xk(x), we define the distri- bution as
∆(x) =span{X1(x),X2(x),· · · ,Xk(x)}.
A vector field X belongs to a distribution ∆ if X(x) ∈ ∆(x), ∀x ∈ Rn. Here we assume all distributions have constant rank.
A codistribution assigns a subspace to the dual space, denoted by (Rn)?. Given a distribution ∆, for each x consider the annihilator of ∆, which is the set of all covectors that annihilates all vectors in∆(x)(see [Isidori, 1995, Chapter 1])
∆⊥ ={
ω∈ (Rn)?| hω,Xi=0, ∀X∈∆}
In this chapter, we assume that each individual agent’s dynamics are described by the following general form (i.e. affine nonlinear control system)
˙ pi = fi,0+ l
∑
j=1 fi,jui,j (9.1)where pi ∈ Rni is the state of agent i, ni is the dimension of state space for agent i, fi,0 is a smooth drift term, and ui,j is thescalar control input associated with the smooth vector field fi,j, and lis the number of vector field functions. Such an affine nonlinear control system (9.1) with a drift term 1 is a very general tool to describe many different types of real-life control systems, including control systems with an underactuation property or nonholonomic constraints.
9.2.2 Motivating examples
The paper [Tabuada et al., 2005] introduced the concept of motion feasibility problem for multi-agent formations. The discussions in [Tabuada et al., 2005] were restricted to the coordination control ofdrift-freecontrol systems (i.e. fi,0=0) in the form of
˙ pi = l
∑
j=1 fi,jui,j (9.2)However, the above model is not general enough to describe many real-life non- linear control systems. In contrast, the control system model in (9.1) encompasses a larger number of practical models and is modelling the most popular nonlinear control system [Isidori, 1995]. As an example, the unicycle-type agent with constant- speed constraints is one such nonlinear control system with drift terms that can be described by (9.1) but not by (9.2). Such system dynamics can be described as
˙ xi =vi cos(θi) ˙ yi =vi sin(θi) (9.3) ˙ θi =ui
wherexi ∈R,yi ∈Rare the coordinates in the real plane andθi is the heading angle for agenti. The agent has a fixed cruising speedvi >0, which could be different for distinct agents;ui is the control input to be designed for steering the orientation.
Introducing the vector fields as
fi,0= vicos(θi) visin(θi) 0 ,fi,1= 0 0 1 , (9.4)
we can rewrite the system (9.3) as ˙
pi = [x˙i, ˙yi, ˙θi]> = fi,0+ fi,1ui (9.5) which has the form of (9.1).
1A statement by Roger Brockett in his recent survey paper [Brockett, 2014]: “almost all real systems
§9.2 Preliminaries, problem formulation and motivating examples 141
9.2.3 Problem formulation: formation feasibility with kinematics and for- mation constraints
In this chapter, we assume a networked multi-agent control system modelled by an undirected graph G, in which we useV to denote its vertex set and E to denote the edge set. The vertices consist ofnheterogeneous agents each modelled by the general dynamical equation (9.1). The graph consists of m edges, each with an inter-agent formation constraint.
A family of formation constraints C is indexed by the edge set, denoted as CE ={cij}(vi,vj) with (vi,vj)∈ E. For each edge(vi,vj),cij is a vector function defin- ing the formation constraints between agents i and j and the constraint is enforced if cij(pi,pj) = 0. Such formation constraints can be used to describe very general coordinate control problems, such as formation shape control, formation tracking, coverage control, etc. For example, in formation shape control, the constraint vector functioncij can be a function of desired relative position, or desired bearings, or de- sired distances between agentsiandjthat can be used to describe a target formation (for example, see [Oh et al., 2015]).
The formation feasibility problem is stated as follows:
Problem 1. Given a formation graph F = [V,E,C], determine whether there are feasible
trajectories pi(t)(or equivalently, feasible agents’ motions p˙i(t)) for all agents whose kine- matics are modelled by (9.1) with possible drift terms, such that the trajectories pi(t)also meet formation constraints CE = 0, where CE = [· · · ,c>ij,· · ·]> for all (vi,vj) ∈ E and t ∈ I where I is a specified time interval.
In the case that there exist feasible motions, we further consider the motion gen- eration problem formulated as below.
Problem 2. Given a formation graph F= [V,E,C]with feasible agents’ motions, determine
an equivalent control system that generates feasible motions for the networked heterogeneous multi-agent system.
Remark 27. A prerequisite of solving the above motion feasibility problem is that the for-
mation constraint for the distributed edge set should be non-conflicting, and the overall con- straint for all the edges should be realizable at least in the full Euclidean space. For example, if the formation is described by relative position vectors which are conflicting, then the for- mation is unachievable and this may lead to unexpected flocking motion [Dimarogonas and Kyriakopoulos, 2008]. As another example, if the formation is described by distance con- straints to realize a rigid shape in a 3-agent group, then the set of formation distances should satisfy the triangle inequality [Sun et al., 2014c]. Thus, in order to well define the feasibil- ity problem, we need to first assume that the formation constraints are non-conflicting and realizable.