Tercera problemática: tentativas homosexuales

In this section we consider several simulation examples on rigid formation control in the 3-D space. The target formation shape is supposed to be a double tetrahedron formation with 9 edges, with the desired distances for each edge being 6. The initial positions and initial velocities for each agent are chosen randomly, but we also ensure that the initial formation shape is close to the target one. The formation stabilization result is shown in Figure 8.1, which illustrates the trajectories of each agent, together with the initial shape and final shape, and the trajectories of each distance error. The formation flocking result is shown in Figure 8.2, which illustrates the trajectories of 8_{Another way to perform the system transformation is to define the velocity disagreement vector}

with respect to theformation centroid velocity. Such a system transformation also gives rise to a reduced- order double-integrator formation system, as discussed in [Deghat et al., 2016].

§8.6 Illustrative examples on convergence 133 −5 0 5 10 −2 0 2 4 6 8 −2 0 2 4 6 0 1 2 3 4 5 6 7 −40 −30 −20 −10 0 10 20 30 t

squared distance errors

Figure 8.1: Simulation on shape stabilization control of a double tetrahedron forma- tion in the 3-D space with double-integrator systems.

each agent, the flocking behavior, and the trajectories of each distance error. It can also be seen from simulations that the convergence of each distance error vector in both cases isexponentially fast.

Note that the above examples only show local convergence to the target formation shape. The following simulation shows a comparison of convergence results between single-integrator formation models and double-integrator formation models. We as- sume the same target formation shape as described above, and choose the initial positions for all agents as p1(0) = [4, 4, 0]>, p2(0) = [0, 0, 0]>, p3(0) = [4, 0, 0]>, p4(0) = [5, 0, 0]> and p5(0) = [−4,−10, 0]>. The simulation result with single- integrator model is depicted in Figure 8.3, which shows the convergence to an in- correctformation shape in a degenerate 2-D plane when agents’ initial positions are chosen in that plane (which has measure zero in the whole spaceR3). The converged equilibrium in this case is p∗₁ = [3.2992, 3.8112, 0]>, p₂∗ = [−1.5078, 0.2206, 0]>, p∗₃ = [4.0052,−2.1471, 0]>, p∗₄ = [4.0052,−2.1471, 0]> and p∗₅ = [−0.8018,−5.7376, 0]>, which span the same plane that contains the initial positions. The convergence to an incorrect equilibrium (more specifically, a degenerate incorrect equilibrium) is due to the rank-preserving property of single-integrator rigid formation systems as discussed in [Sun et al., 2015a].

We then use the same initial positions (and random initial velocities) to perform the simulation with double-integrator models, and observe the convergence to the correct formation shape as shown in Figure 8.4. Actually, as proved in [Sun et al., 2015a], the incorrect equilibrium formation illustrated in Figure 8.3 is a saddle point. Such an equilibrium point is unstable for both single-integrator formation systems and double-integrator formation systems (see Theorem 20). However, this simula- tion also shows one advantage of using double-integrator formation models against

0 10 20 30 −5 0 5 10 15 20 −5 0 5 10 15 20 25 0 1 2 3 4 5 6 7 −40 −30 −20 −10 0 10 20 30 40 t

squared distance errors

Figure 8.2: Simulation on formation flocking control of a double tetrahedron forma- tion in the 3-D space with double-integrator systems.

single-integrator formation models. For double-integrator formation systems, even if one chooses initial positions that live in a lower-dimensional space, the forma- tion system will not always live in that degenerate space. That is, double-integrator formation systems can escape from degenerate positions (e.g., collinear positions or coplanar positions) and thus will avoid the convergence to an incorrect degener- ate equilibrium even if agents’ positions are initially placed in a lower dimensional space. This is due to the fact that therank-preservingproperty does not hold for rigid formation systems modelled by double-integrator dynamics.

8.7

Concluding remarks

In this chapter, we have considered formation control systems modelled by double integrators, which include the formation stabilization model and flocking control model. Due to the multi-equilibrium property caused by the nonlinear formation controller, a complete analysis of the convergence is quite challenging. This chap- ter serves as a further step to understand the dynamical behavior of such formation control systems. Novel properties of the system dynamics and convergence analysis for different equilibria are discussed by analyzing certain properties of the linearized systems and a parameterized Hamiltonian-like system. Some key results are sum- marized as follows:

• We discuss the measurement requirement for individual agents and establish the independence of a global coordinate frame for implementing the formation controller.

• Certain properties of the Jacobian matrix (e.g. the rank, null spaces and eigen- values) for double-integrator formation systems are characterized.

§8.7 Concluding remarks 135 −5 0 5 −10 −5 0 5 −1 −0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 −40 −30 −20 −10 0 10 20 30 40 t

squared distance errors

Figure 8.3: Simulation on formation shape control of a double tetrahedron formation in the 3-D space with single-integrator systems. The initial conditions are chosen to be in a plane in the 3-D space, and the formation converges to an incorrect formation

shape that lives in that plane (i.e. an incorrect planar equilibrium formation).

−5 0 5 10 −5 0 5 10 −5 0 5 10 0 1 2 3 4 5 6 7 −40 −30 −20 −10 0 10 20 30 40 t

squared distance errors

Figure 8.4: Simulation on formation shape control of a double tetrahedron forma- tion in the 3-D space with double-integrator systems. The formation converges to a correct formation shape, even if one chooses degenerate coplanar initial positions.

• Invariance principles concerning the equilibrium set and local stability are es- tablished for a family of parameterized Hamiltonian-like system, which builds the link between single-integrator formation systems and double-integrator formation systems. Thus, available results on equilibrium analysis in single- integrator rigid formation systems reported in the vast literature can be ex- tended to double-integrator formation systems. Note that the rank-preserving property, as discussed in Chapter 3 for single-integrator formation systems, does not hold for double-integrator formation systems.

• Several criteria for determining the stability and convergence of equilibrium sets for double-integrator formation systems are proposed.

Chapter9

Formation feasibility and motion

generation of networked

heterogeneous systems

Chapter summary

In this chapter, we discuss a general problem of formation feasibility for multi-agent coordination control when individual agents have kinematics constraints modelled by affine nonlinear control systems with possible drift terms. Such dynamics mod- els include the single-integrator model and double-integrator model considered in previous chapters, as well as other commonly-used models such as unicycle models discussed in many papers. In this problem, all agents need to work cooperatively to maintain a global formation task described by edge constraints. For such a multi- agent group, we assume that different agents may have totally different dynamics, which brings the problem of coordination control of networked heterogeneous sys- tems. Based on the concepts of (affine) distribution and codistribution, we propose a unified framework and an algebraic condition to determine the existence of feasible motions under both kinematic constraints and formation constraints. In the case that feasible motions exist, we propose a systematic procedure to obtain an equivalent dynamical system which generates all types of feasible motions. Several examples involving coordination control of constant-speed agents and heterogeneous agents are provided to demonstrate the application of this coordination control framework.

9.1

Introduction

9.1.1 Background, motivation and related work

Collective coordination control of networked multi-agent systems has received con- siderable attention in recent years, partly motivated by its applications in many areas [Cao et al., 2013; Knorn et al., 2016]. A particular class of cooperative tasks for multi- agent coordination is formation control, in which the control objective is to form or maintain a prescribed geometric relationship for a group of spatially distributed

agents [Oh et al., 2015]. Maintaining a formation is important in multi-agent coordi- nation and in some cases would be a prerequisite for agents to perform additional tasks such as surveillance, coverage, target detection, etc ([Egerstedt and Hu, 2001; Mesbahi and Egerstedt, 2010]).

Given a predefined formation task assigned to a group of distributed agents, a fundamental problem is to determine whether there exists a feasible trajectory for such an agent group to maintain the formation task subject to both kinematics and formation constraints. Formation feasibility was firstly discussed by P. Tabuada et. al. in [Tabuada et al., 2005]. They employed tools in differential geometry and de- rived an elegant criterion to analyze whether a networked agent group has feasible motions to maintain the formation constraint. Later papers along this direction in- clude [Maithripala et al., 2008], which proposed a similar geometric approach to discuss some real-time formation control problems, including radar deception, for- mation keeping and formation reconfiguration. As revealed in [Tabuada et al., 2005] and [Maithripala et al., 2008], the central idea in the formation feasibility analysis is the interplay of agents’ kinematic constraints and formation task constraints in controlling a formation and in generating a feasible trajectory for all the agents.

The concept of formation feasibility in this chapter builds on the analysis in [Tabuada et al., 2005], while here we present several extensions and generalizations to this fundamental networked control problem. First, in contrast to the control system model discussed in [Tabuada et al., 2005], in this chapter we consider agents’ dynam- ics modelled by affine nonlinear control systems with possibly additionaldrift terms. This is motivated by the fact that most real-life control systems have drift terms. We note that affine nonlinear control systems with drift terms are very general in system modelling and are also popular choices for nonlinear control system analysis in the control community [Isidori, 1995]. One motivating example to consider agent dy- namics with drift terms is the coordination control of constant-speedagents. Coordi- nation control and collective circular motion with unit-speed agents or non-identical constant-speed agents was reported in e. g. [Sepulchre et al., 2008], [Seyboth et al., 2014]. More recently, the tracking control problem for multiple constant-speed agents was discussed in [Sun et al., 2015c] which also revealed the performance limitations in this coordination control problem. However, a general analysis on performance feasibility and its relationship to the speed constraint is still lacking. The results in this chapter will shed new insights to this coordination control problem.

Secondly, we further consider the case that individual agents in the networked system may have totally different dynamics, a possibility which includes fully ac- tuated agents, under-actuated agents, or agents with a nonholonomic constraint and/or drift terms. Under this framework, we develop a fairly checkable condition for the existence of feasible motions by including both heterogeneous agent dynam- ics and formation constraints. Thus, the coordination control framework developed in this chapter is general enough to encompass many coordination control problems and presents a unified approach in the coordination feasibility analysis of networked heterogeneoussystems.