1. Introduction

In this chapter, we present an approach to constrained multi-agent control on the unit sphere; by applying consensus theory and constrained attitude control (CAC) via semidefinite programming. Global navigation can be modeled by control on the unit sphere and such algorithms have applications in: aerial navigation [1]; sea navigation and ocean sampling [2]; space navigation and satellite cluster positioning [3, 4]. For example, the algorithm presented in this chapter will find practical application in aircraft horizontal separation.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

Most path-planning work generally focus on two-dimensional (2D) [5, 6], and three- dimensional motion planning (3D) [7–10]. However, both path planning models are limited when the motion is constrained to evolve on a sphere.

Notation Meaning

ui

x j

vi; vj

n Number of vehicles i Vehicle number i

xi Position vector of vehicle i

, x\_<sup>i</sup> Control input of vehicle i

obs Obstacle vector number <sup>j</sup>

<sup>b</sup><sup>x</sup> Unit vector corresponding to vector <sup>x</sup>

xij � �off Offset vector between vehicles i and j wij Angle between vehicle i and obstacle j θij Angle between vehicles i and j

x Stacked vector of n position vectors u, x\_ Stacked vector of n control inputs L Laplacian matrix, L ¼ D<sup>G</sup> � A<sup>G</sup> L,L<sup>i</sup> Laplacian-like stochastic matrix 0 A vector consisting of all zeros ⊗ Kronecker multiplication operator

SEð Þ3 Special Euclidean group

I<sup>n</sup> The n � n identity matrix

G Graph

Table 1. Frequently used notation in this chapter.

V Set of vertices of G E Set of edges of G vi Vertex vi ∈V

� � Endpoint or edge vi; vj

<sup>N</sup><sup>i</sup> Neighbors of vi; <sup>N</sup><sup>i</sup> <sup>¼</sup> vj <sup>∈</sup><sup>V</sup> : vi; vj

<sup>A</sup><sup>G</sup> Adjacency matrix of <sup>G</sup>; <sup>A</sup><sup>G</sup> <sup>¼</sup> aij � � <sup>D</sup><sup>G</sup> Out-degree matrix of <sup>G</sup>; <sup>D</sup><sup>G</sup> <sup>¼</sup> dij � � S A vector or matrix in the Schur's inequality R A positive definite matrix in the Schur's inequality Q A symmetric matrix in the Schur's inequality

M A positive definite matrix variable G A positive semidefinite matrix

<sup>S</sup><sup>m</sup> The set of <sup>m</sup> � <sup>m</sup> positive definite matrices

Λ A positive definite matrix variable, Λ ∈Sm

<sup>C</sup> The consensus space for x, <sup>C</sup> <sup>¼</sup> <sup>x</sup>jx<sup>1</sup> <sup>¼</sup> <sup>x</sup><sup>2</sup> <sup>¼</sup>; <sup>⋯</sup>; <sup>¼</sup> <sup>x</sup><sup>n</sup> � �

� �∈E

Multi-Path Planning on a Sphere with LMI-Based Collision Avoidance

http://dx.doi.org/10.5772/intechopen.71216

27

� � ∈E � �

α<sup>i</sup> Minimum angular separation from obstacle number i βij Minimum angular separation between vehicles i and j

Looking at the main works that have been done on control on a sphere, [11] applied Lie algebra to develop a model of self-propelled particles (as point masses) which move on the surface of a unit sphere at constant speed. Circular formations of steady motions of the particles around a fixed small circle on the sphere were identified as relative equilibria of the model using a Lie group representation. The paper also provided mathematically justified shape control laws that stabilize the set of circular formations. They also proposed a shape control to isolate circular formations of particles with symmetric spacing by using Laplacian control. Further work on this is presented in [12].

The works [11, 12] are based on [5, 13], where a geometric approach to the gyroscopic control of vehicle motion in planar and three-dimensional particle models was developed for formation acquisition and control with collision avoidance in free space. They discovered three possible types of relative equilibria for their unconstrained gyroscopic control system on SEð Þ3 : (i) parallel particle motion with arbitrary spacing; (ii) circular particle motion that has a common radius, axis and direction of rotation, and arbitrary along-axis spacing; (iii) helical particle motion that has a common radius, axis and direction of rotation, along-axis speed (pitch) and arbitrary along-axis spacing. This approach is effective in formation control of multiple systems in unconstrained spaces and for formations that conform to the three types of relative equilibria described above.

Therefore, it is necessary to consider consensus on a sphere, which can be applied to the more general motion control problem involving: (i) constrained spaces which contain static obstacles such as clutter; (ii) speed constrained vehicles; (iii) other arbitrary formations which are different from the relative equilibria described above. We apply consensus theory to collective motion of a team of communicating vehicles on the sphere and the concept of constrained attitude control (CAC) to generate collision avoidance behavior among the vehicles as they navigate to arbitrary formations [14].

We assume that each individual vehicle can communicate with neighbors within its sensor view. Each vehicle can therefore use the Laplacian matrix of the communication graph L in a semidefinite program to plan consensus trajectories on the sphere. Then the concept of CAC is used to incorporate collision avoidance, by maintaining specified minimum angles between vectors of vehicle positions. The algorithm presented here is applicable to motion control in both constrained and unconstrained spaces on the sphere, e.g. for planning consensus trajectories around static obstacles or adversarial non-cooperative obstacles on the sphere. The approach can also be applied to constrained vehicle motion of non-constant velocities. It is also possible to generate formations on the sphere that are different from circular motion.

The rest of this chapter is organized as follows. In Section 2, we present the mathematical basis of consensus theory, while the problem statement is presented Section 3. In Section 4, the solution and convergence analysis are presented. This is followed by simulation results in Section 5 and references in Section 6. Table 1 lists frequently used notation in this chapter.

#### Multi-Path Planning on a Sphere with LMI-Based Collision Avoidance http://dx.doi.org/10.5772/intechopen.71216 27


Table 1. Frequently used notation in this chapter.

Most path-planning work generally focus on two-dimensional (2D) [5, 6], and three- dimensional motion planning (3D) [7–10]. However, both path planning models are limited when the

Looking at the main works that have been done on control on a sphere, [11] applied Lie algebra to develop a model of self-propelled particles (as point masses) which move on the surface of a unit sphere at constant speed. Circular formations of steady motions of the particles around a fixed small circle on the sphere were identified as relative equilibria of the model using a Lie group representation. The paper also provided mathematically justified shape control laws that stabilize the set of circular formations. They also proposed a shape control to isolate circular formations of particles with symmetric spacing by using Laplacian

The works [11, 12] are based on [5, 13], where a geometric approach to the gyroscopic control of vehicle motion in planar and three-dimensional particle models was developed for formation acquisition and control with collision avoidance in free space. They discovered three possible types of relative equilibria for their unconstrained gyroscopic control system on SEð Þ3 : (i) parallel particle motion with arbitrary spacing; (ii) circular particle motion that has a common radius, axis and direction of rotation, and arbitrary along-axis spacing; (iii) helical particle motion that has a common radius, axis and direction of rotation, along-axis speed (pitch) and arbitrary along-axis spacing. This approach is effective in formation control of multiple systems in unconstrained spaces and for formations that conform to the three types of

Therefore, it is necessary to consider consensus on a sphere, which can be applied to the more general motion control problem involving: (i) constrained spaces which contain static obstacles such as clutter; (ii) speed constrained vehicles; (iii) other arbitrary formations which are different from the relative equilibria described above. We apply consensus theory to collective motion of a team of communicating vehicles on the sphere and the concept of constrained attitude control (CAC) to generate collision avoidance behavior among the vehicles as they

We assume that each individual vehicle can communicate with neighbors within its sensor view. Each vehicle can therefore use the Laplacian matrix of the communication graph L in a semidefinite program to plan consensus trajectories on the sphere. Then the concept of CAC is used to incorporate collision avoidance, by maintaining specified minimum angles between vectors of vehicle positions. The algorithm presented here is applicable to motion control in both constrained and unconstrained spaces on the sphere, e.g. for planning consensus trajectories around static obstacles or adversarial non-cooperative obstacles on the sphere. The approach can also be applied to constrained vehicle motion of non-constant velocities. It is also possible

The rest of this chapter is organized as follows. In Section 2, we present the mathematical basis of consensus theory, while the problem statement is presented Section 3. In Section 4, the solution and convergence analysis are presented. This is followed by simulation results in Section 5 and references in Section 6. Table 1 lists frequently used notation in this chapter.

to generate formations on the sphere that are different from circular motion.

motion is constrained to evolve on a sphere.

26 Advanced Path Planning for Mobile Entities

control. Further work on this is presented in [12].

relative equilibria described above.

navigate to arbitrary formations [14].
