Collision Avoidance of Multiagent Systems on Riemannian Manifolds

Abstract

This paper studies the reduction by symmetry of variational problems on Lie groups and Riemannian homogeneous spaces. We derive the reduced equations of motion in the case of Lie groups endowed with a left-invariant metric, and on Lie groups that admits a bi-invariant metric. We repeated this analysis for Riemannian homogeneous spaces, where we derive the reduced equations by considering an alternative variational problem written in terms of a connection on the horizontal bundle of the underlying Lie group. We study also the case that the underlying Lie group admits a bi-invariant metric, and consider the special case that the homogeneous space is in fact a Riemannian symmetric space. These ideas are applied to geodesics for a rigid body on SO(3) to derive geodesic equations on the dual of its Lie algebra (a vector space), the heavy-top in SE(3) to derive reduced equations of motion on the unit sphere S 2 , geodesics on S 2 as a Riemannian symmetric space endowed with a bi-invariant metric and optimal control problems for applications to robotic manipulators.

A Coordinate-Free Variational Approach to Fourth-Order Dynamical Systems on Manifolds: A System and Control Theoretic Viewpoint

“…Fourth-order (or jounce) dynamical systems on manifolds and their control were studied in detail in a series of contributions, among which we can cite [31][32][33]. We also mention that the special case, where σ = 1, φ = 0, gradV = 0 and f = 0, is related to the theory of 'cubic polynomials' (for a review and connections with non-linear control, readers might consult [34]).…”

Variational problems on Riemannian manifolds with constrained accelerations

“…Remark 1. A generalization of this problem was studied in [5,6], where the authors add a potential into the picture to study dynamical interpolation problems for obstacle avoidance and further used in [2,14,13,16,17,18] to provide necessary and sufficient conditions of collision avoidance of multi-agent systems. There one studies curves in Ω satisfying the constraints (2) and minimizing the functional…”

“…Motivated by this connection and applications to dynamic interpolation on manifolds [22], Crouch and Silva Leite [15] started the development of an interesting geometric theory of generalized cubic polynomials on a Riemannian manifold M , in particular on compact connected Lie groups endowed with a left-invariant metric. Further extensions appears in the context of obstacle avoindance problems [6,4], regression problems on Lie groups [3], collision avoidance problems [16], and sub-Riemannian geometry, with connections with non-holonomic mechanics and control, studied by Bloch and Crouch [7,8]. These sub-Riemannian problems are determined by additional constraints on a non-integrable distribution on M .…”

Aggregation of Multi-Agent System on Manifolds With Riemannian Geometry

“…systems on various manifolds, an extended Cucker-Smale (C-S) model is provided in [11], [12], and [13]. The collisions avoidance task of a multi-agent system on general manifolds has been considered in [14], [15], [16], [17], and [18], and an artificial potential on a general manifold is designed to ensure that agents will avoid collisions within some desired tolerance. In general, these models cannot describe all of the significant features of synergistic behaviors of multiagent systems entirely, and some of their applications are limited.…”