Natural frames and interacting particles in three dimensions

Abstract

Abstract-Motivated by the problem of formation control for vehicles moving at unit speed in three-dimensional space, we are led to models of gyroscopically interacting particles, which require the machinery of curves and frames to describe and analyze. A Lie group formulation arises naturally, and we discuss the general problem of determining (relative) equilibria for arbitrary G-invariant controls (where G = SE(3) is a symmetry group for the control law). We then present global convergence (and non-collision) results for specific two-vehicle interaction laws in three dimensions, which lead to specific formations (i.e., relative equilibria). Generalizations of the interaction laws to n vehicles is also discussed, and simulation results presented.

Coordinated patterns of unit speed particles on a closed curve

“…For underwater gliders, our results are applied by projecting the three dimensional motion onto the plane [7]. New developments have been made in [33] to use a natural frame setting to model three dimensional motion. The resulting steering laws are similar to those derived in the planar setting.…”

Stabilization of collective motion in three dimensions: A consensus approach

“…Model (1) characterizes particles dynamics with forcing only in the directions normal to velocity, i.e.r k = u a k ×ṙ k . An alternative to (1) is to provide each particle with an orthonormal frame and to write the system dynamics in a curve framing setting [3]:…”

“…The resulting closed-loop dynamics evolve in a quotient manifold called shape space and the equilibria of the reduced dynamics are called relative equilibria. Relative equilibria of the model (2) have been characterized in [3]. The equilibria, depicted in Figure 1, are of three types: i) Parallel motion: all particles move in the same direction with arbitrary relative positions; ii) Circular motion: all particles draw circles with the same radius, in planes orthogonal to the same axis of rotation; iii) Helical motion: all particles draw circular helices with the same radius, pitch, axis and axial direction of motion.…”

“…In Section II we define the kinematic model for a group of steered particles moving in three-dimensional Euclidean space with unitary velocity. In Section III we review some concepts from the theory of screws and we present a general methodology to stabilize relative equilibria on SE (3). In Section IV we derive control laws that stabilize relative equilibria in the presence of all-to-all communication.…”

Steering laws for distributed motion coordination of kinematic agents in three dimensions

“…Extension of the flocking results to three dimensions is nontrivial, because of higher degrees of freedom and a larger number of inputs for each robot. Some of the early works on controlling formation of rigid bodies in three-dimensions are [1], [11]. In [1] formation control for a team of nonholonomic aircraft was studied.…”