Safe Positively Invariant Sets for Spacecraft Obstacle Avoidance

Abstract

This paper presents an obstacle avoidance method for spacecraft relative motion control. In this approach, a connectivity graph is constructed for a set of relative frame points, which form a virtual net centered around a nominal orbital position. The connectivity between points in the virtual net is determined based on the use of safe positively invariant sets for guaranteed collision free maneuvering. A graph search algorithm is then applied to find a maneuver that avoids specified obstacles and adheres to specified thrust limits. As compared to conventional open-loop trajectory optimization, this approach enables the handling of bounded disturbances, which can represent the effects of perturbing forces and model uncertainty, while rigorously guaranteeing that nonconvex and possibly time-varying obstacle avoidance constraints are satisfied. Details for handling a single stationary obstacle, multiple stationary obstacles, moving obstacles, and bounded disturbances are reported and illustrated with simulation case studies.

Path planning using positive invariant sets

“…i . The following corollary shows that (12) is equivalent to (11) with a fixed feedback gain F i . Corollary 2: Let F i be fixed.…”

“…3(b) respectively. The projected PI sets CO i for the local controllers designed by the semi-definite program (11) are shown in Fig. 3(c).…”

“…In [11] a path planner, based on the use of positively invariant sets, was developed for the spacecraft obstacle avoidance problem. Unlike the aforementioned planners, the focus is not on optimal trajectory planning or precise C. Danielson reference tracking.…”

“…Unlike the aforementioned planners, the focus is not on optimal trajectory planning or precise C. Danielson reference tracking. Instead, the planner in [11] implicitly finds a state trajectory from the initial state to the target state that explicitly satisfies the system dynamics and constraints. The algorithm uses a graph to switch between a collection of local feedback controllers.…”

Hopf Bifurcation, Positively Invariant Set, and Physical Realization of a New Four‐Dimensional Hyperchaotic Financial System

“…Normal form of a financial system with delaying has been derived, which is associated with Hopf and double Hopf bifurcations and makes the financial system more complicated [10,11]. In addition, the positively invariant sets of the dynamic system have a basic significance in the state constraints and control constraints, which are widely applied in different kinds of field: general 3-body problem [12], delay 2 Mathematical Problems in Engineering differential-equations [13], stability of dynamic system [14], robust attitude control schemes [15], relative motion control of the spacecraft [16], interconnected and time-delay systems [17], and permanent magnet synchronous motor system [18].…”