Optimal Trajectory Design of Formation Flying based on Attractive Sets

Abstract

This paper presents a new method of optimal trajectory design for formation flying. Under linearized assumptions and a quadratic performance index, we introduce an attractive set of optimal control based on the linear quadratic regulator theory. The attractive set is defined as a set of all initial states to reach a desired state for a given cost. In particular, we consider attractive sets for two problems: a fixed final-state, fixed final-time problem and an infinite-time problem, and the optimal initial state is found based on the geometry of the attractive set. The optimal trajectories for two problems are evaluated in terms of L 1-norm of control input and termination time.

Formation-Flying Trajectory Design Based on Attractive Sets of Tschauner–Hempel Equations

“…From the geometry of the sets, the minimum values of the performance index occur at the tangent points between the periodic orbit and the attractive sets, which overlaps the contour of * . These results suggest that the asymptotic behavior of the attractive sets to reflect the free motion and to become less sensitive to the initial state as presented in [15] for circular case holds for the TH equations.…”

“…Then all initial state of Γ( 1 , ) is parameterized by the size parameter > 0 and the initial true anomaly 0 ≤ 0 ≤ 2 so that the corresponding initial position ( 0 , 0 ) of the chaser is uniquely computed from Eqs. (15) and (16). Corresponding initial velocity ( 0 , 0 ) is also uniquely determined on Γ( 1 , ) from Eqs.…”

“…Corresponding initial velocity ( 0 , 0 ) is also uniquely determined on Γ( 1 , ) from Eqs. (15) and (16). Then the attractive sets can be drawn in two-dimensional plane of ( 0 , 0 ) as in the circular case.…”

“…This motivates to study the geometric interpretation of the solutions of the ARE and DRE for formation flying by introducing the concept of attractive set of optimal control. In [15], attractive sets for optimal rendezvous problem along a circular orbit was investigated based on the solution of the ARE. It was shown that the optimal initial positions can be obtained as the tangent point between the largest attractive set inscribed in the periodic orbit and the periodic orbit.…”