Optimal rigid-body rotational maneuvers

Abstract

Optimal rigid-body angular maneuvers are investigated, using restricted control moments—a problem inspired in the context of rotational maneuvers for <i>super-maneuverable</i> aircraft. Most of the analysis is based on the formulation with no direct control over the roll component of angular velocity. The present research effort is conducted in two phases. In the first phase, optimal control of angular <i>rates</i> is closely examined. The second phase deals with the problem of optimal <i>attitude</i> control.

Optimal rigid-body angular <i>rate</i> control is first examined via an <i>approximate</i> dynamic model. The proposed model admits analytical solutions of the optimality conditions. The analysis reveals that over a large range of boundary conditions, there are, in general, <i>several</i> distinct extremal solutions. Second-order necessary conditions are investigated to establish local optimality of candidate minimizers. Global optimality of the extremal solutions is discussed.

Next, the optimal angular <i>rate</i> problem is studied using the <i>exact</i> dynamic model. Numerical solutions of optimality conditions are obtained which corroborate and extend the findings of the <i>approximate</i> problem. The qualitative feature of <i>multiple extremal solutions</i> is retained. Several of these extremal solutions did not satisfy the Jacobi necessary condition. The choice of <i>minimizing</i> solution could be narrowed down to two sub-families of extremal solutions. A locus of Darboux Points is obtained which demarcates the domain over which these two sub-families are globally minimal.

The above studies look at <i>minimum control effort</i> families of extremal solutions. As a next step, we examine the <i>minimum time</i> control of angular rates, with prescribed hard bounds on available control. Existence of singular subarcs in time-optimal trajectories is explored. Qualitative features exhibited by the <i>exact</i> problem are preserved. In addition, the control space is deformed to allow roll control and its effect on extremal solutions is investigated.

In the next phase, we introduce the <i>kinematics</i> into the optimal control problem. Minimum time <i>attitude</i> control of a rigid-body is investigated with prescribed hard bounds on available control. The attitude of the rigid-body is defined using Euler parameters. Existence of singular subarcs in time-optimal trajectories is explored. A numerical survey of first-order necessary conditions reveals that there are <i>several</i> distinct extremal solutions. The character of extremal solutions depend whether <i>pitch</i> or <i>yaw</i> motion assumes the dominating role in controlling <i>roll</i> motion. Moreover, certain <i>spatial symmetries</i> are identified. Maneuvers such as a <i>Roll Around the Velocity Vector</i> and <i>Fuselage Pointing</i> are analyzed.