Analytical Solving the Optimal Control Problem of Spacecraft’s Slew Maneuver with Minimal Energy of Rotation

Dynamic problem of optimal reorientation from an arbitrary initial attitude into the given final angular position with restricted control which minimizes kinetic energy of spacecraft rotation was solved. Termination time of maneuver is known. Quadratic criterion of quality is applied for finding the optimal control program. Use of integral index in special form concerning angular velocity has helped solve the formulated problem by analytical way. Control law was written down in explicit form. Designing the optimal control is based on quaternion variables and models. It is shown that during optimal turn, the controlling moment is parallel to the straight line which is immobile in the inertial space, and direction of spacecraft’s angular momentum in the process of rotation is constant relative to the inertial coordinate system. Special control regime was studied in detail, and conditions of the impossibility of occurrence of this regime are formulated. It is proven that spacecraft rotates by inertia in special control regime if it exists. The formalized equations and computational expressions for determining the optimal rotation program and duration of acceleration and braking were written. A dependence of control variables on phase coordinates is presented also. The proposed control algorithm allows the spacecraft’s reorientation to be carried out within the fixed time period with minimal angular kinetic energy. Analytical expressions for computing the time characteristics of reorientation maneuver are given, and condition for determination of the moment of the beginning of the braking, based on factual kinematic parameters of motion judging by principles of terminal control is formulated, that provides high accuracy of orientation. A comprehensive solution to the control problem is presented for a dynamically symmetric spacecraft: the dependences as explicit functions of time for the control variables are obtained, and relations for calculating the key parameters of the turn maneuver’s control law are given also. A numerical example and the results of mathematical simulation of spacecraft’s motion with optimal control are presented, which demonstrate the practical feasibility of the designed method for controlling the spacecraft attitude. Presence of ready formulas for synthesis of optimal motion program during reorientation maneuver does the executed research as practically significant and suitable for direct use in practice of space flights.

spacecraft, attitude, quaternion, energy of rotation, maximum principle, control function, boundary-value problem

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