Program Control of the Spatial Motion of a Solid Body, Optimal in the Sense of a Minimum Integral Quadratic Functional Relative to the Accelerations, Using Dual Quaternions

The problem of optimal program control of the spatial motion of a free solid body (in particular, a spacecraft) in an inertial coordinate system using dual quaternions (parabolic Clifford biquaternions) is considered. The dual vector control function (dual composition of angular and linear accelerations of a body), constructed using the Pontryagin maximum principle, and is not limited in the dual module. The integral quadratic functional with respect to angular and linear accelerations, cha- racterizing the energy costs of transferring a body from a given initial state to a given final state in a fixed time is minimized. The spatial motion of a body is equivalent to the composition of angular (rotational) and translational (orbital) movements (Chasles’ theorem). The boundary conditions for angular and linear positions, as well as for angular and linear velocities of the body, are arbitrary. The translational (orbital) motion of a body together with the rotation of a body around its center of mass is described using two new biquaternion differential equations. The laws of change of the control force and the control moment are obtained using the constructed optimal laws of change of angular and linear accelerations of a body according to algebraic formulas using the concept of solving inverse problems of dynamics. After applying the maximum principle (to the construction of optimal program accelerations), the control problem under study was reduced to a twenty-eighth order nonlinear differential boundary value problem with a movable right end of the trajectory, which was solved numerically using the Levenberg-Marquardt method. The case of a large deviation in the angular measure between the initial and final orientations of the spacecraft in the presence of a small linear translational displacement of the spacecraft (the problem of optimal spatial maneuvering of the spacecraft) is considered. In this case, the mass distribution of the spacecraft corresponds to a spherically symmetric solid body or the International Space Station (ISS), or the Space Shuttle spacecraft. Graphs of changes in the components of the dual quaternion (biquaternion) describing the orientation of the spacecraft and its location in the inertial coordinate system, the components of angular and linear velocity vectors, the component of angular and linear accelerations vectors (optimal controls), the component of the control moment and the control force vectors are constructed. The obtained numerical solutions are analyzed; the features and patterns of the process of optimal spatial motion of spacecraft are established. А table of values of the components of the control moment vector in the coordinate system associated with the body at the beginning, middle and end of motion for all three bodies in the presence of translational displacement is obtained.