Abstract <p>The problem of the optimal, in terms of speed, program control of the spatial motion of a rigid body of an arbitrary dynamic configuration (in particular, a spacecraft (SC) considered as a free rigid body), equivalent to a composition of angular (rotational) and translational (orbital) motion, is studied. The boundary conditions for the angular and linear position of a rigid body, as well as for the angular and linear velocities of the body, are arbitrary. The dual vector function of the optimized control (the dual composition of the vector of absolute angular acceleration of a rigid body and the vector equal to the local derivative of the vector of absolute velocity of the center of mass of a rigid body (the component of the vector of absolute linear acceleration of the center of mass of the body)) is limited in dual modulus. The vectors of the program control force and program control moment are in accordance with the concept of solving inverse problems of dynamics. To solve the problem, we used new biquaternion equations of the spatial motion of a rigid body, which we proposed, in which the Clifford parabolic biquaternion (dual quaternion) is used to describe spatial motion, as well as equivalent quaternion equations in which two Hamilton quaternions are used to describe spatial motion. Using Pontryagin’s maximum principle, a differential boundary value problem of the 28th order is obtained. Examples of numerical solutions of boundary value problems are given for cases where the mass distribution of a solid body corresponds to a spherically symmetric body and to the international space station as an arbitrary solid body. In this case, the difference between the initial and final orientations of a rigid body is large in angular measure and small in linear displacement (the problem of spatial maneuvering). An analysis of the obtained numerical solutions is given.</p>