Abstract <p>A quaternion solution of the problem on optimal rotation of a rigid body (spacecraft) from an arbitrary initial to a specified angular position with constraints on the control variables is presented. A combined quality functional has been used to optimize the control process. It combines in a given proportion the sum of time and control efforts spent on the rotation and the integral of the kinetic energy of rotation during the rotation. Based on L.S. Pontryagin’s maximum principle and quaternion models of controlled motion of a rigid body, a solution of the problem is obtained. The properties of optimal motion are disclosed in an analytical form. Formalized equations and calculation formulas are written to construct the optimal rotation program. Analytical equations and relations for finding optimal control are given. Key relations that determine the optimal values of the parameters of the rotation control algorithm are given. A constructive scheme for solving the boundary value problem of the maximum principle for arbitrary rotation conditions (initial and final positions and moments of inertia of the rigid body) is also given. In the case of a dynamically symmetric rigid body, a solution of the reorientation problem in closed form is obtained. A numerical example and the results of mathematical modeling, confirming the practical feasibility of the developed method for controlling the orientation of a spacecraft, are presented.</p>