The dynamic problem of the rotation of a rigid body (for example, a spacecraft) from an arbitrary initial to the required final angular position in the presence of control restrictions is considered and solved. The end time of the maneuver is known. To optimize the rotation control program, a quadratic quality criterion is used, the minimized functional characterizes energy costs. The construction of optimal turn control is based on quaternion variables and the L. S. Pontryagin maximum principle. The features of optimal motion are studied in detail. Key properties of the optimal solution are formulated in an analytical form. It is shown that in the case of limited control, the moment of forces in the process of optimal rotation is parallel to a straight line that is stationary in inertial space, and during rotation of a rigid body (spacecraft) the direction of the kinetic moment is constant relative to the inertial coordinate system. Optimal control is presented in the form of synthesis—the synthesizing function is found and the dependence of the control variables on the phase coordinates is given. Formalized equations and calculation expressions are obtained to determine the optimal rotation program. The 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 a solid body) is also described. An example and results of mathematical modeling of the motion of a spacecraft as a solid with optimal control are presented, demonstrating the practical feasibility of the developed method for controlling the spatial orientation of the spacecraft. For a dynamically symmetric solid, a complete solution of the reorientation problem in closed form is given, control variables and the optimal trajectory of motion as functions of time are presented in an analytical form.

中文翻译：

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四元数对刚体（航天器）重新定向最优控制问题的解析解

考虑并解决了刚性主体（例如航天器）在存在控制限制的情况下从任意初始角度位置旋转到所需最终角度位置的动力学问题。机动的结束时间是已知的。为了优化旋转控制程序，使用了二次质量标准，最小化功能体现了能源成本。最佳转向控制的构建基于四元数变量和LS Pontryagin最大原理。详细研究了最佳运动的特征。最佳解决方案的关键特性以分析形式制定。结果表明，在有限控制的情况下，最佳旋转过程中的力矩平行于在惯性空间中静止的直线，在刚体（航天器）旋转期间，动力矩的方向相对于惯性坐标系是恒定的。最优控制以合成的形式呈现-找到了合成函数，并给出了控制变量对相位坐标的依赖性。获得形式化的方程式和计算表达式，以确定最佳的旋转程序。还描述了用于解决任意旋转条件（实体的初始和最终位置以及惯性矩）的最大原理的边值问题的构造方案。给出了具有最优控制的航天器作为固体运动的数学模型的示例和结果，证明了所开发的控制航天器空间方向的方法的实际可行性。对于动态对称的实体，给出了封闭形式的重新定向问题的完整解决方案，并以分析形式给出了控制变量和作为时间函数的最佳运动轨迹。