Abstract—

The method of analytical construction of the control of the spatial motion of a rigid body (control of screw motion, equivalent to the composition of angular (rotational) and translational movements) is being developed in a nonlinear dynamic setting using Clifford biquaternions and dual matrices. The controls provide asymptotic stability in large for any selected programmed spatial motion in the inertial coordinate system and the desired dynamics of the controlled spatial motion of a rigid body. To construct control laws, biquaternionic and dual matrix models of the spatial motion of a rigid body, the concept of solving inverse problems of dynamics, the principle of feedback control and reduction of the constructed nonlinear differential equations of the perturbed spatial motion of a rigid body to the reference linear stationary differential forms of the selected structure by using the proposed nonlinear feedbacks in control laws.

Biquaternion models of the spatial motion of a rigid body are considered, a statement of the problem of control of the motion of a rigid body is given, various forms of nonlinear differential equations of the perturbed spatial motion of a rigid body in biquaternion and screw variables are presented, which are convenient for constructing control laws. Various dual matrix (screw) laws of control of the spatial motion of a rigid body are proposed, for which the nonlinear nonstationary differential equations of the perturbed spatial motion of a rigid body take the form of linear stationary dual matrix differential equations of the second order (with respect to the screw part of the biquaternion of the rigid body position error), invariant with respect to any chosen programmed spatial motion of a rigid body. Constant coefficients (scalar duals or matrix duals) of these equations are the amplification coefficients of nonlinear feedbacks in the proposed dual control laws that ensure the required quality of transient control processes. The determination of the amplification factors of nonlinear feedbacks and the properties of the controlled motion of a rigid body are discussed.

中文翻译：

---

使用双四元数和对偶矩阵控制刚体的空间运动

摘要-

在使用克利福德双四元数和对偶矩阵的非线性动态设置中，正在开发一种控制刚体的空间运动（螺旋运动的控制，等效于角（旋转）和平移运动的组成）的分析构造方法。对于惯性坐标系中任何选定的编程空间运动以及刚体受控空间运动的所需动力学，这些控件都可提供较大的渐近稳定性。为构造刚体的空间运动的控制定律，双四元数和对偶矩阵模型，解决动力学反问题的概念，

考虑了刚体空间运动的双四元数模型，给出了对刚体运动的控制问题的陈述，双四元数和螺旋变量中刚体的扰动空间运动的各种形式的非线性微分方程介绍了这些信息，这些信息对于构造控制定律很方便。提出了控制刚体空间运动的各种双矩阵（螺旋）律，针对这些定律，刚体扰动空间运动的非线性非平稳微分方程采取二阶线性平稳双矩阵微分方程的形式。 （关于刚体位置误差的双四元数的螺杆部分）（相对于刚体的任何选定的编程空间运动）是不变的。这些方程的常数系数（标量对偶或矩阵对偶）是所提出的对偶控制定律中非线性反馈的放大系数，可确保所需的瞬态控制过程质量。讨论了非线性反馈放大因子的确定以及刚体受控运动的性质。