旋转张量的表达及有限转动次序交换定理的证明

Expressions of rotation tensor and proof of finite rotation order exchange theorem

旋转张量在火箭发射和航天飞行中使用广泛,在航天器姿态描述、轨迹追踪和变质量研究等方面有重要应用。讨论了旋转张量的相关问题:从坐标变换和运动学角度出发,分别推导了转动张量的基本表达形式和指数映射形式,对旋转张量公式的来源进行了补充和完善;由旋转张量的指数映射形式,得出对于转轴的二阶反对称张量(-r·ε)与虚数i有相似的幂次规律,并明确其几何意义;从特定张量合成的角度出发,推导了旋转张量用转动平面内夹角为转角的任意两向量表达的新形式,在此基础上,给出了旋转张量合成的几何意义;最后,根据向量和张量转动的概念,给出两次有限转动次序交换定理的新的更简洁的证明方法,并通过实例验证上述结论。

The rotation tensor is widely used for rocket launch and space flight, which has important applications in spacecraft attitude description, trajectory tracking and variable mass research.The problems related to the rotation tensor were discussed: from the perspective of coordinate transformation and kinematics, the basic expression form and the exponential mapping form of the rotation tensor were derived respectively, and the source of rotation tensor formula was supplemented and improved;from the exponential mapping form of the rotation tensor, it was derived that the second-order anti-symmetric tensor for the rotation axis has a similar power law with the imaginary number i, and its geometric meaning was defined;from the perspective of specific tensor synthesis, a new form of the rotation tensor expressed by any two vectors with intersecting angle as rotation angle in rotation plane was derived, on this basis, the geometric meaning of rotation tensor synthesis was given;finally, according to the concept of vector and tensor rotation, a new and more concise proof method of the two finite rotation order exchange theorem was given, and the above conclusion was verified by an example.