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A Jacobian-based algorithm for planning the motion of an underactuated rigid body undergoing forward and reverse rotations

机译：基于雅可比矩阵的算法，用于计划欠驱动刚体进行正向和反向旋转的运动

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A Jacobian-based algorithm that is useful for planning the motion of a floating rigid body operated using two input torques is addressed in this paper. The rigid body undergoes a four-rotation fully reversed (FR) sequence of rotations which consists of two initial rotations about the axes of a coordinate frame attached to the body and two subsequent rotations that undo the preceding rotations. Although a Jacobian-based algorithm has been useful in exploring the inverse kinematics of conventional robot manipulators, it is not apparent how a correct FR sequence for a desired orientation could be found because the Jacobian of FR sequences is singular as well as being a null matrix at the identity. To discover the FR sequences that can synthesize the desired orientation circumventing these difficulties, the Jacobian algorithm is reformulated and implemented from arbitrary orientations where the Jacobian is not singular. Due to the insufficient degrees-of-freedom of four-rotation FR sequences required to achieve all possible orientations, the rigid body cannot achieve certain orientations in the configuration space. To best approximate these infeasible orientations, the Jacobian-based algorithm is implemented in the sense of least squares. As some orientations can never be attained by a single four-rotation FR sequence, two different four-rotation FR sequences are exploited alternately to ensure the convergence of the proposed algorithm. Assuming the orientation is supposed to be manipulated using three input torques, the switching Jacobian algorithm proposed in this paper has significant practical importance in planning paths for aerospace and underwater vehicles which are maneuvered using only two input torques due to the failure of one of the torque-generation mechanisms.
机译：本文提出了一种基于雅可比算法的算法，该算法可用于计划使用两个输入转矩操作的浮动刚体的运动。刚体经历四转完全反转（FR）旋转序列，该过程包括围绕附接到该刚体的坐标框架的轴的两个初始旋转以及撤消之前的旋转的两个后续旋转。尽管基于Jacobian的算法在探索常规机器人操纵器的逆运动学中很有用，但由于FR序列的Jacobian是奇异且为空矩阵，因此如何找到所需方向的正确FR序列尚不清楚在身份。为了发现可以合成所需方向的FR序列，克服了这些困难，对雅可比算法进行了重新构造，并从雅可比不单数的任意方向进行了实现。由于实现所有可能的方向所需的四转FR序列的自由度不足，因此刚体无法在配置空间中实现某些方向。为了最好地近似这些不可行的方向，基于最小二乘的意义实施了基于雅可比行列的算法。由于单个四转FR序列永远无法获得某些方向，因此交替使用两个不同的四转FR序列以确保所提出算法的收敛性。假设应该使用三个输入扭矩来操纵方向，则本文提出的开关雅可比算法在规划航空航天和水下航行器的路径时具有重要的实际意义，这些路径由于其中一个扭矩的失败而仅使用两个输入扭矩进行操纵生成机制。

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