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

SUMMARYA 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.

Low-background single-crystal silicon sample holders for neutron powder diffraction

Neutron diffraction measurements on weakly scattering or highly absorbing samples may demand custom mounting solutions. Two low-background sample holders based on inexpensive single-crystal silicon are described. One uses a conventional cylindrical geometry and is optimized for weakly scattering materials, while the other has a large-area flat-plate geometry and is designed for use with highly absorbing samples. Both holders yield much lower backgrounds than more conventional null-matrix or null-scattering materials and are essentially free from interfering Bragg peaks.

A Jacobian-Based Algorithm for the Attitude Control of a Rigid Body Undergoing Fully-Reversed Sequences of Rotations

Jacobian-based control algorithms for the attitude control of a rigid body undergoing a series of forward and reverse rotations are proposed in this paper. Unlike the Jacobian of conventional systems such as a robot manipulator, the Jacobian of the system manipulated through FR rotations is singular as well as a null matrix at the identity that makes the conventional Jacobian formulation break down. In order to handle the singularities involved in FR rotations, the Jacobian algorithm is reformulated and implemented so as to synthesize the FR rotation for a desired orientation change. We introduce the single-step and multiple-step Jacobian methods in this paper. The single-step Jacobian method synthesizes a specific FR rotation that enables the rigid body to reach a given desired orientation through a single step. The multiple-step Jacobian method synthesizes physically feasible FR rotations on an incremental optimal path to a desired orientation. A comparison with existing control algorithms for FR motions verifies the fast convergence property of the Jacobian-based algorithm and the accuracy of the solutions.