Because of the topology structure of orientation space, singularity of orientation of rigid body moving in space is unavoidable for any of the three parameters representation. There all exists the second-order kinematic singularity configuration in 12 kinds Euler angles parameters when the relationships of angular velocity and angular acceleration between joint space and configuration space are obtained. After gaining all kinds of the first-order and second-order singularity configuration showed by Euler angles , in the engineering control proper Euler angles are chosen to avoid these singularity configuration according to the actual need . So it can avoid the confusion between the output momentum and the torque for the driving mechanism.
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温立书. 欧拉角参数表示下姿态的二阶运动奇异性[J]. 科学技术与工程, 2012, 12(3): . wenlishu. The Second-order Kinematic Singularity of Orientation in Euler Parameters Representation[J]. Science Technology and Engineering,2012,12(3).