The axial offset joint has two rotating axes that do not intersect but have a specific offset in space. It is used widely in parallel manipulators (PMs). The offset-joint workspace can directly affect the PM workspace. This study performed a theoretical derivation and workspace analysis of a class of axial offset joints. First, a theoretical parametric model describing the rotation range of the offset joint is established that considers the interference of the offset joint because of the contact between the upper- and lower-joint brackets during movement. Second, the analytical expressions of the offset-joint workspace are formulated based on the coordinate system transformation. The offset-joint workspace is theoretically calculated in this study using formulations. Then, through a comparative analysis, the superiority of the offset joint compared with the universal joint is verified. The theoretical formulations in this paper can be used to calculate the workspace of a class of axial offset joints. Finally, based on a workspace analysis of three types of PMs using offset, universal, and spherical joints, the offset-joint PM workspace is much larger than those of the other two types.

This paper addresses the issue of how to endow robots with motion skills, flexibility, and adaptability similar to human arms. It innovatively proposes a hybrid-primitive-frame-based robot skill learning algorithm and utilizes the policy improvement with a path integral algorithm to optimize the parameters of the hybrid primitive framework, enabling robots to possess skills similar to human arms. Firstly, the end of the robot is dynamically modeled using an admittance control model to give the robot flexibility. Secondly, the dynamic movement primitives are employed to model the robot’s motion trajectory. Additionally, novel stiffness primitives and damping primitives are introduced to model the stiffness and damping parameters in the impedance model. The combination of the dynamic movement primitives, stiffness primitives, and damping primitives is called the hybrid primitive framework. Simulated experiments are designed to validate the effectiveness of the hybrid-primitive-frame-based robot skill learning algorithm, including point-to-point motion under external force disturbance and trajectory tracking under variable stiffness conditions.

Offset joints are widely used in robotics, and literature has demonstrated that axial offset joints can expand the workspace. However, the hybrid offset joint, which incorporates offsets in three orthogonal directions (x, y, and z axes), provides a more flexible and comprehensive range of motion compared to traditional axial offset joints. Therefore, a comprehensive understanding of the workspace of hybrid offset joints with three-directional offsets is essential. First, through a parameter model, the interference motion of hybrid offset joints is studied, considering three different directional offsets and obtaining analytical expressions. Next, based on coordinate transformations, the workspace of this joint is investigated, resulting in corresponding theoretical formulas. In addition, the influence of offset amounts in various directions on the joint’s workspace is examined. Finally, the application of hybrid offset joints in parallel manipulators (PMs) is introduced, highlighting their practical engineering value. Through comparative analysis, it is found that lateral offsets on the x- and y-axes adjust the maximum rotation angles, while the z-axis offset expands the rotational range of these joints. Moreover, by increasing the limit rotation angle of the passive joint in a specific direction, the application of hybrid offset joints in PMs can impact the workspace. These findings offer valuable insights for the design of hybrid offset joints and their applications in robotics.