Dynamic Modeling and Analysis of Robotic Manipulators With Mixed Rigid–Elastic Joints

Rigid joints are popular in traditional robotic manipulator designs due to their fast-response ability and high position control precision, while elastic joints are well known for their safety. Robotic manipulators using both rigid and elastic joints are developed to harness the benefits and mitigate the drawbacks of these joint types. The integration of mixed rigid-elastic joints in robotic manipulators offers significant potential for enhancing adaptability, precision, and resilience in dynamic and unstructured environments. This approach is particularly advantageous in applications that demand a balance between strength and compliance. This paper presents a systematic, tutorial-like approach to modelling and analysis of robotic manipulators with mixed rigid-elastic joints. Firstly, the dynamic equations are developed using the Lagrangian method. In this paper, the constraint on motor positions is relaxed, which means the motors can be placed on any link on the robotic manipulator. In addition, properties of the mathematical model, including skew-symmetry and linearity with respect to dynamic parameters, are presented. The representation of the dynamic matrices and vectors with respect to the dynamic parameters is introduced. Secondly, the recursive Newton-Euler algorithm is adopted to derive the model of robots with mixed rigid-elastic joints. An inverse dynamic algorithm is presented to calculate the forces/torques from the provided accelerations, velocities, and positions of the joints. Then, a forward dynamic algorithm is proposed to calculate the accelerations, velocities, and positions of the joints from the provided forces/torques. In addition, the impact of varying joint stiffness and the placement of elastic joints is analysed. Finally, an example and simulation are provided to demonstrate the modelling process of robotic manipulators with mixed rigid-elastic joints.