Stylized Computational Method for Reduced Multibody System Transfer Matrix Method Based on Joint Axis Vectors

The reduced multibody system transfer matrix method is a computationally efficient approach for the dynamics of multi-rigid and flexible body systems that facilitates program implementation. When modeling rotational joint elements, the relative motion direction is conventionally defined as the z-axis of the local coordinate system. These local coordinate systems are body-fixed references attached to the corresponding body elements. Consequently, several body-fixed coordinate systems with different orientations are typically established on a single rigid body to characterize its position and orientation relative to the global inertial frame, as well as the relative position and orientation between adjacent body elements. Under the tree topology representation, forward kinematic calculations at position, velocity, and acceleration levels necessitate projecting the absolute kinematic quantities of each body-fixed coordinate system (relative to the global inertial frame) back into their respective local coordinate systems. This process introduces redundant computational overhead in kinematic projection operations. To address this limitation, this paper proposes a joint axis vector based stylized computation method. By directly describing relative motion relationships between adjacent rigid bodies through joint axis vectors, the method effectively eliminates multi coordinate system projection operations. Validation through simulations in quadruped robot multibody dynamics systems demonstrates the correctness of the proposed approach.