Modeling musculoskeletal kinematic and dynamic redundancy using null space projection

The coordination of the human musculoskeletal system is deeply influenced by its redundant structure, in both kinematic and dynamic terms. Noticing a lack of a relevant, thorough treatment in the literature, we formally address the issue in order to understand and quantify factors affecting the motor coordination. We employed well-established techniques from linear algebra and projection operators to extend the underlying kinematic and dynamic relations by modeling the redundancy effects in null space. We distinguish three types of operational spaces, namely task, joint and muscle space, which are directly associated with the physiological factors of the system. A method for consistently quantifying the redundancy on multiple levels in the entire space of feasible solutions is also presented. We evaluate the proposed muscle space projection on segmental level reflexes and the computation of the feasible muscle forces for arbitrary movements. The former proves to be a convenient representation for interfacing with segmental level models or implementing controllers for tendon driven robots, while the latter enables the identification of force variability and correlations between muscle groups, attributed to the system’s redundancy. Furthermore, the usefulness of the proposed framework is demonstrated in the context of estimating the bounds of the joint reaction loads, where we show that misinterpretation of the results is possible if the null space forces are ignored. This work presents a theoretical analysis of the redundancy problem, facilitating application in a broad range of fields related to motor coordination, as it provides the groundwork for null space characterization. The proposed framework rigorously accounts for the effects of kinematic and dynamic redundancy, incorporating it directly into the underlying equations using the notion of null space projection, leading to a complete description of the system.