Trajectory Planning for Continuum Space Manipulators Based on Generalized Jacobian Operator

Continuum manipulators (CMs) hold significant promise for space applications due to their flexibility and adaptability in extreme environments. However, similar to traditional articulated space manipulators (A-SMs), CMs face challenges arising from the dynamic coupling between the manipulator and the spacecraft. A key difficulty with CMs lies in their infinite degrees of freedom (DOFs) and the absence of conventional joint structures, which complicates the development of differential kinematics models similar to those of A-SMs using Jacobian matrix. To address this issue, we introduce the concept of the Jacobian operator to overcome the difficulties in establishing the differential kinematics of CMs. Building on this foundation, we further introduce the concept of the generalized Jacobian operator, an extension of the generalized Jacobian matrix in A-SMs, to develop the differential kinematics of continuum space manipulators (C-SMs). Based on the derived differential kinematics, we then propose a trajectory planning method for C-SMs. Additionally, utilizing the null space of the Jacobian operator, a base disturbance-free trajectory planning approach is developed, ensuring smooth operation under dynamic coupling conditions. Finally, the validity and effectiveness of the proposed planning methods are demonstrated through simulation cases. This framework offers a robust solution to the kinematic challenges of C-SMs, paving the way for more efficient trajectory planning and control in space applications.