Continuum Mechanics Modeling of Flexible Spring Joints in Surgical Robots

Helical extension springs are used as flexible joints in surgical robots to achieve large bending angles. As the spring wire undergoes complex forces, continuum mechanics is essential for characterizing the spring deformation for precise joint bending control. In this work, a mechanical model of a tendon‐actuated bending spring joint is developed based on Cosserat rod theory. A reduced model is further derived to achieve dimension reduction. The contact between adjacent spring coils complicates the mechanical modeling. To address this issue, implicit variables of contact positions and forces are introduced to express geometric constraints at the contact points. The resultant model consists of Cosserat differential equations and algebraic constraint equations, and is solved by the shooting method. The Levenberg–Marquardt algorithm and numerical integration are used to find optimal solutions numerically. Simulation results show the feasibility of the reduced model. Experiments are conducted to calibrate spring parameters and validate the mechanical model. Results show that the model can predict nonlinear relationships of tendon displacements, actuation forces, and bending angles of the spring joint.