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Kinematic models that account for deformation due to applied loads have recently been developed for a variety of continuum robots. In these cases, a set of nonlinear differential equations with boundary conditions must often be solved to obtain the robot shape. Thus, computing manipulator Jacobians and compliance matrices efficiently is not straight forward. In this paper, we propose a method for obtaining an arc length parametrized Jacobian and compliance matrix. Our approach involves obtaining an augmented Jacobian by propagating the necessary partial derivatives through the model equations, resulting in a new set of differential equations. These equations can be solved as an initial value problem, via a single numerical integration. Our method can be generally applied to various continuum robot architectures, regardless of the specific actuation system used. We provide a specific case study using this method to obtain the Jacobian for a concentric-tube robot.
Rucker et al. (Sun,) studied this question.