Inspired by nature, continuum robots show their potential in human-centered environments due to the compliant-to-obstacle features and dexterous mobility. However, there are few such robots successfully implemented outside the laboratory so far. One reason is believed to be due to the real time control challenge for soft robots, which require a highly efficient, highly accurate dynamic model. This paper presents a new systematic methodology to formulate the dynamics of constant curvature continuum robots. The new approach builds on several new techniques: 1) using the virtual work principle to formulate the equation of motion, 2) using specifically selected kinematic representations to separate integral variables from the non-integral variables, and 3) using vector representations to put the integral in a compact form. By doing so, the hard-to-solve integrals are evaluated analytically in advance and the accurate inverse dynamics are established accordingly. Numerical simulations are conducted to evaluate the performances of the newly proposed model.

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