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This paper introduces a method for computing the shape of a continuously-flexible (continuum) robot in 3-D space which includes gravity loading by applying Cosserat rod theory to a continuum robot. With this theory, the shape of the rod can be determined using force-torque balance equations obtained from a simple free body diagram that represents the continuum robot. Real-time performance of 125 Hz makes this approach viable for the control of a continuum robot, enabled by avoiding boundary-value conditions in the solution.
Jones et al. (2009) studied this question.