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Continuum robots (CR) offer excellent dexterity and compliance in contrast torigid-link robots, making them suitable for navigating through, and interactingwith, confined environments. However, the study of path planning for CRs whileconsidering external elastic contact is limited. The challenge lies in the factthat CRs can have multiple possible configurations when in contact, renderingthe forward kinematics not well-defined, and characterizing the set of feasiblerobot configurations as non-trivial. In this paper, we propose to solve thisproblem by performing quasi-static path planning on an implicit manifold. Wemodel elastic obstacles as external potential fields and formulate the robotstatics in the potential field as the extremal trajectory of an optimal controlproblem obtained by the first-order variational principle. We show that the setof stable robot configurations is a smooth manifold diffeomorphic to asubmanifold embedded in the product space of the CR actuation and base internalwrench. We then propose to perform path planning on this manifold usingAtlasRRT*, a sampling-based planner dedicated to planning on implicitmanifolds. Simulations in different operation scenarios were conducted and theresults show that the proposed planner outperforms Euclidean space planners interms of success rate and computational efficiency.

Kinetostatic Path Planning for Continuum Robots By Sampling on Implicit Manifold