Clarke Transform and Clarke Coordinates -- A New Kid on the Block for State Representation of Continuum Robots

For almost all tendon-driven continuum robots, a segment is actuated by three or four tendons constrained by its mechanical design. For both cases, methods to account for the constraints are known. However, for an arbitrary number of tendons, a disentanglement method has yet to be formulated. Motivated by this unsolved general case, we explored state representations and exploited the two-dimensional manifold. We found that the Clarke transformation, a mathematical transformation used in vector control, can be generalized to address this problem. We present the Clarke transform and Clarke coordinates, which can be used to overcome the troublesome interdependency between the tendons, simplify modeling, and unify different improved state representations. Further connection to arc parameters leads to the possibility to derive more generalizable approaches applicable to a wider range of robot types.