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

Abstract

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.

Figures (3)

• Figure 1: Physical interpretation of the Clarke coordinates. The magenta line lies within the bending plane. The length difference to the arc length is the virtual displacement. The yellow arrows are the projected virtual displacements and lie within the respective projected plane corresponding to $xz$-plane and $yz$-plane of the base.
• Figure 2: Spaces and their mappings. To map between the actuator, e.g., motor angles to joint values, e.g., tendon displacements, $f_\text{dyn}$ and $f_\text{dyn}^{-1}$ are used. The robot-dependent mapping is denoted by $f_\text{dep}$, whereas robot-independent mapping is denoted by $f_\text{ind}$. Their inverses are $f_\text{dep}^{-1}$ and $f_\text{ind}^{-1}$.
• Figure 3: Commutative diagram-like overview of the joint space disentanglement usage. To circumvent the explicit consideration of the actuator constraints, i.e., $\sum \rho_i = 0$, methods can act on the 2dof manifold embedded in the $n\,$dof joint space. For the transform, linear maps $\boldsymbol{M}_\mathcal{P}$ and $\boldsymbol{M}_\mathcal{P}^{-1}$ are used and any output denoted by $\left(\rho_\text{Re}^*, \rho_\text{Im}^*\right)$ can subsequently mapped back leading to $\rho^*$.