Multiple aspects grasp quality evaluation in underactuated grasp of tendon-driven continuum robots

Abstract

Continuum robots (CRs) have shown a wide range of potential applications from the medical industry to rescue missions. CRs form curves with continuous tangent vectors and have unlimited degrees of freedom (DOFs), which make them underactuated structures. The underactuation and compliance of CRs cause the adaptability and robustness of CR-based grasps. However, fundamental issues related to the grasp quality evaluation of CRs have yet to be investigated. This paper synthesizes an underactuated grasp of CRs on an arbitrary-shaped grasped object using a constant curvature model. The technique relies on parallel global grasp quality measures based on the grasp configuration and grasp tasks. Then, a numerical example of an underactuated tendon-driven CR-based grasp is investigated to show the validity of the grasp synthesis approach. Finally, the experimental results are reported in which a practical grasp quality measure (i.e., grasp success rate (GSR)) was used for validation. To this purpose, Taguchi method is applied to design the experiments by reducing the number of trials. The designed tests are performed using our robot-assisted catheterization systems (RACS), Althea I and II. Also, the effectiveness of using Taguchi method instead of the full factorial experiments in designing the verification process is presented and discussed.

Appendices

Appendix A: The Jacobian components of each CR

The components of Jacobian of the i^th CR, $J_{jk}$ are as follows.

$$ \begin{aligned} J_{12} & = {\text{s}} \theta_{2} (d_{4} {\text{c}} \theta_{3} + d_{7} {\text{s}} (\theta_{3} - \theta_{5} )) \\ J_{22} & = - {\text{c}} \theta_{2} (d_{4} {\text{c}} \theta_{3} + d_{7} {\text{s}} (\theta_{3} - \theta_{5} )) \\ J_{13} & = {\text{c}} \theta_{2} (d_{4} {\text{s}} \theta_{3} - d_{7} {\text{s}} (\theta_{3} - \theta_{5} )) \\ J_{23} & = {\text{s}} \theta_{2} (d_{4} {\text{s}} \theta_{3} - d_{7} {\text{s}} (\theta_{3} - \theta_{5} )) \\ J_{33} & = d_{4} {\text{c}} \theta_{3} + d_{7} {\text{s}} (\theta_{3} - \theta_{5} )) \\ J_{15} & = d_{7} {\text{c}} (\theta_{3} - \theta_{5} ){\text{c}} \theta_{2} \\ J_{25} & = d_{7} {\text{c}} (\theta_{3} - \theta_{5} ){\text{s}} \theta_{2} \\ J_{35} & = - d_{7} {\text{s}} (\theta_{3} - \theta_{5} ) \\ J_{46} & = - {\text{s}} (\theta_{3} - \theta_{5} ){\text{c}} \theta_{2} \\ J_{56} & = - {\text{s}} (\theta_{3} - \theta_{5} ){\text{s}} \theta_{2} \\ \end{aligned} $$

(A1)

where j and k are number of rows and columns of the J component.

Appendix B: Kinetics and Equilibrium

To investigate the kinetics of the CR-based grasp problem, the equations of motion for both robot and object are considered as follows

$$ {\mathbf{M}}_{r} ({\mathbf{q}}){\ddot{\mathbf{q}}} + {\mathbf{b}}_{r} ({\mathbf{q}},{\dot{\mathbf{q}}}) + {\mathbf{J}}^{T} {\mathbf{f}} = {\mathbf{T}}, $$

(A2)

$$ {\mathbf{M}}_{obj} ({\mathbf{p}}){\dot{\mathbf{t}}} + {\mathbf{b}}_{{{\text{obj}}}} ({\mathbf{p}},{\mathbf{t}}) - {\mathbf{Gf}} = {\mathbf{w}}_{{{\text{ext}}}} , $$

(A3)

where ${\mathbf{M}}_{r}$ and ${\mathbf{M}}_{obj}$ express inertia matrices for the engaged robot and object, respectively, and ${\mathbf{b}}_{r}$ and ${\mathbf{b}}_{obj}$ indicate their velocity-product terms. Moreover, contact forces, generalized joint forces, and total wrench applied on the object are shown with f, ${\mathbf{T}}$, and ${\mathbf{w}}_{{\rm ext}}$, respectively.

Equations (A2) and (A3) represent the dynamics of the Co-manipulative CR and object without regard for the kinematic constraints imposed by the contact models in Eq. (17). Enforcing them, and considering rigid body kinematics of the manipulation system, the dynamic model of the system will be:

$$ \left( {\begin{array}{*{20}c} {{\mathbf{J}}^{T} } \\ { - {\mathbf{G}}} \\ \end{array} } \right){\mathbf{f}} = \left( {\begin{array}{*{20}c} {{\mathbf{T}} - {\mathbf{M}}_{r} ({\mathbf{q}}){\ddot{\mathbf{q}}} - {\mathbf{b}}_{r} ({\mathbf{q}},{\dot{\mathbf{q}}})} \\ {{\mathbf{w}}_{{\rm ext}} - {\mathbf{M}}_{{{\text{obj}}}} ({\mathbf{p}}){\dot{\mathbf{t}}} - {\mathbf{b}}_{{{\text{obj}}}} ({\mathbf{p}},{\mathbf{t}})} \\ \end{array} } \right). $$

(A4)

2.1 Quasi-static model

A quasi-static model assuming a slow-motion is used to simplify the equations as follows

$$ \left( {\begin{array}{*{20}c} {{\mathbf{J}}^{T} } \\ { - {\mathbf{G}}} \\ \end{array} } \right){\mathbf{f}} = \left( {\begin{array}{*{20}c} {\mathbf{T}} \\ {{\mathbf{w}}_{{{\text{ext}}}} } \\ \end{array} } \right), $$

(A5)

disregarding the time-dependent and inertia terms. Finally, Eqs. (8) and (A5) lead to.

$$ \begin{aligned} & {{\varvec{\Omega}}} - {\mathbf{S}}^{\text{T}} {\mathbf{T}} = {\mathbf{0}}, \\ & {\mathbf{T}} - {\mathbf{J}}^{T} {\mathbf{f = }}{\mathbf{0}}, \\ & {\mathbf{w}}_{{\rm ext}} + {\mathbf{Gf}} = {\mathbf{0}}, \\ \end{aligned} $$

(A6)

which can be written in a compact form as

$$ \left[ {\begin{array}{*{20}c} {\mathbf{I}} & {\mathbf{0}} & {\mathbf{G}} \\ {\mathbf{0}} & {\mathbf{I}} & { - {\mathbf{S}}^{\text{T}} {\mathbf{J}}^{T} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\mathbf{w}}_{{\rm ext}} } \\ \begin{gathered} {{\varvec{\Omega}}} \hfill \\ {\mathbf{f}} \hfill \\ \end{gathered} \\ \end{array} } \right] = {\mathbf{0}}. $$

(A7)

These relationships will lead to the basic grasp classification concept [40], which will be the topic of our future work.