Kinematics, statics modeling and workspace analysis of a cable-driven hybrid robot

Abstract

Compared with single rigid serial robots (RSRs) and cable-driven parallel robots (CDPRs), cable-driven hybrid robots (CDHRs) have the advantages of both CDPRs and RSRs, which can improve the deficiencies of a single robot in one aspect. Due to a larger force-enclosed workspace (FEW), CDHRs can accomplish more complex tasks, which have a wide range of applications in industrial picking, disaster rescue, construction renovation, etc. However, as its structural complexity increases, the coupling modeling and workspace analysis will become more difficult. Based on this, this paper provides a coupled kinematics, statics modeling, and workspace analysis method of CDHRs. Firstly, the forward/inverse kinematic equations of the series/parallel coupling are derived, and the corresponding solutions are given according to this composite structure with high redundancy. Secondly, the static equations of the CDHR are further derived based on the coupled kinematics model and solved by the quadratic programming method (QPM). Further, a FEW for the CDHR is established based on collision constraints, containing the workspaces of both the RSR and the CDPR. To evaluate the quality of the workspace for the CDHR, a method for solving the configuration flexibility is proposed. Finally, the workspace and the configuration flexibility of the CDHR are analyzed by the built simulation system. And then, the tracking methods of several typical trajectories are verified by the model.

Appendix:  Calculation method of the cable tension distribution

Let the working point of the end-effector (i.e., $z_{\mathrm{ee}}$) lie in the bottommost $XY$ plane of $z_{\mathrm{pf}}=z_{\min}$, that is:

$$ \textstyle\begin{array}{l} z_{\min} = \mathop{\arg\min}\limits _{\boldsymbol {X}_{\mathrm{ee}}}z_{\mathrm{ee}}, \\ z_{\mathrm{ee}} \in \boldsymbol {X}_{\mathrm{ee}},\boldsymbol {X}_{\mathrm{ee}} \in U_{\mathrm{pf}} . \end{array} $$

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Take the workspace that satisfies $z_{\mathrm{pf}} = z_{\min}$, i.e.:

$$ \textstyle\begin{array}{l} \left ( x_{\mathrm{pf}}^{\left ( k \right )},y_{\mathrm{pf}}^{\left ( k \right )},\gamma _{\mathrm{pf}}^{\left ( k \right )} \right ) = \mathrm{random}\left (\ \right ) ,\\ \boldsymbol {Q}_{1}^{\left ( k \right )} = \left [ x_{\mathrm{pf}}^{\left ( k \right )},y_{\mathrm{pf}}^{\left ( k \right )},z_{\min},0,0,\gamma _{\mathrm{pf}}^{\left ( k \right )},\theta _{1},\theta _{2}, \ldots ,\theta _{j}, \ldots ,\theta _{n} \right ]^{\mathrm{T}} . \end{array} $$

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During the filling of $U_{s}$, record the generalized coordinates $\boldsymbol {Q}_{1}^{\left ( k \right )}$ corresponding to the cable tension (i.e., $\boldsymbol {T}_{\mathrm{e}}^{\left ( k \right )}$) and the maximum cable tension (i.e., $\max \left ( \boldsymbol {T}_{\mathrm{c}}^{\left ( k \right )} \right )$), the equation of the heat map can be written as

$$ \textstyle\begin{array}{l} \mathit{heat} = \max \left ( \boldsymbol {T}_{\mathrm{c}}^{\left ( k \right )} \right ), \\ x_{\mathrm{h}} = x_{\mathrm{ee}}^{\left ( k \right )},y_{\mathrm{h}} = y_{\mathrm{ee}}^{\left ( k \right )}, \end{array} $$

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where $\left ( x_{\mathrm{h}},y_{\mathrm{h}} \right )$ is the coordinate of the point on the heat map, and $\mathit{heat}$ is the heat value of the corresponding coordinate point.