Cable path optimization by fixing multiple guides on one link for industrial robot arms

Abstract

Several methods have been developed in recent years to automatically design cable paths for a given robot motion. However, an optimal solution may not be found when applied to complex motions such as inspection or assembly processes. Observation of experts’ works reveals that multiple guides are empirically fixed on one link to avoid cable collisions with peripheral devices and to lessen cable bending fatigue. This study proposes a cable path optimization method with multiple guides on one link to obtain the optimal solution for complex motions. Furthermore, it experimentally demonstrates its effectiveness. However, fixing multiple guides on one link may increase the computational burden because the solution search space will remarkably expand. Therefore, a combination of Hermite interpolation and interpolated coordinate system calculation using rotation-minimizing frames was introduced to reduce the significant burden of connected geometry computation. The connected geometry is the geometry of the cable segment which connects the guides at both ends and is used as an initial geometry in the physical simulation considering gravity to determine the convergent geometry of the cable segment.

Appendices

Appendix 1. PV candidate set ${{\varvec{D}}}_{0,{\varvec{m}}}$

${D}_{0,m}$ [16] is given by

$${D}_{0,m}=\left\{{(S}_{m,{i}_{m} },{P}_{m,{j}_{m}},{P}_{m + 1,{j}_{m + 1}})\right|1\le {i}_{m}\le {I}_{m}, 1\le {j}_{m}\le {J}_{m},1\le {j}_{m + 1}\le {J}_{m + 1},1\le m<M\},$$

(5)

where {${S}_{m,{i}_{m}}| 1\le {i}_{m}\le {I}_{m}\}$ denotes a candidate set of sub-segment numbers in ${C}_{m}$ that connects ${G}_{m}$ and ${G}_{m+1}$. ${I}_{m}$ denotes the candidate number of sub-segment numbers. {${S}_{m,{i}_{m}}$} are prepared as an arithmetic series of constant interval $\Delta S$. Furthermore, {${P}_{m,{j}_{m}}\left| 1\le {j}_{m}\le {J}_{m}\right\}$ denotes a candidate set of configurations of ${G}_{m}$, and ${J}_{m}$ represents the candidate number of configurations of ${G}_{m}$. {${P}_{m + 1,{j}_{m + 1}}\left| 1\le {j}_{m + 1}\le {J}_{m + 1}\right\}$ indicates a candidate set of configurations of ${G}_{m + 1}$. ${J}_{m+1}$ represents the candidate number of configurations of ${G}_{m+1}$.

Appendix 2. PV candidate set ${{\varvec{D}}}_{1,{\varvec{m}}}$

${D}_{1,m}$ Satisfying stress constraints in the attachment test is given by

$$\begin{array}{c}{D}_{1,m}=\left\{{(S}_{m,{i}_{m} },{P}_{m,{j}_{m}},{P}_{m + 1,{j}_{m + 1}})\in {D}_{0,m}\right|\\ {\mathrm{ Impulse}}_{\mathrm{th}}\ge {\mathrm{Impulse}}_{({S}_{m,{i}_{m}},{P}_{m,{j}_{m}},{P}_{m + 1,{j}_{m + 1}})}\left(\theta \right),\\ \begin{array}{c}{\mathrm{ Stretch}}_{\mathrm{th}}\ge {\mathrm{Stretch}}_{\left({S}_{m,{i}_{m}},{P}_{m,{j}_{m}},{P}_{m + 1,{j}_{m + 1}}\right)}\left(\theta \right), \\ {\mathrm{ Curvature}}_{\mathrm{th}} \le {\mathrm{Curvature}}_{\left({S}_{m,{i}_{m}},{P}_{m,{j}_{m}},{P}_{m + 1,{j}_{m + 1}}\right)}\left(\theta \right),\\ for \forall \theta \in {{\Theta }^{\mathrm{^{\prime}}}}_{m}\},\end{array}\end{array}$$

(6)

where ${\mathrm{Impulse}}_{({S}_{m,{i}_{m}},{P}_{m,{j}_{m}},{P}_{m + 1,{j}_{m + 1}})}\left(\theta \right)$ represents the maximum impulse received by each sub-segment of ${C}_{m},$ when the joint angles are set to ${\mathrm{\Theta {\prime}}}_{m}$ that contains only the initial pose as one of the arbitrary poses. Similarly, ${\mathrm{Stretch}}_{({S}_{m,{i}_{m}},{P}_{m,{j}_{m}},{P}_{m + 1,{j}_{m + 1}})}\left(\theta \right)$ and ${\mathrm{Curvature}}_{({S}_{m,{i}_{m}},{P}_{m,{j}_{m}},{P}_{m + 1,{j}_{m + 1}})}\left(\theta \right)$ are the maximum stretch and minimum curvature radius of the sub-segments, respectively.

Appendix 3. PV candidate set ${{\varvec{D}}\boldsymbol{^{\prime}}}_{0,{\varvec{m}}}$

${D{\prime}}_{0,m}$ Satisfying the adjacency condition between ${D}_{0,m}$ and ${D}_{2,m-1}$ in the attachment test [16] is given by

$$\begin{array}{c}{D{\prime}}_{0,m}=\left.\left\{({S}_{m,{i}_{m}},{P}_{m,{j}_{m}},{P}_{m + 1,{j}_{m + 1}})\right.\right|\\ \left({S}_{m-1,{i}_{m-1}},{P}_{m-1,{j}_{m-1}},{P}_{m,{j}_{m}}\right)\in {D}_{2,m-1},\\ \left. \left({S}_{m,{i}_{m}},{P}_{m,{j}_{m}},{P}_{m + 1,{j}_{m + 1}}\right)\in {D}_{0,m}\right\},\end{array}$$

(7)

Compared to ${D}_{0,m}$, ${D{\prime}}_{0,m}$ includes fewer PVs. Thus, the burden can be reduced by applying the adjacency condition in CSAT [16].