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Implementing the homotopy continuation method in a hybrid approach to solve the kinematics problem of spatial parallel robots

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Abstract

In this paper, first the application of homotopy continuation method (HCM) in numerically solving kinematics problem of spatial parallel manipulators is investigated. Using the HCM the forward kinematics problem (F-Kin) of a six degrees of freedom (DOFs) 6–3 Stewart platform and the inverse kinematics problem (I-Kin) of a 3-DOF 3-PSP robot are solved. The governing equations of the kinematics problems of the robots are developed and embedded in the homotopy continuation function. The HCM is utilized in order to solve the nonlinear system of equations derived from the kinematics analysis of the robots. Then, to represent the real case application an initial guess far from the correct answer is selected. It is shown that, comparing with the Newton–Raphson method (NRM), the F-Kin calculation time for the Stewart robot is decreased by 43%. Therefore, using the HCM a hybrid method is suggested to solve the F-Kin of the Stewart robot. Furthermore, the HCM, as an innovative method, relieves other downsides of the conventional numerical methods, including a proper initial guess requirement as well as the problems of convergence.

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Authors and Affiliations

  1. Ferdowsi University of Mashhad, Mashhad, Iran

    Naser Mostashiri, Alireza Akbarzadeh & Amir Rezaei

Authors
  1. Naser Mostashiri
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  2. Alireza Akbarzadeh
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  3. Amir Rezaei
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Corresponding author

Correspondence to Alireza Akbarzadeh.

Appendix

Appendix

Extracting Eq. (34) and substituting rotation matrix components, constraint equations, which are nine nonlinear algebraic equations, can be obtained as follows,

$$\begin{aligned} {\varvec{\psi }} \left( \mathbf{q } \right) =\mathbf 0 \end{aligned}$$
(39)

where

$$\begin{aligned} \psi _{1}= & {} {x}_{\mathrm{P}}+{b}_{1}\left( \cos (\varphi )\cos \left( {\lambda } \right) \right) -{h}\left( \sin (\theta )\sin (\lambda )\right. \nonumber \\&\left. +\,\sin (\varphi )\cos (\theta )\cos (\lambda ) \right) -{a}=0\nonumber \\ \psi _{2}= & {} {y}_{\mathrm{P}}+{b}_{1}\left( \cos (\varphi )\sin (\lambda ) \right) -{h}\left( -\sin (\theta )\cos (\lambda )\right. \nonumber \\&\left. +\,\sin (\varphi )\cos (\theta )\sin (\lambda ) \right) =0\nonumber \\ \psi _{3}= & {} {z}_{\mathrm{P}}+{b}_{1}\left( -\sin (\varphi ) \right) -{h}\left( \cos (\varphi )\cos (\theta ) \right) -{q}_{1}=0\nonumber \\ \psi _{{4}}= & {} {x}_{\mathrm{P}}-\frac{1}{2}\,\,{b}_{2}\left( \cos (\varphi )\cos \left( {\lambda } \right) \right) +\frac{\sqrt{3} }{2}\,\,{b}_{2}\left( -\cos \left( {\theta } \right) \sin \left( {\lambda } \right) \right. \nonumber \\&\left. +\,\sin (\varphi )\sin (\theta )\cos (\lambda ) \right) -{h}\left( \sin (\theta )\sin (\lambda )\right. \nonumber \\&+\,\left. \sin (\varphi )\cos (\theta )\cos (\lambda ) \right) +\frac{1}{2}\,\,{a}=0\nonumber \\ \psi _{5}= & {} {y}_{\mathrm{P}}-\frac{1}{2}\,\,{b}_{2}\left( \cos (\varphi )\sin (\lambda ) \right) +\frac{\sqrt{3} }{2}\,\,{b}_{2}\left( \cos (\theta )\cos (\lambda )\right. \nonumber \\&\left. +\,\sin (\varphi )\sin (\theta )\sin (\lambda ) \right) -{h}\left( -\sin (\theta )\cos (\lambda )\right. \nonumber \\&+\,\left. \sin (\varphi )\cos (\theta )\sin (\lambda ) \right) -\frac{\sqrt{3} }{2}\,\,{a}=0\nonumber \\ \psi _{6}= & {} {z}_{\mathrm{P}}-\frac{1}{2}\,\,{b}_{2}\left( -\sin (\varphi ) \right) +\frac{\sqrt{3} }{2}\,\,{b}_{2}\left( \cos (\varphi )\sin (\theta ) \right) \nonumber \\&\quad -\,{h}\left( \cos (\varphi )\cos (\theta ) \right) -{q}_{2}=0\nonumber \\ \psi _{\mathrm{7}}= & {} {x}_{\mathrm{P}}-\frac{1}{2}\,\,{b}_{3}\left( \cos \left( {\varphi } \right) \cos \left( {\lambda } \right) \right) -\frac{\sqrt{3} }{2}\,\,{b}_{3}\left( -\cos \left( {\theta } \right) \sin \left( {\lambda } \right) \right. \nonumber \\&\left. +\,\sin \left( {\varphi } \right) \sin \left( {\theta } \right) \cos \left( {\lambda } \right) \right) -{h}\left( \sin (\theta )\sin (\lambda )\right. \nonumber \\&\left. +\,\sin (\varphi )\cos (\theta )\cos (\lambda ) \right) +\frac{1}{2}\,\,{a}=0\nonumber \\ \psi _{{8}}= & {} {y}_{\mathrm{P}}-\frac{1}{2}\,\,{b}_{3}\left( \cos (\varphi )\sin (\lambda ) \right) -\frac{\sqrt{3} }{2}\,\,{b}_{3}\left( \cos (\theta )\cos (\lambda )\right. \nonumber \\&\left. +\,\sin (\varphi )\sin (\theta )\sin (\lambda ) \right) -{h}\left( -\sin (\theta )\cos (\lambda )\right. \nonumber \\&\left. +\,\sin (\varphi )\cos (\theta )\sin (\lambda ) \right) +\frac{\sqrt{3} }{2}\,\,{a}=0\nonumber \\ \psi _{{9}}= & {} {z}_{\mathrm{P}}-\frac{1}{2}\,\,{b}_{3}\left( -\sin (\varphi ) \right) -\frac{\sqrt{3} }{2}\,\,{b}_{3}\left( \cos (\varphi )\sin (\theta ) \right) \nonumber \\&-\,{h}\left( \cos (\varphi )\cos (\theta ) \right) -{q}_{3}=0 \end{aligned}$$
(40)
$$\begin{aligned}&x_{3}-50ax_{3}-50hx_{1}+x_{1}^{2}x_{3}-50ax_{1}^{2}x_{3}-50hx_{1}x_{3}^{2}=0\nonumber \\&3{x}_{1}+1.732{x}_{2}+3{x}_{1}^{3}{x}_{3}^{2}+3 {x}_{1}^{3}{x}_{2}^{4}-1.732{x}_{1}^{4}{x}_{2}^{3}-1.732 {x}_{2}^{3}{x}_{3}^{2}\nonumber \\&\quad -\,3.464{x}_{1}{x}_{3}-2 {x}_{2}{x}_{3}-150{ax}_{1}^{3} +86.602{ax}_{2}^{3}+173.205{hx}_{1}^{2}\nonumber \\&\quad -\,173.205{hx}_{2}^{2}-3 {x}_{1}{x}_{3}^{2}-3{x}_{1}{x}_{2}^{4}-1.732 {x}_{2}{x}_{3}^{2}-3.464{x}_{1}^{3}{x}_{3}\nonumber \\&\quad -\,1.732{x}_{1}^{4}{x}_{2}+2{x}_{2}^{3}{x}_{3}-3 {x}_{1}^{3}+1.732{x}_{2}^{3}+450{ax}_{1}{x}_{2}^{2} \nonumber \\&\quad +433.013{ax}_{1}^{2}{x}_{2} -150{ax}_{1}{x}_{2}^{4} +173.205{ax}_{1}^{3}{x}_{3}-86.602{ax}_{1}^{4}{x}_{2}\nonumber \\&\quad -\,100{ax}_{2}^{3}{x}_{3}+200{hx}_{1}{x}_{2}^{3}-200{hx}_{1}^{3}{x}_{2} +20.785{x}_{1}{x}_{2}^{2}{x}_{3}\nonumber \\&\quad +\,20{x}_{1}^{2}{x}_{2}{x}_{3} -3.464{x}_{1}{x}_{2}^{4}{x}_{3}-2{x}_{1}^{4}{x}_{2}{x}_{3} -433.013{ax}_{1}^{2}{x}_{2}^{3}\nonumber \\&\quad +\,450{ax}_{1}^{3}{x}_{2}^{2}+150{ax}_{1}^{3}{x}_{3}^{2}-86.602{ax}_{2}^{3}{x}_{3}^{2}\nonumber \\&\quad -\,692.82{hx}_{1}^{2}{x}_{2}^{2}+173.205{hx}_{1}^{2}{x}_{3}^{2}+173.205{hx}_{1}^{2}{x}_{2}^{4}+200{hx}_{1}^{3}{x}_{2}^{3}\nonumber \\&\quad -\,173.205{hx}_{2}^{2}{x}_{3}^{2}-173.205{hx}_{1}^{4}{x}_{2}^{2}-20{x}_{1}^{2}{x}_{2}^{3}{x}_{3}+20.785 {x}_{1}^{3}{x}_{2}^{2}{x}_{3}\nonumber \\&\quad +\,3{x}_{1}{x}_{2}^{4} {x}_{3}^{2}+1.732{x}_{1}^{4}{x}_{2}{x}_{3}^{2}-3.464 {x}_{1}^{3}{x}_{2}^{4}{x}_{3} +2{x}_{1}^{4} {x}_{2}^{3}{x}_{3}\nonumber \\&\quad -\,346.41{ax}_{1}{x}_{3}-200{ax}_{2}{x}_{3}-200{hx}_{1}{x}_{2}-3 {x}_{1}^{3}{x}_{2}^{4}{x}_{3}^{2}\nonumber \\&\quad +\,1.732{x}_{1}^{4} {x}_{2}^{3}{x}_{3}^{2} +433.013{ax}_{1}^{2}{x}_{2}^{3}{x}_{3}^{2} -450{ax}_{1}^{3}{x}_{2}^{2}{x}_{3}^{2}\nonumber \\&\quad -\,692.82{hx}_{1}^{2}{x}_{2}^{2}{x}_{3}^{2}+173.205{hx}_{1}^{2}{x}_{2}^{4} {x}_{3}^{2}+200{hx}_{1}^{3}{x}_{2}^{3}{x}_{3}^{2}\nonumber \\&\quad -\,173.205{hx}_{1}^{4}{x}_{2}^{2}{x}_{3}^{2}+519.615{ax}_{1}{x}_{2}^{2}{x}_{3}+500{ax}_{1}^{2}{x}_{2}{x}_{3}\nonumber \\&\quad +\,173.205{ax}_{1}{x}_{2}^{4} {x}_{3}+100{ax}_{1}^{4}{x}_{2}{x}_{3} -200{hx}_{1}{x}_{2}{x}_{3}^{2}\nonumber \\&\quad -\,450{ax}_{1}{x}_{2}^{2}{x}_{3}^{2}-433.013{ax}_{1}^{2}{x}_{2}{x}_{3}^{2}-500{ax}_{1}^{2}{x}_{2}^{3}{x}_{3}\nonumber \\&\quad +\,519.615{ax}_{1}^{3}{x}_{2}^{2}{x}_{3} +150{ax}_{1}{x}_{2}^{4}{x}_{3}^{2}+86.602{ax}_{1}^{4} {x}_{2}{x}_{3}^{2}\nonumber \\&\quad -\,346.41{ax}_{1}^{3}{x}_{2}^{4} {x}_{3}+200{ax}_{1}^{4}{x}_{2}^{3}{x}_{3}+200{hx}_{1}{x}_{2}^{3}{x}_{3}^{2}\nonumber \\&\quad -\,200{h}{x}_{1}^{3}{x}_{2}{x}_{3}^{2}=0\nonumber \\&1.732{x}_{2}-3{x}_{1}-3{x}_{1}^{3}{x}_{3}^{2}-3{x}_{1}^{3}{x}_{2}^{4} -1.732{x}_{1}^{4}{x}_{2}^{3}-1.732{x}_{2}^{3}{x}_{3}^{2}\nonumber \\&\quad -\,3.464 {x}_{1}{x}_{3}+2{x}_{2}{x}_{3}+150{ax}_{1}^{3} +86.602{ax}_{2}^{3}+173.205{hx}_{1}^{2}\nonumber \\&\quad -\,173.205{hx}_{2}^{2}+3{x}_{1}{x}_{3}^{2}+3{x}_{1}{x}_{2}^{4}-1.732{x}_{2}{x}_{3}^{2}-3.464 {x}_{1}^{3}{x}_{3}\nonumber \\&\quad -\,1.732{x}_{1}^{4}{x}_{2}-2 {x}_{2}^{3}{x}_{3}+3{x}_{1}^{3}+1.732 {x}_{2}^{3}-450{ax}_{1}{x}_{2}^{2}\nonumber \\&\quad +\,433.013{ax}_{1}^{2}{x}_{2}+150{ax}_{1}{x}_{2}^{4} +173.205{ax}_{1}^{3}{x}_{3}-86.602{ax}_{1}^{4} {x}_{2}\nonumber \\&\quad +\,100{ax}_{2}^{3}{x}_{3}-200{hx}_{1}{x}_{2}^{3}+200{hx}_{1}^{3}{x}_{2}+20.785{x}_{1}{x}_{2}^{2}{x}_{3}\nonumber \\&\quad -\,20{x}_{1}^{2}{x}_{2}{x}_{3}-3.464 {x}_{1}{x}_{2}^{4}{x}_{3}+2{x}_{1}^{4}{x}_{2}{x}_{3}-433.013{ax}_{1}^{2}{x}_{2}^{3}\nonumber \\&\quad -\,450{ax}_{1}^{3}{x}_{2}^{2}-150{ax}_{1}^{3}{x}_{3}^{2}-86.602{ax}_{2}^{3}{x}_{3}^{2} -692.82{hx}_{1}^{2}{x}_{2}^{2}\nonumber \\&\quad +\,173.205{hx}_{1}^{2}{x}_{3}^{2}+173.205{hx}_{1}^{2}{x}_{2}^{4}-200{hx}_{1}^{3}{x}_{2}^{3}\nonumber \\&\quad -\,173.205{hx}_{2}^{2}{x}_{3}^{2}-173.205{hx}_{1}^{4}{x}_{2}^{2}+20{x}_{1}^{2}{x}_{2}^{3}{x}_{3}+20.785 {x}_{1}^{3}{x}_{2}^{2}{x}_{3}\nonumber \\&\quad -\,3{x}_{1}{x}_{2}^{4} {x}_{3}^{2}+1.732{x}_{1}^{4}{x}_{2}{x}_{3}^{2}-3.464 {x}_{1}^{3}{x}_{2}^{4}{x}_{3}-2{x}_{1}^{4}{x}_{2}^{3}{x}_{3}\nonumber \\&\quad -\,346.41{ax}_{1}{x}_{3}+200{ax}_{2}{x}_{3}+200{hx}_{1}{x}_{2}+3{x}_{1}^{3}{x}_{2}^{4} {x}_{3}^{2}\nonumber \\&\quad +\,1.732{x}_{1}^{4}{x}_{2}^{3}{x}_{3}^{2} +433.013{ax}_{1}^{2}{x}_{2}^{3}{x}_{3}^{2}+450{ax}_{1}^{3}{x}_{2}^{2}{x}_{3}^{2}\nonumber \\&\quad -\,692.82{hx}_{1}^{2}{x}_{2}^{2}{x}_{3}^{2}+173.205{hx}_{1}^{2}{x}_{2}^{4} {x}_{3}^{2}-200{hx}_{1}^{3}{x}_{2}^{3}{x}_{3}^{2}\nonumber \\&\quad -\,173.205{hx}_{1}^{4}{x}_{2}^{2}{x}_{3}^{2}+519.615{ax}_{1}{x}_{2}^{2}{x}_{3}-500{ax}_{1}^{2}{x}_{2}{x}_{3}\nonumber \\&\quad +v173.205{ax}_{1}{x}_{2}^{4} {x}_{3}-100{ax}_{1}^{4}{x}_{2}{x}_{3} +200{hx}_{1}{x}_{2}{x}_{3}^{2}+450{ax}_{1}{x}_{2}^{2}{x}_{3}^{2}\nonumber \\&\quad -\,433.013{ax}_{1}^{2}{x}_{2}{x}_{3}^{2}+500{ax}_{1}^{2}{x}_{2}^{3}{x}_{3}+519.615{ax}_{1}^{3}{x}_{2}^{2}{x}_{3}\nonumber \\&\quad -\,150{ax}_{1}{x}_{2}^{4}{x}_{3}^{2}+86.602{ax}_{1}^{4} {x}_{2}{x}_{3}^{2}-346.41{ax}_{1}^{3}{x}_{2}^{4} {x}_{3}\nonumber \\&\quad -\,200{ax}_{1}^{4}{x}_{2}^{3}{x}_{3}-200{hx}_{1}{x}_{2}^{3}{x}_{3}^{2}+200{hx}_{1}^{3}{x}_{2}{x}_{3}^{2}=0 \end{aligned}$$
(41)

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Mostashiri, N., Akbarzadeh, A. & Rezaei, A. Implementing the homotopy continuation method in a hybrid approach to solve the kinematics problem of spatial parallel robots. Intel Serv Robotics 10, 257–270 (2017). https://doi.org/10.1007/s11370-017-0222-0

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