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Development of a new spinning gait for a planar snake robot using central pattern generators

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Abstract

In this paper, we first present dynamic equation of n-link snake robot using Lagrange’s method in a simplified matrix form and verify them experimentally. Next, we introduce a new locomotion mode called spinning gait. Central pattern generators (CPGs) are used for online gait generation. To realize spinning gait, genetic algorithm is used to find optimal CPG network parameters. We illustrate both theoretically, using derived robot dynamics and experimentally that the CPG-based online gait generation method allows continuous and rather smooth transitions between gaits. Lastly, we present an application where the snake robot is guided from an initial to final position while avoiding obstacles by changing CPG parameters.

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Acknowledgments

The authors would like to thank Ferdowsi University of Mashhad for their financial support in this project

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Correspondence to Alireza Akbarzadeh.

Appendix

Appendix

The detailed forms of \(M, H, B\) and \(F\) in Eq. 11 are presented as

$$\begin{aligned} {M}_{{ij}}&\!=\!&\left\{ {\begin{array}{ll} [m_{j} {d}_{j} {l}_{i} \! +\! (\sum \limits _{{k = j + 1}}^{n} {m_{k} )l_{i} l_{j} ]\cos (\theta _{i} \! -\! \theta _{j}})&{} \quad i \!< \!j, 1 \!\le \! j\! \le \! n \\ {I}_{i} \! +\! {m}_{i} d_{i}^{2} \!+\! {l}_{i}^{2} (\sum \limits _{{j = i + 1}}^{n} {m_{j}} )&{}\quad i \!=\! j,\,\,1\! \le \! j\! \le \! n \\ M_{{ji}}&{}\quad i \!>\! j,\,\,1 \!\le \! j \!\le \! n\,\,\, \\ \end{array}} \right. \end{aligned}$$
$$\begin{aligned}&{M}_{n+1,j} \!\!=\! \!-\!\sin \theta _{j} \left[ \!m_{j} d_{j} \!+\!\left( \sum _{k=j+1}^{n} m_{k}\right) {l}_{j}\right] (j\!=\!1,\ldots ,n) \nonumber \\&{M}_{n+2,j} \!=\!\cos \theta _{j} \left[ \!m_{j} d_{j}\! +\!\left( \sum _{k=j+1}^{n} m_{k}\right) l_{j}\!\right] (j\!\!=\!1,\!\ldots ,\!n)\nonumber \\&{M}_{n+1,n+1} \!=\!{M}_{n+2,n+2}\! =\!\sum _{i=1}^n {{m}_{i}} \\&{M}_{n+1,n+2} = {M}_{n+2,n+1} =0 \nonumber \end{aligned}$$
(25)
$$\begin{aligned} {H}_{i}&= {l}_{i}\sum _{j=i}^{n} \left\{ \left[ m_{j} d_{j} +l_{j} \left( \sum _{k=j+1}^{n} {m_{k}}\right) \right] \sin (\theta _{i} -\theta _{j} ) \dot{\theta }_{j} ^{2} \right\} \nonumber \\&+\sum _{j=1}^{i-1} \left\{ \!\left[ \!{m}_{i} {d}_{i} \!+\!{l}_{i} \left( \sum _{k=j+1}^{n} {{m}_{k}}\!\right) \!\right] {l}_{j}\sin (\theta _{i} -\theta _{j} )\dot{\theta }_{j} ^{2} \!\right\} \nonumber \\&\times (i=1,\ldots ,n) \\ {H}_{n+1}&= -\sum _{i=1}^n {\cos \theta _{i} \left[ {m}_{i} {d}_{i} +\left( \sum _{k=i+1}^n {{m}_{k}}\right) {l}_{i} \right] \dot{\theta }_{i} ^{2}} \nonumber \\ {H}_{n+2}&= -\sum _{i=1}^{n} {\sin \theta _{i} \left[ m_{i} d_{i} + \left( \sum _{k=i+1}^{n} {{m}_{k}}\right) {l}_{i}\right] \dot{\theta }_{i} } \nonumber \end{aligned}$$
(26)
$$\begin{aligned} B = \left[ \begin{array}{l} D_{{n \times n - 1}} \\ \,\,\,\,\,0_{{}} \\ \end{array} \right] ,\quad {\mathrm{where}}\,D_{{ij}} = \left\{ \begin{array}{l} - 1\quad i = j \\ 1\quad i = j + 1 \\ 0\quad \mathrm{others} \\ \end{array} \right. \end{aligned}$$
(27)
$$\begin{aligned} F&= \left[ {\begin{array}{l} {^p{f}}_{n\times 1} \\ {^q{f}}_{2\times 1} \\ \end{array}} \right] \mathrm{where} \nonumber \\ {^p{f}}_{j}&= {d}_{j}({f}_{xj} \sin \theta _{j} -f_{yj} \cos \theta _{j} )\nonumber \\&+l_{j} \left[ \sin \theta _{j} \sum _{i=j+1}^{n} {(f_{xi} )-\cos \theta _{j} \sum _{i=j+1}^n {(f_{yi} )}} \right] \\ ^{q}f&= \left[ \begin{array}{l} - \sum \limits _{{i = 1}}^{n} {(f_{{xi}} )} \\ - \sum \limits _{{i = 1}}^{n} {(f_{{yi}} )} \\ \end{array} \right] \nonumber \end{aligned}$$
(28)

The detailed final dynamic equation, Eq. 11, has a simplified matrix format and can easily be expanded for any number of links.

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Hasanzadeh, S., Akbarzadeh, A. Development of a new spinning gait for a planar snake robot using central pattern generators. Intel Serv Robotics 6, 109–120 (2013). https://doi.org/10.1007/s11370-013-0129-3

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