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Development of a compliant legged quadruped robot

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Abstract

This paper presents the detailed steps for design and development of a compliant legged fault tolerant quadruped robot where each leg has two links and two motorized revolute joints for locomotion. The body and upper links of legs are rigid whereas the lower link of each leg is compliant. Amble gait is demonstrated on the developed robot. Safety and reliability are the most critical issues for the quadruped robot. During the failure of any joint, performance of quadruped robot is affected. In this paper, locked joint failure is also discussed. Strategies for fault tolerant control of the quadruped are developed and experimentally validated. The developed robot can be used for various hardware-in-the-loop controller prototyping such as reconfiguration, fault tolerant control, and posture control, etc. pertaining to quadruped robots.

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Acknowledgements

The research work presented here was funded by the Department of Science and Technology (DST), India under Indo–Korea Joint Research in Science and Technology vide Grant No. INT/Korea/P–13. Mr M M Gor is thankful to G. H. Patel College of Engineering and Technology, Gujarat, India for allowing him to carry out research work at IIT Roorkee. The work of Mr J M Yang and Mr S W Kwak was supported by the Korea government (MEST) vide the National Research Foundation of Korea Grant No. NRF–2011–0027705.

Author information

Authors and Affiliations

  1. Indian Institute of Technology Roorkee, Roorkee, India

    M M Gor, P M Pathak, K Alam, P Kumar, D Anand, P Vijay & R Sarkar

  2. Indian Institute of Technology Kharagpur, Kharagpur, India

    A K Samantaray

  3. Kyungpook National University, Daegu, South Korea

    J-M Yang

  4. Keimyung University, Daegu, South Korea

    S W Kwak

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  1. M M Gor
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  2. P M Pathak
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  3. A K Samantaray
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  4. K Alam
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  5. P Kumar
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  6. D Anand
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  7. P Vijay
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  8. R Sarkar
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Corresponding author

Correspondence to M M Gor.

Appendices

Appendix A

Equation of motions for back stance phase and front stance phase

$$ \begin{aligned} \tau_{1} & = \left[ { - l_{1} \left( {m_{b} + m_{1} + m_{2} } \right)\sin \left( {\theta_{2} } \right)} \right]\ddot{l}_{2} + [ - l_{1}^{2} \left( {m_{b} + m_{1} + m_{2} } \right) \\ & \quad - l_{2}^{2} \left( {m_{b} + 2m_{1} + m_{2} } \right) - Ll_{1} \left( {m_{b} + 2m_{1} + 2m_{2} } \right)\sin (\theta_{1} ) \\ & \quad - 2l_{1} l_{2} \left( {m_{b} + m_{1} + m_{2} } \right)\cos \left( {\theta_{2} } \right) + \left( {m_{1} + m_{2} } \right)l_{1}^{2} \cos \left( {\theta_{1} + \theta_{3} } \right) \\ & \quad + \left( {m_{1} + m_{2} } \right)l_{1} l_{2} \cos \left( {\theta_{1} - \theta_{2} + \theta_{3} } \right) + m_{2} l_{1} l_{0} \cos (\theta_{1} + \theta_{3} - \theta_{4} ) \\ & \quad + m_{2} l_{2} l_{0} \cos (\theta_{2} - \theta_{1} + \theta_{4} - \theta_{3} ) - Ll_{2} \left( {m_{b} + 2m_{1} + 2m_{2} } \right) \\ & \quad \sin \left( {\theta_{1} - \theta_{2} } \right)]\ddot{\theta } + [l_{1}^{2} \left( {m_{b} + m_{1} + m_{2} } \right) + m_{1} r_{1}^{2} \\ & \quad + l_{2}^{2} \left( {m_{b} + 2m_{1} + m_{2} } \right) + 2l_{1} l_{2} \left( {m_{b} + m_{1} + m_{2} } \right)\cos \left( {\theta_{2} } \right)]\ddot{\theta }_{1} \\ & \quad + [ - l_{2}^{2} \left( {m_{b} + 2m_{1} + m_{2} } \right) - l_{1} l_{2} \left( {m_{b} + m_{1} + m_{2} } \right)\cos \left( {\theta_{2} } \right)]\ddot{\theta }_{2} \\ & \quad - 2\left( {m_{b} + 2m_{1} + m_{2} } \right)l_{2} \dot{l}_{2} \left( {\dot{\theta } + \dot{\theta }_{2} - \dot{\theta }_{1} } \right) + \left( {m_{b} + 2m_{1} + 2m_{2} } \right) \\ & \quad \left[ { - Ll_{1} \dot{\theta }^{2} \cos \left( {\theta_{1} } \right) - Ll_{2} \dot{\theta }^{2} \cos \left( {\theta_{1} - \theta_{2} } \right)} \right] + \left( {m_{b} + m_{1} + m_{2} } \right)l_{1} \\ & \quad \left[ {\dot{l}_{2} \left( {2\dot{\theta }_{2} - 2\dot{\theta }_{1} + 2\dot{\theta }} \right)\cos (\theta_{2} ) + l_{2} \dot{\theta }_{2} \left( {\dot{\theta }_{2} - 2\dot{\theta }_{1} + 2\dot{\theta }} \right)\sin (\theta_{2} )} \right] \\ & \quad + \left( {m_{1} + m_{2} } \right)\left[ { - l_{1}^{2} \dot{\theta }^{2} \sin \left( {\theta_{1} + \theta_{3} } \right) - l_{1} l_{2} \dot{\theta }^{2} \sin \left( {\theta_{1} - \theta_{2} + \theta_{3} } \right)} \right] \\ & \quad + m_{2} \left[ { - l_{1} l_{0} \dot{\theta }^{2} \sin (\theta_{1} + \theta_{3} - \theta_{4} ) + l_{0} l_{2} \dot{\theta }^{2} \sin (\theta_{2} - \theta_{1} + \theta_{4} - \theta_{3} )} \right] \\ & \quad + g\left[ { - \left( {m_{b} + m_{1} + m_{2} } \right)l_{1} \sin \left( {\theta_{1} - \left( {\theta + \theta_{p} } \right)} \right)} \right. \\ & \quad \left. { + \left( {m_{b} + 2m_{1} + m_{2} } \right)l_{2} \sin (\theta_{2} - \theta_{1} + \theta + \theta_{p} )} \right] \\ \end{aligned} $$
$$ \begin{aligned} \tau_{2} & = [\left( {m_{b} + 2m_{1} + m_{2} } \right)l_{2}^{2} + \left( {m_{b} + 2m_{1} + 2m_{2} } \right)Ll_{2} \\ & \quad \sin \left( {\theta_{1} - \theta_{2} } \right) + \left( {m_{b} + m_{1} + m_{2} } \right)l_{1} l_{2} \cos \left( {\theta_{2} } \right) \\ & \quad - \left( {m_{1} + m_{2} } \right)l_{1} l_{2} \cos \left( {\theta_{1} - \theta_{2} + \theta_{3} } \right) \\ & \quad - m_{2} l_{0} l_{2} \cos (\theta_{2} - \theta_{1} + \theta_{4} - \theta_{3} )]\ddot{\theta } \\ & \quad + [ - l_{2}^{2} \left( {m_{b} + 2m_{1} + m_{2} } \right) - l_{1} l_{2} \left( {m_{b} + m_{1} + m_{2} } \right) \\ & \quad { \cos }\left( {\theta_{2} } \right)]\ddot{\theta }_{1} + [l_{2}^{2} \left( {m_{b} + 2m_{1} + m_{2} } \right) + m_{2} r_{2}^{2} ]\ddot{\theta }_{2} \\ & \quad + \left( {m_{b} + 2m_{1} + 2m_{2} } \right)Ll_{2} \dot{\theta }^{2} \cos \left( {\theta_{1} - \theta_{2} } \right) \\ & \quad + \left( {m_{b} + m_{1} + m_{2} } \right)l_{1} l_{2} \left( {\dot{\theta } - \dot{\theta }_{1} } \right)^{2} \sin \left( {\theta_{2} } \right) \\ & \quad + \left( {m_{1} + m_{2} } \right)\left[ {l_{1} l_{2} \dot{\theta }^{2} \sin \left( {\theta_{1} - \theta_{2} + \theta_{4} } \right)} \right] \\ & \quad + m_{2} l_{0} \left[ { - l_{2} \dot{\theta }^{2} { \sin }(\theta_{2} - \theta_{1} + \theta_{4} - \theta_{3} )} \right] \\ & \quad - g\left[ {\left( {m_{b} + 2m_{1} + m_{2} } \right)l_{2} \sin (\theta_{2} - \theta_{1} + \theta + \theta_{p} )} \right] \\ & \quad + 2\left( {m_{b} + 2m_{1} + m_{2} } \right)l_{2} \dot{l}_{2} \left( {\dot{\theta } + \dot{\theta }_{2} - \dot{\theta }_{1} } \right) \\ \end{aligned} $$
$$ \begin{aligned} \tau_{3} & = \left[ { - l_{1} \left( {m_{b} + m_{1} + m_{2} } \right)\sin \left( {\theta_{2} } \right)} \right]\ddot{l}_{2} + [ - l_{1}^{2} \left( {m_{b} + m_{1} + m_{2} } \right) \\ & \quad - l_{2}^{2} \left( {m_{b} + 2m_{1} + m_{2} } \right) - Ll_{1} \left( {m_{b} + 2m_{1} + 2m_{2} } \right)\sin (\theta_{3} ) \\ & \quad + 2l_{1} l_{2} \left( {m_{b} + m_{1} + m_{2} } \right)\cos \left( {\theta_{4} } \right) + \left( {m_{1} + m_{2} } \right)l_{1}^{2} \cos \left( {\theta_{3} + \theta_{1} } \right) \\ & \quad + \left( {m_{1} + m_{2} } \right)l_{1} l_{2} \cos \left( {\theta_{4} - \theta_{3} - \theta_{1} } \right) - m_{2} l_{1} l_{0} \\ & \quad \cos (\theta_{3} + \theta_{3} - \theta_{4} ) - m_{2} l_{2} l_{0} \cos (\theta_{4} - \theta_{3} + \theta_{2} - \theta_{1} ) \\ & \quad - Ll_{2} \left( {m_{b} + 2m_{1} + 2m_{2} } \right)\sin \left( {\theta_{3} - \theta_{4} } \right)]\ddot{\theta } \\ & \quad + \left[ {l_{1}^{2} \left( {m_{b} + m_{1} + m_{2} } \right) + m_{1} r_{1}^{2} + l_{2}^{2} \left( {m_{b} + 2m_{1} + m_{2} } \right)} \right. \\ & \quad \left. { + l_{1} l_{2} \left( {m_{b} + m_{1} + m_{2} } \right)} \right]\ddot{\theta }_{3} + \left[ { - l_{2}^{2} \left( {m_{b} + 2m_{1} + m_{2} } \right)} \right. \\ & \quad \left. { - l_{1} l_{2} \left( {m_{b} + m_{1} + m_{2} } \right)} \right]\ddot{\theta }_{4} + [ - 2\left( {m_{b} + 2m_{1} + m_{2} } \right)l_{2} \dot{l}_{2} \\ & \quad \left( {\dot{\theta } + \dot{\theta }_{4} - \dot{\theta }_{3} } \right) + \left( {m_{b} + 2m_{1} + 2m_{2} } \right)\left[ { - Ll_{1} \dot{\theta }^{2} \cos \left( {\theta_{3} } \right)} \right. \\ & \quad \left. { - Ll_{2} \dot{\theta }^{2} \cos \left( {\theta_{3} - \theta_{4} } \right)} \right] + \left( {m_{b} + m_{1} + m_{2} } \right)l_{1} \\ & \quad \left[ { - \dot{l}_{2} \left( {2\dot{\theta }_{4} - \dot{\theta }_{3} + \dot{\theta }} \right)\cos (\theta_{4} ) + l_{2} \dot{\theta }_{4} \left( {\dot{\theta }_{4} - \dot{\theta }_{3} + \dot{\theta }} \right)\sin (\theta_{4} )} \right] \\ & \quad + \left( {m_{1} + m_{2} } \right)\left[ { - l_{1}^{2} \dot{\theta }^{2} \sin \left( {\theta_{3} + \theta_{1} } \right) + l_{1} l_{2} \dot{\theta }^{2} \sin \left( {\theta_{4} - \theta_{3} - \theta_{1} } \right)} \right] \\ & \quad + m_{2} \left[ {l_{1} l_{0} \dot{\theta }^{2} \sin (\theta_{3} + \theta_{1} - \theta_{2} ) + l_{0} \dot{l}_{2} \dot{\theta }\cos \left( {\theta_{4} - \theta_{3} - \theta_{2} + \theta_{1} + 2\left( {\theta - \theta_{p} } \right)} \right)} \right. \\ & \quad \left. { - l_{0} l_{2} \dot{\theta }^{2} { \sin }(\theta_{4} - \theta_{3} + \theta_{2} - \theta_{1} )} \right] + g\left[ { - \left( {m_{b} + m_{1} + m_{2} } \right)l_{1} \sin \left( {\theta_{3} - \theta + \theta_{p} } \right)} \right. \\ & \quad \left. { + \left( {m_{b} + 2m_{1} + m_{2} } \right)l_{2} \sin (\theta_{4} - \theta_{3} + \theta - \theta_{p} )} \right] \\ \end{aligned} $$
$$ \begin{aligned} \tau_{4} & = [\left( {m_{b} + 2m_{1} + m_{2} } \right)l_{2}^{2} + \left( {m_{b} + 2m_{1} + m_{2} } \right)Ll_{2} \sin \left( {\theta_{3} - \theta_{4} } \right) \\ & \quad + \left( {m_{b} + m_{1} + m_{2} } \right)l_{1} l_{2} \cos \left( {\theta_{4} } \right) - \left( {m_{1} + m_{2} } \right)l_{1} l_{2} \\ & \quad \cos \left( {\theta_{4} - \theta_{3} - \theta_{1} } \right) + m_{2} l_{0} l_{2} \cos (\theta_{4} - \theta_{3} + \theta_{2} - \theta_{1} )]\ddot{\theta } \\ & \quad + \left[ { - l_{2}^{2} \left( {m_{b} + 2m_{1} + m_{2} } \right) - l_{1} l_{2} \left( {m_{b} + m_{1} + m_{2} } \right)\cos \left( {\theta_{4} } \right)} \right]\ddot{\theta }_{1} \\ & \quad + \left[ {l_{2}^{2} \left( {m_{b} + 2m_{1} + m_{2} } \right)} \right]\ddot{\theta }_{4} + 2\left( {m_{b} + 2m_{1} + m_{2} } \right)l_{2} \dot{l}_{2} \left( {\dot{\theta } + \dot{\theta }_{4} - \dot{\theta }_{3} } \right) \\ & \quad + \left( {m_{b} + 2m_{1} + m_{2} } \right)L\dot{l}_{2} \dot{\theta }\sin \left( {\theta_{3} - \theta_{4} } \right) + \left( {m_{b} + 2m_{1} + 2m_{2} } \right) \\ & \quad Ll_{2} \dot{\theta }\cos \left( {\theta_{3} - \theta_{4} } \right) + \left( {m_{b} + m_{1} + m_{2} } \right)l_{1} l_{2} \left( {\dot{\theta }_{3} - \dot{\theta }} \right)^{2} \sin \left( {\theta_{4} } \right) \\ & \quad + \left( {m_{b} + 2m_{1} + 2m_{2} } \right)L\left( {l_{2} \dot{\theta }\left( {\dot{\theta } + \dot{\theta }_{4} - \dot{\theta }_{3} } \right)\cos \left( {\theta_{3} - \theta_{4} } \right) - \dot{l}_{2} \dot{\theta }\sin \left( {\theta_{3} - \theta_{4} } \right)} \right) \\ & \quad + (m_{1} + m_{2} )\left[ { - l_{1} l_{2} \dot{\theta }^{2} \sin (\theta_{4} - \theta_{3} - \theta_{1} )} \right] + m_{2} l_{0} \left[ {\dot{l}_{2} \dot{\theta }\cos \left( {\theta_{4} - \theta_{3} + \theta_{2} - \theta_{1} } \right)} \right. \\ & \quad \left. { - \dot{l}_{2} \dot{\theta }\cos \left( {\theta_{4} - \theta_{3} - \theta_{2} + \theta_{1} + 2\left( {\theta - \theta_{p} } \right)} \right) + l_{2} \dot{\theta }^{2} { \sin }(\theta_{4} - \theta_{3} + \theta_{2} - \theta_{1} )} \right] \\ & \quad - g\left[ {\left( {m_{b} + 2m_{1} + m_{2} } \right)l_{2} \sin (\theta_{4} - \theta_{3} + \theta - \theta_{p} )} \right] \\ \end{aligned} $$

Appendix B

Coefficients used in equations to determine the required efforts at the joints of the two-DOF planar robot

Coefficient

Equation

β 10

\( \begin{aligned} & \left( {\sin \left( \theta \right)\sin \left( {\theta_{P1} } \right) + \sin \left( \psi \right)\cos \left( \theta \right)\cos \left( {\theta_{P1} } \right)} \right)l_{P1} \\ & \quad + \left[ { - \left( { - \sin \left( \theta \right)\cos \left( {\theta_{P1} } \right) + \sin \left( \psi \right)\cos \left( \theta \right)\sin \left( {\theta_{P1} } \right)} \right)\sin \left( {\theta_{P2} } \right)} \right. \\ & \quad \left. { + \left( {\sin \left( \theta \right)\sin \left( {\theta_{P1} } \right) + \sin \left( \psi \right)\cos \left( \theta \right)\cos \left( {\theta_{P1} } \right)} \right)\cos \left( {\theta_{P2} } \right)} \right]l_{P2} \\ \end{aligned} \)

β 11

\( \begin{aligned} & \left[ { - \cos \left( \theta \right)\sin \left( \phi \right)\sin \left( {\theta_{P1} } \right) + \left( {\sin \left( \psi \right)\sin \left( \theta \right)\sin \left( \phi \right) + \cos \left( \psi \right)\cos \left( \phi \right)} \right)\cos \left( {\theta_{P1} } \right)} \right]l_{P1} \\ & \quad + \left[ { - \left( {\cos \left( \theta \right)\sin \left( \phi \right)\cos \left( {\theta_{P1} } \right) + \left( {\sin \left( \psi \right)\sin \left( \theta \right)\sin \left( \phi \right) + \cos \left( \psi \right)\cos \left( \phi \right)} \right)\sin \left( {\theta_{P1} } \right)} \right)\sin \left( {\theta_{P2} } \right)} \right. \\ & \quad \left. { + \left( { - \cos \left( \theta \right)\sin \left( \phi \right)\sin \left( {\theta_{P1} } \right) + \left( {\sin \left( \psi \right)\sin \left( \theta \right)\sin \left( \phi \right) + \cos \left( \psi \right)\cos \left( \phi \right)} \right)\cos \left( {\theta_{P1} } \right)} \right)\cos \left( {\theta_{P2} } \right)} \right]l_{P2} \\ \end{aligned} \)

β 12

\( \begin{aligned} & [\left( { - \cos \left( \theta \right)\cos \left( \phi \right)\sin \left( {\theta_{P1} } \right) + \sin \left( \psi \right)\sin \left( \theta \right)\cos \left( \phi \right)\cos \left( {\theta_{P1} } \right) - \cos \left( \psi \right)\sin \left( \phi \right)\cos \left( {\theta_{P1} } \right)} \right)l_{P1} \\ & \quad + \left( { - \left( {\cos \left( \theta \right)\cos \left( \phi \right)\cos \left( {\theta_{P1} } \right) + \left( {\sin \left( \psi \right)\sin \left( \theta \right)\cos \left( \phi \right) - \cos \left( \psi \right)\sin \left( \phi \right)} \right)\sin \left( {\theta_{P1} } \right)} \right)\sin \left( {\theta_{P2} } \right)} \right. \\ & \quad \left. { + \left( { - \cos \left( \theta \right)\cos \left( \phi \right)\sin \left( {\theta_{P1} } \right) + \left( {\sin \left( \psi \right)\sin \left( \theta \right)\cos \left( \phi \right) - \cos \left( \psi \right)\sin \left( \phi \right)} \right)\cos \left( {\theta_{P1} } \right)} \right)\cos \left( {\theta_{P1} } \right)} \right]l_{P2} \\ \end{aligned} \)

β 13

\( \begin{aligned} & \left[ { - \left( { - \sin \left( \theta \right)\cos \left( {\theta_{P1} } \right) + \sin \left( \psi \right)\cos \left( \theta \right)\sin \left( {\theta_{P1} } \right)} \right)\sin \left( {\theta_{P2} } \right) + } \right. \\ & \quad \left. {\left( {\sin \left( \theta \right)\sin \left( {\theta_{P1} } \right) + \sin \left( \psi \right)\cos \left( \theta \right)\cos \left( {\theta_{P1} } \right)} \right)\cos \left( {\theta_{P2} } \right)} \right]l_{P2} \\ \end{aligned} \)

β 14

\( \begin{aligned} & \left[ { - \left( {\cos \left( \theta \right)\sin \left( \phi \right)\cos \left( {\theta_{P1} } \right) + \left( {\sin \left( \psi \right)\sin \left( \theta \right)\sin \left( \phi \right) + \cos \left( \psi \right)\cos \left( \phi \right)} \right)\sin \left( {\theta_{P1} } \right)} \right)\sin \left( {\theta_{P2} } \right)} \right. \\ & \quad \left. { + \left( { - \cos \left( \theta \right)\sin \left( \phi \right)\sin \left( {\theta_{P1} } \right) + \left( {\sin \left( \psi \right)\sin \left( \theta \right)\sin \left( \phi \right) + \cos \left( \psi \right)\cos \left( \phi \right)} \right)\cos \left( {\theta_{P1} } \right)} \right)\cos \left( {\theta_{P2} } \right)} \right]l_{P2} \\ \end{aligned} \)

β 15

\( \begin{aligned} & \left[ { - \left( {\cos \left( \theta \right)\cos \left( \phi \right)\cos \left( {\theta_{P1} } \right) + \left( {\sin \left( \psi \right)\sin \left( \theta \right)\cos \left( \phi \right) - \cos \left( \psi \right)\sin \left( \phi \right)} \right)\sin \left( {\theta_{P1} } \right)} \right)\sin \left( {\theta_{P2} } \right)} \right. \\ & \quad \left. { + \left( { - \cos \left( \theta \right)\cos \left( \phi \right)\sin \left( {\theta_{P1} } \right) + \left( {\sin \left( \psi \right)\sin \left( \theta \right)\cos \left( \phi \right) - \cos \left( \psi \right)\sin \left( \phi \right)} \right)\cos \left( {\theta_{P1} } \right)} \right)\cos \left( {\theta_{P2} } \right)} \right]l_{P2} \\ \end{aligned} \)

Nomenclature

L :

Half the distance between the hip joints

m b :

Body mass

r b :

Body radius of gyration

k :

Spring stiffness

b :

Damping coefficient

lo:

Free leg length (zero spring force)

l 1 :

Link 1 length

l 2 :

Link 2 length

m 1 :

Mass of link 1

m 2 :

Mass of link 2

θ p :

Inclination of plane w.r.to horizontal

θ :

Body angle w.r.to inclined plane

θ 1 :

Angle of rotation of back leg (link 1)

θ 2 :

Angle of rotation of back leg (link 2)

θ 3 :

Angle of rotation of front leg (link 1)

θ 4 :

Angle of rotation of front leg (link 2)

\( \tau_{1} \) :

Torque applied at back leg (link 1)

\( \tau_{2} \) :

Torque applied at back leg (link 2)

\( \tau_{3} \) :

Torque applied at front leg (link 1)

\( \tau_{4} \) :

Torque applied at front leg (link 2)

\( \theta_{P1} \) :

Angle of rotation of frame {P1} of two DOF planar robot

\( \theta_{P2} \) :

Angle of rotation of frame {P2} of two DOF planar robot

\( \phi , \theta , \psi \) :

Z-Y-X Euler angles

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Gor, M.M., Pathak, P.M., Samantaray, A.K. et al. Development of a compliant legged quadruped robot. Sādhanā 43, 102 (2018). https://doi.org/10.1007/s12046-018-0918-7

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  • DOI: https://doi.org/10.1007/s12046-018-0918-7

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