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A spatial single loop kinematotropic mechanism used for biped/wheeled switchable robots

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Abstract

In this paper, we propose a novel biped walking robot on the basis of a spatial eight-bar mechanism. The kinematotropic characteristic of the mechanism is investigated and the geometric constrains deriving the kinematotropy are specified. The mechanism possesses two motion branches with different degrees of freedom. A detailed kinematic analysis is presented for both motion branches. A biped/wheeled switchable robot is further designed. In the motion branch I, the mechanism is used as a biped walking mechanism. In the motion branch II, it is utilized as a wheeled robot and completes the switch process between two modes. At last, we demonstrate the gait motion of the robot in the biped mode and the switch process.

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Abbreviations

d :

Degree of freedom of the motion space

f i :

Mobility of the ith joints

F :

Degree of freedom of a mechanism

g :

Number of joints

h i :

The number of constraint screws of the ith limb

k :

The dimension of the wrench system excluding common constraints

l i :

Direction cosine in the x-axis

L i :

Length of link i

m i :

Direction cosine in the y-axis

n :

Number of links including the base

n i :

Direction cosine in the z-axis

p :

The limb number of a parallel mechanism

p i :

Moment in the x-axis

q i :

Moment in the y-axis

r i :

Moment in the z-axis

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r}_{i} \) :

Position vector

R i :

Axis of revolute joint i

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {r}_{i} \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {s}_{i} \) :

Moment of the screw axis

\( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {s}_{i} \) :

Unit vector

s :

Equation parameter

$ i :

Unit screw

\( \hat{S} \) :

Twist system of mechanism

$r :

Common constraint screw

$ i Bj :

Twist i in motion branch j

t :

Equation parameter

f2 f1 T :

Transfer matrix transferring from frame f1 to frame f2

v :

The number of redundant constraints

u,v :

Unit vector

θ i :

Angle of joint i

θ ij :

Sum of joint angle θ i and joint angle θ j , θ ij  = θ i  + θ j

θ ijk :

Sum of joint angle θ i , joint angle θ j and joint angle θ k , θ ijk  = θ i  + θ j  + θ k

ρ i :

Position variable

λ :

Number of independent common constraints

φ :

Twist matrix

σ i :

Position variable

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Acknowledgments

This work was supported by the National Nature Science Foundations of China [Grant Number 51175029].

Author information

Authors and Affiliations

  1. Robotics Research Lab, School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Naiyue Wang, Yuefa Fang & Dan Zhang

  2. Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, Oshawa, ON, L1H 7K4, Canada

    Dan Zhang

Authors
  1. Naiyue Wang
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  2. Yuefa Fang
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  3. Dan Zhang
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Corresponding author

Correspondence to Yuefa Fang.

Appendix

Appendix

$$ \rho_{1} = - \frac{{\delta z_{A} z_{C} (x_{B} z_{A} - x_{A} z_{B} )}}{{ - x_{C} y_{A} (z_{A} + z_{B} ) - \delta z_{A} z_{C} (x_{A} + x_{B} )}} $$
$$ \sigma_{1} = - \frac{{z_{A} x_{C} (x_{B} z_{A} - x_{A} z_{B} )}}{{x_{C} y_{A} (z_{A} - z_{B} ) + \delta z_{A} z_{C} (x_{A} - x_{B} )}} $$
$$ \rho_{2} = \frac{{y_{A} z_{C} (z_{A} - z_{B} ) + \delta z_{A} z_{C} (z_{A} - z_{B} )}}{{ - x_{C} y_{A} (z_{A} + z_{B} ) - \delta z_{A} z_{C} (x_{A} + x_{B} )}} $$
$$ \sigma_{2} = \frac{{z_{A} ( - x_{C} z_{A} + x_{C} z_{B} + x_{A} z_{C} - x_{B} z_{C} )}}{{x_{C} y_{A} (z_{A} - z_{B} ) + \delta z_{A} z_{C} (x_{A} - x_{B} )}} $$
$$ \delta = \frac{{ - n_{7} x_{C} y_{B} + n_{7} x_{C} y_{C} + l_{7} y_{B} z_{C} - l_{7} y_{C} z_{C} }}{{z_{C} (n_{7} x_{B} - n_{7} x_{C} - l_{7} z_{B} + l_{7} z_{C} )}} $$
$$ \rho '_{1} = - z_{A} z_{C} (y_{B} - y_{C} )(x_{B} z_{A} - x_{A} z_{B} ) $$
$$ \sigma '_{1} = - \eta x_{C} z_{A} (x_{B} z_{A} - x_{A} z_{B} )(z_{B} - z_{C} ) $$
$$ \rho '_{2} = - \eta y_{A} z_{C} (z_{A} - z_{B} )(y_{C} z_{B} - y_{B} z_{C} ) $$
$$ \sigma '_{2} = \eta z_{A} z_{C} (x_{A} - x_{B} )( - y_{C} z_{B} + y_{B} z_{C} ) $$
$$ \rho '_{3} = - \frac{{z_{C} }}{{x_{C} }} + \frac{{\eta z_{A} z_{C} }}{{x_{C} }}(y_{B} - y_{C} )[x_{C} (z_{A} - z_{B} ) + z_{C} ( - x_{A} + x_{B} )] $$
$$ \sigma '_{3} = - \eta z_{A} (z_{B} - z_{C} )[x_{C} (z_{A} - z_{B} ) + z_{C} ( - x_{A} + x_{B} )] $$
$$ \eta = \left[ {x_{C} y_{A} (z_{A} - z_{B} )(z_{B} - z_{C} ) - z_{A} z_{C} (x_{A} - x_{B} )(y_{B} - y_{C} )} \right]^{ - 1} $$
$$ q_{3} = 2Lc\theta_{2} \;r_{3} = 2Ls\theta_{2} \;p_{4} = - 2Lc\theta_{2} \;r_{4} = 2L $$
$$ p_{5} = L[ - 2c\theta_{2} + c(\theta_{23} - \theta_{4} ) + c\theta_{234} - 2s\theta_{23} ] $$
$$ r_{5} = 2L - 2Ls\theta_{4} \;p_{8} = - 2Lc\theta_{1} \;r_{8} = 2Ls\theta_{1} $$
$$ l_{7} = Lc(\theta_{1} - \theta_{8} )\;n_{7} = - Ls(\theta_{1} - \theta_{8} )\;p_{7} = - 2L^{2} s(\theta_{1} - \theta_{8} ) $$
$$ q_{7} = 2L^{2} c\theta_{8} \;r_{7} = - 2L^{2} c(\theta_{1} - \theta_{8} ) $$
$$ p_{6} = - 2L^{2} (c\theta_{23} - s\theta_{2} + c\theta_{4} s\theta_{23} )s(\theta_{1} - \theta_{8} ) $$
$$ \begin{gathered} q_{6} = 2L^{2} [c\theta_{3} c(\theta_{1} - \theta_{8} )s\theta_{2} \hfill \\ \;\;\;\;\;\; - c\theta_{2} c(\theta_{1} - \theta_{8} )( - 1 + c\theta_{3} c\theta_{4} - s\theta_{3} ) \hfill \\ \;\;\;\;\; + c\theta_{4} c(\theta_{1} - \theta_{8} )s\theta_{2} s\theta_{3} + s(\theta_{1} - \theta_{8} ) - s\theta_{4} s(\theta_{1} - \theta_{8} )] \hfill \\ \end{gathered} $$
$$ r_{6} = - 2L^{2} c(\theta_{1} - \theta_{8} )(c\theta_{23} - s\theta_{2} + c\theta_{4} s\theta_{23} ) $$
$$ q_{3}^{\prime} = 2Lc\theta_{2} \;r_{3}^{\prime} = 2Ls\theta_{2} \;p_{4}^{\prime} = - 2Lc\theta_{2} \;r_{4}^{\prime} = 2L $$
$$ p_{5}^{,} = L( - 2c\theta_{2} + c(\theta_{23} - \theta_{4} ) + c\theta_{234} - 2s\theta_{23} ) $$
$$ r_{5}^{\prime} = 2L - 2Ls\theta_{4} $$
$$ p_{8}^{\prime} = - 2Lc\theta_{1} \;r_{8}^{\prime} = 2Ls\theta_{1} \;q_{7}^{,} = 2Lc\theta_{1} \;r_{7}^{\prime} = - 2L $$
$$ q_{6}^{\prime} = - L( - 2c\theta_{2} + c(\theta_{23} - \theta_{4} ) + c\theta_{234} - 2s\theta_{23} ) $$
$$ r_{6}^{\prime} = - 2L(c\theta_{23} - s\theta_{2} + c\theta_{4} s\theta_{23} ) $$

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Wang, N., Fang, Y. & Zhang, D. A spatial single loop kinematotropic mechanism used for biped/wheeled switchable robots. Int J Mech Mater Des 11, 287–299 (2015). https://doi.org/10.1007/s10999-014-9274-x

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