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Dynamics equations of a mobile robot provided with caster wheel

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Abstract

Kinematics and dynamics of a mobile robot, consisting of a platform, two conventional wheels and a crank that controls the motion of a free rolling caster wheel, are analyzed in the paper. Based on several matrix relations of connectivity, the characteristic velocities and accelerations of this non-holonomic mechanical system are derived. Using the principle of virtual work, expressions and graphs for the torques and the powers of the two driving wheels are established. It has been verified the results in the framework of the second-order Lagrange equations with their multipliers. The study of the dynamics problems of the wheeled mobile robots is done mainly to solve successfully the control of the motion of such systems.

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Abbreviations

Ox 0 y 0 z 0 :

inertial reference frame with origin in the ground surface

a k,k−1 :

orthogonal transformation matrix

\(\vec{u}_{1},\vec{u}_{2},\vec{u}_{3}\) :

three right-handed orthogonal unit vectors

θ 1,θ 2 :

rotation angles of two driving wheels

θ 3 :

rotation angle of the caster wheel

ψ :

rotation angle of the crank PO 3

l :

distance between the wheel centers

a+b :

height of the triangular platform

r :

radius of each driving wheel

r 0 :

radius of the caster wheel

\(\vec{r}_{21}^{A},\vec{r}_{21}^{B},\vec{r}_{32}^{C}\) :

relative position vectors of wheel centers

θ,x 10,y 10,H :

orientation angle and center coordinates of the moving platform

\(\vec{\omega}_{k,k-1}\) :

relative angular velocity of T k rigid body

\(\vec{\omega}_{k0}\) :

absolute angular velocity of T k

\(\tilde{\omega}_{k,k-1}\) :

skew-symmetric matrix associated with the angular velocity \(\vec{\omega}_{k,k-1}\)

\(\vec{\varepsilon}_{k,k-1}\) :

relative angular acceleration of T k

\(\vec{\varepsilon}_{k0}\) :

absolute angular acceleration of T k

\(\tilde{\varepsilon}_{k,k-1}\) :

skew-symmetric matrix associated with the angular acceleration \(\vec{\varepsilon}_{k,k-1}\)

\(\vec{r}_{k}^{C}\) :

position vector of the mass center of T k

\(m_{2}^{A},m_{2}^{B},\hat{J}_{2}^{A},\hat{J}_{2}^{B}\) :

mass and symmetric matrix of tensor of inertia of each driving wheel

\(m_{2}^{C},\hat{J}_{2}^{C}\) :

mass and tensor of inertia of the crank

\(m_{3}^{C},\hat{J}_{3}^{C}\) :

mass and tensor of inertia of the caster wheel

M 1,M 2 :

torques applied by two electric motors to the wheels jointed at A 2,B 2

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  1. Department of Mechanics, University “Politehnica” of Bucharest, 313 Splaiul Independentei, Bucharest, Romania

    Stefan Staicu

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Staicu, S. Dynamics equations of a mobile robot provided with caster wheel. Nonlinear Dyn 58, 237–248 (2009). https://doi.org/10.1007/s11071-009-9474-3

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