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What Is The Last Step To Equalize The Linear Velocity Vector Of A Standard Wheel

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The last step is to equalize the linear velocity vector of the standard wheel to that of equation 3. ( ■(s(α+β)&-c(α+β)&-cβd@c(α+β)&s(α+β)&sβd@0&0&0) )( ■(x ̇@y ̇@θ ̇ ) )=( ■(rψ ̇@0@0) ) (4)
In case of the steerable standard wheel, equation 4 can be used in the same way, if the fixed angle β is replaced by a function β(t). This equation could also be applied to the spherical wheel (because of the forces which affect the wheel and change β(t), only a linear velocity in the rolling direction exists). If the wheel is a castor wheel, the y-component of the velocity vector is depending on the angular velocity β ̇ and the length of the rod (see equation 5). ( ■(s(α+β)&-c(α+β)&-cβd@c(α+β)&s(α+β)&sβd@0&0&0) )( ■(x ̇@y ̇@θ ̇ ) )=( ■(rψ ̇@-d_c β …show more content…

The vector q ⃗ describes the configuration (position and orientation) of the robot at any time. q ⃗=[■(x&y&ϕ)]^T (7) where x, y are the coordinates of the robot (point o) in the inertial frame. If the linear speed and angular velocity of the robot are v and ω, respectively, assuming no-slip on the wheels, the velocity components can be written as x ̇=v cos⁡ϕ (8) y ̇=v sin⁡ϕ (9) ϕ ̇=ω (10)
The kinematics of motion of the robot can be written as follows: q ⃗ ̇= [■(cos⁡ϕ&0@sin⁡ϕ&0@0&1)] u ⃗ (11)
Where, q ̇ is the time derivative of configuration q and u ⃗=[■(v&w)]^T is the control input given to the robot for navigation. The angular and linear velocity of the robot can be written in terms of the linear velocities of the right and the left wheel centers (v_r,v_l) ω=(v_r-v_l)/2d (12) v=(v_r+v_l)/2 (13)
Here for the right wheel, the Newton-Euler equation is expressed as F_r - f_r = m_w x ̈_r (14) τ_r - F_r .r = J_w ω ̇_r (15) where F_r is the reaction force applied to the right wheel by the rest of the robot; f_r is the friction force between the right wheel and the ground; m_w is the mass of the wheel; τ_r is the torque acting on the right wheel provided by the right motor; r is the radius of the wheel; and J_w is the inertia of the wheel. Note that the Coriolis part had been deleted since it is negligible due to the fact that the wheel inertia is much smaller than the robot inertia.
Similarly, for the left wheel we can write F_l - f_l = m_w

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