Editing Differential wheeled robot (section)

== Kinematics of Differential Drive Robots ==
[[File:Differential Drive Kinematics of a Wheeled Mobile Robot.svg|thumb|300x300px|Differential Drive Kinematics]]
The illustration on the right shows the differential drive kinematics of a mobile wheeled robot. The variables are expressed using the following notation: <math display="inline">X</math> and <math display="inline">Y</math> are the global coordinate system. Using the point midway between the wheels as the origin of the robot, one can define <math display="inline">X_{B}</math> and <math display="inline">Y_{B}</math> as the locale body coordinate system. The orientation of the robot with respect to the global coordinate system is the angle <math display="inline">\varphi</math>. The radius of the wheels is <math display="inline">r</math> and the width of the vehicle <math display="inline">b</math>. Assuming that the wheels are at any time in contact with the ground (there is no slip), the wheels describe arcs in the plane in such a way that the vehicle always rotates around a point (referred to as <math display="inline">ICR</math>⁣ - [[Instant centre of rotation|instantaneous center of rotation]]). The ground contact speed of the left wheel <math display="inline">v_{L}</math> and the right wheel <math display="inline">v_{R}</math> lead to a rotation of the vehicle by the angular velocity <math display="inline">\omega</math>. Following the definition of [[angular velocity]], one obtains:<math display="block">\begin{align} \omega \cdot (R+b/2) &= v_{R}  \\ \omega \cdot (R-b/2) &= v_{L} \\ \end{align}</math>Solving these two equations for <math display="inline">\omega</math> and <math display="inline">R</math>, while the latter is defined as the distance from <math display="inline">ICR</math> to the center of the robot<math display="block">\begin{align} \omega &= (v_{R}-v_{L})/b   \\ R &= b/2 \cdot (v_{R}+v_{L}) / (v_{R}-v_{L}) \\ \end{align}</math> Using the equation for the [[angular velocity]], the instantaneous velocity <math>V</math> of the point midway between the robot's wheels is given by<math display="block">V = \omega \cdot R = \frac{v_{R}+v_{L}}{2}</math>The wheel tangential velocities can also be written as<math display="block">\begin{align}v_{R} & = r \cdot \omega_{R} \\ v_{L} & = r \cdot \omega_{L}  \end{align}</math> where <math>\omega_{R}</math> and <math>\omega_{L}</math> are the left and the right angular velocities of the wheels around their axes. The robot kinematics in local body coordinates can thus be written as<math display="block">\begin{bmatrix}
 \dot{x}_{B}\\
 \dot{y}_{B}\\
\dot{\varphi}
\end{bmatrix}
= 

\begin{bmatrix}
v\, {x}_{B} \\
v\, {y}_{B} \\
\omega
\end{bmatrix}
\overbrace{=}^{v=r \omega}

\begin{bmatrix}
\frac{r}{2} & \frac{r}{2}\\
0 &0\\
-\frac{r}{b}        & \frac{r}{b}  
\end{bmatrix}
\begin{bmatrix}
\omega_{L}\\
\omega_{R}
\end{bmatrix}</math>Using a coordinate transformation ([[Rotation of axes]]), the robot's kinematic model in global coordinates can finally be obtained<math display="block">\begin{bmatrix}
 \dot{x}\\
 \dot{y}\\
\dot{\varphi}
\end{bmatrix}
= 

\begin{bmatrix}
\cos{\varphi} & 0\\
\sin{\varphi} &0\\
0        & 1  
\end{bmatrix}
\begin{bmatrix}
V\\
\omega
\end{bmatrix}</math> where <math>V</math> and <math>\omega</math> are the control variables.<ref name=":0" /><ref>{{Cite book |url=https://www.worldcat.org/oclc/272306791 |title=Springer handbook of robotics |date=2008 |publisher=Springer |others=Bruno Siciliano, Oussama Khatib |isbn=978-3-540-30301-5 |location=Berlin |oclc=272306791}}</ref>

=== Differential Drive Controller ===
One might face a situation where the velocity <math display="inline">V</math> and the angular velocity <math display="inline">\omega</math> are given as inputs, and the angular velocities of the left <math display="inline">\omega_{L}</math>and right wheels <math display="inline">\omega_{R}</math> are sought as control variables (see figure above). In this case, the already mentioned equation can be easily reformulated. Using the relations <math display="inline">R = V/\omega</math> and <math display="inline">\omega_{R} = v_{R}/r</math> in

<math display="block">\omega \cdot (R+b/2) = v_{R}</math> one obtains the equation for the angular velocity of the right wheel <math display="inline">\omega_{R}</math>

<math display="block">\omega_{R} =   \frac{V+\omega \cdot b/2}{r}</math>

The same procedure can be applied to the calculation of the angular velocity of the left wheel <math display="inline">\omega_{L}</math>

<math display="block">\omega_{L} = \frac{V-\omega \cdot b/2}{r}</math>