Differential wheeled robot

Kinematics of Differential Drive Robots

The illustration on the right shows the differential drive kinematics of a mobile wheeled robot. The variables are expressed using the following notation: $X$ and $Y$ are the global coordinate system. Using the point midway between the wheels as the origin of the robot, one can define $X_B$ and $Y_B$ as the locale body coordinate system. The orientation of the robot with respect to the global coordinate system is the angle $\phi$. The radius of the wheels is $r$ and the width of the vehicle $b$. 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 $ICR$⁣ - instantaneous center of rotation). The ground contact speed of the left wheel $v_L$ and the right wheel $v_R$ lead to a rotation of the vehicle by the angular velocity $\omega$. Following the definition of angular velocity, one obtains:$${\displaystyle \begin{array}{cc}\omega \cdot (R+b/2)& =v_R\\ \omega \cdot (R-b/2)& =v_L\end{array}}$$Solving these two equations for $\omega$ and $R$, while the latter is defined as the distance from $ICR$ to the center of the robot$${\displaystyle \begin{array}{cc}\omega & =(v_R-v_L)/b\\ R& =b/2\cdot (v_R+v_L)/(v_R-v_L)\end{array}}$$ Using the equation for the angular velocity, the instantaneous velocity ${\displaystyle V}$ of the point midway between the robot's wheels is given by$${\displaystyle V=\omega \cdot R=\frac{v_R+v_L}{2}}$$The wheel tangential velocities can also be written as$${\displaystyle \begin{array}{cc}{v}_R& =r\cdot {\omega }_R\\ v_L& =r\cdot {\omega }_L\end{array}}$$ where ${\displaystyle {\omega }_R}$ and ${\displaystyle {\omega }_L}$ 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$${\displaystyle \left[\begin{array}{c}{\dot{x}}_B\\ {\dot{y}}_B\\ \dot{\phi }\end{array}\right]=\left[\begin{array}{c}v{x}_B\\ v{y}_B\\ \omega \end{array}\right]\stackrel{v=r\omega }{\overbrace{=}}\left[\begin{array}{cc}\frac{r}{2}& \frac{r}{2}\\ 0& 0\\ -\frac{r}{b}& \frac{r}{b}\end{array}\right]\left[\begin{array}{c}{\omega }_L\\ {\omega }_R\end{array}\right]}$$Using a coordinate transformation (Rotation of axes), the robot's kinematic model in global coordinates can finally be obtained$${\displaystyle \left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\phi }\end{array}\right]=\left[\begin{array}{cc}\cos \phi & 0\\ \sin \phi & 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}V\\ \omega \end{array}\right]}$$ where ${\displaystyle V}$ and ${\displaystyle \omega }$ are the control variables.

Differential Drive Controller

One might face a situation where the velocity $V$ and the angular velocity $\omega$ are given as inputs, and the angular velocities of the left ${\omega }_L$and right wheels ${\omega }_R$ are sought as control variables (see figure above). In this case, the already mentioned equation can be easily reformulated. Using the relations $R=V/\omega$ and ${\omega }_R=v_R/r$ in

$${\displaystyle \omega \cdot (R+b/2)=v_R}$$ one obtains the equation for the angular velocity of the right wheel ${\omega }_R$

$${\displaystyle {\omega }_R=\frac{V+\omega \cdot b/2}{r}}$$

The same procedure can be applied to the calculation of the angular velocity of the left wheel ${\omega }_L$

$${\displaystyle {\omega }_L=\frac{V-\omega \cdot b/2}{r}}$$

Examples of differential wheeled robots

• Alice
• Alvik
• Armadillo, Robotican
• Big Trak
• Edison
• Giraff
• Khepera and e-puck
• Magni
• Pioneer, PeopleBot, PowerBot & PatrolBot- the ActivMedia Robotics mobile robots.
• Roomba
• RoboMow
• Sbot
• SCUTTLE An open-source payload-capable robot for hobby and university research.
• Shonkbot, a minimal-cost, Arduino-based teaching robot
• Thymio