Two-Wheel Mobile Robot

A 2-wheeled robot typically has the following form:

• Center of wheels $P$

The forward velocity $v$ is tangent to the path:

$$\begin{array}{rl}\dot{x}& =v\cos \theta \\ \dot{y}& =v\sin \theta \\ \dot{\theta }& =\omega \end{array}$$

Or:

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{\theta }\end{array}\right]=\left[\begin{array}{cc}\cos \theta & 0\\ \sin \theta & 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}v\\ \omega \end{array}\right]$$

• Usually, $(v,\omega )$ are treated as “inputs” and $(x,y,\theta )$ are treated as “state variables”
• The above constraint is an example of so non-holonomic constraints

Kinematics for wheel control

What are the wheel speeds $(u_r,u_l)$ to move the robot’s center point with $v$ and $\omega$?

$$\begin{array}{rl}R& =\frac{v}{\omega },\\ v_r^w& =\omega \left(R+\frac{T}{2}\right)=v+\frac{T\omega }{2}\\ v_{\ell }^w& =\omega \left(R-\frac{T}{2}\right)=v-\frac{T\omega }{2}\end{array}$$

• $R$ is the instantaneous center of velocity
• $T$ is the track width
• $v_r^w,v_l^w$ are the tangential linear speeds at the wheel rims (ground speed at contact points with no slip)
• In a left turn ($\omega >0$), the right wheel traces a large arc and must spin faster and vice versa

Converting to wheel angular speeds:

$$\begin{array}{r}{u}_r=\frac{1}{r}{v}_r^w=\frac{1}{r}\left(v+\frac{T\omega }{2}\right)\\ u_{\ell }=\frac{1}{r}{v}_{\ell }^w=\frac{1}{r}\left(v-\frac{T\omega }{2}\right)\end{array}$$

• $r$ is the wheel radius

Working backward, we can then find $v$ and $\omega$ based on our wheel speeds:

$$\begin{array}{r}v=r\frac{u_r+u_{\ell }}{2}\\ \omega =r\frac{u_r-u_{\ell }}{T}\end{array}$$

and in turn:

$$\begin{array}{rl}\dot{x}& =r\frac{u_r+u_l}{2}\cos \theta \\ \dot{y}& =r\frac{u_r+u_l}{2}\sin \theta \\ \dot{\theta }& =r\frac{u_r-u_{\ell }}{T}\end{array}$$

Example