[mobile robot] principle of wheel odometer

Reference material

• Mobile robot wheel odometer

Principle analysis

The robot can deduce the track based on the code disk data and the chassis kinematics model , So we can get the trajectory of the robot .

Here we have Differential drive robot Take an example for analysis .

Its kinematic equation is very simple , as follows ：
$$\begin{array}{rl}{\dot{x}}_r& =v{\cos }_{{\theta }_r}\\ {\dot{y}}_r& =v{\sin }_{{\theta }_r}\\ {\dot{\theta }}_r& ={\omega }_r\end{array}\text{\hspace{1em}(1)}$$

1. Linear velocity derivation

Suppose there is a speed measurement unit for each of the left and right wheels （ Code disk ）, The current rotation speeds of the left and right wheels of the trolley can be obtained in real time through the rotation speed measurement unit $a_l,a_r$, Unit is ( circle /s). Suppose the radius of the wheel is $r$, The perimeter of $S$, Further, the speed of the left and right wheels is ：
$$\begin{gathered} v_l=a_{l}S \\ {v}_r=a_{r}S\text{\hspace{1em}(2)} \end{gathered}$$

We take the speed at the center of the left and right wheel axles as the vehicle speed , It can be calculated as
$$v=\frac{v_l+v_r}{2}\text{\hspace{1em}(3)}$$

2. Angular velocity derivation （ Direction of travel ）

For mobile robots , Assume that the initial starting position of the robot is known , In an environment where the driving dimension is one dimension （ for example ： One way highway 、 One way guideway ）, The robot can estimate where it is in this one-dimensional driving environment in real time according to the number of turns of the wheel .

In the case of a two-dimensional plane , The number of revolutions of the left and right wheels can be deduced through the speed measurement unit of the left and right wheels , obviously , When the number of turns of the left and right wheels is different , The mobile robot will turn left or right . for example , When the speed of the left wheel is lower than that of the right wheel , The left wheel turns less than the right wheel , Therefore, it can be inferred that the robot turns left at this time .

Suppose the mobile robot travels a certain distance at this time , Pictured above , The distance traveled by the left wheel is $s_l$, The distance traveled by the right wheel is $s_r$, Wheel spacing is $L$. By the geometric relation of the arc $s=\theta R$ You know
$$\{\begin{array}{c}{s}_l={\theta }_{r}R\\ s_r={\theta }_r(R+L)\end{array}\text{\hspace{1em}(4)}$$
Take the derivative of time on both sides of the equation , Further :
$$\{\begin{array}{c}{v}_l={\omega }_{r}R\\ v_r={\omega }_r(R+L)\end{array}\text{\hspace{1em}(5)}$$
The angular velocity of the mobile robot is
$$\omega =\frac{v_r-v_l}{L}\text{\hspace{1em}(6)}$$

Sum up , Through the formula (3) And the formula (6), After knowing the linear velocity and angular velocity , Then, according to the formula (1) Get its position and posture on the two-dimensional plane （ Yaw angle ）.

This is the general principle of the wheel odometer .