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Odemetry - Notes

Table Of Contents

What is Odometry ?

Estimates the robot position relative to a starting location.

Odometry of a robot - the distance traveled by each of the wheels is identical, but the final positions are different.

How it works ?

  graph LR
    A[Input]-->B[Odometry Model]
    B-->C[Output]
  graph LR
    A[Initial position/location]-->B[Integration of <br>velocity measurements]
    B-->C[Computing of <br> new position/location]

This method is sensitive to errors due to the integration of velocity measurements over time. But we can use dynamic control model to reduce the errors (Ex : inner-loop technique etc) 

  where : 
    - Initial position/location : O(x_0, y_0, theta_0), center coordinates
    - New position/location : O(x, y, theta), position and orientation

Kinematic Model

  • Allows describe the movement of the robot
  • calculate the global movement of the robot from the odometric measurements
  • the robot movement is controlled by the speed/velocity differential btw two drive wheels.

Where :

  • $d_{l}$ , $d_{r}$ : the respective movements of the left and right wheels
  • $v_{l}, v_{r}$ : the respective speeds of the left and right wheels
  • $x, y, \theta$ : the coordinates of the robot (position and orientation)
  • $d$ : the movement of the robot
  • $v$ : the speed of the robot
  • $e$ : the gap between the two wheels;
  • ${\displaystyle x_{0}}, {\displaystyle y_{0}}$ and $R$ : center coordinates $O$ of the trajectory circle and its radius.

Direct Model

If we assume that the trajectory of the robot is a circle of radius $R$ traveled at angular rate $\omega$ :

$\omega = \frac {\mathrm {d} \theta }{\mathrm {d} t}$ ($R > 0$, if the circle is traversed counterclockwise)

Notice : $v = R\omega$ , then we have:

$\displaystyle v=R{\frac {\mathrm {d} \theta }{\mathrm {d} t}}$

then wheel speeds :

$$ \begin{matrix} v_{l}&=&\displaystyle (R-{\frac {e}{2}})\;{\frac {\mathrm {d} \theta } {\mathrm {d} t}}&=&\displaystyle (R-{\frac {e}{2}}){\frac {v}{R}} \ v_{r}&=&\displaystyle ( R+{\frac {e}{2}})\;{\frac {\mathrm {d} \theta }{\mathrm {d} t}}&=&\displaystyle (R+{\frac {e}{2 }}){\frac {v}{R}} \end{matrix} $$

Inverse Model

  • The inversion of the previous system gives:

$$ \begin{matrix} v&=&\displaystyle {\frac {v_{l}+v_{r}}{2}} \ R&=&\displaystyle {\frac {e}{2}} \,{\frac {v_{r}+v_{l}}{v_{r}-v_{l}}} \end{matrix} $$

Calculation of Odometry

$$ \displaystyle \mathrm {d} \theta={\frac {d}{R}} $$

$$ \begin{matrix} x_{O}&=&x-R\,\cos(\theta -{\frac {\pi }{2}}) \ y_{O}&= &y-R\,\sin(\theta -{\frac {\pi }{2}}) \end{matrix} $$

Update the position of the robot :

$$ \begin{matrix} \theta & \leftarrow &\theta +\mathrm {d} \theta \ x&\leftarrow &x_{O}+R\,\cos(\theta -{\ frac {\pi }{2}}) \ y&\leftarrow &y_{O}+R\,\sin(\theta -{\frac {\pi }{2}}) \end{matrix} $$

Applications

  • Robotics
  • Self-Driving Cars
  • Trains
  • ...

Example of Model Implementation in Matlab - Control Law

fprintf('@TODO\n');

Example of Model Implementation in C language

  • Position

printf("the new position is : x=%d m and y=%d m\n", x, y);
- Orientation 
printf("the new orientation is : Theta=%d degree\n", theta);
@TODO

Example Machine Learning Modeling of Kinematic Model

@TODO

Tools & frameworks

References

en : - Odometry : - https://en.wikipedia.org/wiki/Odometry - Differential wheeled robot : - https://en.wikipedia.org/wiki/Differential_wheeled_robot - Robo-Rats Locomotion Page: - https://groups.csail.mit.edu/drl/courses/cs54-2001s/locomotion.html

fr - wiki : - https://fr.wikipedia.org/wiki/Odom%C3%A9trie - https://fr.wikipedia.org/wiki/Portail:Robotique -

  • Positionnement du robot (OLIVIER COCHELIN (COCO) :
  • http://manubatbat.free.fr/doc/positionning/Positionning.html
  • http://manubatbat.free.fr/doc/positionning/node5.html

Additional reading and tools

  • https://en.wikipedia.org/wiki/Classical_mechanics
  • https://en.wikipedia.org/wiki/Kinematics
  • https://en.wikipedia.org/wiki/Rotation_around_a_fixed_axis
  • https://en.wikipedia.org/wiki/Coordinate_system
  • https://en.wikipedia.org/wiki/Control_engineering
  • https://en.wikipedia.org/wiki/Dynamical_system
  • https://en.wikipedia.org/wiki/Algebra
  • https://en.wikipedia.org/wiki/Mathematical_analysis
  • https://en.wikipedia.org/wiki/Trigonometry

YT - Configuration and Velocity Constraints - Holonomic vs Nonholonomic