Direct Kinematics of Wheeled Platforms

# Kinematics of Wheeled Platforms

The study of motion in wheeled robotic platforms. An analysis of the position, velocity, and acceleration of the platform and how it relates to the wheel setups. There are two major approaches to deriving the direct kinematics, an overview approach and a wheel by wheel approach.

## Methods of Modeling Direct Kinematics

When deriving the direct (also referred to as forward) kinematics there are two major approaches. They two approaches are the frame of reference (or overview) approach. The second is the wheel type approach. Both approaches are commonly used, however, most books on robot kinematics use a wheel by wheel approach when setting up the direct kinematic equations for a wheeled platform.

An important note that affects both approaches is that only fixed wheels and steerable wheels have an impact on the robot chassis kinematics. Therefore only these two types of wheels require consideration when computing the robot's kinematic constraints.

### Frame of Reference Approach

In simple cases $\dot \xi_{R} = R(\frac{\pi}{2})\dot \xi_{I}$ is sufficient to capture the direct kinematics of a mobile robot. It is important to note that:

• subscript $I$ refers to the global reference frame
• subscript $R$ refers to the robot's frame
• $\xi_{I} = \left[ \begin{array}{c} x \\ y \\ \theta \end{array} \right]$

Next we look at mapping and realize that it is a function of the robot's pose in its current state and can be represented by the following orthogonal matrix:

(1)
\begin{align} R(\theta) = \left[ \begin{array}{ccc} cos\theta & sin\theta & 0\\ -sin\theta & cos\theta & 0\\ 0 & 0 & 1 \end{array} \right] \end{align}

This orthogonal matrix can then be used to map the motion in reference to the global frame. Then $\{X_{I}, Y_{I}\}$ can be mapped in terms of the robots frame (local frame) $\{X_{R},Y_{R}\}$. This mapping operation can be denoted by $R(\theta)\dot\xi_{I}$. [1][2]

### Wheel Type Approach

It can be said that a robot has $N$ standard wheels where $N = N_{f} + N_{s}$. Where $N_{f}$ refers to fixed wheels and $N_{s}$ refers to steerable wheels. You can then denote the variable steering angles of all the $N_{s}$ wheels with $\beta _{s}(t)$. Another useful notation is the orientation of the fixed wheels, $N_{f}$, which is represented by $\beta _{f}$.

Both types of wheels, $N$ wheels, are subject to wheel spin. This means that the rotational position is a function of time. These positions are contained in an aggregate matrix, $\theta (t)$. The matrix is of the form:

(2)
\begin{align} \theta (t) = \left[ \begin{array}{c} \theta _{f}(t) \\ \theta _{s}(t) \end{array} \right] \end{align}

This matrix contains the positions of the fixed and steered wheels, $\theta _{f}(t)$ and $\theta _{s}(t)$ respectively.

Then you can combine all the rolling constraints into a single equation.

(3)
\begin{align} J_{1}(\beta _{s})R(\theta)\dot\xi _{I} + J_{2}\dot\theta = 0 \end{align}

It should be noted that this equation follows the constraint that all standard wheels must spin around their horizontal axis so that rolling occurs at the ground point.

## References

Bibliography
1.
2.
3. Canfield, Stephen. Lecture notes: “Mobile Robot Kinematics -A Primer- * So Easy an ECE Could Do It*”
4. Spong, et.al. Robot Modeling and Control.
5. Mandow, et.al."Experimental kinematics for wheeled skid-steer mobile robots"