3.3 Differential Kinematics

Two links is well and good, but what if theres more DoF and more complex robot geometries?

On mobile robots we care about the temporal evolution of robot configurations, because going straight and then turning is not the same as turning and then going straight.

3.3.1 Forward Differential Kinematics

Let the transitional speed of a robot be given by

$v=\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\end{array}\right]$ Let the angular velocity of a robot be given by

$\omega =\left[\begin{array}{c}\dot{{\omega }_x}\\ \dot{{\omega }_y}\\ \dot{{\omega }_z}\end{array}\right]$ We can now write the generalized velocity vector $v={\left[\begin{array}{cc}v& \omega \end{array}\right]}^T=\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\\ \dot{{\omega }_x}\\ \dot{{\omega }_y}\\ \dot{{\omega }_z}\end{array}\right]$ Let the generalized configuration in joint space be $q={\left[\begin{array}{c}{q}_1,...,q_n\end{array}\right]}^T$ Therefore the set of joint speed is $\dot{q}={\left[\begin{array}{c}\dot{q_1},...,\dot{q_n}\end{array}\right]}^T$ a simple derivation of eq 3.1 (find) with respect to time gives

$v={\left[\begin{array}{cc}v& \omega \end{array}\right]}^T=J(q)\cdot \dot{q}$ where $J(q)$ is the Jacobian matrix. It is a function of the joint configuration $q$ and contains all of the partial derivatives of $f$, relating every joint angle to every velocity. Together, this describes how the robot moves with respect to time.

3.3.2 Forward kinematics of a Differential Wheeled Robot

Holonomic definiton:

A system is non-holonomic when closed trajectories in its configuration space may not return it to its original state. A robot arm is holonomic because each joint position corresponds to a unique position in space (Figure 3.4, bottom): a generic joint-space trajectory that comes back to the starting point will position the robot’s end-effector at the exact same position in operational space.

A differential wheeled robot is non-holonomic, which means we need to care about speed and direction with respect to time in order to have a chance at preforming forward kinematics.

Let’s establish $\{I\}$ as a world coordinate system (i.e. the inertial frame). Let $\{R\}$ be a coordinate system on the robot centered on between its two wheels, with the x-axis pointing toward the default driving direction and z upwards.

Let’s define the robots speed as a vector $\text{prescript}R\dot{\xi }=\left[\begin{array}{c}\text{prescript}R\dot{x}\\ \text{prescript}R\dot{y}\\ \text{prescript}R\dot{\theta }\end{array}\right]$ where $\text{prescript}R\dot{x}$ and $\text{prescript}R\dot{y}$ refer to the speed along the $x$ and $y$ directions, and $\text{prescript}R\dot{\theta }$ refers to the rotation velocity around the z-axis.

We need to relate $\text{prescript}R\dot{\xi }$ to $\text{prescript}I\xi =\left[\begin{array}{c}\text{prescript}Ix\\ \text{prescript}Iy\\ \text{prescript}I\theta \end{array}\right]$ , or the $x$, $y,$ and $\theta$ position in the inertial or world frame in order to be of particular use.

One complication. A movement along the x-axis in $\{R\}$ might be a movement along the x-axis and y-axis in $\{I\}$. Here are the components for $x$ $y$ and $\theta$ in $\{I\}$

${\dot{x}}_{I,y}=cos(\theta ){\dot{x}}_R-sin(\theta ){\dot{y}}_R$ ${\dot{y}}_{I,y}=sin(\theta ){\dot{x}}_R+cos(\theta ){\dot{y}}_R$ ${\dot{\theta }}_I={\dot{\theta }}_R$ which is true because the Robot’s z-axis and the world’s z-axis are shared.

So ${\dot{\xi }}_I=\text{prescript}IRT(\theta ){\dot{\xi }}_R$ We’re almost there! Now, how to calculate ${\dot{\xi }}_R$? We can take advantage over the kinematics of wheels to find out. First, ${\dot{y}}_R$ is always zero since we aligned the x-axis with the default forward motion of the robot. Then for $x$ we can find

$\dot{x}=\dot{\varphi }r$ where $\varphi$ is the angle the wheel has rolled by (in radians) and $\dot{\varphi }$ is the rate of change of that angle.

We can also define $\dot{{\varphi }_r}$ and $\dot{{\varphi }_l}$ as the individual speeds of each wheel.

${\dot{x}}_R=\frac{1}{2}(r\dot{{\varphi }_l}+r\dot{{\varphi }_r})$ Now we only need to calculate the rotation of the robot around the z-axis. Given an axle diameter of $d$, we can write

$\dot{{\omega }_r}=\frac{\dot{{\varphi }_r}r}{d}$ where ${\omega }_r$ is the angle of rotation around the left wheel. This equation assumes a “limp” accessory wheel. The equation is the same for the left wheel. Adding in rotation speeds finds

$\dot{\theta }=\frac{\dot{{\varphi }_r}r}{d}-\frac{\dot{{\varphi }_l}r}{d}$ Putting it all together..

$\left[\begin{array}{c}{\dot{x}}_I\\ {\dot{y}}_I\\ {\dot{\theta }}_I\end{array}\right]=\left[\begin{array}{ccc}cos(\theta )& -sin(\theta )& 0\\ sin(\theta )& cos(\theta )& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}\frac{r\dot{{\varphi }_r}}{2}+\frac{r\dot{{\varphi }_l}}{2}\\ 0\\ \frac{r\dot{{\varphi }_r}}{d}-\frac{r\dot{{\varphi }_l}}{d}\end{array}\right]$ And we’ve found ${\dot{\xi }}_I=\text{prescript}IRT(\theta ){\dot{\xi }}_R$ !!!!

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3.3.3 Forward kinematics of Car-like steering

Skipped.