ZED tracking coordinate-based kart control

[arya1106]

Use the information provided by #6 and a series of target points to plan and follow a given path. Output control values of:

• Braking
• Steering
• Throttle

[arya1106]

Discord Channel

[arya1106]

Planning to use LQR implementation with forward kinematics from here. Another example implementation. Simulation code here.

[arya1106]

Symbol definitions:

$x$ – x-coordinate relative to world frame
$\dot{x}$ – x-coordinate change
$y$ – y-coordinate relative to world frame
$\dot{y}$ – y-coordinate change
$\theta$ – yaw in degrees relative to world frae
$\dot{\theta}$ – change in yaw
$\phi$ – current steering angle of car
$\dot{\phi}$ – current steering angle of car
$t_0$ – current time
$t_1$ – time we are trying to predict
$dt$ – discrete time step
$\beta$ – Brake (0-255)
$\tau$ – Throttle (0-255)
$\pi$ – Steering Angle (0-255)
$u_1$ - velocity of kart
$u_2$ - turning rate of kart

What we get from the kart:

$x, y, \theta$ – Coordinates from #6
$\phi$ - Control memory

Forward kinematics

State vector

$$\begin{bmatrix} x \\\ y \\\ \phi \\\ \theta \end{bmatrix}$$

Inputs:

• $u_1$
• $u_2$

Software (us) Controls (inputs):

• $\beta$
• $\tau$
• $\pi$

Conversions:

$u_1$ = $c_1\beta + c_2\tau$
$u_2$ = $\frac{\pi - \phi}{dt}$

$c_1$ and $c_2$ to be found experimentally.

Differential State Matrix

$$\begin{bmatrix} \dot{x} \\\ \dot{y} \\\ \dot{\phi} \\\ \dot{\theta} \end{bmatrix} = \begin{bmatrix} u_1\cos\theta \\\ u_1\sin\theta \\\ u_2 \\\ u_1\cdot\frac{1}{L}\cdot\tan\phi \end{bmatrix}$$

State prediction

$$\begin{bmatrix} x_t \\\ y_t \\\ \phi_t \\\ \theta_t \end{bmatrix} + \begin{bmatrix} \dot{x} \\\ \dot{y} \\\ \dot{\phi} \\\ \dot{\theta} \end{bmatrix} = \begin{bmatrix} x_{t+1} \\\ y_{t+1} \\\ \phi_{t+1} \\\ \theta_{t+1} \end{bmatrix}$$

[arya1106]

Just found this....

Worth a try to implement ourselves for the challenge, but we can use the reference here if we struggle.