Robot maths and control

March 2016

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• Wheeled machines
• Control

Coordinate frames

$x_{\text{X}w}=x_{\text{X}r}\cos\theta_c - y_{\text{X}r}\sin\theta_c +x_c$ $y_{\text{X}w}=x_{\text{X}r}\sin\theta_c + y_{\text{X}r}\cos\theta_c +y_c$

Coordinate frames.

$x_{\text{X}w}=x_{\text{X}r}\cos\theta_c - y_{\text{X}r}\sin\theta_c +x_c$ $y_{\text{X}w}=x_{\text{X}r}\sin\theta_c + y_{\text{X}r}\cos\theta_c +y_c$ $\left[\begin{array}{c} x_{\text{X}w}\\ y_{\text{X}w}\\ 1 \end{array}\right] = \left[\begin{array}{ccc}\cos\theta_c&-\sin\theta_c&x_c\\ \sin\theta_c&\cos\theta_c&y_c\\ 0&0&1\\ \end{array}\right] \left[\begin{array}{c} x_{\text{X}r}\\ y_{\text{X}r}\\ 1 \end{array}\right]$

Coordinate Frames

Robot to world frame conversion $\left[\begin{array}{c} x_{\text{X}w}\\ y_{\text{X}w}\\ 1 \end{array}\right] = {}^w_rT \left[\begin{array}{c} x_{\text{X}r}\\ y_{\text{X}r}\\ 1 \end{array}\right]$ World to robot coordinate frame conversion $$\left[\begin{array}{c} x_{\text{X}r}\\ y_{\text{X}r}\\ 1 \end{array}\right] = {}^r_wT \left[\begin{array}{c} x_{\text{X}w}\\ y_{\text{X}w}\\ 1 \end{array}\right] = {}^w_rT^{-1} \left[\begin{array}{c} x_{\text{X}w}\\ y_{\text{X}w}\\ 1 \end{array}\right]$$

Robot Kinematics of two independent wheels

$$v_x=v \cos\theta={dx \over dt}$$ $$v_y=v \sin\theta={dy \over dt}$$ $$\omega={d\theta \over dt}$$

where $v$ is the forward velocity of the robot, angled at $\theta$ to the $x$ axis.

Wheel equations

$$v={(\omega_R+\omega_L)R\over 2}$$ $$\omega={(\omega_R-\omega_L)R\over L}$$

where $R$ is the radius of the wheels and $L$ is the distance between wheels

• Requires numerical integration techniques
• Usually forward or backward Euler
• Errors accumulate
• Wheel slip
• Inacurate parameter estimates (e.g. for R and L - flat tires?)

Inverse Kinematics

Also known as non holonomic planning Multiple solutions e.g. $v=3$ $$\theta=\begin{cases} 0,& \text{if 0\le t<0.4}\\ \frac{5\pi}{2}\left(t-0.4\right),& \text{if 0.4\le t<0.6}\\ \frac\pi2, & \text{if 0.6\le t<1} \end{cases}$$

Manipulator Kinematics

• Cartesian robots have simple kinematics
• Consider a planar 2R serial chain robot.

$$x=a_1\cos\theta_1 + a_2\cos(\theta_1+\theta_2)$$ $$y=a_1\sin\theta_1 + a_2\sin(\theta_1+\theta_2)$$ $$\theta=\theta_1+\theta_2$$

Inverse kinematics

• Elbow in/elbow out solutions

$$x^2+y^2=a_1^2+a_2^2+2a_1a_2\cos\theta_2$$

 hence

$$\theta=\arccos\left({(x^2+y^2)-(a_1^2+a_2^2) \over 2a_1a_2}\right)$$

• no solution if the expression is not between $-1$ and $1$.

$$\tan(\alpha+\theta_1)={y \over x}$$ $$\sin\alpha={a_2\sin\theta_2\over \sqrt{x^2+y^2}}$$

Videos

(What not to do with 6R industrial robots) Upenn quadcopter ETH quadcopter