Twist

Definition

A twist describes a rigid body motion (rotation + translation). Twist coordinates: $\theta \xi =\theta (v_1,v_2,v_3,{\omega }_1,{\omega }_2,{\omega }_3)$. As a $4\times 4$ matrix:

$$\hat{\xi }=\left[\begin{array}{cc}\hat{\omega }& v\\ 0& 0\end{array}\right],\dot{\mathbf{p}}=\hat{\xi }\mathbf{p},G=\left[\begin{array}{cc}R& \mathbf{t}\\ 0& 1\end{array}\right]=\exp (\theta \hat{\xi })$$

The exponential has a closed form, like the Rodrigues formula for rotations.

Product of exponentials (kinematic chain)

$$G_{sb}(\Theta )=e^{{\hat{\xi }}_1{\theta }_1}{e}^{{\hat{\xi }}_2{\theta }_2}\cdots e^{{\hat{\xi }}_n{\theta }_n}{G}_{sb}(0)$$

Each joint $i$ has an axis ${\xi }_i$ and an angle ${\theta }_i$. $G_{sb}(0)$ is the rest pose.

Trap

$\exp ({\theta }_i{\hat{\xi }}_i)$ is not the mapping from segment $i+1$ to segment $i$. It is the relative motion of joint $i$.

Inverse kinematics: find the joint angles so that a body point $X_A$ reaches a goal $X_B$: $\arg {\min }_{\theta }\Vert \exp ({\theta }_1{\hat{\xi }}_1)\cdots \exp ({\theta }_n{\hat{\xi }}_n)X_A-X_B{\Vert }^2$. It is non-linear and is linearized with the articulated Jacobian.

In the comparison table of the lecture, twists rank first for number of parameters, human constraints and optimization.

Appears in

• Lecture 2.2, Twists
• Lecture 2.2, Articulation: product of exponentials, inverse kinematics
• Lecture 2.2, Which representation

Related concepts

• Axis-Angle and Exponential Map
• Rotation Matrix