Product of Exponentials

The product of exponentials describes an arm with two things: its pose at home, with all joints at zero, and one screw axis per joint, written in the fixed base frame. Each joint motion is a twist about its screw, and chaining the twists moves the home pose.

θ₁ θ₂ θ₃ Show the home pose

Drag the 3D view to look from another angle.

The formula

With home pose $M$ and space-frame screw axes $\mathcal{S}_i = (\boldsymbol{\omega}_i, \mathbf{v}_i)$, where $\mathbf{v}_i = -\boldsymbol{\omega}_i \times \mathbf{q}_i$ for a point $\mathbf{q}_i$ on the axis:

$$T(\boldsymbol\theta) = e^{[\mathcal{S}_1]\theta_1}\, e^{[\mathcal{S}_2]\theta_2}\, e^{[\mathcal{S}_3]\theta_3}\, M$$

Each exponential of a revolute screw is a rotation about a line in space:

$$e^{[\mathcal{S}]\theta} = \begin{bmatrix} e^{[\boldsymbol\omega]\theta} & \left(I\theta + (1-\cos\theta)[\boldsymbol\omega] + (\theta - \sin\theta)[\boldsymbol\omega]^2\right)\mathbf{v} \\ 0 & 1 \end{bmatrix}$$

What to notice

The three amber lines are the screw axes, fixed in the base frame. They are read off the arm at home and never change. That is the appeal: no per-link frames to place, only one pose and a list of axes. The readout shows that the result matches the DH table on the previous page exactly. This form is used in Lynch and Park's Modern Robotics.