Forward Kinematics: Product of Lie Groups
The Product of Lie Groups method computes the forward kinematics of the end-effector relative to the world frame by multiplying relative frame transformations:
\[g_{we} = g_{w1} \, g_{12} \, g_{23} \, \cdots \, g_{ne}\]Each of these transformations is computed using the displacement and rotation from the last frame to the next frame (with each subsequent frame attached to the link immediately after the joint). These transformations are in homogeneous form:
\[g_{ij} = \begin{bmatrix} R_{ij} & p_{ij} \\ 0 & 1 \end{bmatrix}\]For all of the demos, we will consider the planar 3-DOF manipulator. This manipulator has the individual transformations defined as follows:
\[\begin{align*} g_{w1} &= \begin{bmatrix} R_z(\theta_1) & \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \\ 0 & 1 \end{bmatrix}, \quad g_{12} = \begin{bmatrix} R_z(\theta_2) & \begin{bmatrix} L_1 \\ 0 \\ 0 \end{bmatrix} \\ 0 & 1 \end{bmatrix}, \quad g_{23} = \begin{bmatrix} R_z(\theta_3) & \begin{bmatrix} L_2 \\ 0 \\ 0 \end{bmatrix} \\ 0 & 1 \end{bmatrix}, \quad g_{3E} = \begin{bmatrix} I & \begin{bmatrix} L_3 \\ 0 \\ 0 \end{bmatrix} \\ 0 & 1 \end{bmatrix}. \end{align*}\]Typically we use these only to compute the end-effector transformation ( g_{we} ) as:
\[g_{we} = g_{w1} \, g_{12} \, g_{23} \, g_{3E}\]However, we can also compute the transformation of each intermediate frame relative to the world frame as:
\[g_{w2} = g_{w1} \, g_{12}, \quad g_{w3} = g_{w2} \, g_{23}, \quad g_{wE} = g_{w3} \, g_{3E}\]The following interactive GUI demonstrates these calculations being computed in real time.