Product of exponentials formula in the space frame

The following steps can be taken to determine the forward kinematics (\(\FromTo{b}{s}{\HomogeneousTransformationMatrix}(\Configuration)\)) based on the joint’s unit twists (screw axes) expressed in the space frame \(\CoorSys{s}\).

1. We start with the transformation \(\FromTo{b}{s}{\HomogeneousTransformationMatrix}(\Configuration)\) for \(\Configuration=0\). This transformation, \(\FromTo{b}{s}{\HomogeneousTransformationMatrix}(0)=M\), is the pose of frame \(\CoorSys{b}\) expressed in frame \(\CoorSys{s}\) when the robot is at its zero configuration.

2. Then we find the screw axes \(\Screw_1,\ldots,\Screw_n\) for all of the \(n\) joints, expressed in the space frame \(\CoorSys{s}\), also in the robot’s zero configuration (\(\Configuration=0\)).

3. Finally when we have a configuration \(\Configuration\) for which we want to determine the forward kinematics \(\FromTo{b}{s}{\HomogeneousTransformationMatrix}(\Configuration)\), we construct the product of exponentials (PoE) formula in the space frame, and evaluate it for the given configuration \(\Configuration\):

   \[\FromTo{b}{s}{\HomogeneousTransformationMatrix}= \exp({\tilde{\mathcal{S}}_1\GeneralizedCoordinate_1}) \exp({\tilde{\mathcal{S}}_2\GeneralizedCoordinate_2}) \cdots \exp({\tilde{\mathcal{S}}_n\GeneralizedCoordinate_n}) M\]