# A.5 Spatial velocities and forces

Given two 3D coordinate frames A and B, the spatial velocity, or twist, of B with respect to A is given by the 6D composition of the translational velocity of the origin of B with respect to A and the angular velocity :

 (A.24)

Similarly, the spatial force, or wrench, acting on a frame B is given by the 6D composition of the translational force acting on the frame’s origin and the moment , or torque, acting through the frame’s origin:

 (A.25)
Figure A.9: Two frames A and B rigidly connected within a rigid body and moving with respect to a third frame C.

If we have two frames and rigidly connected within a rigid body (Figure A.9), and we know the spatial velocity of with respect to some third frame , we may wish to know the spatial velocity of with respect to . The angular velocity components are the same, but the translational velocity components are coupled by the angular velocity and the offset between and , so that

is hence related to via

where is defined by (A.22).

The above equation assumes that all quantities are expressed with respect to the same coordinate frame. If we instead consider and to be represented in frames and , respectively, then we can show that

 (A.26)

where

 (A.27)

The transform is easily formed from the components of the rigid transform relating to .

The spatial forces and acting on frames and within a rigid body are related in a similar way, only with spatial forces, it is the moment that is coupled through the moment arm created by , so that

If we again assume that and are expressed in frames and , we can show that

 (A.28)

where

 (A.29)