Drake

Spatial vectors are 6element quantities that are pairs of ordinary 3vectors.
That is, Drake spatial vectors are logically elements of R³×R³, not R⁶; that is, these are not Plücker vectors! However, we can still operate on them as 6element column vectors so that we don't have to distinguish rotational from translational operations. Elements 02 are always the rotational quantity, elements 35 are translational. Here are the spatial vectors we use in Drake:
Spatial: Velocity Acceleration Force    Eigen access 0       rotation 1  ω   α   τ  .head<3>() 2          3       translation 4  v   a   f  .tail<3>() 5          Symbol: V A F
When we need to refer to the underlying 3vectors in a spatial vector, we use the following symbols, with English alphabet substitutions for their Greek equivalents:
Code  Rotational quantity  Code  Translational quantity  

w  ω  angular velocity  v  linear velocity  
alpha  α  angular acceleration  a  linear acceleration  
t  τ  torque  f  force 
While the rotational component of a spatial vector applies to a rigid body or frame as a whole, the translational component refers to a particular point rigidly fixed to that same body or frame. When assigned numerical values for computation, both subvectors must be expressed in the same frame, which may be that body's frame or any other specified frame. Thus, unambiguous notation for spatial vectors must specify both a point and an expressedin frame. Motion quantities must also state the reference frame with respect to which the motion is measured.
Example spatial quantity  At  Exp  Typeset  Code  Full 

Body B's spatial velocity in A  Bo  A  \(^AV^B \)  V_AB  V_ABo_A 
Same, but expressed in world  Bo  W  \([^AV^B]_W \)  V_AB_W  V_ABo_W 
B's spatial acceleration in W  Bcm  W  \(^WA^{B_{cm}} \)  A_WBcm  A_WBcm_W 
Spatial force acting on body B  Bcm  W  \([F^{B_{cm}}]_W\)  F_Bcm_W  F_BBcm_W 
Spatial force acting on body A  Q  W  \([F^{A/Q}]_W \)  F_AQ_W  — 
In the above table "At" is the point at which the translational activity occurs; "Exp" is the expressedin frame in which both vectors are expressed. The expressedin frame defaults to the reference (left) frame and the point defaults to the target (right) frame origin. The "Code" column shows the notation to use in code, using the available defaults; "Full" shows the code notation with the defaults shown explicitly.
For spatial forces we need to identify the body (actually a frame) on which the force is acting, as well as a point rigidly fixed to that body (or frame). When the body is obvious from the point name (such as Bo or Bcm above), the body does not need to be specified again. However, when the body is not clear it should be listed before the point as in the last line of the table above. There it can be read as "the point of body A coincident in space with point Q", where point Q might be identified with a different body. You should use fullyexpanded symbols, and helpful comments, if there is any chance of confusion.
Next topic: Spatial Mass Matrix (Spatial Inertia)