Drake

Quantities of interest in multibody dynamics have distinct types.
For example, a rotation matrix is denoted with R
and a position vector with p
. New quantities can be created by differentiation of an existing quantity (see Time Derivatives of Multibody Quantities).
Most quantities have a reference and target, either of which may be a frame, basis, or point, that specify how the quantity is to be defined. In computation, vector quantities are expressed in a particular basis to provide numerical values for the vector elements. For example, the velocity of a point P moving in a reference frame F is a vector quantity with target point P and reference frame F. In typeset this symbol is written as \(^Fv^P\). Here v is the quantity type, the left superscript F is the reference, and the right superscript P is the target. In computation, this vector is expressed in a particular basis. By default, the assumed expressedin frame is the same as the reference frame, so in this case, the assumed expressedin frame is frame F's basis. Alternately, to use a different expressedin frame, say W, typeset with the bracket notation: \([^Fv^P]_W\).
The typeset symbol \(^Fv^P\) is translated to monogram notation as v_FP
. The quantity type always comes first, then an underscore, then left and right superscripts. The symbol v_FP
implies the vector is expressed in frame F. Alternately, to express in frame W, we typeset as \([^Fv^P]_W\) and use the monogram notation v_FP_W
(adding a final underscore and expressedin frame W). We adhere to this pattern for all quantities and it is quite useful once you get familiar with it. As a second example, consider the position vector of point Bcm (body B's center of mass) from point Bo (the origin of frame B), expressed in B. In full typeset, this is \([^{B_o}p^{B_{cm}}]_B \) whereas in implicit typeset this can be abbreviated \(^Bp^{B_{cm}}\) (where the leftsuperscript B denotes Bo and the expressedin frame is implied to be B). The corresponding monogram equivalents are p_BoBcm_B
and p_BBcm
, respectively.
Here are some more useful multibody quantities.
Quantity  Symbol  Typeset  Code  Meaning † 

Rotation matrix  R  \(^BR^C\)  R_BC  Frame C's orientation in B 
Position vector  p  \(^Pp^Q\)  p_PQ  Position from point P to point Q 
Transform/pose  X  \(^BX^C\)  X_BC  Frame C's transform (pose) in B 
Angular velocity  w  \(^B\omega^C\)  w_BC  Frame C's angular velocity in B † 
Velocity  v  \(^Bv^Q\)  v_BQ  Point Q's velocity in B 
Spatial velocity  V  \(^BV^C\)  V_BC  Frame C's spatial velocity in B 
Angular acceleration  alpha  \(^B\alpha^C\)  alpha_BC  Frame C's angular acceleration in B 
Acceleration  a  \(^Ba^Q\)  a_BQ  Point Q's acceleration in B 
Spatial acceleration  A  \(^BA^C\)  A_BC  Frame C's spatial acceleration in B 
Torque  t  \(\tau^{B}\)  t_B  Torque on a body (or frame) B 
Force  f  \(f^{P}\)  f_P  Force on a point P 
Spatial force  F  \(F^{P}\)  F_P  Spatial force (torque/force) †† 
Inertia matrix  I  \(I^{B/Bo}\)  I_BBo  Body B's inertia matrix about Bo 
Spatial inertia  M  \(M^{B/Bo}\)  M_BBo  Body B's spatial inertia bout Bo † 
† In code, a vector has an expressedinframe which appears after the quantity.
Example: w_BC_E
is C's angular velocity in B, expressed in frame E, typeset as \([^B\omega^C]_E \).
Similarly, an inertia matrix or spatial inertia has an expressedinframe.
Example: I_BBo_E
is body B's inertia matrix about Bo, expressed in frame E, typeset as \([I^{B/Bo}]_E\).
For more information, see Spatial Mass Matrix (Spatial Inertia)
†† In mechanical systems, it is often useful to replace a set of forces by an equivalent set with a force fᴾ placed at an arbitrary point P (fᴾ is equal to the set's resultant), together with a torque t
equal to the moment of the set about P. A spatial force Fᴾ containing t
and fᴾ can be useful for representing this replacement.
Next topic: Time Derivatives of Multibody Quantities