Drake
Collaboration diagram for Multibody Quantities:

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 expressed-in frame is the same as the reference frame, so in this case, the assumed expressed-in frame is frame F's basis. Alternately, to use a different expressed-in 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 expressed-in 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 left-superscript B denotes Bo and the expressed-in 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 SymbolTypeset 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 accelerationalpha \(^B\alpha^C\)alpha_BCFrame C's angular acceleration in B
Acceleration a \(^Ba^Q\) a_BQ Point Q's acceleration in B
Spatial accelerationA \(^BA^C\) A_BC Frame C's spatial acceleration in B
Torque t \(\tau\) t Torque
Force f \(f^{P}\) f_P Force on a point P
Spatial force F \(F^{P}\) F_P Spatial force (torque/force) ††

† In code, a vector has an expressed-in-frame which appears after the quantity.
Example: w_BC_E is C's angular velocity in B, expressed in frame E, typeset as \([^B\omega^C]_E \).

†† 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