Drake
Multibody Quantities
Collaboration diagram for Multibody Quantities:


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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 $$\tau^{B}$$ 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 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 $$[^B\omega^C]_E$$.
Similarly, an inertia matrix or spatial inertia has an expressed-in-frame.
Example: I_BBo_E is body B's inertia matrix about Bo, expressed in frame E, typeset as $$[I^{B/Bo}]_E$$.
†† 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.