Time Derivatives of Multibody Quantities

**Scalar quantities**: The ordinary first time-derivative of the scalar x is denoted xdot or xDt whereas the ordinary second time-derivative of x is denoted xddot or xDDt.

**Vector quantities** (Advanced topic): The ordinary time-derivative of a vector v (such as position or velocity) is different than the derivative of a scalar. A vector has direction whereas a scalar does not. The derivative of a vector requires a frame in which the derivative is being taken. The typeset notation for the ordinary time-derivative in frame \( G \) of a vector \( v \) is \( \frac{^Gd}{dt}\,v \) and its monogram notation is `DtG_v`

. Since the derivative of a vector is a vector, we need to specify an expressed-in frame E. The typeset notation is \( [\frac{^Gd}{dt}\,v]_E \) whereas the monogram notation is `DtG_v_E`

. In unicode comments (e.g., in a header or source file), use `[ᴳd/dt v]_E`

or `DtG(v)_E`

(see below).†

Important note: The derivative operator applies to the vector, *not* the computational representation of the vector. It is misleading to include an expressed-in frame in the symbol name for the vector v. The expressed-in frame applies only to the final derived quantity. For example, consider `V_BC`

, frame C's spatial velocity in frame B (a spatial velocity contains two vectors, namely angular velocity and velocity). In code, you may express `V_BC`

in frame E as `V_BC_E`

. The definition of the spatial acceleration `A_BC`

is the derivative in frame B of `V_BC`

(the derivative **must** be in B). However, the resulting expressed-in frame is arbitrary, e.g., a frame F. The expressed-in frame E for `V_BC`

does not impact the final result. The monogram notation for this derivative is `DtB_V_BC_F`

which is interpreted as \( [\frac{^Bd}{dt}\,^BV^C]_F \); the `_F`

goes with the result, not the quantity being differentiated. The resulting vector happens to be `A_BC_F`

, but that is only because the derivative was taken in frame B. If the derivative was taken in F (or C or E or any frame other than B), there is **no** conventional spatial acceleration name or notation for the result `DtF_V_BC`

.

When using this DtFrame derivative notation in code, the expressed-in frame is *always* specified at the end of the symbol. However there is *never* an expressed-in frame specified for the quantity being differentiated. For example, given a vector v, the expression `DtG(v_E)_F`

is misleading. Instead, use `DtG(v)_F`

.

† In unicode comments for the derivative in frame A of a vector v, use `ᴬd/dt v`

(preferred if available) or `DtA(v)`

. Although the former is preferred, not all uppercase letters are available as superscripts in unicode. Consider choosing frame names to accommodate this strange quirk in unicode.

Next topic: Spatial Algebra