Notation and Symbols#
Notation |
Meaning |
|---|---|
\(x\) |
a scalar |
\(\vec{x}\) |
a vector |
\(x^y\) |
a scalar raised to a power, where \(y\) is a scalar |
\(\vec{x}^c\) |
a vector in the coordinate system \(c\) |
\(\vec{x}_{B/A}\) |
a vector from point A to point B (“B with respect to A”) |
\({^r \dot{\vec{x}}}\) |
the derivative of a vector taken in reference frame \(\mathcal{F}_r\) |
\(x_k\) |
a variable at index \(k\) of a sequence of length \(K\) |
\(x^{(n)}\) |
element \(n\) of a set of \(N\) elements |
\(\mat{X}_{M \times N}\) |
a matrix with \(M\) rows and \(N\) columns |
\(\mat{X}^z\) |
a matrix exponential, where \(z\) is a scalar |
\(\left| x \right|\) |
absolute value of a scalar |
\(\norm{\vec{x}}\) |
Euclidean norm of a vector |
\(\left| \mat{X} \right|\) |
determinant of a matrix |
\(\mat{C}_{b/a}\) |
the directed cosine matrix that transforms vectors from coordinate system \(a\) into coordinate system \(b\) |
\(\vec{q}_{b/a}\) |
a quaternion that encodes the relative orientation of coordinate system \(b\) relative to coordinate system \(a\) |
\(\vec{\omega}_{b/a}\) |
angular velocity vector of frame \(\mathcal{F}_b\) with respect to frame \(\mathcal{F}_a\) |
\(f(\cdot)\), \(func(\cdot)\), etc |
functions, where |
Another notation which is useful when building systems of equations involving matrices is the cross-product matrix operator, so that \(\crossmat{\vec{v}} \vec{x} \equiv \vec{v} \times \vec{x}\):
By their nature, vectors require the most intricate notation, since a fully specified vector might include all of:
A reference frame
A coordinate system
A fixed point (if it’s a bound vector)
For simplicity, Table 1 only shows examples of each distinct element of a vector encoding. In practice, vectors may appear quite complex; for some realistic examples taken from [12]: