4. Notation

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During this course several mathematical expressions are used. A description of these expressions is given in the table below. Although extensive symbols and accents can be used to indicate what certain variables represent, in this work, the context will give equal indication, to improve readability.

Symbol	Description
\(\Vert\Sigma^{-1}\Vert\)	Inverse covariance norm (Mahalanobis distance).
\(\lvert \langle\dots\rangle \rvert\)	1-norm, also known as absolute value norm.
\(\lvert \langle\dots\rangle \rvert_2\)	2-norm, also known as Euclidean norm.
\(\lbrace\langle\dots\rangle\rbrace\)	Used to refer to a certain coordinate system.
\(\mathcal{V}\)	Twist, generalized velocity consisting of angular part and linear part.
\(\mathcal{F}\)	Wrench, generalized force consisting of moment and linear force part.
\(^{\langle\dots\rangle}{\langle\dots\rangle}_{\langle\dots\rangle}\)	Sub- and superscripts, has various usages.
\(\wedge\)	Cross product operator.
\(R\)	Rotation matrix, \(R \in SO(3)\).
\(T\)	Homogeneous transformation matrix, \(T \in SE(3)\).
\(\mathrm{Ad}_{\langle\dots\rangle}\)	Big adjoint of the transformation matrix \(\langle\dots\rangle\).
\(\mathrm{ad}_{\langle\dots\rangle}\)	Little adjoint of the velocity twist \(\langle\dots\rangle\).
\(\langle\dots\rangle^{\mathsf{T}}\)	Transposed of matrix \(\langle\dots\rangle\), obtained by flipping along the diagonal.
\(\operatorname{trace}\langle\dots\rangle\)	Trace of matrix \(\langle\dots\rangle\), obtained by summing the diagonal elements.
\(\det \langle\dots\rangle\)	Determinant of matrix \(\langle\dots\rangle\). See also Matrices.
\(\langle\dots\rangle^\dagger\)	Pseudoinverse of the matrix \(\langle\dots\rangle\). The matrix can be singular, or non-square.
\(x \sim \mathcal{N}(\mu, \sigma^2) \)	Indicates that \(x\) is a normally distributed random variable with mean \(\mu\) and variance \(\sigma^2\).
\(\langle\dots\rangle^{\mathsf{H}}\)	Conjugated transpose of complex matrix \(\langle\dots\rangle\), obtained by flipping along the diagonal and taking the complex conjugate of all elements. For real matrices, this is the same as transposing.
\(\in\)	In the set.