This work is based on the following notation:

  • Scalars are denoted with regular characters, such as a, \(\alpha \), V, or \(\Psi \).

  • Vectors are denoted by bold lower case characters, such as \(\boldsymbol{v}_\text {c}\) or \(\boldsymbol{\tau }\). The only exception are wrenches , which are a combination of a Cartesian force \(\boldsymbol{f}\in \mathbb {R}^3\) and torque \(\boldsymbol{\tau }\in \mathbb {R}^3\).

  • Matrices are denoted by bold upper case characters, such as \(\boldsymbol{M}\) or \(\boldsymbol{\Lambda }\).

  • Round and square brackets are used to combine symbols into vectors and matrices, respectively. For instance, a vector is given by  and a matrix by .

  • The Euclidean norm of a vector is denoted by \(\left\| \bullet \right\| \), as for instance \(\left\| \boldsymbol{g}_0 \right\| = \sqrt{ \boldsymbol{g}_0^T \, \boldsymbol{g}_0}\).

In general, vectors and matrices can be expressed with respect to the base vectors of any arbitrary frame. In order to indicate the choice of base vectors, the notation is extended by a leading superscript: For instance, \(^A\boldsymbol{\lambda }\) and \(^B\boldsymbol{\lambda }\) indicate that the vector \(\boldsymbol{\lambda }\) is given with respect to the base vectors of frame \(\mathcal {F}_A\) or \(\mathcal {F}_B\), respectively. For simplicity of notation, all position, velocity, force, and torque vectors are expressed in world coordinates, that is with respect to the base vectors of the world frame \(\mathcal {W}\), unless otherwise mentioned. The superscript \({^\mathcal {W}\bullet }\) will be dropped.