Wrench vector

Suppose that in a reference frame \(\{a\}\) there is a linear force \(f_a \in \mathbb{R}^3\) acting on a rigid body at point \(r_a \in \mathbb{R}^3\). The torque/moment \(\tau_a \in \mathbb{R}^3\) is \(\tau_a = r_a \times f_a\). Similar to twists, we can put the moment and force in a single vector in the \(\{a\}\) frame \(\mathcal{F}_a\) called the spatial force, or wrench, where \(\mathcal{F}_a = \begin{bmatrix} \tau_a\\ f_a \end{bmatrix} \in \mathbb{R}^6\). If more than one wrench acts on a rigid body, the total wrench is their sum, assuming that they are all represented in the \(\{a\}\) frame. When \(f_a = \mathbf{0}\), the wrench is called a pure moment.