Duality

Statics

The concept of statics in robotics is to find the relationship between force applied to the end-effector and the generalized torques applied to the joints when the robot arm is at equilibrium. We will apply the principle of virtual work to determine this relationship.

The principle of virtual work

The principle of virtual work states that in equilibrium the virtual work of the forces applied to a system is zero

Let \(\boldsymbol{\gamma}_{e}=\begin{bmatrix} \boldsymbol{f}_{e}\\ \boldsymbol{\mu}_{e} \end{bmatrix}\in \mathbb{R}^{r}\) be the wrench applied to the robot end-effector. It includes the linear force \(\boldsymbol{f}_{e}\) and moment \(\boldsymbol{\mu}_{e}\). The visual work by this end-effector wrench:

\[d W_{\gamma}=\boldsymbol{f}_{e}^{T} d \boldsymbol{p}_{e}+\boldsymbol{\mu}_{e}^{T} \boldsymbol{\omega}_{e} d t =\boldsymbol{f}_{e}^{T} \boldsymbol{J}_{P}(\boldsymbol{q}) d \boldsymbol{q}+\boldsymbol{\mu}_{e}^{T} \boldsymbol{J}_{O}(\boldsymbol{q}) d \boldsymbol{q} =\boldsymbol{\gamma}_{e}^{T} \boldsymbol{J}(\boldsymbol{q}) d \boldsymbol{q}\]

where \(d \boldsymbol{p}_{e}\) is the linear virtual displacement and \(\boldsymbol{\omega}_{e} d t\) is the angular virtual displacement of the end-effector.

Let \(\boldsymbol{\tau}\in\mathbb{R}^{n}\) be the joint torques. The visual work by the joint torques:

\[d W_{\tau}=\boldsymbol{\tau}^{T} d \boldsymbol{q}\]

where \(d \boldsymbol{q}\) is the joint virtual displacement.

According to the principle of virtual work, the robot arm is at static equilibrium if and only if

\[\delta W_{\tau}=\delta W_{\gamma}\]

This leads to the statics equation:

\[\boldsymbol{\tau}=\boldsymbol{J}^{T}(\boldsymbol{q}) \boldsymbol{\gamma}_{e}\]

stating a relationship between the end-effector forces and joint torques when the robot is at its equilibrium.

Kineto-Statics Duality

The kineto-statics duality states that

\[\begin{aligned} \boldsymbol{v}_{e}=\boldsymbol{J}(\boldsymbol{q}) \dot{\boldsymbol{q}} \quad\quad\quad \boldsymbol{\tau}=\boldsymbol{J}^{T}(\boldsymbol{q}) \boldsymbol{\gamma}_{e} \end{aligned}\]

• The range space \(\mathcal{R}\left(\boldsymbol{J}^{T}\right)\) is the subspace in \(\mathbb{R}^{n}\), where the the joint torques that can balance the end-effector wrench.

• The null space \(\mathcal{N}\left(\boldsymbol{J}^{T}\right)\) is the subspace in \(\mathbb{R}^{r}\) of the end-effector wrench that does not require any balancing joint torques. This means that the end-effector wrenches \(\gamma_{e} \in \mathcal{N}\left(\boldsymbol{J}^{T}\right)\) are entirely absorbed by the mechanical structure of the robot arm.

Fig. 60 Mapping between the end-effector force space and the joint torque space

From fundamental relationship in linear algebra, we have the following relationships:

\[\mathcal{N}(\boldsymbol{J}) \equiv \mathcal{R}^{\perp}\left(\boldsymbol{J}^{T}\right) \quad \mathcal{R}(\boldsymbol{J}) \equiv \mathcal{N}^{\perp}\left(\boldsymbol{J}^{T}\right)\]