A transformation matrix for wrenches maps forces and moments between coordinate frames. It accounts for both rotation and translation. The matrix combines a rotation matrix with terms involving the relative position vector of the frames, reflecting how forces at a distance create moments. This transformation is crucial in robotics and mechanics for analyzing rigid body equilibrium and motion, enabling the calculation of equivalent forces and moments at different points on a body. It's essential for tasks like robotic manipulation and dynamic simulations.

We have:

$$ \left[\begin{array}{c} { }^j m _j \\ { }^j f _j \end{array}\right]=\left[\begin{array}{cc} { }^j R _i-{ }^j R _i{ }_i \hat{ r }_j \\ 0 _3 & { }^j R _i \end{array}\right]\left[\begin{array}{c} i \\ { }^i m _i \\ { }^i f _i \end{array}\right]={ }^j \overline{ T }_i\left[\begin{array}{c} { }^i m _i \\ { }^i f _i \end{array}\right] $$

Simplified:

$$ \left[\begin{array}{c} { }^j f _j \\ { }^j m _j \end{array}\right]=\left[\begin{array}{cc} { }^j R _i & 0 _3 \\ -{ }^j R _i{ }^i \hat{ r }_j{ }^j R _i \end{array}\right]\left[\begin{array}{c} { }^i f _i \\ { }^i m _i \end{array}\right] $$

The key components of the transformation involve:

Breakdown of the Transformations

1. First Expression:

$$ \left[\begin{array}{c} { }^j m_j \\ { }^j f_j \end{array}\right]=\left[\begin{array}{cc} { }^j R_i-{ }^j R_{i i} \hat{r}_j & 0_3 \\ 0_3 & { }^j R_i \end{array}\right]\left[\begin{array}{c} { }^i m_i \\ { }^i f_i \end{array}\right] $$

Here, the rotation matrix ${ }^j R_i$ is applied to forces directly, while torques are transformed using both rotation and the moment contribution from the force. 2. Alternative Expression:

$$ \left[\begin{array}{c} { }^j f_j \\ { }^j m_j \end{array}\right]=\left[\begin{array}{cc} { }^j R_i & 0_3 \\ { }_{-}{ }^j R_i{ }^i \hat{r}_j{ }^j R_i & { }^j R_i \end{array}\right]\left[\begin{array}{c} { }^i f_i \\ { }^i m_i \end{array}\right] $$

This form is sometimes preferred in robotics and multibody dynamics. 3. Compact Notation:

$$ { }^j w_j={ }^i \bar{T}_j^{T_i} w_i $$

Where the transformation matrix $\bar{T}_j^{T_i}$ relates the wrench representation in different frames.

Interpretation

• Forces transform purely via rotation ${ }^j R_i$.
• Moments include an additional contribution from the force and position offset.
• This transformation is fundamental in robotics, biomechanics, and multibody dynamics, ensuring consistent force-torque relationships across coordinate frames.

🧠Example Application in Robotics: Force Transformation in a Robotic Arm

Scenario Consider a robotic manipulator where a force/torque sensor is mounted at the wrist (frame $i$ ), but we need to express the wrench in the base frame (frame $j$ ). This is necessary for force control, inverse dynamics, or external force estimation.