Tool Transformation

Introduction

A wrist force sensor is calibrated in a frame at the centre of the sensor, called the sensor frame $\{S\}$. But you usually need the force and torque at the tool tip, in the tool frame $\{T\}$. The tool transformation changes the values from $\{S\}$ to $\{T\}$.

How it works

Displacement only

If the tool frame is only displaced by a vector $d$ from the sensor frame, the force is the same, but the moment changes:

$$F^S=F^T$$ $$N^S=N^T+d\times F^T\Rightarrow N^T=N^S-d\times F^S$$

Displacement and rotation

For a full tool transformation with a displacement vector $r$ and a rotation matrix $R_T^S$:

$$F^T=R_T^S{F}^S$$ $$N^T=R_T^S{N}^S+r\times F^T=R_T^S{N}^S+r\times R_T^S{F}^S$$

Key equations

In matrix form:

$$\left[\begin{array}{c}{F}^T\\ N^T\end{array}\right]=\left[\begin{array}{cc}{R}_T^S& 0\\ r\times R_T^S& R_T^S\end{array}\right]\left[\begin{array}{c}{F}^S\\ N^S\end{array}\right]$$

Use a $6\times 1$ force vector $\mathfrak{F}=\left[\begin{array}{c}F\\ N\end{array}\right]$. Then:

$$\boxed{{\mathfrak{F}}^T=T_f{\mathfrak{F}}^S}$$

where $T_f$ is the force moment transformation from the sensor frame to the tool frame.