FTA-Sensor: Combination of Force/Torque and Acceleration - Robotized Transcranial Magnetic Stimulation - page 104

Fig. 6.6 Approximate spatial
relationship between FT
sensor coordinate system and
IMU coordinate system

between FT sensor and IMU. As the translational shift of the IMU can be
neglected, the transform only consists of a rotational matrix.
Due to the system setup (cf. Fig. 6.6 ), we have a coarse knowledge of the
orientation of IMU and FT sensor:

e FT x e A z ð 6 : 6 Þ

e FT y e A y ð 6 : 7 Þ

e FT z e A z ð 6 : 8 Þ

where e denotes the corresponding unit vector. Figure 6.7 illustrates this rela-
tionship with recorded force and acceleration measurements. Also the cosine fits
for each modality are shown. Consequently, we know that a rotation of approxi-
mately 90

around the y-axis is needed to transform accelerations into the FT-

sensor coordinate frame. Furthermore, the remaining phase angles must be

adapted, resulting in the following equation:
FT T IMU R z ð 0 Þ R x ð 0 Þ R y ð p
2

Þ; ð 6 : 9 Þ

where R y describes a rotation around the y-axis, R z and R x around z- and x-axis,
respectively.
Using the phase angles b l , the equation can now be refined as:
FT T IMU ¼ R z ð b F z b A x Þ R x ð b F x b A z Þ R y ð p

2 ð b F y b A y ÞÞ:

ð 6 : 10 Þ

Note that the Eqs. 6.9 and 6.10 can be easily adapted to any other system setup.
The rotational matrices must be changed in accordance with the specific setup.
Additionally, we are applying the calibration matrix FT T V which converts the
voltage readings from the FT sensor into forces and torques. As a result, we
employ