FTA-Sensor: Combination of Force/Torque and Acceleration - Robotized Transcranial Magnetic Stimulation - page 107

T 1
2

T 1
1

T e 1 ¼ T 1 and T e 2 ¼ T 2 : ð 6 : 16 Þ

where T e i are rotational matrices.
The stability is now expressed as the computed rotational error e rot as

e rot ¼ 1
2

ð 6 : 17 Þ

ð j h 1 jþj h 2 j Þ;

using the axis-angle (i.e., a i ; h ð Þ ) representation of the matrices T e i . Note that, as
the calibration of IMU to FT only consists of a rotational part, no translational
error is estimated.

6.1.5.2 Gravity Compensation

To estimate the quality of the independent gravity compensation based on accel-
erations, we mount a weight onto the sensor and estimate the tool's weight and
centroid (cf. Sect. 5.1.2 ). We use these parameters for gravity compensation
(Eqs. 6.11 and 6.12 ). We now move the robot randomly within all spatial axes and
record the gravity compensated forces and torques from the FTA sensor. In this
way, we collect roughly 20 ; 000 data points which we use for evaluation. Note that
we move the robot with the robot controller in order to have no additional impact
to the sensor which would bias the measurements.
To estimate the accuracy of the gravity compensation, we compare the gravity
compensated forces and torques to 0, as the forces and torques in all spatial axes
should be zero for perfectly compensated values.

6.1.5.3 Latency

Furthermore, we measure the maximum latency of the FTA sensor. In this case, we
estimate the maximum time from a detected impact which is stronger than the
security limit to setting the emergency stop. As we cannot use a global timer on the
FTA, we apply the execution counter count instead. The execution counter is
increased after each computation cycle (cf. Fig. 6.2 ). We are now connecting the
FTA sensor to a host computer and continuously query the status of the FTA
sensor including the execution counter. Subsequently, we record the computer's
system time t corresponding to the FTA sensor data. We move the robot in a
random pattern and query at least 10 ; 000 samples from the FTA. For evaluation,
we are calculating the relative time and the relative number of executions between
two consecutive samples i and i þ 1. We estimate the maximum latency lat by
dividing the relative time by the number of executions:

t ð i þ 1 Þ t ð i Þ
count ð i þ 1 Þ count ð i Þ :

lat ¼ ð 6 : 18 Þ