McGhee et al. 2000 presented a non-orthogonal transformation that links body rate angular velocity to Euler angle rate. This transformation matrix that allows one to specify angular velocities in local (body) axes from Euler angle rates calculated in global axes. The formulation is given in Appendix A of the linked paper, and is also copied below in figure form. This is a very powerful transformation matrix that can be used to transform Euler angle rates into body angular rates, which makes direct comparison of angular velocities measured using any Euler/Cardan sequence. A working example is presented here.
At Callaway Golf, I ran multi-sensor swing studies (MS3) using multiple advanced measurement technologies for Product Player Matching initiatives. The projects that I was in charge of included Player Profiling, Digital Human Modeling, and Club Fitting. A central component to these 3 projects was an inertial measurement unit (IMU) on-board diagnostic (OBD) shaft-based data acquisition system, that would link all 3 projects together. One of the more difficult aspects of the project was verifying angular velocity measurements between the modular IMU sensor shuttle and motion capture measurements collected in the Player Performance Bay which were used as inputs to the digital human modeling efforts.
In the Player Performance Bay, we used an active optoelectronic motion capture system to record player swing data and also used instrumented clubs to record club motion. An example of the shaft marker prong used on tested clubs is shown below. Inside the butt end of the shaft of the club was the modular IMU sensor shuttle, also shown below.
In order to synergistically use the data from the 2 very different systems, we needed to be able to get the data outputs into a common coordinate system. The modular IMU sensor shuttle directly measures accelerations and angular rates relative to the local (body) axes defined by the internal board configuration, which was placed inside the shaft of the club. An orientation system had to be used to ensure that the modular shuttles could be inserted the same in every club relative the club head. Similarly, the shaft marker prong was installed on the tested shaft, and an orientation process was used to align the marker and define club axes relative to the club head. There was actually a third coordinate system used in these tests with markers placed on the club head to define club head axes as well, but that is not necessary for this discussion.
As we knew the axes definitions for both the modular IMU sensor shuttle and the shaft marker prong for the instrumented club, it was possible to compare the angular rate measurements between the two systems using the non-orthogonal body rate to Euler rate transformation matrix given below. The IMU club used angular rate sensors that were operating well beyond their technical specs; the sensors were individually bench top tested to calibrate them. It was desired to compare the two systems directly to ensure dynamic operation of the out-of-spec sensors. This was done for each modular shuttle to verify the calibrated dynamic outputs.
Due to confidentiality concerns, I can not show those outputs here. However, we had undertaken a very similar test with SmartSwing instrumented clubs. Their board layout is shown below. A few graphs from that test were presented at a conference so can be displayed here. The main reason for testing the SmartSwing clubs was to evaluate the dynamic response of the accelerometers and the angular rate sensors. As in our internal modular IMU sensor shuttle design, there were no available sensors at that time that had the necessary technical specifications to dynamically measure the angular rates found in the golf swing of a professional golfer. We wanted to test the dynamic response of the SmartSwing club as we had done with our own IMU clubs to evaluate dynamic operation.
Below are plots that show the SmartSwing outputs plotted directly against the motion capture outputs that were transformed into the local SmartSwing axes using the transformation matrix described below. As one can see, we obtained very good dynamic results comparing the two systems. There were some instances where the accelerometers clipped as was expected given their specs. There were also some irregularities with the angular rate outputs near impact at the maximum values. One other interesting note in these plots is the fact that the motion capture sampling rate is much lower, so there were some instances where the absolute peak angular rate is missed with the motion capture system because the sampling rate was not high enough for this very high-speed motion. The instrumented club sampled at 1000 Hz and did not suffer from this problem. This transformation technique was very powerful and allowed us to directly compare results from two entirely different measurement systems in a local body coordinate system.
McGhee et al. 2000 Rigid Body Dynamics, Inertial Reference Frames, and Graphics Coordinate Systems: A Resolution of Conflicting Conventions and Terminology