Functional methods on synthetic data

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Supplementary material to Sangeux M, Peters A, Baker R. Hip joint centre ... (R1: flexion-extension, R2: Ab-Adduction, R3: internal-external rotation) of the ... simulate star movements of various ROM in either plan. ... Journal of Orthopaedic Research. ... centre in adults, children, and patients with cerebral palsy based on ...
Supplementary material to Sangeux M, Peters A, Baker R. Hip joint centre localization: Evaluation on normal subjects in the context of gait analysis. Gait & Posture. 2011;34(3):324-8.

Functional methods on synthetic data In order to check and compare the performance of the algorithms on synthetic data a virtual skeleton made of one pelvis and a left femur was created. The pelvis and the femur had 4 markers attached to them in similar positions to the participant data (see Figure 1 in original paper). Two sizes of the skeleton were considered. They were labeled as adult and child. The skeleton was driven by a set of 9 generalized coordinates (GC) and described by 3 parameters which were the position of the HJC coordinates with respect to the pelvis. The first 6 GC concerned the translations (Tx, Ty, Tz) and rotations (Rx, Ry, Rz) of the pelvis in the world and the last 3 were the rotations (R1: flexion-extension, R2: Ab-Adduction, R3: internal-external rotation) of the femur with respect to the pelvis. Two calibration movements were simulated. The first was a random movement within a cone of angle α similar to Ehrig et al, [16], with: -α < [R1=R2] < +α, R3 = 0. The second was a star movement determined from the functional calibration motion of one of the participants. The model has been registered to one of the participant’s data and the GCs have been determined following a global kinematic fitting procedure similar to [27]. As with all the other processes the complete task was performed from a custom-made Matlab (MathWorks, Natick, MA, USA) package (ViLab) interfaced with Nexus through PECS (Vicon, Oxford, UK). The minFunc, [26], routine was also used to solve the global optimization cost function. R1 and R2 determined from this procedure were scaled to their respective maximum absolute value to set: 1 < [R1unit, R2unit] < 1 (Figure 1). R1unit and R2unit could then be multiplied by respectively α and β to simulate star movements of various ROM in either plan. Figure 1 presents the obtained R1unit and R2unit. Figure 1: Star movement, Left: Skeleton and one marker’s trajectory view from above, Right: Generalized coordinates R1unit (Flexion-Extension) and R2unit (Ab-Adduction) angles during the star movement

The testing procedure followed the methodology of Ehrig et al. The virtual skeleton and GCs were used to create 200 frames of marker trajectories for the two simulation movements with various angles. The 6 first GCs (Tx, Ty, Tz, Rx, Ry, Rz) were set to zero which means that the pelvis was held stationary during the movement. Random Gaussian noise (STD: 1 or 3mm) was applied to all markers from both segments independently, each of the functional calibration algorithms was then run to provide an estimate of the HJC location. This process was repeated 1000 times. The error was expressed as the RMS distance of the 1000 estimates compared to the known reference position. Functional methods simulation results All the RMS results have been averaged to produce the graphs on Figure 2.

Figure 2: Simulation results for the random cone (left) and Star 5 movements (right). RMS results have been averaged for the simulations involving 1 and 3mm noise and Adult or Child marker set size.

The results show that in both cases with a ROM greater than 20° all methods give similar results. At, or below 10° of ROM the algebraic method gave the worst results. In all configurations and with ROM greater than 20° the RMS error was below 1cm. These simulations have partly been performed to reproduce the results provided by Ehrig et al. review paper in order to verify the implementation of methods tested. Figure 2 showed that results were similar for comparable settings. The simulations have also been performed to study the potential impact of type of movement on synthetic results. A simulated star movement that reproduces the clinically performed movement has been tested for various amplitudes in the flexion-extension and ab-adduction planes. Results showed that the various algorithms behaved similarly to the cone movement although it needed slightly greater amplitude to reach the same level of accuracy. The results showed that if the amplitude in one plane was very small (≤10), increasing the amplitude in other planes did not improve results. Overall, the transformation techniques (CTT, SCORE) slightly outperformed the sphere fitting ones by giving excellent accuracy as soon as the amplitude was greater than 20° in one plane.

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