Dynamic calibration of six-component wheel
dynamometers for biomechanical studies of manual
wheelchair locomotion

Christophe Sauret, Michel Dabonneville, Yves Couétard, Nicolas de Saint Rémy, Philippe Kauffmann, Mariano Cid, Philippe Vaslin

► To cite this version:

Christophe Sauret, Michel Dabonneville, Yves Couétard, Nicolas de Saint Rémy, Philippe Kauffmann, et al.. Dynamic calibration of six-component wheel dynamometers for biomechanical studies of manual wheelchair locomotion. ASSISTH'2011: 2° Conférence Internationale sur l'Accessibilité et les Systèmes de Suppléance aux Personnes en Situations de Handicaps, Nadine Vigouroux & Philippe Gorce, Jan 2011, Paris, France. pp.209-218. hal-05010566

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# Dynamic calibration of six-component wheel dynamometers for biomechanical studies of manual wheelchair locomotion.

Christophe Sauret¹, Michel Dabonneville¹, Yves Couétard², Nicolas de Saint Rémy¹, Philippe Kauffmann¹, Mariano Cid² and Philippe Vaslin¹

¹ LIMOS, UMR 6158 CNRS, Clermont Université, F-63000 Clermont-Ferrand, France
² LMP, UMR 5469 CNRS, Université Bordeaux 1, F-33400 Talence, France
vaslin@isima.fr

**Abstract:** A six-component wheel dynamometer is a dynamometer that rotates around the wheel axle during measurements. For this kind of dynamometer, a static calibration is not sufficient because the weight and the inertial forces measured by the sensors vary according to the angular position and velocity of the dynamometer. The dynamic calibration of a wheelchair wheel dynamometer presented in this paper is based on the direct correction of the sensors signals using Fourier's polynomials that take into account the influences of both the handrim weight distribution on the sensors with respect to the wheel angular position, and the wheel angular velocity. When these corrections were applied on the signals produced by the sensors while the wheel dynamometer was rotating and no effort was exerted on the handrim, the calculated forces and torques remained close to zero, as expected. From these results, the wheel dynamometer can thus be confidently used for studying manual wheelchair locomotion in various real conditions.

**Keywords:** Wheelchair, Rehabilitation, Biomechanics, Wheel dynamometer; Calibration.

## Introduction

For evaluating some motor skills of manual wheelchair (MWC) users during the rehabilitation process, and the impact of rehabilitation programs on the improvement of these skills, clinicians have always needed and used various measurement devices, like roller ergometers, motor driven treadmills or wheelchair simulators. Most of these devices have been first designed by researchers in the aim of improving scientific knowledge on MWC locomotion and especially quantifying the efforts produced or sustained by users during propulsion. In the last two decades, some laboratories developed new-sophisticated wheel dynamometers (WD) for directly
measuring the six components of the wrench (i.e., three forces: Fₓ, Fᵧ, Fᵧ, and three torques: Tₓ, Tᵧ, Tᵧ) applied by the user on rear wheel handrim [1], [2], [3], [4], [5], [6]. These WD were mounted on different MWC and have been used in several studies for calculating various dynamic or kinetic parameters during propulsion within the laboratory either on roller ergometers [3], [7], [8], [9], [10], [11] or directly on the ground [1], [5], [6] [12], [13].

Prior to these experimentations, WD had to be calibrated both statically and dynamically. Static calibration is usually performed along the same method as that used for any six-component dynamometer like a force-plate, i.e. several sets of known forces and torques are applied on different precise points of the handrim and the results are used to compute the coefficients of the calibration matrix. The sensitivity matrix, which is the inverse of the calibration matrix, then allows calculating the six components of the effort applied on the handrim by the user during static experiments [14], [15]. Unlike a force-plate, however, which basis is usually firmly bolted on a horizontal concrete floor, a WD is designed for rotating around the axle of the rear wheel on which it is fixed. During this rotation, the force sensors measure a varying part of the handrim weight, and likely the centrifugal force due to the handrim misalignment or a heterogeneous distribution of the handrim mass, due to the fixation of a data-logger or batteries, for instance.

Accounting for these mechanical phenomena, previous WD have been dynamically calibrated along different methods, more or less complex and precise. Some authors [2] determined the dynamic characteristics of their system “by spinning the wheel with no load on the pushrim at accelerations similar to normal wheelchair propulsion”, which they assumed to be small, and concluded that these “inertia/acceleration components” were “negligible within the resolution of the device” (p. 1322). Nevertheless, they conceded that “a computer program could be developed to minimize their effects”, without giving any information on the method that should be used. Other authors [5] used a program for subtracting the forces and torques offsets stored in a “zero load offset data file” from the analog data recorded during experiments. That operation was performed “by first matching the encoder value at each time increment with the corresponding encoder value in the zero load offset data file” (p. 84). Although these authors gave the resolution of their optical encoder (8 bits = 256 steps), they did not mention if the offset values were measured statically or at different WD angular velocities.

The only real dynamic calibration of a WD was performed by Wu *et al.* [6] who mounted their instrumented wheel vertically and spun it with “an electric motor at eight different constant speeds, and at a variable speed, ranging from 0.96 to 12.51 rad/s” (p. 534). This experiment was realized first with no load and then with two masses (0.451 kg and 1.484 kg) attached on the handrim. As their results “from the no-load baseline trials” showed “a minimal variation for the prescribed speed range”, the authors could estimate the relationship between each force and torque component (y) and the wheel angular position (θ) by a simple sine function (y = a sin (θ + φ) + b). They also found a “quadratic relationship between inertia forces and angular velocity of the wheel (mrω²)” from the dynamic calibration with the 0.451 kg mass.
This method of forces and torques correction relies on classical mechanical principles and quite simple mathematical functions. However, in some WD, each force and torque component is computed from the measurements of several force transducers (e.g. six in our case), which signals are all involved simultaneously. So, it could thus be hypothesized that it would be better to apply the corrections directly on the input data of the equation of wrench components computation (i.e. sensors signals) than on the output data (i.e. forces and torques).

In the aim of improving the measurements of the efforts applied on the handrims or sustained by the MWC user during various actual displacements, an alternative method to that previously described [2], [5], [6], and based on the correction of sensors signals, is proposed for dynamically calibrating wheel dynamometers.

# Material and methods

## Wheels dynamometers

The WD (Fig. 1, TSR-Mesures, France) [18] used in this study is built along an original mechanical principle [19], [20], which has already been applied for measuring the six components of the wrench exerted by the pole-and-vaulter's system in the vaulting box during pole vault [21], [22]. It is composed of only three two-component strain gauges force transducers directly fixed on the felly 120° apart between wheel spokes. Each transducer measures one component of the resultant force in the plane of the wheel and the other perpendicularly to that plane. The sensing elements of the transducers are linked together by a flat aluminum ring on which the handrim is firmly bolted. The flat ring and the handrim are assumed to be rigid, within the range of efforts applied by the subject, and figure the top plate of a force platform vertically oriented and able to freely rotate around the axis passing through its center (i.e., rear wheel axle). The angular positions of the rear wheel are measured by a rotating potentiometer fixed on its axle (Spectrol 601-1045, Vishay, USA - Resolution: 0.5 deg.).
<figure><img src="image_2.png" /><figcaption>Fig. 1. View of a wheel dynamometer mounted in lieu of the rear wheel of a Wireless Wheelchair Ergometer.</figcaption></figure>

## Static calibration

The WD had been previously calibrated statically along a classical process: the WD was fixed in different successive positions and orientations on a rigid frame and several sets of known vertical forces were applied on 13 known points of the handrim following an increasing-decreasing process. The applied vertical forces were measured with a force sensor mounted on a steel cable and ranged from 0 to 300 N in the handrim plane (x- and y-axes) and from 0 to 100 N along the transverse direction (z-axis). The mean loading velocity was 6.67 N/s and the sensors signals were sampled at a 12 Hz frequency. These data have been then used to calculate the thirty-six coefficients of the WD calibration matrix [$M_C$] through a multiple linear regression. The inverse of the calibration matrix gave the sensitivity matrix [$M_S$] that was then used during laboratory or field experiments for computing the six components of the wrench measured by the WD [W] from the signals [S] produced by the WD force sensors.

$$ [W] = [M_S][S] \qquad (1) $$

## Dynamic calibration

For the dynamic calibration, the WD axle was horizontally fixed so that the wheel could freely rotate in the vertical plane with no load hung on the handrim. Then, the
WD was manually spun at different velocities and it was let decelerating under the only action of the bearing resistance. Signals of the force transducers and the rotating potentiometer were sampled at 1000 Hz with an A/D converter (NI USB-6218, National Instruments, USA) connected to a mini-computer (UMPC AK7, Akori, France). Thirty trials were performed and data were recorded a few seconds after the beginning of the WD rotation until it completely stopped. Force data were then gathered according to the WD angular velocity that was split into height levels between 1.67 to 13.33 rad/s, which corresponded to linear velocities ranging from 0.5 to 4.0 m/s with a step of 0.5 m/s.

For each velocity, the mean signal of each force transducer was computed with respect to the WD angular position (θ). The obtained curve was then approximated by a Fourier's polynomial, which coefficients (a, b<sub>k</sub> and c<sub>k</sub>) were calculated by increasing the order (n) of the Fourier's polynomial until the mean absolute error becomes lower than 0.05 mV. According to [6], the possible influence of the centrifugal force was also taken into account by a quadratic term (dω<sup>2</sup>) including the WD angular velocity (ω) and a coefficient (d) that was computed using a linear least square regression on the drift of the WD signal with respect to the WD angular velocity. Finally, the signal of each force transducer (m) with respect to angular position (θ) and velocity (ω) was approximated by the following generic mathematical function:

$$m = a + \sum_{k=1}^{n} \left[ b_k \sin(k\theta) + c_k \cos(k\theta) \right] + d\omega^2 \quad (2)$$

The signal (m) was then subtracted from the experimental signal of each force transducer before computing the six-components of the applied wrench.

## Results

For each sensor, a low signal drift was observed at low speeds (i.e. between 1.67 and 5.0 rad/s) but this drift only became significant at high speeds (Fig. 2). The quadratic term (dω<sup>2</sup>) showed a good relationship with the observed drift on experimental data of five sensors (r<sup>2</sup> ranged from 0.83 to 0.98) but a weak relationship (r<sup>2</sup> = 0.34) for one transverse sensor.

After the drift due to the angular velocity had been subtracted, the 75 coefficients of a 37th order Fourier's polynomial were computed on each sensor signal. Then, according to WD angular position (θ), the signal (m) of each force transducer was simulated using Fourier's polynomials from the first (n=1) to the 37th order (n=37). The mean absolute error (ε) of the computed signal (m) with regard to experimental data felt under 0.05 mV for Fourier's polynomials of at least the 5th order for sensor 1 and up to the 15th order for sensor 3, for instance (Fig. 3). On the other hand, the improvement of the m signal became marginal beyond the 20th order.
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Evolution of sensor 1 signal with respect to the WD angular position at three angular velocities.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Evolution of the mean absolute error (ε) of two typical sensors with respect to the Fourier's polynomial order.</figcaption></figure>

Within this interval of Fourier's polynomial orders (5th to 15th), the regression analysis between simulated signal and experimental data yielded coefficients of determination (r²) ranging from 0.931 to 0.999 (mean: 0.976) with the lowest values obtained for the three transverse sensors, which were the least solicited in this experimentation. Besides, neglecting WD angular velocity gave r² ranging from 0.923 to 0.998 (mean: 0.973), which were very close to the former ones.
After the Fourier's polynomials had been subtracted from the experimental signals, the six components of the applied wrench were computed and remained close to zero within the ranges of ± 1.1 N for forces and ± 1.4 N.m for torques components (Fig. 4).

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Evolution of Fₓ component with respect to the WD angular position, computed from all experimental data recorded during several turns.</figcaption></figure>

Discussion

The dynamic calibration of a six-component WD carried out in this study was based on the hypothesis that it should be better to directly correct the signal of each force transducer of a WD than each force and torque component calculated from its sensitivity matrix. This approach was thus focused on the signals actually produced by the force transducers when the WD was spun at different angular velocities, and no hypothesis was formulated on the theoretical outputs of the six-component dynamometer.

One main result of this study is the negligible influence of the WD angular velocity on the six force transducers signals at low MWC linear velocities like those used in daily locomotion (around 1 m/s). Indeed, the angular velocity could only explain from 0.1 to 0.8 % of the force transducer signals. These results and this analysis are supported by the fact that no additional mass (e.g., data-logger) was hung on the handrim of our WD. Nevertheless, even in this case, the proposed mathematical model would allow correcting the sensors signals with respect to the WD angular velocity.

Secondly, it appeared that the influence of the handrim weight with respect to the WD angular position can be well approximated with a mean absolute error below 0.05 mV by Fourier's polynomials of 5th order or more. These results demonstrate that a single sine function like that used in [6] was not sufficient to dynamically calibrate
this WD. On the other hand, it appears that it is not useful to increase the Fourier's polynomial order above the 20th order because the mean absolute error is then not significantly improved. It must be noticed, however, that these calibration functions are specific to the chosen camber angle (null in this study) and that they should be calculated again for any other rear wheel camber.

The theoretical signals (m) computed with the Fourier's polynomials were very close to the experimental data and allowed correcting the force sensors signals in such a good way that the six components of the wrench calculated from these data remained close to zero, as it was expected in this experimental setting where no effort was applied on the handrim.

Beyond the technical differences between our WD and that of Wu et al. [6], and the slight methodological divergences in our respective calibration methods, it seems that, until now, they are the only wheelchair WD that have actually been dynamically calibrated.

## Conclusion

This study has shown that the dynamic calibration of a wheelchair six-component wheel dynamometer requires to take into account both its angular position and velocity, even if this latter could be neglected for MWC velocities ranging from 0.5 to 4 m/s, corresponding to those currently adopted by MWC users during most of their daily displacements. The Wireless Wheelchair Ergometer equipped with this wheel dynamometer is thus particularly well suited for continuously evaluating the efforts produced or the constraints sustained by wheelchair users during locomotion in various actual conditions (e.g., rolling and turning on different types of indoor or outdoor floors, climbing inclines or clearing obstacles and thresholds).

## Conflict of interest

None

## Acknowledgments

This study was a part of the SACR-FRM project that was financially supported by the French National Research Agency (ANR-06-TecSan-020).

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