Towards an accurate rolling resistance: Estimating intra-cycle load distribution between front and rear wheels during wheelchair propulsion from inertial sensors

Figures & data

Figure 1. Image of “actual” rolling resistance (black line) and rolling resistance based on a static drag test (blue line). This image is adapted from the image reported by Sauret et al. (2013). Note that – in this example – the drag test-based rolling resistance is similar to the average “actual” rolling resistance. However, depending on the upper body pose during deceleration or drag test, this value may be higher or lower.

Figure 2. Schematic overview of measurements during different sessions. The “original” treadmill session refers to the condition with no added mass (0 kg) and fully inflated rear wheel tyres (5.25 bar). Mass (i.e., total mass of participant and wheelchair) was assessed on a 1.0 × 1.0-m force plate.

Table 1. Overview of the abbreviations and calculation method of all predictor features and outcome feature that were used in the predictive model.

	Predictor feature	Determined by
$v_{\mathit{wc}}$	Wheelchair velocity in m/s^−1	(Gyroscope signal of IMU around wheel axis ⋅ rear wheel diameter ⋅ π)/360 (van der Slikke et al., 2015)^a
$a_{\mathit{wc}}$	Wheelchair acceleration in m/s^−2	Derivative of $v_{\mathit{wc}}$
${\phi }_{\mathit{tr}}$	Trunk inclination angle in rad	Based on extended Madgwick filter (van Dijk, Kok, et al., 2021), with β-value being 0.0015 (if |wheelchair acceleration| <0.1 m · s^−2 for at least five consecutive samples) or 0.9635 (otherwise)
${\dot{\phi }}_{\mathit{tr}}$	Angular velocity of trunk (around sagittal axis) in rad/s^−1	Gyroscope signal (around sagittal axis) of trunk-mounted IMU
$\mathit{\phi }\mathit{tr}$	Angular acceleration of trunk (around sagittal axis) in rad/s^−2	Derivative of ${\dot{\phi }}_{\mathit{tr}}$
$a_{\mathit{tr},\mathrm{\perp }}$	Trunk acceleration perpendicular to the frontal plane of the trunk in m/s^−2	Acceleration signal (directed perpendicular to frontal plane) of trunk-mounted IMU
$a_{\mathit{tr},\parallel }$	Trunk caudal-cranial acceleration in m/s^−2	Acceleration signal (in caudal-cradial direction) of trunk-mouned IMU
$\left|a_{\mathit{tr}}\right|$	Magnitude of trunk acceleration vector in m/s^−2	Euclidean norm of the three-dimensional acceleration signal of the trunk-mounted IMU subtracted by 9.81
	Outcome feature	Determined by
${\hat{F}}_{\mathit{N},\mathit{f}}$	Normalized relative front wheel-load in %	Force data from the front wheels’ load pins, calculated as $F_{\mathit{N},\mathit{f}}$/$F_{\mathit{N},\mathit{tot}}$ × 100%

^aIn case of curves and turns, the linear velocity obtained from the wheel should be corrected for turning using the algorithm described by van der Slikke et al. (2015).

Figure 3. Typical example of the relative load on the front wheels of the wheelchair (expressed as percentage of the total mass of the wheelchair and user) of a representative subject in the original wheelchair condition during one propulsion cycle (0% representing the start of the push) for propulsion style 1 (no trunk motion at 1.2 m · s⁻¹), style 2 (unrestricted trunk motion at 1.2 m · s⁻¹) and style 3 (unrestricted trunk motion at 1.7 m · s⁻¹).

Table 2. Mean (variation) and range of relative front wheel loads (expressed as percentage (%) of the total load) for all participants per pushing style (characterized by the amount of trunk motion, i.e., TM) in the training and validation set and test set. The range consists of the average front wheel load of the participant with the lowest load and the average front wheel load of the participant with the highest load. The variation is determined by the average of the standard deviations per person. The values are based on 60 s of propulsion per pushing style per person.

	Style 1: No trunk motion	Style 2: Moderate trunk motion	Style 3: Full trunk motion
Mean (training and validation set) (%)	20.0 (4.9)	24.9 (6.9)	27.1 (9.2)
Range (training and validation set) (%)	12.8–26.3	14.0–32.1	17.5–35.4
Mean (test set) (%)	20.7 (4.4)	24.0 (6.4)	24.9 (7.6)
Range (test set) (%)	15.2–26.2	16.3–31.5	17.0–31.2

Table 3. The results in terms of mean error (ME), mean absolute error (MAE) and root-mean-squared error (RMSE) of the five model types that were trained to predict the front wheel load as a percentage of the total mass of wheelchair and user (i.e., RFWL). Models were trained on the training set (with default hyperparameters) and evaluated on the validation set for which the results are shown.

	Evaluated on validation set					Evaluated on test set
	LR	RFR	MLP	LSTM	GRU	LSTM (final)	LSTM (final) + noise
ME (RFWL)	1.9	1.8	1.6	1.0	1.2	0.5	0.5
MAE (RFWL)	4.4	3.8	3.4	2.7	2.9	3.8	3.8
RMSE (RFWL)	5.9	5.1	4.6	3.5	3.6	4.4	4.4
Comp. time E^−03 (s)	0.0	0.1	0.4	1.8	1.8	-	-

Figure 4. Gold standard rolling resistance (black line), load distribution model-based rolling resistance (purple line) and drag test-based rolling resistance (dotted line) for the “original” condition, condition with −3.5 bar tyre pressure and condition with +5 kg added mass.

Table 4. Comparisons between the accuracies of the load distribution model-based rolling resistance (F_{roll,LD model}) and the drag test-based rolling resistance (F_roll,drag) based on the “original” condition-data from the test set (pushing style: full trunk motion). Accuracies were determined by comparison with the “gold standard” load pin-based rolling resistance (F_{roll,gold standard}) and expressed in mean error (ME), mean absolute error (MAE) and root-mean-squared error (RMSE). The actual gold standard rolling resistance values are also reported. The measures are given for each subject in the test set.

	Froll,gold standard (N)	Froll,LD model − Froll,gold standard (%)			Froll,drag − Froll,gold standard (%)
		ME	MAE	RMSE	ME	MAE	RMSE
Subject 1	8.6 (1.0)	0.5	0.7	0.9	−1.1	1.4	1.8
Subject 2	9.9 (1.0)	−0.4	1.0	1.3	−0.4	3.3	4.0
Subject 3	9.4 (0.9)	0.3	1.0	1.4	−2.3	2.8	3.9
Mean	9.3	0.1	0.9**	1.2*	−1.3	2.5	3.2

* p < 0.05, ** p < 0.01.

Table 5. Comparisons between the accuracies of the load distribution model-based rolling resistance (F_{roll,LD model}) and the drag test-based rolling resistance (F_roll,drag) based on the “−3.5 bar” and “+5 kg” condition-data from the test set (for each pushing style). Accuracies were determined by comparison with the “gold standard” load pin-based rolling resistance (F_{roll,gold standard}) and expressed in mean error (ME), mean absolute error (MAE) and root-mean-squared error (RMSE). The actual gold standard rolling resistance values are also reported.

Pushing style	Froll,gold standard (N)	Froll,LD model − Froll,gold standard (%)			Froll,drag − Froll,gold standard (%)
		ME	MAE	RMSE	ME	MAE	RMSE
−3.5 bar condition
Style 1	8.5 (0.7)	−0.5	0.8	1.0	0.4	1.5	1.9
Style 2	8.7 (0.8)	−0.4	1.0	1.3	−1.1	2.3	2.8
Style 3	8.7 (0.8)	−0.1	1.0	1.3	−1.2	2.7	3.4
+5 kg condition
Style 1	8.4 (0.5)	0.2	1.1	1.3	1.6	2.7	3.3
Style 2	8.5 (0.5)	−0.1	1.5	2.0	−0.2	3.2	3.8
Style 3	8.6 (0.6)	0.2	1.4	1.8	−0.9	3.7	4.5

Sauret, C., Vaslin, P., Lavaste, F., de Saint Remy, N., & Cid, M. (2013). Effects of user’s actions on rolling resistance and wheelchair stability during handrim wheelchair propulsion in the field. Medical Engineering & Physics, 35(3), 289–297. https://doi.org/10.1016/j.medengphy.2012.05.001

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Data availability statement

The raw data supporting the conclusions of this article are publicly available at 10.4121/bc9a8588-5e50-4dff-aa77-5114ff7626f7.v2. The machine learning model, including instructions on how to use it, is available at 10.4121/c533f919-1a44-48d5–8543-5c7f8be29bb0.