# Laboratory Assessment of Manual Wheelchair Propulsion
## 2 Experimental Testing Methods
### Determining Wheelchair Trajectory Based on Dynamometer Data  
Using fixed wheelchair dynamometers in biomechanical testing can limit the ability to capture the dynamic movement of the human-wheelchair system during propulsion. Traditional wheelchairs employ differential steering, where individual wheel speeds control direction and movement [46]. Replicating this user-controlled steering in laboratory settings can be challenging, especially when users lack sensory feedback, which may disrupt the study. To address this, it is crucial to inform participants about the intended wheelchair trajectory during testing. A method for determining such a trajectory, known as the trapezoid method, is described in detail in two articles, _Methods of Determining Trajectory for Wheelchair with Manual Pushrims Drive_[47] and _Bezier curve based trajectory planning for an intelligent wheelchair to pass a doorway_[48], and is called the trapezoid method. This method divides the wheelchair's movement into trapezoidal segments based on the differential steering principle. The turning radius (\(R\)) is determined by the speed difference between the left (\(v_{\mathrm{L}}\)) and right (\(v_{\mathrm{P}}\)) drive wheels and the wheelbase (\(L\)) (Figure 8). When measuring the kinematic parameters of the wheelchair, the left wheel path \(s_{\mathrm{L}}\) and right wheel path \(s_{\mathrm{P}}\) were measured independently in the same unit of time \(t\), dividing the movement of the wheelchair into equal intervals and modeling these intervals as trapezoids, in which the base is equal to the wheelbase \(L\) and the sides are equal to the lengths of the distance traveled by the left wheel \(s_{\mathrm{L}}\) and the right wheel \(s_{\mathrm{P}}\) (Figure 9A,B). According to the trapezoid method, the wheelchair movement should first be divided into equal time intervals for which the paths of the left wheel \(s_{L}\) and the right wheel \(s_{\mathrm{P}}\) are known (Figure 6a,b). Then, it is required to calculate the angle \(\alpha\) of inclination relative to the horizontal straight line connecting the ends of the left wheel path \(s_{L}\) and the right wheel \(s_{P}\) (1). \[\alpha=atan\left(\frac{s_{L}^{i}-s_{P}^{i}}{L}\right) \tag{1}\]  
Figure 8: Determination of the turning radius \(R\), using the trapezoid method based on known values of left wheel speed \(v_{\mathrm{L}}\), right wheel speed \(v_{\mathrm{P}}\), and wheelbase \(L\) (source: [47]). Figure 7: Relative error (**A**) and number of detection points (**B**) as a function of marker speed (adapted from [45]). The trajectory of a wheelchair can be described as a change in the position in the space of a point that is the center of the rear axle of the driving wheels. Therefore, for each trapezoid describing the path in equal time intervals t, you should determine the path of the center of the axis \(s_{C}\), which is the average of the paths traveled by the left wheel \(s_{L}\) and the right wheel \(s_{P}\) (2). \[s_{C}^{i}=\frac{s_{P}^{i}-s_{L}^{i}}{2} \tag{2}\]  
To model wheelchair movement using the trapezoidal method, the entire wheelchair path is approximated by a series of trapezoids. The initial trapezoid, which encloses the wheels' paths and the axle's center, has a base parallel to the x-axis. The point representing the center of the axle at time \(t=0\) serves as the origin of the coordinate system. Subsequent trapezoids are constructed such that their bases align with the line connecting the endpoints of the left \(s_{L}\) and right wheel \(s_{P}\) paths from the preceding propulsion cycle. Additionally, the center points of the driving wheel axle from the previous interval's end and the new interval's beginning must coincide, as illustrated in point B of Figure 9.  
To specify the trajectory of a wheelchair movement, it is sufficient to determine the coordinates of the position of the center point of the drive wheel axle for the end of each of the separated propulsion compartments (Figure 6d). The coordinates \(x_{0}=0\) and \(y_{0}=0\) are taken as the first point.