# Duty Factor Reflects Lower Limb Kinematics of Running
## 2 Materials and Methods
### Biomechanical Parameters  
From the marker set, a full-body biomechanical model with six degrees of freedom and 15 rigid segments was constructed. Segments included the head, upper arms, lower arms, hands, thorax, pelvis, thighs, shanks, and feet. In Visual3D software, segments were treated as geometric objects. Segments were assigned inertial properties and center of mass locations based on their shape [48] and attributed relative mass based on standard regression equations [49]. Kinematic parameters were calculated using rigid-body analysis and whole-body COM location was calculated from the parameters of all 15 segments (COM was directly provided by Visual3D). Lower limb joint angles (hip, knee, and ankle) were defined as the orientation of the distal segment relative to the proximal one [50] and computed using an _x-y-z_ Cardan sequence [51,52]. For example, the knee joint angle was computed as the orientation of the shank relative to the thigh. Using the proximal segment as the reference segment makes the _x-y-z_ Cardan sequence equivalent to the joint coordinate system [52,53], leading to rotations with functional and anatomical meaning (flexion-extension, abduction-adduction, and internal-external rotation). Segment angles (pelvis and foot) were defined as the orientation of a specific segment relative to the laboratory coordinate system and were also computed using an _x-y-z_ Cardan sequence. These segment angles have also functional meaning that are retroversion-anteversion, negative drop-positive drop, and external-internal rotation for the pelvis and forefoot-rearfoot angle, pronation-supination, and toe-out-toe-in for the foot. Running events were derived from the kinematic data. More explicitly, mid-toe and mid-foot landmarks were created midway between the first and fifth toe markers, and the heel marker and mid-toe landmark, respectively. The mid-toe landmark was rescaled by subtracting its respective global minimum. Moreover, heel and mid-toe accelerations were calculated as the second derivative of the heel marker and mid-toe landmark, respectively. As for each running trial, FS was defined as the first acceleration spike between the heel marker and mid-toe landmark acceleration spikes. TO was defined as the instance when the mid-toe reached 1 cm on ascent. \(t_{c}\) and swing time (\(t_{s}\)) were defined as the time from FS to TO and from TO to FS of the same foot, respectively, and \(t_{f}\) as the time from TO of a given foot to FS of the contralateral foot. MS was defined as the instant when the COM reached its lowest vertical position during \(t_{c}\) while MF was defined as the instant when the COM reached its highest vertical position during \(t_{f}\). The end of the IA phase was set at 50 ms after FS for every runner due to several reasons: (1) the passive impact peak of forefoot strikers is not identifiable without a force platform, (2) runners showing a passive impact peak have a large variability in the timing of this peak around 25 ms, and (3) this peak was shown to occur around 25 ms for speeds ranging from 3 to 5.5 m/s [28]. Passive peak attenuation time (\(t_{a}\)) was defined as the time from FS to IA, i.e., 50 ms. All events were verified to ensure correct identification and were manually adjusted when required. Values for \(t_{c}\), \(t_{a}\), \(t_{b}\), \(t_{p}\), \(t_{f}\), \(t_{c}\), \(t_{d}\), and \(t_{s}\) were calculated based on FS, IA, MS, TO, and MF events, and DF was calculated as follows [54]  
\[\text{DF}=\frac{t_{c}}{t_{c}+t_{s}} \tag{1}\]  
All COM positions and displacements were expressed as a percentage of participant's height. Temporal characteristics and COM positions and displacements were given as the average between right and left values. The joint and segment angles (\(\theta\)) were calculated at FS, IA, MS, and TO together with their range of motions during the attenuation phase (\(\Delta\theta_{a}\); from FS to IA), braking phase (\(\Delta\theta_{b}\); from FS to MS),and pushing phase (\(\Delta\theta_{p}\); from MS to TO).