## 3 Results
#### Forward modeling  
We first considered the condition of hip kinematics unchanged (full compensation at the hip by the hip muscles, Figure 3). This of course is a hypothetical situation that only can be reproduced in simulation, but it helps understanding the separate effect of RF at the knee joint. With the RF in place (100% RF force and residual knee moments applied), the model performed a normal step as expected. After removal of the RF and without any compensation from other knee extensor muscles (0% RF force and residual moments applied), exaggerated knee flexion and foot rising occurred in swing, which prevented the foot from getting in contact with the ground at the end of the stride cycle (Figure 3, left panel). The same occurred at 50% of RF force. At 75% of RF force, the step was still incomplete. In clinical terms this would correspond to abnormal foot contact occurring with the toes (initial toe contact). At 90% and 95% of the RF force, the foot recovered a position, at the end of the stride cycle, that was compatible with foot plant initial contact and heel initial contact, respectively. Figure 3: Left panel: kinematics resulting from decreasing RF force; right panel: kinematics resulting from increasing RF force. The model is represented in the position achieved at the end of stride time (100% stride time). The red and blue curves represent the trajectory of a point located on the heel of, respectively, the right side (unaffected, blue line) and the left side (our test limb, red line). The hip joint was fully compensated (hip kinematics imposed). When the RF was hyperactive, a reduced knee flexion was observed (Figure 3, right panel) that in some cases prevented the foot clearance (the forefoot moved below the ground level). As shown in Figure 3, knee extension was faster than at basic condition, so that at the time corresponding to stride duration the knee was hyperextended and the forward displacement of the foot was exaggeratedly enhanced. Figure 4 depicts the knee flexion angles for all the simulated percentages of RF force compensation (50%, 75%, 90%, 95%, 100%) and RF hyperactivation (105%, 110% 125%, 150%). It clearly reveals that knee flexion increases as far as the RF force decreases, and, at the contrary, knee flexion decreases while the RF force increases. Similar effects of RF removal and hyper-activation were obtained when the hip joint as well as the knee joint was left free to rotate under the effect of the RF force and residual hip moment (hip kinematics were not imposed). The entire lower limb kinematics dramatically changed for large reductions in RF force (condition 50% and 75% in Figure 5). Less apparent deviations were observed for RF force enhancement. For example, the increase in stride length seemed not too big at 150% RF (Figure 5, right panel). However, it has to be noted that the foot largely moves below ground level, and this is related to reduced movement at both the hip and knee. In clinical terms, this would correspond to great difficulty to obtain foot clearance. Figure 6 shows the changes in hip and knee joint angles. Decreasing the RF force produced an increase in both hip and knee flexion; the opposite occurred with increasing RF force. Compared to Figure 4, the reduction in knee flexion was more pronounced in this case in which the RF also had an effect at the hip. Figure 4: Changes of the knee joint angle with increasing and decreasing RF force. The hip joint was fully compensated (hip kinematics imposed). Figure 5: Left: kinematics resulting from decreasing RF force; right panel: kinematics resulting from increasing RF force. The model is represented in the position achieved at the end of stride time (100% stride time). The red and blue curves represent the trajectory of a point located on the heel of, respectively, the right side (unaffected, blue line) and the left side (our test limb, red line). The hip joint flexion was not imposed (no compensation for RF force changes). Figure 6: Changes of the hip and knee joint angles with increasing and decreasing RF force.