We display in Figure 5 the hip horizontal displacement and hip forward velocity for a complete, periodic, double step with an impulsive liftoff force. In order that the various walking trajectories may be more easily compared, we plot on the horizontal axis the normalized time for a step (normalized so that the final time for every step is 1). The stick walking figures on the top portion of the figure indicate for the two plots beneath it which part of the walking step the plotted points correspond to. The plots begin with the swing leg leaving the ground, and ends with the same leg about to leave the ground once again.
The solid (with ankle torques) and dashed (without ankles) lines indicate optimal walking solutions. The dashdot (with ankles) and dotted (without ankles) have an additional parameter fixed which is that of the average forward velocity set at a much faster 50 m/min. The first vertical line indicates the moment of collision, the second line is the time when the second step begins with the other leg lifting off the ground, and the last line is once again a collision of the swing leg with the ground.
From the velocity plot at the bottom of Figure 5 we see the biped loses a considerable amount of forward velocity at the moment of collision, in particular for the faster walking speed. The faster walking speed serves to exaggerate the overall effects as it is also more apparent that the biped needs to slow down quite a bit near the middle of the swing phase rather being able to maintain a more constant forward velocity. With the inclusion of a liftoff force, our experiments have shown that we are able to obtain more consistent forward motion and the large variation evident in the velocity plot is substantially reduced over the case when liftoff forces are not included. We speculate that without the explicit modeling of the foot, the biped cannot efficiently operate at higher speeds.
We may observe the vertical movement of the hip and knee in Figure 6. The height of the hip and knees stays roughly at the same level during the slower 12m/min globally optimal walk while during faster walks, the well-known sinusoidal motion effect of the hip is more apparent [10].