Swing leg retraction helps biped walking stability
In human walking, the swing leg moves backward just prior to ground contact, i.e. the relative angle between the thighs is decreasing. We hypothesized that this swing leg retraction may have a positive effect on gait stability, because similar effects have been reported in passive dynamic walking models, in running models, and in robot juggling. For this study, we use a simple inverted pendulum model for the stance leg. The swing leg is assumed to accurately follow a time-based trajectory. The model walks down a shallow slope for energy input which is balanced by the impact losses at heel strike. With this model we show that a mild retraction speed indeed improves stability, while gaits without a retraction phase (the swing leg keeps moving forward) are consistently unstable. By walking with shorter steps or on a steeper slope, the range of stable retraction speeds increases, suggesting a better robustness. The conclusions of this paper are therefore two-fold; (1) use a mild swing leg retraction speed for better stability, and (2) walking faster is easier
[Show description] [Hide description] DESCRIPTION: Swing-leg retraction in walking is the slowing or reversal of the forward rotation of the swing leg at the end of the swing phase prior to ground contact. For retraction, a hip torque is often applied to the swing leg at about the same time as stance-leg push-off. Due to mechanical coupling, the push-off force affects leg swing, and hip torque affects the stance-leg extension. This coupling makes the energetic costs of retraction and push-off depend on their relative timing. Here, we find the energy-optimal relative timing of these actions. We first use a simplified walking model with non-regenerative actuators, a work-based energetic-cost, and impulsive actuations. Depending on whether the late-swing hip torque is retracting or extending (pushing the leg forward), we find that the optimum is obtained by applying the impulsive hip torque either following or prior to the impulsive push-off force, respectively. These trends extend to other bipedal models and to aperiodic gaits, and are independent of step lengths and walking speeds. In one simulation, the cost of a walking step is increased by 17.6% if retraction torque comes before push-off. To consider non-impulsive actuation and the cost of force production, we add a force-squared (F^2) term to the work cost. We show that this cost promotes simultaneous push-off force and retracting torque, but does not change the result that any extending torque should come prior to push-off. A high-fidelity optimization of the Cornell Ranger robot is consistent with the swing-retraction trends from the models above. Robotica, first view, Cambridge Journals, Available online: http://journals.cambridge.org/repo_A99bB3BUGQE.HQ
- "Depending on the energetic cost properties of the actuators, this can reduce the net cost of the gait by reducing both the cost of active negative work performed during heel-strike and the cost of positive work to be performed elsewhere to make up for the heel-strike energy loss and maintain motion [5, 16]. Relative timing of push-off and retraction, a potential energy saver: Although gravity alone might retract the swing leg at slow walking speeds (e.g. in passive walkers [6, 17]), in general a hip torque is needed at the end of the swing phase to mediate the retraction and to bring the leg rotation rate to an optimal value. In walking, the stance-leg preemptive push-off, a key feature of energy efficient walking [13, 14], also takes place before heel-strike, at almost the same time as swing-retraction torque. "
[Show abstract] [Hide abstract] ABSTRACT: In biped walking, swing-leg retraction can reduce the energy loss at heel-strike by reducing the foot speed relative to the ground at touch-down. However, it also takes extra effort to brake and then accelerate the swing leg in the rearward direction. This energetic trade-off of retraction is influenced by the mechanical coupling with the stance leg preemptive push-off, and changes with different step lengths and speeds. Previously it has not been clear under which circumstances a retracting hip torque has a net energetic benefit (if ever). Here, using a simple biped model probed with numerical and analytic methods, we show how the effectiveness of leg retraction is influenced by actuator efficiencies for positive and negative work. As the efficiency of negative work decreases, the energetic advantage of retracting hip torque is found (mainly) for longer steps, whereas at shorter steps the hip joint extends (not retracts) prior to heel-strike. For fast walking, however, a braking hip torque is still required at the end of swing phase to ensure heel-strike.
- "Different researchers have used different definitions for the swing retraction rate. In ,  it is defined as the angular speed of the hip joint (angular speed of the swing leg relative to the stance leg), whereas in , ,  it is defined as the absolute angular speed of the swing leg (relative to an inertial reference). To avoid confusion, we refer to these speeds as the hip-joint retraction rate ˙ ψ, and the swing-leg retraction rate ω, respectively. "
[Show abstract] [Hide abstract] ABSTRACT: No legged walking robot yet approaches the high reliability and the low power usage of a walking person, even on flat ground. Here we describe a simple robot which makes small progress towards that goal. Ranger is a knee-less four-legged ‘bipedal’ robot which is energetically and computationally autonomous, except for radio controlled steering. Ranger walked 65.2 km in 186,076 steps in about 31 h without being touched by a human with a total cost of transport [TCOT ≡ P/mgv ] of 0.28, similar to human’s TCOT of ≈ 0.3. The high reliability and low energy use were achieved by: (a) development of an accurate bench-test-based simulation; (b) development of an intuitively tuned nominal trajectory based on simple locomotion models; and (c) offline design of a simple reflex-based (that is, event-driven discrete feed-forward) stabilizing controller. Further, once we replaced the intuitively tuned nominal trajectory with a trajectory found from numerical optimization, but still using event-based control, we could further reduce the TCOT to 0.19. At TCOT = 0.19, the robot’s total power of 11.5 W is used by sensors, processors and communications (45%), motor dissipation (≈34%) and positive mechanical work (≈21%). Ranger’s reliability and low energy use suggests that simplified implementation of offline trajectory optimization, stabilized by a low-bandwidth reflex-based controller, might lead to the energy-effective reliable walking of more complex robots.
- "When this was given as a parameter to the optimization the optimization chose a gait with swing-leg retraction prior to heel-strike and thus a higher vertical velocity of the swing foot before heel-strike. Note that this robustness from leg retraction due to improved state estimation is unrelated to the stability that leg retraction can provide in open loop control, as described elsewhere[45, 57]. The COT=0.28 controller was robust enough to work reliably on a running track where the maximum slopes were about 1 @BULLET and maximum step-to-step variation was a few mm, for the same controller. "