Foot-flip COT optimization

The truncated-heel foot shape was designed to give adequate ground clearance in swing if the swing toes are pointed up towards the shins. For robustness we use this full flip-up at every step, executing the flip-up quickly at the start of swing and lowering the foot quickly once the swing leg is sufficiently forward of the stance leg. We use the same flip-up and flip-down control independent of the walking speed or step length. We found that 0.25 s was fast enough for both flip-up and flip-down. Because the feet are light we assume that leg swing has no effect on ankle torques and currents, and vice versa.

Thus we did a decoupled optimization on the strategy for lifting (and, identically, for lowering) the foot the specified angle in the specified time (see appendix E, section E.1).

Figure 5.1b shows the contours for foot-flip COT after this optimization. As evident in equation 5.6 and from the figure 5.1b, the COT for foot-flip is inversely proportional to the step length and is independent of step velocity. Thus, the foot-flip COT favors walking at big step lengths.

Had we, instead, named a minimal foot clearance, the optimization would find a minimal motion that just makes that clearance. For example, if we only insist that the foot clear the ground the optimization finds a motion where the swing foot barely clears the ground for the whole swing phase. Here, there is a clear trade-off between energy use and robustness, and for this part of the control we choose robustness (big ground clearance by fast early foot lifting and fast late foot lowering).

The numerical flip-up and flip-down optimizations sought an ankle current as a func- tion of time Iaf(t), which starts the ankle motors from rest, turns the feet an angle of 1.7

optimization used the full motor model, accelerating the motor rotor inertia, but ne- glected the mass of the foot, the return spring and the Achilles-tendon compliance (tak- ing the tendon to be inextensible). The strategy for raising is the same as for lowering. Thus our cost function for flip-up-flip-down is,

Efoot-flip = 2 Z t=0.25s t=0 Pfoot-flip + dt (5.5)

where Pfoot-flip is the power in the motor that turns the foot attached to the swing leg in

single stance. The factor of 2 accounts for the inclusion of both flip-up and flip-down. To describe our approximation that the system is non-regenerative (the motors are not efficient at recharging the batteries when doing negative work) we define the function [P]+as follows, [P]+= P, if P ≥ 0 and [P]+= 0, if P < 0.

The optimization is non-trivial because of the relatively complex motor equation 3.9. To solve the problem numerically, we use a discrete approximation for current versus time with 33 intervals of piecewise linearly-varying current (see appendix E, section E.1).

The result of the optimization is a foot-flip cost Econst

foot-flip that is independent of step

length and speed.

Thus the foot-flip cost COT optimization problem from equations 5.2 and 5.3 is

COTfoot-flip =

Econst foot-flip

Mtotg dstep

. (5.6)

The optimized ankle flip-up current is shown between about 0.66 s and 0.9 s in the lower solid red curve in figure 5.2. The profile is approximately explicable in terms of what we would get by minimizing either of the standards for ‘energy’ optimization:R I2dtor R

power dt. The former optimization would give current varying linearly in time and the latter a bang-bang (impulsive) control. Thus we see a ramp with a slight spike at

each end. In addition there is a slight bias that seems to counter mechanical friction. As mentioned, the mirrored current is used for flip-down.

Because the foot-flip energy cost is independent of speed or step length, from equa- tion 5.6, we see that COTfoot-flip is inversely proportional to the step length and indepen-

dent of step velocity. That is, the foot-flip contribution to the TCOT is minimized by maximizing step length. Taking this cost together with the fixed cost and optimizing would give the longest possible steps at the greatest possible forward speed. The opti- mal solution is not maximally fast with maximally long steps, however, because of the swing and push-off costs, both of which penalize speed and step length.

5.2.3

‘Walk’ COT optimization

The first two terms in the expression for energy cost, the electrical overhead fixed cost and the foot flip up/down cost, have been reduced to simple dependencies on step length and step velocity.

The remaining ‘walk’ cost is given by

COTwalk = Ewalk Mtotg dstep = Z t=tstep t=0 P [Pwalk]+dt Mtotg dstep (5.7) where Pwalk are the various motor (plus controller) powers. The summation here is over

all the motors involved in powering the walk excluding the foot flip-up and flip-down. Again we assume that our batteries cannot regenerate (see appendix E, section E.2 for more details).

dence. Walking at frequencies other than those which are close to the ‘natural’ leg frequency are expensive. We found that the minimum for the ‘walk’ COT occurs at step length of 0.13 m and step velocity of 0.21 m/s (step frequency = 1.6 Hz). That the ‘walk’ COT is minimized for a non-zero step length came as a surprise because simple models of walking with work-based cost have shown that zero step lengths are optimum [40, 61, 62, 96].

Figure 5.1c shows a contour plot of the ‘walk’ COT (hip and stance-ankle motors) as a function of prescribed step length and speed. Note that step frequency is step velocity divided by step length. So, for a line through the origin, the reciprocal of the slope is the step frequency. The dotted line in the figure is one such line, a characteristic frequency of this machine. In this case it is the frequency of oscillation of the robot, a double pendulum with a spring at the intermediate joint, oscillating about vertical (but not the real root associated with the unstable falling motion).