Fig 1.
(A.) The “simplest walking model” treats the stance leg as an inverted pendulum, so that the body center of mass (COM) moves in an arc each step. (B.) Stance leg angle is measured with angle θ, with fixed angle 2α between the legs at double support, yielding fixed step length S. Point mass M, leg length L, and gravitational acceleration g serve as base units for non-dimensionalization of model equations. (C.) Nominal gait has a step-to-step transition between pendulum-like arcs. The COM velocity is redirected from forward-and-downward , to forward-and-upward
, through active, impulsive trailing leg push-off (PO), followed by impulsive leading leg collision (CO). (D.) Bump occurs on step i = 0, with an early collision, so that push-off energy is released late, resulting in post-bump velocity
. (E.) Stepping off the bump occurs with pre-emptive push-off. Optimal push-off sequence is computed to minimize total push-off work.
Fig 2.
Recovery strategies from a bump in N controlled steps.
(A.) Cumulative distance vs. time for walking over a basic bump (B = 0.025L) and recovering within N modulated steps (with no delay), with stepping onto the bump occurring at time 0. All gaits except nominal experience an early collision with the bump. The “No Compensation” strategy applies nominal work U for all steps (including mid-stance input at step 0), and eventually regains nominal walking speed, but with a time lag (“Lag”) relative to level walking. (B.) Optimal push-off sequences ui vs. step number for varying N, with the bump occurring at step number 0. (C.) Total relative positive work needed to recover in N steps, equivalent to the cumulative work of (B.) but subtracting the nominal work for the same number of steps on the level, N ⋅ U.
Fig 3.
Effect of delay d on recovery strategy.
(A.) Cumulative distance vs. time for recovery delayed by d steps, making up for lost time within N = 5 steps. (B.) Optimal push-off sequences ui vs. step number for varying d (bump at step number 0). (C.) Total cumulative positive work vs. N for various delay d; cumulative refers to the summed work of sequences from (B.), without subtracting nominal work as in Fig 2C. Open symbols denote steps where model tends to leap off the bump.
Fig 4.
Effect of bump height b on recovery strategy.
(A.) Cumulative distance vs. time for various bump heights, recovering with N ranging 2–8 steps and zero delay (only the first 3 steps shown for legibility). (B.) Optimal push-off sequences ui vs. step number for varying b. (C.) Total positive work as function of b, for recovery within N = 2 and 8 steps; total work is compared to the positive work to take the same number of nominal, level steps. (D.) Step time (τ0) atop the bump, as function of bump height b for N = 2 and 8.
Fig 5.
Recovery strategies for a single up step of height b (as opposed to up-and-down over a bump).
(A.) Cumulative distance vs. time for various number of steps N, with zero delay. (B.) Optimal push-off sequences ui vs. step number for varying N. (C.) Total additional positive work to recover as function of N, for recovery within N ranging 2–16 steps and varying delay d. Curves show total work cost over N steps, minus nominal work (N + d).
Fig 6.
Trade-off between time and work, for two-step recovery from a bump.
The nominal gait is represented by the time vs. positive work needed to take two steps of level, steady walking (Steady-state trade-off), as a function of varying two-step time (varied above and below the nominal 2T, vertical dashed line). These constitute economical options for steady walking at faster or slower speeds. A Transient trade-off curve for the standard bump (height 0.025 L), showing optimal responses for recovering from a bump in two steps (N = 2), taking more or less time than nominal. Various curves show cost for different bump heights. Also shown is the curve for a fully nonlinear model (compare “nonlinear” and “linearized,” both B = 0.025 L).