Static support boundary¶

Goal¶

Test the Iteration 0 prediction for issue #24: does the signed support margin reach zero before the rear feet unload?

Human prediction: a centered ballast keeps the COM centered; a forward ballast moves it toward the front; beyond the front support edge creates a tipping hazard. The human predicted rear-foot load would increase with forward ballast and estimated a 20–30° forward post-tip pitch.

This notebook is the analysis front end. It invokes the focused headless MuJoCo fixture only to obtain contact loads; it does not open a viewer or make a C-1N capability claim.

Setup¶

Key assumptions¶

Three feet remain fixed relative to the torso.
Only an abstract payload position changes.
The model is a support-mechanics fixture, not C-1N morphology.
A two-second rollout is used to let each condition settle or expose loss of rear contact.

Show code

In [1]:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

from static_support_boundary import PAYLOAD_SHIFT_LIMIT, observe

Steps¶

1. Sweep payload position¶

The sweep is coarse by design. Use the first transition to choose a narrower next sweep.

Show code

In [2]:

coarse_shifts = np.round(np.linspace(0.0, 0.9, 10), 2)
near_boundary_shifts = np.array([0.92, 0.94, 0.96, 0.98, 1.00, 1.10, 1.20])
shifts = np.concatenate((coarse_shifts, near_boundary_shifts))
records = []
for shift in shifts:
    observation = observe(float(shift), seconds=2.0)
    loads = observation['foot_normal_loads_n']
    records.append({
        'payload_shift_m': shift,
        'support_margin_m': observation['support_margin_m'],
        'front_load_n': loads[0],
        'rear_left_load_n': loads[1],
        'rear_right_load_n': loads[2],
        'minimum_rear_load_n': min(loads[1:]),
        'com_x_m': observation['com_world_xy_m'][0],
        'roll_deg': np.rad2deg(observation['torso_roll_rad']),
        'pitch_deg': np.rad2deg(observation['torso_pitch_rad']),
        'standing_metric_pass': bool(observation['standing_metric_pass']),
    })
results = pd.DataFrame(records)
results

Out[2]:

	payload_shift_m	support_margin_m	front_load_n	rear_left_load_n	rear_right_load_n	minimum_rear_load_n	com_x_m	roll_deg	pitch_deg	standing_metric_pass
0	0.00	0.130408	7.599474	5.765013	5.765013	5.765013	-0.001359	-3.072152e-18	0.005018	True
1	0.10	0.116914	8.792316	5.168592	5.168592	5.168592	0.029828	1.233045e-17	0.009814	True
2	0.20	0.103424	9.984834	4.572333	4.572333	4.572333	0.061006	-1.787620e-16	0.014506	True
3	0.30	0.089937	11.177012	3.976244	3.976244	3.976244	0.092175	-1.586728e-16	0.019086	True
4	0.40	0.076454	12.368841	3.380329	3.380329	3.380329	0.123335	2.336176e-17	0.023554	True
5	0.50	0.062976	13.560317	2.784591	2.784591	2.784591	0.154485	-1.639864e-16	0.027907	True
6	0.60	0.049501	14.751442	2.189028	2.189028	2.189028	0.185626	1.416759e-16	0.032147	True
7	0.70	0.036030	15.942219	1.593639	1.593639	1.593639	0.216757	5.748162e-17	0.036276	True
8	0.80	0.022564	17.132655	0.998421	0.998421	0.998421	0.247879	9.481841e-17	0.040294	True
9	0.90	0.009101	18.322758	0.403370	0.403370	0.403370	0.278993	3.048139e-18	0.044205	True
10	0.92	0.006409	18.560738	0.284379	0.284379	0.284379	0.285215	-1.422508e-16	0.044975	True
11	0.94	0.003717	18.798711	0.165393	0.165393	0.165393	0.291437	-7.185765e-17	0.045740	True
12	0.96	0.001007	19.038238	0.045629	0.045629	0.045629	0.297705	1.073486e-16	0.046506	True
13	0.98	-0.096543	15.158607	0.000000	0.000000	0.000000	0.545929	2.489353e-14	23.588235	False
14	1.00	-0.095626	15.146059	0.000000	0.000000	0.000000	0.542643	-5.733115e-11	23.934518	False
15	1.10	-0.093017	15.111429	0.000000	0.000000	0.000000	0.541255	-6.283008e-08	25.073569	False
16	1.20	-0.094944	15.138791	0.000000	0.000000	0.000000	0.549230	-1.344223e-07	24.208645	False

Show code

In [3]:

fig, axes = plt.subplots(2, 1, figsize=(8, 6), sharex=True)
axes[0].plot(results.payload_shift_m, results.support_margin_m, marker='o', color='#2f69ad')
axes[0].axhline(0, color='black', linewidth=1)
axes[0].set_ylabel('support margin (m)')
axes[0].set_title('Fixed-foot support boundary sweep')
axes[1].plot(results.payload_shift_m, results.front_load_n, marker='o', label='front')
axes[1].plot(results.payload_shift_m, results.rear_left_load_n, marker='o', label='rear left')
axes[1].plot(results.payload_shift_m, results.rear_right_load_n, marker='o', label='rear right')
axes[1].axhline(0, color='black', linewidth=1)
axes[1].set_xlabel('payload shift (m)')
axes[1].set_ylabel('normal load (N)')
axes[1].legend()
fig.tight_layout()

No description has been provided for this image

Checks¶

Fixture standing metric: all three feet have more than 0.001 N normal load after a two-second rollout, and both roll and pitch remain within ±5°. This is a test-bench metric, not the future C-1N STAND criterion.

A fixed-foot quasi-static boundary should pair an exhausted support margin with unloading of the limiting rear contact. The coarse sweep can bracket the transition, but it cannot establish an exact threshold.

Show code

In [4]:

last_positive_margin = results.loc[results.support_margin_m > 0].tail(1)
first_unloaded_rear = results.loc[results.minimum_rear_load_n <= 1e-3].head(1)
print('Last sampled positive margin:')
display(last_positive_margin)
print('First sampled unloaded rear contact:')
display(first_unloaded_rear)

Last sampled positive margin:

	payload_shift_m	support_margin_m	front_load_n	rear_left_load_n	rear_right_load_n	minimum_rear_load_n	com_x_m	roll_deg	pitch_deg	standing_metric_pass
12	0.96	0.001007	19.038238	0.045629	0.045629	0.045629	0.297705	1.073486e-16	0.046506	True

First sampled unloaded rear contact:

	payload_shift_m	support_margin_m	front_load_n	rear_left_load_n	rear_right_load_n	minimum_rear_load_n	com_x_m	roll_deg	pitch_deg	standing_metric_pass
13	0.98	-0.096543	15.158607	0.0	0.0	0.0	0.545929	2.489353e-14	23.588235	False

Takeaways¶

The centered case passes the fixture metric. At shift = 0.96 m, the margin is +0.0010 m and each rear foot carries 0.046 N; it also passes. At 0.98 m, both rear contacts are unloaded, the metric fails, and pitch is +23.6° after tipping.

The observed model corrects one prediction: forward ballast increases the front-foot load and decreases rear-foot load. The COM-location and forward-pitch predictions held. The support-margin and rear-unloading events are bracketed in the same 0.96–0.98 m interval.

A later C-1N experiment must separately measure changing support geometry and contact set during gait; this fixed-foot fixture cannot answer that dynamic question.