Passive-Dynamic-Walking-Inspired Dynamics Guidance for Energy-Efficient Humanoid Locomotion

Hyeonjin Choi; Joongheon Kim; Daekyum Kim

cs.ROSep 28, 2026

Passive-Dynamic-Walking-Inspired Dynamics Guidance for Energy-Efficient Humanoid Locomotion

Authors: Hyeonjin Choi, Joongheon Kim, Daekyum Kim

Organizations: School of Mechanical Engineering, Korea University, Seoul 02841, Republic of Korea · School of Electrical Engineering, Korea University, Seoul 02841, Republic of Korea · School of Smart Mobility, Korea University, Seoul 02841, Republic of Korea

Abstract

Learning energy-efficient humanoid locomotion requires discovering mechanically economical gait coordination, not merely reducing actuator effort. Reinforcement learning promotes efficiency through effort-related reward penalties, which guide the step-to-step mechanics of walking only indirectly. This article proposes a framework inspired by passive dynamic walking (PDW) that temporarily creates slope-equivalent conditions favorable to economical gait discovery and removes all PDW-specific guidance before nominal-dynamics optimization. During early training, a tilted-gravity field assists sagittal progression on flat collision geometry, complemented by curriculum-coupled reward terms. The core framework requires no reference trajectories, gait phases, or contact schedules. In a five-seed forward-locomotion study on a 29-DoF Unitree G1, the framework reduces mechanical cost of transport by 6.8-15.2% over commanded speeds of 0.5-2.0m/s without degrading velocity tracking. Mechanical-work decomposition attributes the reduction to positive actuator work, and reward-matched comparisons separate the guided regime's faster gait acquisition from the tilt's additional benefit to converged economy. The framework extends to unassisted omnidirectional locomotion, where its benefit persists once a walking-specific motion prior supplies kinematic coordination, the combination reducing speed-matched cost of transport by 18.7%. On hardware, forward cost of transport falls by 16.3% with the motion prior and by 4.5% without it, the latter within the trial-to-trial spread.

Figures & tables

Fig. 1: Overview of the proposed PDW-inspired slope-to-flat curriculum. Phase 1 : early policy search is conducted under a PDW-inspired tilted-gravity field, which supplies gravitational assistance to fore–aft step-to-step progression. Phase 2 : the tilt decays linearly, returning the training dynamics toward nominal. Phase 3 : all PDW-specific guidance is fully withdrawn, and the policy must sustain the induced gait structure under powered flat-ground command tracking. The collision geometry remains flat throughout; the depicted incline illustrates only the energetic effect of the tilted-gravity field.

Fig. 2: Training schedule of the proposed framework. The three curriculum components and the adversarial motion prior are shown against training time, divided into the three phases used throughout this article. Phase 1 : the tilted-gravity field supplies full slope-equivalent assistance, the PDW-inspired reward terms are active alongside the baseline objective, and commands are sagittally dominant. Phase 2 : the tilt decays linearly and the PDW-inspired reward terms are scaled down by the same curriculum factor

\sigma(k)

, while the command distribution expands toward the full omnidirectional set. Phase 3 : all PDW-specific guidance has vanished, and the policy is optimized under nominal flat-ground dynamics, the baseline objective alone, and the complete command distribution. The adversarial motion prior, when enabled, is active throughout training and is not part of the curriculum. Fig. 1 illustrates the mechanical intuition for the tilted-gravity component.

Fig. 3: Synchronized expansion of the translational command sectors and the yaw range over the curriculum. Left ( $k<K_{w}$ ): commands are restricted to two sagittal sectors spanning $\pm 45^{\circ}$ about the forward and backward directions, and no yaw command is issued. Middle ( $K_{w}\leq k<K_{t}$ ): each sector expands continuously toward $\pm 90^{\circ}$ while the yaw range opens. Right ( $k\geq K_{t}$ ): the union of the two sectors covers the full $360^{\circ}$ translational command range at the final yaw range. Green arrows denote admissible translational command directions; blue denotes the commanded yaw rate.

Condition	Tilted gravity	Velocity band	Positive-work
		(VB)	penalty (PW)
Flat (baseline)	—	—	—
Flat + VB	—	$\checkmark$	—
Flat + VB + PW	—	$\checkmark$	$\checkmark$
Slope + VB	$\checkmark$	$\checkmark$	—
Slope + VB + PW	$\checkmark$	$\checkmark$	$\checkmark$

TABLE I: Conditions of the Forward-Locomotion Mechanism Study

Fig. 4: Cost of transport on the sagittal task across commanded speeds (mean over 5 seeds; error bars show the S.D. of total CoT). Total bar height gives the total CoT, split into a positive component (dark) and a negative component (light). The full curriculum (Slope + VB + PW) attains the lowest CoT at every speed and lowers it primarily by shrinking the positive component, while the negative component stays essentially unchanged across all conditions and speeds.

Condition	$1.0\,\mathrm{CoT}^{+}$	$1.0\,\mathrm{CoT}^{-}$	$2.0\,\mathrm{CoT}^{+}$	$2.0\,\mathrm{CoT}^{-}$
Flat (baseline)	$0.3274$	$0.1213$	$0.4229$	$0.1562$
	${\pm}0.0065$	${\pm}0.0053$	${\pm}0.0205$	${\pm}0.0095$
Flat + VB	$0.3250$	$0.1241$	$0.4022$	$0.1557$
	${\pm}0.0083$	${\pm}0.0031$	${\pm}0.0202$	${\pm}0.0099$
Flat + VB + PW	$0.3003$	$0.1219$	$0.3642$	$0.1461$
	${\pm}0.0152$	${\pm}0.0029$	${\pm}0.0143$	${\pm}0.0037$

TABLE II: Mechanical-Work Decomposition,

\mathrm{CoT}^{\pm}

(Mean

\pm

S.D., 5 Seeds)

Fig. 5: Emergence of sustained stepping during training (mean

\pm

s.d. over five seeds). Left: mean foot-contact events per episode. Right: fraction of episodes with at least five step events. The horizontal axis is the PPO policy-update index. The flat and tilted-gravity variants with matched reward terms follow similar trajectories, whereas the conventional baseline reaches sustained stepping substantially later.

$\theta_{\mathrm{init}}$	$W^{+}/d$	$(W^{+}{+}W^{-})/d$	vs. $5^{\circ}$	Surv.	Failed
	[J/m]	[J/m]			seeds
$0^{\circ}$	$108.6\pm 4.2$	$152.8\pm 3.1$	$+4.9$ %	0.982	0/3
$3^{\circ}$	$108.1\pm 0.5$	$151.6\pm 1.1$	$+4.1$ %	0.980	0/3
$4^{\circ}$	$104.6\pm 2.3$	$149.1\pm 1.4$	$+2.4$ %	0.975	0/3
$\mathbf{5^{\circ}}$	$\mathbf{102.2\pm 0.5}$	$\mathbf{145.6\pm 1.9}$	—	0.992	0/3
$6^{\circ}$	$104.8\pm 1.7$	$148.9\pm 3.0$	$+2.3$ %	0.985	0/3

TABLE III: Initial-Tilt-Angle Sweep (3 Seeds)

Speed	CoT	CoT	$\Delta$	Vel. err.	Gait sym.
	(no tilt)	(perm. tilt)		$\Delta$	$\Delta$
0.5	0.4512	0.4882	$\mathbf{+8.2\%}$	$+183.4$ %	$-32.8$ %
1.0	0.4419	0.4554	$\mathbf{+3.1\%}$	$+80.7$ %	$-4.8$ %
1.5	0.4779	0.5388	$\mathbf{+12.8\%}$	$+17.3$ %	$-3.8$ %
2.0	0.5422	1.0848	$\mathbf{+100.1\%}$	$+161.0$ %	$-8.0$ %

TABLE IV: Permanent

5^{\circ}

Tilt vs. No Tilt, Rewards Identical (Paired Over 3 Seeds: 42, 123, 7)

Condition	CoT	Tracking	Stuck
	(speed-matched)	error [m/s]	rate [%]
Baseline	$0.5641{\pm}0.0072$	$0.110{\pm}0.005$	$0.02{\pm}0.03$
Ours	$0.5431{\pm}0.0196$	$0.118{\pm}0.009$	$0.07{\pm}0.12$
Baseline + AMP	$0.4982{\pm}0.0280$	$0.256{\pm}0.035$	$15.95{\pm}10.91$
Ours + AMP	$\mathbf{0.4588{\pm}0.0041}$	$0.244{\pm}0.056$	$\mathbf{4.60{\pm}4.13}$

TABLE V: Forward-Command Evaluation under the Omnidirectional Task (Mean

\pm

S.D., 3 Seeds)

Effect	Raw [%]	Speed-matched [%]
Full PDW-inspired curriculum, no AMP	$-4.70{\pm}2.40$	$-3.75{\pm}2.68$
Motion prior alone	$-7.99{\pm}9.14$	$\mathbf{-11.67{\pm}5.30}$
Both	$-17.03{\pm}2.62$	$-18.66{\pm}1.55$
Additive prediction	$-12.69{\pm}6.75$	$-15.42{\pm}2.64$
Interaction	$-4.34{\pm}4.13$	$\mathbf{-3.24{\pm}1.96}$

TABLE VI:

2\times 2

Decomposition of the CoT Effect (vs. Baseline; Mean

\pm

S.D., 3 Seeds)

Fig. 6: Sagittal joint power over the gait cycle, from ipsilateral foot touchdown to the next touchdown of the same foot, for forward locomotion speed-matched over a realized-speed range of 0.50–1.40 m/s. Curves are means across three independently trained seeds and shaded bands denote

\pm

one standard deviation across seeds; markers on the abscissa indicate each condition’s mean toe-off. Power is normalized by the nominal robot mass

m

. Note that the vertical scale differs between panels: the ankle operates at roughly one quarter of the hip and knee power range.

Method	CoT, Random	CoT, Forward	Vel. Err.
	Cmd. $\downarrow$	Cmd. $\downarrow$	$\downarrow$
Baseline	$0.598{\pm}0.025$	$0.589{\pm}0.009$	$0.139{\pm}0.004$
Ours	$0.577{\pm}0.024$	$0.562{\pm}0.021$	$0.141{\pm}0.005$
$\Delta$	$+3.5$ %	$+4.7$ %	$-1.7$ %
Baseline + AMP	$0.560{\pm}0.042$	$0.542{\pm}0.049$	$0.413{\pm}0.117$
Ours + AMP	$\mathbf{0.481{\pm}0.007}$	$\mathbf{0.489{\pm}0.012}$	$\mathbf{0.317{\pm}0.081}$

TABLE VII: Deployment-Condition CoT in Simulation (Mean

\pm

S.D., 3 Seeds)

Panel of Figure 14 — Fig. 7: Real-robot deployment on the Unitree G1. (a) Time-ordered snapshot sequences over roughly one walking stride for the four policies: Baseline and Ours without the motion prior, and Baseline + AMP and Ours + AMP. An overhead safety sling is attached in all trials and does not support the robot’s weight during walking. (b) Improvement of Ours over Baseline in cost of transport (solid) and in its positive-work component (hatched) by commanded direction, without AMP (green) and with AMP (red); positive values indicate greater efficiency for Ours, and distances are recovered from onboard sensing (Section IV-E ).

AMP	Direction	Baseline	Ours	$\Delta\mathrm{CoT}$	$\Delta\mathrm{CoT}^{+}$
No	Forward	0.581	0.555	$+4.5$ %	$+4.1$ %
	Backward	0.614	0.622	$-1.3$ %	$-2.1$ %
	Lateral	0.587	0.585	$+0.4$ %	$-1.9$ %
Yes	Forward	0.521	0.436	$\mathbf{+16.3}$ %	$\mathbf{+22.3}$ %
	Backward	0.499	0.499	$+0.1$ %	$+3.0$ %
	Lateral	0.489	0.406	$\mathbf{+17.0}$ %	$\mathbf{+13.1}$ %

TABLE VIII: Hardware Cost of Transport by Direction (Measured Distance)

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Simulation and control
Item	Value
Simulator	Isaac Lab 2.3.0 / Isaac Sim 5.1.0 (PhysX)
Physics step / decimation	0.005 s / 4
Control period $\Delta t$	0.02 s (50 Hz)
Episode length	20 s (1000 control steps)
Parallel environments	1024

Appendix

TABLE IX: Simulation, Actuation, and Domain Randomization

Item	Value
Recurrent core	single-layer LSTM, 256 units
MLP head hidden sizes	$[256,128]$
Observation stack	last 5 control steps
Actor input	$96\times 5=480$
Critic input	$99\times 5=495$ (adds base lin. vel.)
Learning rate	$10^{-3}$ (adaptive), target KL 0.01

Appendix

TABLE X: Policy / Critic Network and PPO Hyperparameters

#	Term	Formula $f_{i}$	Weight $w_{i}$
1	Linear-velocity tracking	$\exp\!\big(-\lVert\mathbf{c}_{v}-\mathbf{v}^{b}_{xy}\rVert^{2}/\sigma^{2}\big)$	$+1.0$
2	Angular-velocity tracking (yaw)	$\exp\!\big(-(\omega_{z}^{*}-\omega_{b,z})^{2}/\sigma^{2}\big)$	$+1.0$
3	Alive bonus	$\mathbf{1}[\,\text{not terminated}\,]$	$+0.15$
4	Base angular velocity (roll/pitch)	$\lVert\boldsymbol{\omega}_{b,xy}\rVert^{2}$	$-0.05$
5	Base orientation	$\lVert\mathbf{g}_{b,xy}\rVert^{2}$	$-1.0$
6	Joint velocity	$\sum_{j}\dot{q}_{j}^{2}$	$-1{\times}10^{-3}$

Appendix

TABLE XI: Baseline Locomotion Reward

r^{\mathrm{base}}

, Identical Across All Conditions

Term / parameter	Value	Gate / meaning
Reward terms, each multiplied by $\sigma(k)$
Positive-work penalty	$5\times 10^{-4}$	slope-gated
Velocity-band reward	0.3	slope-gated
lower threshold $v_{\min}$	—	$0.05$ m/s
upper bound (forward study)	—	$1.0$ m/s, absolute
upper bound (omnidir.)	$\eta_{\mathrm{VB}}$	$1.2\,s_{t}^{*}$

Appendix

TABLE XII: PDW-Inspired Reward Terms and Curriculum Schedule

Item	Value
Projected gravity / root height	3 / 1
Root lin. / ang. velocity (yaw frame)	3 / 3
Joint positions / velocities	29 / 29
Pelvis-relative key-body positions	12
Feature total	80
Discriminator MLP	$160$ – $1024$ – $512$ – $1$

Appendix

TABLE XIII: AMP Motion Feature

\Phi

(80-Dim) and Discriminator

Condition	0.5 m/s	1.0 m/s	1.5 m/s	2.0 m/s
Flat (baseline)	$0.4463$	$0.4487$	$0.5017$	$0.5791$
Flat + VB	$0.4508$	$0.4491$	$0.4861$	$0.5579$
Flat + VB + PW	$0.4260$	$0.4222^{*}$	$0.4553^{*}$	$0.5103^{**}$
Slope + VB	$0.4404$	$0.4392$	$0.4765$	$0.5339^{*}$
Slope + VB + PW	$\mathbf{0.4161}^{***}$	$\mathbf{0.4051}^{***}$	$\mathbf{0.4356}^{**}$	$\mathbf{0.4910}^{***}$
Full vs. baseline	$-6.8\%$	$-9.7\%$	$-13.2\%$	$-15.2\%$

Appendix

TABLE XIV: Cost of Transport, Sagittal Task (Mean

\pm

S.D., 5 Seeds)

Bin ( $n$ )	Baseline	Ours	Baseline + AMP	Ours + AMP
forward (368)	$0.573{\pm}0.009$	$0.544{\pm}0.022$	$0.510{\pm}0.038$	$\mathbf{0.470{\pm}0.008}$
backward (55)	$0.544{\pm}0.026$	$0.530{\pm}0.013$	$0.555{\pm}0.064$	$\mathbf{0.441{\pm}0.016}$
lateral (272)	$0.529{\pm}0.030$	$0.514{\pm}0.020$	$0.525{\pm}0.040$	$\mathbf{0.435{\pm}0.011}$
walk_turn (1183)	$0.581{\pm}0.016$	$0.561{\pm}0.018$	$0.562{\pm}0.047$	$\mathbf{0.479{\pm}0.007}$
all moving (1878)	$0.571{\pm}0.017$	$0.550{\pm}0.019$	$0.546{\pm}0.042$	$\mathbf{0.470{\pm}0.006}$

Appendix

TABLE XV: CoT by Commanded Direction (Mean

\pm

S.D., 3 Seeds;

n

= per-seed episodes per bin)

Bin	Ours vs. Baseline	Ours + AMP vs. Baseline + AMP
forward	5.2	8.0
backward	2.5	20.6
lateral	2.9	17.1
walk_turn	3.3	14.7
all moving	3.6	14.0

Appendix

TABLE XVI: Directional CoT Reduction (%)

Fig. 8: Distribution of positive actuator work among the sagittal leg joints during forward locomotion, speed-matched over a realized-speed range of 0.50–1.40 m/s. Each robot bar sums to 100% within the three sagittal joint groups (Section IV-D7 ). Bars are means over three seeds, and the across-seed standard deviation is at most 2.9 percentage points for every segment. The hatched row labels the walking ranges reported by Farris and Sawicki [ 18 ] , with segment widths set by the normalized midpoints of those ranges; those three ranges are reported independently and need not sum to

100\%

, so the row is a set of reference intervals rather than a partition.

Metric	Baseline	Ours	Baseline + AMP	Ours + AMP
Ankle peak [W/kg]	$0.10{\pm}0.08$	$0.17{\pm}0.02$	$0.64{\pm}0.14$	$0.57{\pm}0.03$
Peak phase [%]	$40{\pm}29$	$23{\pm}4$	$63{\pm}2$	$61{\pm}0$
Toe-off [%]	$62.8{\pm}3.6$	$65.0{\pm}0.4$	$69.5{\pm}0.4$	$68.3{\pm}1.0$
Push-off frac. [%]	$14{\pm}19$	$0{\pm}0$	$72{\pm}11$	$62{\pm}4$
$W^{+}_{\mathrm{ank,PO}}$	$0.0045$	$0.0021$	$0.0336$	$0.0324$
	${\pm}0.0033$	${\pm}0.0002$	${\pm}0.0065$	${\pm}0.0020$

Appendix

TABLE XVII: Phase-Resolved Sagittal Joint-Power Metrics (Mean

\pm

S.D., 3 Seeds)

CoT is the raw overall-command value at $m{=}33.34$ kg (Table IX ),
Framework	Policy	CoT	Vel. error [m/s]	Stuck rate
Baseline	MLP	$0.620{\pm}0.018$	$0.169{\pm}0.002$	$0.002{\pm}0.001$
Baseline	LSTM	$0.580$	$0.142$	$0.001$
Ours	MLP	$0.576{\pm}0.005$	$0.154{\pm}0.009$	$0.001{\pm}0.000$
Ours	LSTM	$0.561$	$0.143$	$0.000$
a few percent above the re-binned all-moving CoT of Table XV .

Appendix

TABLE XVIII: Random-Command Evaluation: Feed-Forward (MLP) vs. Recurrent (LSTM) Policy

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