EqC: Equilibrium Energy Control for Introspective Lifelong Robot Learning

Haodong Yang; Runhan Huang; Bowen Zhang; Iofai Lei; Zunzhe Zhang; Xiangcheng Zhang; Shaoting Zhu; Linzhan Mou

cs.ROOct 13, 2026

EqC: Equilibrium Energy Control for Introspective Lifelong Robot Learning

Authors: Haodong Yang, Runhan Huang, Bowen Zhang, Iofai Lei, Zunzhe Zhang, Xiangcheng Zhang, Shaoting Zhu, Linzhan Mou,

Organizations: Tsinghua University · Galaxea AI · Princeton University

Abstract

Long-term robot autonomy depends on more than effective action generation: a policy must recognize when its behavior is no longer sufficiently supported and use assistance to improve. We introduce Equilibrium Energy Control (EqC), a framework for introspective lifelong robot learning built around an explicit conditional energy landscape. EqC casts generative action refinement as equilibrium-seeking optimization, while the geometry of the refinement process reveals whether the generated behavior remains consistent with the learned landscape. This competence signal enables Energy-Gated DAgger, where persistent energy-consistency violations trigger expert takeover and the resulting recovery trajectories are aggregated for subsequent policy updates. EqC integrates naturally into both Diffusion Transformer policies and flow-matching-based vision-language-action (VLA) policies. Across simulation and real-world bimanual manipulation, EqC matches or improves control performance and robustness while enabling timely intervention near the policy's competence boundary. A real-world study further shows how these interventions progressively expand the range of situations handled autonomously, providing a practical path from static generative controllers toward self-improving robot systems.

Figures & tables

Table 1: Success rates (%) on LIBERO with the DiT policy and VLABench with VLA policies.

Table 2: VLA policy success rates (%) on RoboTwin 2.0 (Easy / Hard) and LIBERO-Plus.

Figure 2: Real-world bimanual execution. Top (1–3): Flower Insertion. Middle (1–3): Towel Folding. Bottom (1–3): Debris Sweeping.

Table 3: Control and competence-awareness evaluation. (a) Success and latency on three real-world bimanual tasks (20 trials each). (b) Failure detection on pooled LIBERO and LIBERO-Plus rollouts. “EqC+Diff-DAgger” denotes the Diff-DAgger flow-loss detector applied to the EqC backbone; EqC uses path-wise energy consistency.

Panel of Figure 3 — Figure 3: Real-world failure detection (mean $\pm 1\,\text{std}$ , $n=25$ ). (a) EqC cleanly separates success and failure early for timely intervention, whereas (b) Diff-DAgger exhibits heavy overlap.

Panel of Figure 4 — Figure 4: Real-world lifelong learning evaluation with Energy-Gated DAgger. Top: Quantitative success curves across deployment rounds (averaged over 3 independent runs, with 20 autonomous evaluation trials per checkpoint per run). Bottom: Qualitative rollouts: early drift triggers $U_{\Delta E}$ alarms (red banners) for expert recovery; fine-tuned policies achieve autonomous execution (Rows 3–4). Across both (a) Hierarchical Block Stacking and (b) Garment Folding , EqC achieves 96.7% and 98.3% mean success rates, respectively, decisively outperforming Diff-DAgger and human intervention.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Benchmark	Coverage	Episodes
LIBERO–DiT	4 suites; 40 tasks	50/task
LIBERO–VLA	4 suites; 40 tasks	50/task
RoboTwin 2.0	5 tasks; Easy and Hard	100/task/setting
VLABench	Tracks 1, 2, 3, 4, and 6; 50 tasks	50/task
LIBERO-Plus	7 perturbation families	10030 total

Appendix

Table 4: Evaluation coverage and rollout budgets.

Seed	Goal	Spatial	Object	Long	Avg.
0	95.6	97.2	98.6	85.4	94.3
1	96.0	97.0	98.4	87.0	94.7
2	95.0	96.0	97.6	85.4	93.5
3	95.8	95.2	97.8	84.2	93.3
Mean $\pm$ sample s.d.	$95.60\pm 0.43$	$96.35\pm 0.93$	$98.10\pm 0.48$	$85.50\pm 1.15$	$93.95\pm 0.66$

Appendix

Table 5: Detailed EqC success rates (%) across multiple training seeds on LIBERO.

Method	Mean (%)	Sample s.d. (%)
Fixed RF, 20 steps	89.12	0.80
GeCO	92.30	1.02
EqC (ours)	93.95	0.66

Appendix

Table 6: Overall LIBERO success across multiple training seeds.

Method	Goal	Spatial	Object	Long	Avg.
$\pi_{0}$ + FAST ( Pertsch et al., 2025 )	88.6	96.4	96.8	60.2	85.5
$\pi_{0}$	95.8	96.8	98.8	85.2	94.2
$\pi_{0}$ + GeCO	96.4	96.8	97.0	85.4	93.9
$\pi_{0}$ + EqC (ours)	94.6	97.6	98.8	86.2	94.3

Appendix

Table 7: LIBERO success rates (%) with the VLA backbone.

Table 8: Detailed composition of the shared

10\%

calibration split (

1{,}203

total rollouts).

Table 9: Sensitivity analysis of detection window size

m

under two distinct threshold calibration strategies on the 90% test split.

	FPR-Constrained Calibration			Youden’s $J$ Index Calibration
$m$	Diff-DAgger	EqC + Diff	EqC (ours)	Diff-DAgger	EqC + Diff	EqC (ours)
3	33.6	44.5	45.2	47.9	51.8	53.7
4	31.8	42.5	45.3	39.2	53.8	57.4
5	32.1	42.7	44.5	49.4	51.8	55.5
6	30.6	42.6	46.7	45.2	52.2	51.5
7	32.0	42.8	46.6	47.0	50.2	57.1

Appendix

Table 10: Evaluation of early intervention capability (Time Saved %) on the 90% test split. The operational decision thresholds are calibrated using the 10% calibration split under FPR-Constrained and Youden’s

J

index strategies.

Method	Orig.	Camera	Robot	Lang.	Light	Backgr.	Noise	Layout	W. Avg.
Diff-DAgger ( $m=4$ )	27.7	50.5	50.1	46.4	30.0	34.0	44.1	47.9	46.1
Diff-DAgger ( $m=11$ )	26.5	44.2	46.0	39.9	32.5	32.3	37.3	44.9	41.5
Diff-DAgger ( $m=13$ )	28.9	47.2	48.6	44.1	33.8	34.5	43.1	46.1	44.6
Diff-DAgger ( $M=16,m=4$ )	27.0	45.0	47.5	44.1	33.8	28.0	39.0	44.0	42.7
Diff-DAgger ( $M=16,m=11$ )	23.5	41.3	42.4	37.4	28.5	30.2	35.1	40.6	38.4
Diff-DAgger ( $M=16,m=13$ )	25.9	42.8	43.5	39.0	30.7	30.5	37.1	41.8	39.8

Appendix

Table 12: Breakdown of theoretically optimal inference time saved (%) across LIBERO and LIBERO-Plus suites under representative evaluation setups. Orig. corresponds to the LIBERO tasks, while the remaining seven columns represent environmental perturbation variants in LIBERO-Plus. Diff-DAgger uses

M=10

unless otherwise specified.

Method	Mean Latency (ms)
Diff-DAgger ( $M=10$ )	79.44
Diff-DAgger ( $M=16$ )	91.43
Diff-DAgger ( $M=32$ )	122.11
EqC + Diff-DAgger ( $M=10$ )	171.03
EqC (ours)	108.86

Appendix

Table 13: Comparison of per-replan inference latency (ms) across methods. The latency includes both model action generation and failure-detection signal computation (evaluated at batch size = 1 after JAX warm-up). The optimization step budget

K

for EqC is set to 10.

Table 14: Ablation analysis comparing single-point energy signals against full path-wise EqC across window sizes

m

under two calibration strategies on the 90% test split.

Hyperparameter	Value
Base VLA Backbone	Galaxea G0.5
Optimizer	AdamW
Batch Size	16
Peak Learning Rate	$8\times 10^{-5}$
Warmup Schedule	Linear warmup (500 steps)
Decay Schedule	Cosine decay ( warmup_constant_cosine )

Appendix

Table 15: Unified training and inference hyperparameters for real-robot experiments.

Flower Insertion
Method	Time (min)	Overhead
Galaxea G0.5	11.58	—
Galaxea G0.5 + EqC (ours)	14.58	+25.9%
Towel Folding
Galaxea G0.5	40.48	—
Galaxea G0.5 + EqC (ours)	49.42	+22.1%

Appendix

Table 16: Wall-clock fine-tuning time across real-robot tasks.

Method	Run	R0 (18)	R1 (24)	R2 (30)	R3 (36)	R4 (42)	R5 (48)
Human Intervention	Run 1	0.0	0.0	5.0	15.0	20.0	50.0
	Run 2	0.0	0.0	10.0	35.0	60.0	70.0
	Run 3	0.0	0.0	0.0	20.0	45.0	55.0
	Mean $\pm$ Std	$0.0\pm 0.0$	$0.0\pm 0.0$	$5.0\pm 5.0$	$23.3\pm 10.4$	$41.7\pm 20.2$	$58.3\pm 10.4$
Galaxea G0.5 + Diff-DAgger	Run 1	0.0	0.0	20.0	60.0	70.0	90.0
	Run 2	0.0	0.0	10.0	80.0	70.0	70.0

Appendix

Table 17: Detailed progressive success rates (%) across 3 independent lifelong learning runs on the physical Galaxea bimanual workstation (20 autonomous trials per checkpoint per run). Both individual run outcomes and summary statistics (

\text{Mean}\pm\text{Std}

) are reported across deployment rounds.

Figure 5: Cross-task adaptation on Hierarchical Cup Stacking . Top (Round 0): Zero-shot deployment causes grasp slippage and drift, promptly caught by a persistent $U_{\Delta E}$ failure alarm. Bottom (Round 2): After two rounds of Energy-Gated DAgger, the policy achieves fully autonomous execution, cleanly assembling the three-tier cup tower.

Metric / Task	Round 0 (Base)	Round 1 (+6 Cup)	Round 2 (+6 Cup, +6 Block)
Cumulative Demos	48 (Block only)	54 (48 Block, 6 Cup)	66 (54 Block, 12 Cup)
Block Stacking Success	100.0	60.0	100.0
Cup Stacking Success	0.0	50.0	100.0
Dual-Task Average	50.0	55.0	100.0

Appendix

Table 18: Progressive performance (%) across continual learning rounds under task shift.

Methods	Goal	Spatial	Object	Long	Average
EqC (uniform 15 steps)	95.8	95.2	98.4	83.0	93.1
EqC (uniform 20 steps)	95.4	96.2	98.0	87.2	94.2
EqC (Cosine decay)	95.6	97.2	98.6	85.4	94.3

Appendix

Table 19: Success rate (%) of EqC under different optimization step schedules across the LIBERO suites. The highest values in each column are highlighted in bold.

Step $k$	$0$	$5$	$6$	$7$	$8$	$9$	$14$	$19$	$32$
Mean Gradient Norm $\\|\nabla E\\|$	49.3	44.4	29.9	12.9	6.0	3.6	1.2	0.7	0.3
$E$ Saturation Ratio (%)	0.00	82.51	95.18	99.22	99.79	99.91	99.99	–	–

Appendix

Table 20: Convergence profile of mean gradient norm

\|\nabla E\|

and normalized energy saturation ratio across optimization steps

k

on LIBERO.

Task Suite	Baseline ( $K=32$ )	$g_{\mathrm{min}}=1.0$	$g_{\mathrm{min}}=3.0$
libero_goal	$95.6$ / $32.0$	$97.0$ / $16.9$	$97.0$ / $11.1$
libero_spatial	$96.4$ / $32.0$	$95.2$ / $17.3$	$98.0$ / $11.3$
libero_object	$98.1$ / $32.0$	$98.0$ / $16.2$	$97.6$ / $10.9$
libero_long	$85.5$ / $32.0$	$84.6$ / $16.8$	$84.2$ / $10.9$
Average	$\mathbf{94.0}$ / $\mathbf{32.0}$	$\mathbf{93.7}$ / $\mathbf{16.8}$	$\mathbf{94.2}$ / $\mathbf{11.1}$

Appendix

Table 21: Performance and inference efficiency under adaptive early stopping across LIBERO task suites. Metrics are reported as Success Rate (%) / Average Optimization Steps per action chunk.

Figure 6: Proactive competence-boundary detection of EqC across rigid and deformable manipulation. Persistent $U_{\Delta E}$ violations trigger timely expert recovery at the earliest onset of execution drift, preserving an ample window for teleoperation before irreversible physical compounding occurs. Top row (Hierarchical Block Stacking): Left: Alarm raised when the block is accidentally pushed away, preventing it from being pushed further; Right: Alarm triggered by angular misalignment. Bottom row (Garment Folding): Left: Alarm triggered when the robot attempts to grasp the right sleeve but fails to secure it; Right: Alarm triggered upon failure to fold the garment to the designated position.

Figure 7: Qualitative failure modes of Galaxea G0.5 + Diff-DAgger across rigid and deformable manipulation. Rows 1–2 evaluate Hierarchical Block Stacking under nominal setups (two independent instances per category), while Row 3 evaluates Garment Folding . Left column (Delayed Detection): The detector remains silent throughout compounding drift, alarming only belatedly after the block arrangement is disrupted or a severe angular tilt has already occurred, missing the teleoperation recovery window. Middle column (Spurious Alarm): During nominal execution—including pristine, fully unfolded garment setups (Row 3)—noisy flow-loss expectation triggers false-positive alarms, needlessly aborting valid rollouts. Right column (Irreversible Failure): The detector triggers only after irrecoverable failures have occurred (e.g., mis-sequenced block placement or fabric bunched into an entangled knot), leaving no viable recovery path without a complete manual scene reset.

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