SafeMind: A Risk-Aware Differentiable Control Framework for Adaptive and Safe Quadruped Locomotion

Zukun Zhang; Kai Shu; Mingqiao Mo

cs.ROApr 10, 2026

SafeMind: A Risk-Aware Differentiable Control Framework for Adaptive and Safe Quadruped Locomotion

Authors: Zukun Zhang, Kai Shu, Mingqiao Mo

Organizations: The University of Hong Kong, Pokfulam, Hong Kong, China · Alibaba, Beijing, China · University of Chinese Academy of Sciences, No.1 Yanqihu East Rd, Huairou District, Beijing, China

Abstract

Learning-based quadruped controllers achieve impressive agility but typically lack formal safety guarantees under model uncertainty, perception noise, and unstructured contact conditions. We introduce SafeMind, a differentiable stochastic safety-control framework that unifies probabilistic Control Barrier Functions with semantic context understanding and meta-adaptive risk calibration. SafeMind explicitly models epistemic and aleatoric uncertainty through a variance-aware barrier constraint embedded in a differentiable quadratic program, thereby preserving gradient flow for end-to-end training. A semantics-to-constraint encoder modulates safety margins using perceptual or language cues, while a meta-adaptive learner continuously adjusts risk sensitivity across environments. We provide theoretical conditions for probabilistic forward invariance, feasibility, and stability under stochastic dynamics. SafeMind is deployed on Unitree A1 and ANYmal C at 200~Hz and validated across 12 terrain types, dynamic obstacles, morphology perturbations, and semantically defined tasks. Experiments show that SafeMind reduces safety violations by 3--10x and energy consumption by 10--15% relative to state-of-the-art CBF, MPC, and hybrid RL baselines, while maintaining real-time control performance.

Figures & tables

Fig. 1: Overall architecture of SafeMind , a risk-aware differentiable control framework for adaptive and safe quadruped locomotion. Semantic cues and onboard sensory observations are encoded into task-dependent safety parameters $(h_{s},\alpha,\kappa)$ , which modulate the risk-aware differentiable safety layer. A stochastic Control Barrier Function (CBF) with variance-aware correction enforces the probabilistic safety constraint $L_{f}h+L_{g}h\,u+\alpha h-\kappa\sigma_{h}\geq 0$ through a differentiable quadratic program, yielding the safe control command $u^{\ast}$ . A meta-adaptive learning module continuously updates risk sensitivity according to violation statistics and environmental shifts. The resulting controller operates in real time on quadruped platforms such as Unitree A1 and ANYmal C, enabling robust locomotion across uneven terrain, dynamic obstacles, and semantically defined hazards.

Fig. 2: Safety violation rate (SVR) and tracking RMSE over time for SafeMind and baseline controllers. Curves show mean performance across repeated trials; shaded regions denote 95% confidence intervals. SafeMind maintains near-zero violation rates while achieving the lowest tracking error, significantly outperforming Safety–MPC, Robust–MPC, BarrierNet, and RL–CBF under mixed-terrain conditions.

Method	SVR (%)	RMSE (cm)	$\mathbf{E}\,(\mathrm{N\cdot m}^{2})$	$\mathbf{T_{a}}\,(\mathrm{s})$	Robustness Index
Nominal MPC	7.84 $\pm$ 0.90	3.4 $\pm$ 0.3	1.00 $\pm$ 0.04	2.41 $\pm$ 0.13	0.52 $\pm$ 0.03
RL–CBF	5.21 $\pm$ 0.70	3.8 $\pm$ 0.4	1.09 $\pm$ 0.05	2.03 $\pm$ 0.11	0.56 $\pm$ 0.03
Safety–MPC	2.91 $\pm$ 0.40	3.1 $\pm$ 0.3	1.18 $\pm$ 0.05	1.54 $\pm$ 0.09	0.63 $\pm$ 0.03
Robust–MPC	2.44 $\pm$ 0.30	3.0 $\pm$ 0.3	1.21 $\pm$ 0.05	1.47 $\pm$ 0.08	0.67 $\pm$ 0.03
BarrierNet	1.96 $\pm$ 0.30	3.4 $\pm$ 0.3	1.13 $\pm$ 0.04	1.33 $\pm$ 0.08	0.71 $\pm$ 0.02
SafeMind (ours)	0.58 $\pm$ 0.10	2.8 $\pm$ 0.2	0.97 $\pm$ 0.03	0.88 $\pm$ 0.06	0.89 $\pm$ 0.02

TABLE I: Comprehensive locomotion performance across 12 terrain families. All values are reported as mean

\pm

95% CI across 2400 episodes unless otherwise stated. Lower is better for SVR, RMSE,

E

, and

T_{a}

. Robustness index is normalized, higher is better.

Transition	Method	$\Delta\text{SVR}$ (%)	$T_{\mathrm{rec}}$ (s)
T1 $\rightarrow$ T5	Safety–MPC	3.8	1.72 $\pm$ 0.12
	BarrierNet	4.6	1.41 $\pm$ 0.10
	SafeMind	0.9	0.63 $\pm$ 0.05
T2 $\rightarrow$ T4	Safety–MPC	2.1	1.34 $\pm$ 0.10
	BarrierNet	2.7	1.12 $\pm$ 0.09
	SafeMind	0.6	0.48 $\pm$ 0.04

TABLE II: Safety degradation and recovery across terrain transitions.

T_{\mathrm{rec}}

is reported as mean

\pm

95% CI across repeated transition trials. Proportion-based metrics are reported as point estimates.

\Delta\text{SVR}

is violation rate increase immediately after transition.

T_{\mathrm{rec}}

is time to recover safe-set invariance.

Method	0%	15%	25%	35%
Nominal MPC	3.2	7.4	10.1	15.6
BarrierNet	1.1	3.8	6.3	9.7
Robust–MPC	0.9	2.8	4.2	6.9
Safety–MPC	1.2	3.1	5.0	7.3
SafeMind	0.3	0.9	1.7	3.1

TABLE III: Impact of uncertainty scaling on safety violation rate (%). All values are reported as mean

\pm

95% CI across repeated trials unless otherwise stated.

Method	SVR (%)	RMSE [cm]	Impulse	$T_{a}$ [s]
Nominal MPC	14.2	4.4 $\pm$ 0.3	1.82 $\pm$ 0.07	3.21 $\pm$ 0.18
BarrierNet	6.1	4.8 $\pm$ 0.4	1.63 $\pm$ 0.06	2.34 $\pm$ 0.14
Safety–MPC	4.3	4.2 $\pm$ 0.3	1.51 $\pm$ 0.05	1.89 $\pm$ 0.11
Robust–MPC	3.8	4.6 $\pm$ 0.3	1.49 $\pm$ 0.05	1.73 $\pm$ 0.10
SafeMind (ours)	1.12	3.6 $\pm$ 0.2	1.28 $\pm$ 0.04	1.02 $\pm$ 0.07

TABLE IV: Rocky Terrain Performance (T3, 200 episodes). Continuous valued metrics are reported as mean

\pm

95% CI unless otherwise stated. Proportion-based metrics are reported as point estimates.

Method	SVR (%)	RMSE [cm]	Energy	Jerk
Nominal MPC	16.7	5.2 $\pm$ 0.4	1.00 $\pm$ 0.04	1.00 $\pm$ 0.04
RL–CBF	11.4	5.6 $\pm$ 0.4	1.12 $\pm$ 0.05	1.19 $\pm$ 0.05
BarrierNet	7.9	5.8 $\pm$ 0.4	1.10 $\pm$ 0.05	1.14 $\pm$ 0.05
Robust–MPC	6.3	5.4 $\pm$ 0.4	1.18 $\pm$ 0.05	1.21 $\pm$ 0.05
SafeMind (ours)	1.94	4.3 $\pm$ 0.3	0.93 $\pm$ 0.03	0.87 $\pm$ 0.03

TABLE V: Rubble Field Traversal (T8). Continuous valued metrics are reported as mean

\pm

95% CI unless otherwise stated. Proportion-based metrics are reported as point estimates.

Method	SVR (%)	Success (%)	RMSE [cm]	$T_{a}$ [s]
Nominal MPC	22.5	61.3	5.1 $\pm$ 0.4	4.10 $\pm$ 0.22
BarrierNet	9.7	82.0	4.8 $\pm$ 0.3	2.66 $\pm$ 0.15
Safety–MPC	7.4	88.1	4.6 $\pm$ 0.3	2.21 $\pm$ 0.13
Robust–MPC	5.6	90.8	4.9 $\pm$ 0.3	2.08 $\pm$ 0.12
SafeMind (ours)	1.85	97.2	3.7 $\pm$ 0.2	1.12 $\pm$ 0.07

TABLE VI: Gap Crossing Performance (T9). Continuous valued metrics are reported as mean

\pm

95% CI unless otherwise stated. Proportion-based metrics are reported as point estimates.

Method	SVR (%)	Lateral RMSE [cm]	Energy	Jerk
Nominal MPC	19.8	4.2 $\pm$ 0.3	1.00 $\pm$ 0.04	1.00 $\pm$ 0.04
BarrierNet	8.9	3.8 $\pm$ 0.3	1.13 $\pm$ 0.05	1.21 $\pm$ 0.05
Safety–MPC	6.3	3.5 $\pm$ 0.2	1.17 $\pm$ 0.05	1.27 $\pm$ 0.05
Robust–MPC	4.1	3.6 $\pm$ 0.3	1.22 $\pm$ 0.05	1.25 $\pm$ 0.05
SafeMind (ours)	1.02	2.4 $\pm$ 0.2	0.91 $\pm$ 0.03	0.83 $\pm$ 0.03

TABLE VII: Narrow Corridor Navigation (T11). Continuous valued metrics are reported as mean

\pm

95% CI unless otherwise stated. Proportion based metrics are reported as point estimates.

Method	SVR (%)	RMSE [cm]	$E$	$T_{a}$ [s]
Nominal MPC	14.8	4.1 $\pm$ 0.3	1.00 $\pm$ 0.04	3.88 $\pm$ 0.21
BarrierNet	5.3	4.4 $\pm$ 0.3	1.11 $\pm$ 0.05	2.41 $\pm$ 0.14
Safety–MPC	3.7	4.0 $\pm$ 0.3	1.18 $\pm$ 0.05	1.97 $\pm$ 0.12
Robust–MPC	2.9	4.2 $\pm$ 0.3	1.21 $\pm$ 0.05	1.62 $\pm$ 0.10
SafeMind	0.77	3.3 $\pm$ 0.2	0.95 $\pm$ 0.03	0.89 $\pm$ 0.06

TABLE VIII: Mixed-Adversarial Terrain Transitions. Continuous valued metrics are reported as mean

\pm

95% CI unless otherwise stated. Proportion-based metrics are reported as point estimates.

Method	SVR (%)	RMSE [cm]	$E$	Success (%)
Nominal MPC	16.2	6.4	1.00	58.1
BarrierNet	8.3	6.1	1.11	72.4
Safety–MPC	5.6	5.4	1.14	78.6
Distributional RL	4.1	6.8	1.21	74.2
Ensemble–CBF	3.7	5.9	1.17	81.0
SafeMind	1.12	4.3	0.98	92.7

TABLE IX: Morphology Variation and Structural Disturbance.

Method	SVR (%)	RMSE (cm)	Energy
Nominal MPC	11.2	4.1	1.00
BarrierNet	5.3	4.0	1.08
Safety–MPC	3.9	3.7	1.12
Robust–MPC	3.0	4.2	1.15
SafeMind	0.88	3.1	0.96

TABLE X: Low–Friction Indoor Environment (20 trials).

Method	SVR (%)	Pitch Var (deg 2 )	Energy
BarrierNet	4.8	2.31	1.13
Safety–MPC	3.7	2.08	1.15
Robust–MPC	2.5	1.84	1.19
SafeMind	0.64	1.12	1.04

TABLE XI: Gravel and Soft Soil (ANYmal C).

Method	Clear. Viol. (%)	RMSE (cm)	Energy
Safety–MPC	3.4	3.2	1.16
BarrierNet	2.8	3.4	1.12
Ensemble–CBF	2.1	3.1	1.15
SafeMind	0.47	2.6	1.07

TABLE XII: Industrial Stairs – Clearance Violations.

Method	Wall Contact (%)	Yaw Var (deg 2 )	Energy
Nominal MPC	14.7	5.1	1.00
RL–CBF	8.3	4.4	1.04
BarrierNet	3.2	3.9	1.09
SafeMind	0.36	2.8	1.03

TABLE XIII: Narrow Corridor Safety.

Method	$\mathbf{d<1.5\,\mathrm{m}}$ Events	Tracking RMSE	Energy
Nominal MPC	17	3.8	1.00
RL–CBF	9	4.2	1.11
BarrierNet	6	3.4	1.09
SafeMind	0	3.1	1.05

TABLE XIV: Human–Robot Safety Distance Maintenance.

Method	SVR (%)	Pitch Var	Energy
BarrierNet	4.1	2.7	1.12
Safety–MPC	2.3	2.2	1.14
SafeMind	0.77	1.6	1.06

TABLE XV: Outdoor Grass under Weather Variation.

Method	Trip Incidents	Clear. Viol.	Energy
Safety–MPC	4	2.5%	1.16
BarrierNet	3	2.1%	1.14
SafeMind	0	0.6%	1.10

TABLE XVI: Wrinkled Carpet Experiments.

Method	Hazard Entered	RMSE	Energy
RL–CBF	13	4.3	1.11
BarrierNet	4	3.8	1.13
Safety–MPC	2	3.5	1.12
SafeMind	0	3.2	1.07

TABLE XVII: Semantic Region Avoidance.

Method family	Key properties / limitations
BarrierNet	Differentiable CBF–QP layer with fixed barrier parameters; cannot adapt risk to time-varying stochasticity
Safety– / Robust– / Stochastic MPC	Strong guarantees in deterministic or worst-case settings; high online cost and limited suitability for 200–250 Hz on-board control; not end-to-end differentiable
Ensemble–CBF	Uses model ensembles to capture epistemic uncertainty; still enforces deterministic CBF constraints without probabilistic invariance
RL–CBF	RL policy filtered by CBF QP; improves safety during learning but breaks differentiability and lacks explicit stochastic barrier modeling
Distributional RL	Learns risk-sensitive value distributions; no mechanism for state-wise safety invariance or hard constraints
Nominal MPC	High tracking performance but no safety guarantees; used as performance upper bound and safety lower bound

TABLE XVIII: Summary of baseline families and structural characteristics.

Protocol element	Description
Randomization	Terrains, friction, mass/inertia, sensor noise, external pushes, semantic layouts
Control frequencies	250, 200, 100, 50 Hz
Episode horizons	12–15 s (short), up to 600 s (long-horizon)
Metrics	SVR, tracking RMSE, $E$ , jerk, impact impulse, $T_{a}$ , semantic compliance
Baselines	Differentiable CBF layer, MPC variants, ensemble- and RL-based controllers
Statistical reporting	Mean $\pm$ std., Wilson 95% CIs, paired bootstrap significance tests

TABLE XIX: Unified evaluation protocol used for all controllers.

Configuration	SVR (%)	RMSE (cm)	Energy
Full SafeMind	0.58	2.9 $\pm$ 0.2	0.96 $\pm$ 0.03
w/o Stochastic Term	3.74	3.3 $\pm$ 0.2	1.04 $\pm$ 0.04
w/o Risk Tightening $(\kappa=0)$	5.92	3.2 $\pm$ 0.2	1.03 $\pm$ 0.04
w/o Meta-Adaptation	1.48	3.1 $\pm$ 0.2	1.07 $\pm$ 0.04
w/o Semantic Encoder	2.61	3.5 $\pm$ 0.3	1.01 $\pm$ 0.03
Fixed $(\alpha,\kappa)$	1.83	3.1 $\pm$ 0.2	1.02 $\pm$ 0.03

TABLE XX: Extended ablation across all simulation domains. Continuous valued metrics are reported as mean

\pm

95% CI unless otherwise stated. Proportion-based metrics are reported as point estimates.

Parameter Scaling	SVR (%)	RMSE (cm)	Energy
Mass $+15\%$	0.72	3.1 $\pm$ 0.2	0.99 $\pm$ 0.03
Mass $-15\%$	0.69	3.0 $\pm$ 0.2	0.98 $\pm$ 0.03
Friction $-40\%$	1.31	3.4 $\pm$ 0.2	1.02 $\pm$ 0.03
Latency $+20$ ms	1.12	3.7 $\pm$ 0.3	1.05 $\pm$ 0.04
Control Frequency 100 Hz	1.84	3.3 $\pm$ 0.2	1.11 $\pm$ 0.04
Control Frequency 50 Hz	3.02	3.8 $\pm$ 0.3	1.20 $\pm$ 0.05

TABLE XXI: Sensitivity of SafeMind to parameter scaling. Continuous valued metrics are reported as mean

\pm

95% CI unless otherwise stated. Proportion-based metrics are reported as point estimates.

Terrain Transition	Mean $\kappa$	Std	Peak
Flat $\rightarrow$ Slippery	2.48 $\pm$ 0.14	0.91	4.83 $\pm$ 0.27
Soft $\rightarrow$ Rubble	3.07 $\pm$ 0.18	1.12	5.14 $\pm$ 0.31
Rubble $\rightarrow$ Gap	2.81 $\pm$ 0.16	0.97	4.55 $\pm$ 0.28
Gap $\rightarrow$ Flat	1.21 $\pm$ 0.09	0.41	2.19 $\pm$ 0.15

TABLE XXII: Statistics of risk sensitivity

\kappa

across terrain transitions. Mean and Peak are reported as mean

\pm

95% CI; Std denotes the empirical standard deviation of

\kappa

over the transition window.

Constraint	T2	T4	T5	T8
Foot Clearance	22.3	18.1	13.8	31.4
Body Tilt	11.8	33.2	24.9	29.5
Joint Limits	4.3	5.9	4.8	6.4
Semantic Boundaries	0.0	2.7	4.6	7.1

TABLE XXIII: Constraint activation frequencies (percentage of active cycles). All values are reported as point estimates across repeated runs.

Stress Scenario	SVR (%)	Failure Rate	Recovery Time (s)
50 N Push	1.14	0.0 (0/200)	1.25 $\pm$ 0.09
Friction Collapse	2.38	0.0 (0/200)	1.91 $\pm$ 0.13
300 ms Perception Drop	1.07	0.0 (0/200)	1.48 $\pm$ 0.10
10 cm Drop-off	1.44	0.0 (0/200)	1.72 $\pm$ 0.12

TABLE XXIV: Aggregated results under rare-event stress tests. Recovery time is reported as mean

\pm

95% CI. Proportion-based metrics are reported as point estimates. Failure rate is reported as observed episode-level failure percentage with count in parentheses.

Controller	150 s	300 s	450 s	600 s
Nominal MPC	4.2%	7.3%	9.7%	12.4%
Safety–MPC	1.9%	3.1%	4.8%	5.6%
BarrierNet	1.6%	2.8%	3.7%	4.2%
SafeMind	0.41%	0.58%	0.65%	0.71%

TABLE XXV: Long-horizon accumulated violation probability. All values are reported as mean

\pm

95% CI unless otherwise stated.

$\kappa$	SVR (%)	Energy	RMSE (cm)
0.5	1.82	0.91 $\pm$ 0.03	2.4 $\pm$ 0.2
1.5	0.93	0.95 $\pm$ 0.03	2.7 $\pm$ 0.2
3.0	0.58	0.96 $\pm$ 0.03	2.9 $\pm$ 0.2
5.0	0.41	1.04 $\pm$ 0.04	3.3 $\pm$ 0.2

TABLE XXVI: Safety–performance Pareto frontier for varying

\kappa

. Continuous valued metrics are reported as mean

\pm

95% CI across repeated trials. Proportion-based metrics are reported as point estimates.

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