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CALOS: Control-Affine Lyapunov On-manifold Safety Layer for Safe Deep Reinforcement Learning for Quadrotors
Fabrizio Cesareo, Sebastiano Mengozzi, Nicola Mimmo, Andrea Acquaviva
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
Last extracted: 9/17/2026, 3:51:13 AM
Summary
The paper introduces CALOS (Control-Affine Lyapunov On-manifold Safety), a runtime safety layer for Deep Reinforcement Learning (DRL) in quadrotor control. CALOS unifies attitude tilt constraints and Lyapunov stability conditions into a single quadratic program (QP) to compute minimum-norm torque corrections. Evaluated in NVIDIA Isaac Lab, CALOS reduces lateral tracking error by 55-60% compared to unconstrained PPO, achieves zero attitude constraint violations on training trajectories, and accelerates training convergence by restricting exploration to safe state regions.
Entities (7)
Relation Signals (6)
CALOS → appliesto → Quadrotor
confidence 96% · Deep Reinforcement Learning has demonstrated remarkable capability in quadrotor control... We present CALOS... a runtime safety layer that enforces attitude constraints on a quadrotor
CALOS → uses → Quadratic Program
confidence 95% · CALOS formulates four tilt-angle inequalities and a Lyapunov descent condition as a single quadratic program whose solution is the minimum-norm correction to the nominal torque output of the policy.
CALOS → enforces → Tilt Constraints
confidence 94% · CALOS formulates four tilt-angle inequalities and a Lyapunov descent condition as a single quadratic program
CALOS → enforces → Lyapunov Stability
confidence 94% · CALOS formulates four tilt-angle inequalities and a Lyapunov descent condition as a single quadratic program
CALOS → evaluatedin → NVIDIA Isaac Lab
confidence 92% · Evaluated on trajectory-tracking tasks in NVIDIA Isaac Lab
CALOS → improves → Proximal Policy Optimization
confidence 90% · Evaluated on trajectory-tracking tasks in NVIDIA Isaac Lab, CALOS reduces lateral tracking error by 55-60% relative to an unconstrained Proximal Policy Optimization baseline
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Abstract
Abstract:Deep Reinforcement Learning has demonstrated remarkable capability in quadrotor control, yet learned policies offer no guarantee of respecting safety constraints during training or deployment. We present CALOS (Control-Affine Lyapunov On-manifold Safety), a runtime safety layer that enforces attitude constraints on a quadrotor without modifying the underlying learning algorithm. CALOS formulates four tilt-angle inequalities and a Lyapunov descent condition as a single quadratic program whose solution is the minimum-norm correction to the nominal torque output of the policy. The quadratic program is solved exactly via active-set enumeration over the three-dimensional torque space, with a computational cost low enough to enforce constraints in real time across thousands of parallel simulation environments, as required by modern massively parallel Deep Reinforcement Learning training. Evaluated on trajectory-tracking tasks in NVIDIA Isaac Lab, CALOS reduces lateral tracking error by 55-60% relative to an unconstrained Proximal Policy Optimization baseline while achieving zero attitude-constraint violations on the training trajectory. By restricting exploration to safe regions of the state space, the safety layer also accelerates training convergence and improves data efficiency without producing suboptimal policies.
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- Source: https://arxiv.org/abs/2609.17758v1
- Canonical: https://arxiv.org/abs/2609.17758v1
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CALOS: Control-Affine Lyapunov On-manifold Safety Layer for Safe Deep Reinforcement Learning for Quadrotors Fabrizio Cesareo Sebastiano Mengozzi Nicola Mimmo Andrea Acquaviva Affiliation: Department of Electrical, Electronic, and Information Engineering “Guglielmo Marconi” — DEI Affiliation: University of Bologna, Italy Abstract Deep Reinforcement Learning has demonstrated remarkable capability in quadrotor control, yet learned policies offer no guarantee of respecting safety constraints during training or deployment. We present CALOS (Control-Affine Lyapunov On-manifold Safety), a runtime safety layer that enforces attitude constraints on a quadrotor without modifying the underlying learning algorithm. CALOS formulates four tilt-angle inequalities and a Lyapunov descent condition as a single quadratic program whose solution is the minimum-norm correction to the nominal torque output of the policy. The quadratic program is solved exactly via active-set enumeration over the three-dimensional torque space, with a computational cost low enough to enforce constraints in real time across thousands of parallel simulation environments, as required by modern massively parallel Deep Reinforcement Learning training. Evaluated on trajectory-tracking tasks in NVIDIA Isaac Lab, CALOS reduces lateral tracking error by 55–60% relative to an unconstrained Proximal Policy Optimization baseline while achieving zero attitude-constraint violations on the training trajectory. By restricting exploration to safe regions of the state space, the safety layer also accelerates training convergence and improves data efficiency without producing suboptimal policies. I Introduction Deep Reinforcement Learning (DRL) policies trained in simulation can push quadrotors to the edge of their flight envelope [1], yet the resulting controllers inherit no classical stability guarantee [2]. Safe DRL [3] addresses this gap through two broad strategies: modifying the optimization criterion via Constrained Markov Decision Processes (CMDPs) [4, 5], or modifying the action at runtime via shielding [6] or control barrier functions [7]. A persistent tension remains: CMDP solvers can destabilize learning [8], while runtime filters applied during training risk correcting actions so aggressively that the gradient signal is degraded and the policy fails to improve. A further limitation is architectural: many runtime approaches enforce a single certificate type and compose them sequentially. ATACOM [9, 10] projects actions onto the constraint manifold tangent space but lacks energy-based certificates, offering no guarantee on rotational energy dissipation. Lyapunov-based safe policy optimization [11] enforces energy dissipation but treats constraints independently, so satisfying one may violate another. The Barrier-Lyapunov Actor-Critic [12] integrates both but requires constrained actor updates and a backup controller, coupling safety enforcement to the learning algorithm. None jointly optimize over tilt and Lyapunov constraints in a single step, leaving sequential composition as the only option, with the feasibility issues this entails. This paper presents CALOS (Control-Affine Lyapunov On-manifold Safety), a runtime safety layer that addresses these limitations. CALOS unifies attitude-tilt constraints and a Lyapunov stability condition into a single quadratic program (QP), producing the minimum-norm torque correction at each control step. The layer is transparent to the learning algorithm, here Proximal Policy Optimization (PPO) [13], and compatible with any policy gradient method. I Problem Formulation We consider quadrotor trajectory tracking: a learning-based controller must follow a time-varying reference pd(t)p_d(t) while respecting attitude constraints. The quadrotor uses the Combined Thrust and Body Torques (CTBT) representation [14]: the translational and rotational dynamics are: mv˙ m v =mge3−TRe3+Fd, =mge_3-TRe_3+F_d, (1) JΩ˙ J =τ−Ω×JΩ, =τ- × J , (2) where m is the mass, v the inertial velocity, R the rotation matrix from body to world frame, FdF_d the aerodynamic drag, J∈ℝ3×3J ^3× 3 the inertia matrix, Ω∈ℝ3 ^3 the body angular velocity, and τ∈ℝ3τ ^3 the body torque vector. The CTBT is input-affine which is essential for linear constraints on the input. The DRL policy outputs u0=(T,τ0)u_0=(T,\, _0), where T is the collective thrust and τ0∈ℝ3 _0 ^3 is the nominal body torque. The safety layer receives τ0 _0 and the current state x=(gb,Ω)x=(g_b, ), comprising the body-frame gravity vector and angular velocity, and returns a corrected torque: τ=(x,τ0),τ=S(x, _0), (3) where S denotes the safety operator, leaving thrust unmodified to preserve vertical authority and avoid coupling with altitude regulation. I Safety Layer Design I-A Tilt constraints and Predictive Tilt Projection Attitude tilt is expressed via the gravity direction in body frame, gb=R⊤e3g_b=R e_3, where the subscript b denotes the body-frame representation, avoiding Euler-angle singularities. We impose roll and pitch limits ϕmax=θmax=60∘ _ = _ =60 . Using a symplectic Euler discretization of g˙b=gb×Ω g_b=g_b× and the rotational dynamics, the predicted gb,t+1g_b,t+1 depends affinely on τ [9]: gb,t+1≈gconst(xt)+M(xt)τ,g_b,t+1≈ g_const(x_t)+M(x_t)\,τ, (4) where gconstg_const collects all τ-independent terms, and M(xt)=Δt2[gb]×J−1M(x_t)= t^2[g_b]_×J^-1 is the sensitivity matrix mapping torque to changes in the body-gravity vector; [gb]×[g_b]_× is the skew-symmetric matrix of gbg_b. Imposing |gb,x|≤sinθmax|g_b,x|≤ _ , |gb,y|≤sinϕmax|g_b,y|≤ _ yields four linear inequalities: APTP(x)τ≤bPTP(x),A_PTP(x)\,τ≤ b_PTP(x), (5) where APTP=[Mx−MxMy−My]∈ℝ4×3,bPTP=[sinθmax−gconst,xsinθmax+gconst,xsinϕmax−gconst,ysinϕmax+gconst,y]∈ℝ4,A_PTP= bmatrixM_x\\ -M_x\\ M_y\\ -M_y bmatrix\! ^4× 3, b_PTP= bmatrix _ -g_const,x\\ _ +g_const,x\\ _ -g_const,y\\ _ +g_const,y bmatrix\! ^4, (6) with MxM_x and MyM_y denoting the first and second rows of M(xt)M(x_t). The vector bPTPb_PTP encodes the tilt bounds offset by gconstg_const. The Predictive Tilt Projection (PTP) enforces (5) via an ATACOM-inspired [9] iterative correction. Let v+(τ)=maxAPTPτ−bPTP, 0v_+(τ)= \A_PTP\,τ-b_PTP,\,0\; the torque is corrected as: τ←τ−APTP†kv+(τ)τ←τ-A_PTP \,k\,v_+(τ) (7) where k>0k>0 is a tunable correction gain and APTP†=APTP⊤(APTPAPTP⊤+λI)−1A_PTP =A_PTP (A_PTPA_PTP +λ I)^-1 is the damped pseudo-inverse of APTPA_PTP, with damping factor λ>0λ>0 ensuring numerical conditioning. I-B Lyapunov constraint and scaling We define a Lyapunov candidate V(x)=12qg‖eg‖2+12qω‖Ω‖2V(x)= 12q_g\|e_g\|^2+ 12q_ω\| \|^2 (8) on tilt error eg=gb−gd,be_g=g_b-g_d,b and angular velocity Ω , and enforce V˙(x,τ)≤−cV(x) V(x,τ)≤-cV(x) with decay rate c>0c>0. Since the rotational dynamics are control-affine, V˙ V is affine in τ, yielding: AL(x)τ≤bL(x),A_L(x)\,τ≤ b_L(x), (9) where AL=qωΩ⊤J−1∈ℝ1×3A_L=q_ω J^-1 ^1× 3 captures the sensitivity of rotational energy to applied torque, and bL=−cV(x)−BL(x)∈ℝb_L=-cV(x)-B_L(x) , with BL(x)=qgeg⊤(gb×Ω)−qωΩ⊤J−1(Ω×JΩ)B_L(x)=q_g\,e_g (g_b× )-q_ω\, J^-1( × J ) collecting the tilt-error coupling and the gyroscopic terms. As a standalone operator, the Lyapunov scaling, enforces (9) by reducing torque magnitude: τ=sτ0τ=s\, _0, s∈[0,1]s∈[0,1]. I-C Sequential composition A sequential Cascade composition of tilt projection, Lyapunov scaling and tilt projection check, does not guarantee joint feasibility. When the tracking error is large, V(x)V(x) grows and scaling drives s→0s→ 0, nullifying the torque when the controller needs maximum to recover the large error. This motivates our unified formulation. I-D CALOS: unified formulation CALOS stacks (5) and (9) into a single system of m=5m=5 linear constraints: A(x)=[APTP(x)AL(x)]∈ℝ5×3,b(x)=[bPTP(x)bL(x)]∈ℝ5.A(x)= bmatrixA_PTP(x)\\ A_L(x) bmatrix\! ^5× 3, b(x)= bmatrixb_PTP(x)\\ b_L(x) bmatrix\! ^5. (10) The safety layer solves: τ⋆=argminτ∈ℝ312‖τ−τ0‖2s.t.A(x)τ≤b(x),τ = _τ ^3\; 12\|τ- _0\|^2 .t. A(x)τ≤ b(x), (11) which admits a unique solution whenever the feasible set is non-empty. CALOS can be solved in two ways. CALOS-P is a projected approximation that solves (11) approximately by applying the same damped pseudo-inverse correction (7) to the unified system (A,b)(A,b), enforcing all five constraints jointly without guaranteeing the exact minimum-norm solution, trading exactness for speed in large batches, which directly reduces training time when the safety layer must be evaluated across thousands of parallel environments at every timestep. CALOS-QP is an exact solver that exploits the fact that τ∈ℝ3τ ^3 is three-dimensional, so at most three linearly independent constraints can be simultaneously active. This yields ∑k=03(5k)=26 _k=0^3 5k=26 candidate active sets. For each candidate I, the Karush–Kuhn–Tucker optimality conditions give τ=τ0−AI⊤λτ= _0-A_I λ and (AIAI⊤)λ=AIτ0−bI(A_IA_I )λ=A_I _0-b_I [15]. A candidate is feasible if the dual variables satisfy λ≥0λ≥ 0 and the resulting torque satisfies all constraints Aτ≤bAτ≤ b; among all feasible candidates, we select the one minimizing ‖τ−τ0‖2\|τ- _0\|^2.If none exists, the uncorrected policy action τ0 _0 is applied and the actuators are clamped to their physical limits, ensuring that the system remains within hardware-safe bounds even when the QP has no feasible solution. The enumeration is fully parallelizable, essential for modern DRL training where the safety layer must scale to thousands of environments without becoming a bottleneck for the training pipeline. IV Experiments IV-A Setup All agents are trained with PPO [13] using the DRL library SKRL on 4096 parallel environments in NVIDIA Isaac Lab for ∼109 10^9 timesteps. The policy is a two-layer MLP 64 units each, with ELU activation and Gaussian output. The observation ot∈ℝ15o_t ^15 comprises body-frame velocity vbv_b, angular velocity Ωb _b, projected gravity gbg_b, desired position pd,bp_d,b, and desired velocity vd,bv_d,b, all in ℝ3R^3. A reward curriculum progressively tightens the tracking precision from 0.50 m to 0.15 m throughout training, requiring increasingly accurate position following as the policy improves. Six variants are compared: PPO, PTP, Lyapunov scaling, Cascade, CALOS-P, CALOS-QP. We evaluated the six variants over 200 episodes on three different trajectories. The first is the training Lissajous ℒ1L_1, with amplitude of 1 m and period of 5 s. The second is a harder Lissajous ℒ2L_2, with amplitude of 1.5 m and 43% higher peak velocity than the training reference. The third is a circle C with radius r=1r=1 m and angular rate ωc=1.777 _c=1.777 rad/s, chosen so that the tangential speed matches the peak velocity of the training reference.ℒ2L_2 and C are never seen during training and are tested with a large initial position offset. A mean lateral error exceeding 0.20 m is considered a tracking failure. IV-B Tracking Performance TABLE I: Mean lateral error e¯xy e_xy [m] across evaluation conditions. Agent ℒ1L_1 ℒ2L_2 (unseen) C (unseen) PPO 0.157 0.171 0.165 PTP 0.081 0.107 0.040 Lyapunov 0.127 0.256 0.569 Cascade 0.108 0.243 0.352 CALOS-P 0.062 0.076 0.043 CALOS-QP 0.069 0.113 0.042 Table I summarizes tracking accuracy. CALOS-P and CALOS-QP reduce lateral error by 55–60% on ℒ1L_1 relative to PPO. On the unseen trajectories with large initial offset, Lyapunov and Cascade fail to track the reference due to the torque nullification discussed in Section I, while both CALOS variants maintain errors below 0.08 m. Figure 1 shows results on the circular trajectory C . The constant curvature of this reference produces a steady-state attitude regime once the initial transient has settled, making it straightforward to distinguish the transient recovery phase from nominal tracking and to identify differences in constraint enforcement across methods. The trajectory plot (top) confirms that CALOS-QP tracks the reference closely while PPO exhibits visible drift. In the attitude plot (bottom), PTP is also shown to highlight the effect of the Lyapunov component: CALOS-QP extends PTP by adding the energy dissipation constraint (9) to the unified QP, and this additional term visibly accelerates the convergence of roll and pitch toward the steady-state regime. PTP, which enforces only tilt bounds, exhibits larger oscillations and a slower settling, because it has no mechanism to actively dissipate rotational energy during the recovery phase. CALOS-QP, by jointly enforcing both tilt and Lyapunov constraints, damps the angular transient more aggressively while still respecting the attitude bounds. Fig. 1: Circular trajectory C with large initial offset. Top: xyxy-plane tracking (reference dashed). Bottom: roll and pitch; dashed lines mark the ±60∘± 60 constraint. PTP is included to show the effect of the Lyapunov constraint: CALOS-QP settles faster due to active energy dissipation. IV-C Safety Analysis Table I reports attitude safety metrics on ℒ1L_1 evaluated with a large initial position offset, a condition never encountered during training, designed to stress-test the safety layer under extreme tracking error. The roll and pitch constraint enforced by CALOS is set at 60∘60 . TABLE I: Roll-violation statistics on ℒ1L_1 with large initial offset (200 episodes, 60∘ limit). Avg and Max refer to consecutive steps in violation. Agent max|ϕ| |φ| Viol. eps. Avg steps Max steps PPO 116.9∘ 4096 5.0 6 PTP 68.7∘ 1122 1.8 3 Lyapunov 180.0∘ 2988 5.2 27 Cascade 175.4∘ 1599 3.2 14 CALOS-P 71.5∘ 4096 2.6 3 CALOS-QP 99.9∘ 3069 4.3 9 CALOS-P violations last at most 3 consecutive steps, occurring only during the first simulation steps when the controller is correcting the large initial error; once the attitude enters the feasible region, no further violations occur. In contrast, Lyapunov violations reach a maximum tilt of 180∘180 , lasting up to 27 consecutive steps. On ℒ1L_1 without an initial offset, CALOS-QP achieves zero roll-violation episodes, demonstrating that the safety layer fully enforces the attitude constraints under nominal conditions. The residual violations observed under large initial offsets could be further reduced by including such high-error conditions in the training distribution. IV-D Internalized Safe Behavior The policy trained with CALOS retains safer and more accurate behavior even without the safety layer at test time, as shown in Table I: the mean lateral error is 55–74% lower than PPO. The constraint projection prevents unrecoverable attitudes during training, so all samples come from states where meaningful learning occurs. This restricts exploration to safe regions, accelerating convergence and improving data efficiency. Moreover, CALOS formulation preserves sufficient freedom within the feasible set, avoiding suboptimal policies: the policy internalizes the constraint boundaries as part of its nominal behavior while matching or exceeding the tracking performance of the unconstrained baseline. TABLE I: Tracking with the safety layer disabled at test time. Traj. Agent e¯xy e_xy [m] R¯ R ℒ1L_1 PPO 0.157 93.2 CALOS-P (layer off) 0.062 100.9 CALOS-QP (layer off) 0.067 100.9 C PPO 0.165 90.1 CALOS-P (layer off) 0.043 99.0 CALOS-QP (layer off) 0.041 100.3 IV-E Why Sequential Methods Fail The failure of Lyapunov scaling and Cascade under large tracking errors is due to the multiplicative structure: τ=sτ0τ=s _0 with s→0s→ 0 when V(x)V(x) is large, nullifying the torque precisely when maximum authority is needed. CALOS encodes the Lyapunov condition as AL(x)τ≤bL(x)A_L(x)τ≤ b_L(x), defining in ℝ3R^3 the set of all torque vectors whose component along ALA_L does not exceed the dissipation budget bLb_L. The QP can project the torque to satisfy dissipation while preserving a large component along the recovery direction. Across all experimental conditions—all agents, trajectories, with and without initial offset—the CALOS-QP solver always found a feasible solution to (11). The fallback to τ0 _0, i.e., applying the uncorrected policy action, was never triggered, confirming that the joint tilt-Lyapunov constraint set remained non-empty throughout all tests. V Conclusions We presented CALOS, a runtime safety layer that unifies tilt and Lyapunov constraints into a single low-dimensional QP for quadrotor attitude control under DRL.The method enforces constraints in real time with minimal intervention—achieving zero attitude violations on the training trajectory and at most 3 consecutive violation steps under large perturbations—while reducing lateral tracking error by 55–74% relative to unconstrained PPO. The QP feasible set was never empty across all tests, meaning the fallback to the uncorrected policy action was never triggered. Moreover, policies trained with CALOS internalize safe behavior, retaining comparable tracking improvement even with the safety layer disabled, confirming that the constrained exploration improves convergence and data efficiency without sacrificing optimality. Future work includes: (i) learned, task-aware Lyapunov certificates [16, 17]: the current quadratic certificate drives toward the zero-tilt equilibrium, causing unnecessary interventions when the drone tracks at a non-zero but feasible attitude; a learned certificate that decreases toward the current reference would eliminate this; (i) extensions to non-control-affine dynamics, since higher-fidelity models with rotor-speed dynamics or blade-flapping effects break the affine assumption that CALOS requires; (i) sim-to-real transfer with domain randomization and data-driven safety filters [18] to handle model uncertainty on physical hardware. References [1] E. Kaufmann, L. Bauersfeld, A. Loquercio, M. Müller, et al. 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