A Constraint-Based Safety Evaluation Model for Low-Impact Separation of Combined UAVs

Abstract

For the wingtip-connected combined UAV considered here, the proposed constraint-based assessment demonstrates that the clearance margin changes sign between the sampled 5° and 6° angles of attack, thereby bracketing the clearance transition within this interval. This study presents a deterministic, constraint-based assessment framework for the separation of wingtip-connected combined unmanned aerial vehicles (UAVs). The previously developed torque-driven compliant interface is treated as the existing physical platform rather than as a new mechanism contribution. Structural-strength, roll-control, and collision-clearance requirements are formulated as individual limit-state margins and linked by a non-compensatory minimum operator, so that failure of one quantified constraint cannot be offset by favorable performance in another. Previously reported aerodynamic, finite-element, multibody-dynamics, and ground-test records are reanalyzed as case-study inputs; they are not presented as independent validation of the complete classifier. The verified stress contours show that parametric refinement reduces the maximum equivalent von Mises stress from 17.2 MPa to 10.4 MPa (39.5%). Ground measurements acquired at 1000 Hz yield R² = 0.96 for a descriptive sinusoidal fit, supporting response smoothness but not proving the complete low-impact safety hypothesis. The framework therefore provides a traceable requirement-checking route; with the presently retained records, its demonstrated implementation is a clearance-decision template rather than a numerically complete three-channel safety index.

1. Introduction

Wingtip-connected aircraft can cruise as a high-aspect-ratio assembly and later recover independent flight capability through controlled separation [1,2,3,4,5,6,7,8,9,10]. The safety problem is therefore not limited to whether a connector unlocks. During the short transition from a constrained multibody configuration to two free bodies, structural capacity, release-induced rolling moment, and local collision clearance must remain acceptable at the same time. A mechanism may exhibit smooth motion while still producing an unsafe trajectory, whereas a collision-free trajectory does not by itself demonstrate adequate structural or control margin.

Previous studies have addressed multibody flight dynamics, wingtip docking, aeroelastic coupling, and separation devices [11,12,13,14,15,16,17,18,19,20]. Recent work has also examined UAV operational-risk modelling, sense-and-avoid performance, safety spacing, and collision-risk assessment [21,22,23,24,25]. In structural and system reliability, limit-state functions distinguish safe and failed domains, while a series system fails when any required component limit state is violated [26,27,28]. Formal aerospace safety assessment additionally uses functional-hazard and failure-mode analyses to relate failure conditions, severity classes, and probability objectives [29,30]. Those certification-oriented processes require traceable failure data, uncertainty models, and system evidence. The present work does not replace them, propose a new general theory of limit states, or estimate a probability of failure. Instead, it adapts non-compensatory series-system logic to a specific coupled-UAV separation problem and organizes heterogeneous structural, control, and clearance evidence into a deterministic requirement check.

The scope relative to Ref. [26] is as follows: The interface configuration, working principle, sinusoidal groove law, multibody response, and original ground-test records originate from Ref. [26]. Their reuse and the derived clearance margins in the present paper are explicitly treated as reprocessing or reinterpretation. The present contribution is limited to defining the three application-specific limit states, establishing the non-compensatory decision logic, tracing a failed decision to the violated constraint, and evaluating threshold sensitivity and evidential limitations. The mechanism and its original dynamic characteristics are not reclaimed as new contributions.

The principal contribution Is an application-specific assessment workflow rather than the minimum operator itself. It defines normalized limit-state margins for structural strength, roll-control authority, and geometric clearance and combines them non-compensatorily, ensuring that violation of one constraint cannot be offset by favorable performance in another. The normalization scales affect margin magnitudes and the identification of the active constraint but not the sign of each margin. The stationary-pin refinement is included only as a structural-channel case result.

Section 2 describes the existing platform and assessment inputs. Section 3 formulates and applies the deterministic limit states, evaluates clearance-threshold sensitivity, and distinguishes calculation verification from independent physical validation. Section 4 summarizes the findings and limitations.

2. Existing Separation Platform and Assessment Inputs

2.1. Previously Developed Mechanism and Scope Boundary

The assessment object is the torque-driven wingtip interface developed in the authors’ earlier work [26]. It comprises a synchronized drive, a rotary separation disk, rolling elements, and stationary pins. Rotation of the disk converts the prescribed groove path into an outward release stroke while the stationary pins successively leave their locating holes. Figure 1 shows the existing combined-UAV platform and docking layout used as the assessment object. Figure 2 shows the existing compliant interface adopted as the physical evaluation platform. Only the geometry, loading path, and response variables required by the new safety model are summarized here; the detailed synthesis and original verification of the mechanism are available in [26].

In the present paper, the sinusoidal groove is not reintroduced as a new design. Its role is limited to defining a smooth reference input for the dynamic-response channel. This separation of prior hardware development from current model development reduces duplication and makes the incremental contribution explicit.

2.2. Aero-Frictional Loading Inputs

The assessment model requires aerodynamic loads, interface friction, and the resulting unlocking resistance as input quantities.

Table 1. Key design and environmental parameters for separation analysis.

Category	Parameter	Symbol	Value	Unit
Geometric	Total Length of Guide Groove	L	150	mm
	Total Height of Guide Groove	H	40	mm
Material	Material Grade	Al-7075	-	-
	Young’s Modulus	E	71.7	GPa
Aerodynamic	Cruise Speed	V	60	m/s
	Air Density	ρ	1.225	kg/m3

lists the geometric, material, and environmental parameters retained for this purpose. Unlike the previous mechanism-centered analysis [26], the present use of these quantities is to calculate constraint margins and identify the limiting safety channel.

The retained aerodynamic input was generated using a CFD model with approximately five million polyhedral cells, wall-resolution values of Y+ below 1, and the SST k–ω turbulence model. The archived record does not include a mesh-convergence sequence or a separate CFD validation dataset; these settings therefore document the adopted input model rather than establish mesh independence. The resulting aerodynamic load is coupled with friction at the stationary pins. The resisting friction torque is

Tf = Σ_i=1ⁿ μ_i N_i r_i

(1)

where μ_i, N_i, and r_i are the friction coefficient, normal contact force, and equivalent moment arm of the i-th contact pair, respectively. The required unlocking torque is then written as

Treq = Tf + Ta + Ts

(2)

where Ta is the aerodynamic-load-induced torque and Ts is the resistance associated with local elastic deformation of the interface. Equation (2) adopts a conservative worst-case superposition in which Tf, Ta, and Ts are taken at their maximum adverse values and assumed to act simultaneously about the unlocking axis; therefore, Treq represents a design-envelope torque rather than a time-synchronized instantaneous value. The unlocking process of the separation mechanism is shown in Figure 3.

2.3. Incremental Structural Optimization and Strength Margin

A parametric finite-element model is used to improve the strength margin of the stationary-pin region before the assessment is performed. The wing root is constrained and the retained CFD pressure field is mapped to the load-bearing interface. The optimization minimizes the maximum equivalent von Mises stress subject to the original geometric envelope and functional stroke constraints. The archived record does not include a separate finite-element mesh-convergence sequence; consequently, the stress result is treated as a retained structural-channel case input and not as independent validation of the complete separation classifier.

Min_x σeq,max(x),       subject to xL ≤ x ≤ xU

(3)

The gradient-based interior-point procedure converged after 45 iterations. Consistent with the verified contour maxima in Figure 4 and the values in

Table 2. Quantitative comparison of load-bearing performance before and after geometric optimization.

Performance Indicator	Baseline Configuration	Refined Configuration	Relative Variation
Maximum Equivalent von Mises Stress σeq,max	17.2 MPa	10.4 MPa	39.5%
Torque Ripple Rate	12.50%	8.20%	34.40%

, the maximum equivalent von Mises stress decreases from 17.2 Mpa to 10.4 Mpa, corresponding to a 39.5% reduction. These changes are interpreted as improvements in component-level structural performance rather than as independent validation of the complete three-channel assessment.

The stress contours in Figure 4 show that the refined geometry redistributes the local load away from the pin roots. The corresponding peak stress is subsequently used in the strength limit state of the safety evaluation model.

3. Constraint-Based Separation Safety Evaluation and Case Application

3.1. Dynamic Response as an Evaluation Input

The optimized interface is simulated using a multibody contact model to obtain the displacement, velocity, and acceleration histories required by the low-impact response assessment. Hertzian normal contact with stiffness and damping terms represents the roller–groove interaction. The model is retained as an input generator; the safety decision itself is made by the limit-state formulation in Section 3.3.

$$\mathit{Fn}=k{\delta }^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+c\dot{\delta }$$

(4)

where Fn is the normal contact force, k is the equivalent contact stiffness, c is the damping coefficient, δ is the local contact deformation, and $\dot{\delta }$ is the deformation rate.

For an equivalent linearized contact mode, the characteristic period is Tc = 2π√(m_eff/k). With k = 2.5 × 10⁶ N/m and Δt = 0.001 s, the corresponding temporal resolution is Tc/Δt = 3.97√(m_eff) steps per period, with m_eff expressed in kilograms.The specific parameters are listed in

Table 3. Summary of dynamics simulation parameters for the separation mechanism.

Parameter	Symbol	Value	Unit
Contact Stiffness	k	2.5 × 106	N/m
Damping Coefficient	c	450	N·s/m
Friction Coefficient (Rolling)	μr	0.005	-
Simulation Time Step	Δt	0.001	s

. Because the retained model record does not specify m_eff, a numerical resolution ratio cannot be reconstructed independently. Accordingly, Δt is reported as a reused model setting, while the response-smoothness assessment is supported by the 1000 Hz ground-test records rather than by the timestep selection alone.

Figure 5 shows four recorded runs in the existing response dataset. Tests 1–4 identify the individual runs; the markers denote sampled acceleration values and the dashed curves denote the corresponding descriptive fits. These records illustrate the repeatability and smoothness of the measured response, but they are not used as independent evidence of collision clearance or roll-control sufficiency.

3.2. Roll-Control Constraint

Release of one wing changes the lateral lift distribution and produces a transient rolling moment. The relevant safety question is whether this disturbance remains within the available control authority, not simply whether the rolling moment is small in absolute terms.

Mroll = ΔL · y

(5)

Here, ΔL denotes the separation-induced change in the resultant aerodynamic lift represented by the adopted load model. The moment arm y is the shortest lateral distance from the aircraft longitudinal roll axis (centerline) to the line of action of that resultant load; it is not automatically equal to the wingtip distance or semispan. Both quantities must be extracted in the same body-fixed coordinate system. The calculated disturbance moment is then compared with a separately specified available roll-control moment in the normalized control margin.

Figure 6 provides the existing disturbance-moment trend used by the control channel. Here, Mctrl is defined as the maximum corrective roll moment available about the body-fixed longitudinal axis at the specified flight condition, after accounting for trim demand, actuator limits, control allocation, and any required control reserve. It must be evaluated over the same transient and sign convention as Mroll. Because the retained record does not provide the control-effectiveness and actuator-limit data needed to reconstruct Mctrl for every

Table 4. Existing flight-condition results interpreted using the trajectory limit state of the proposed evaluation model.

AOA (α)	Speed (V)	Min. Clearance (dmin)	Safety Status
3°	60 m/s	12.4 mm	Clearance pass
4°	60 m/s	14.8 mm	Clearance pass
5°	60 m/s	5.2 mm	Clearance pass
6°	60 m/s	−1.5 mm	Collision Risk
7°	60 m/s	−2.1 mm	Collision Risk

case, the revised paper neither infers missing values nor claims independent numerical validation of the positive control margin.

3.3. Constraint-Normalized Framework and Clearance-Decision Template

Three limit-state functions are defined for strength, roll control, and collision clearance. A positive value indicates compliance with the corresponding requirement, zero denotes the boundary, and a negative value indicates failure:

g_σ = σallow − σeq,max,       gM = Mctrl − |Mroll|, gd = dmin − dreq

(6a)

Because the three responses have different physical units, direct addition would be arbitrary. The margins are therefore normalized by their governing capacity or by a characteristic geometric length:

ησ = gσ/σallow,      ηM = gM/Mctrl,       ηd = gd/dref

(6b)

For a fully parameterized application, the composite separation-safety function is defined by the least favorable normalized margin:

Ssep = min(ησ, ηM, ηd)

(6c)

This minimum-margin function represents a deterministic series requirement: every quantified constraint must be satisfied. It removes the need to select empirical coefficients for a weighted average, but it does not remove all modeling choices because the normalization scales influence the relative magnitudes of positive margins and the reported active constraint. The safe/unsafe sign decision is invariant to a positive rescaling of an individual margin, whereas cross-channel ranking is scale-dependent. The decision rule is

Ssep > 0: safe;      Ssep = 0: boundary;       Ssep < 0: unsafe

(6d)

In Equation (6a–d), σ_allow is a material- and design-specific allowable stress, σ_eq,max is the calculated peak equivalent stress, Mctrl is the available corrective roll moment defined in Section 3.2, Mroll is the release-induced rolling moment, dmin is the minimum body-to-body clearance, and dreq is the required engineering clearance. In the present case application, σ_eq,max refers to the refined value of 10.4 MPa, whereas the baseline value of 17.2 MPa is used only for comparison in Section 2.3 and Table 2. The framework does not generate σallow, Mctrl, or dreq. In an engineering application, dreq should be specified from the clearance budget, including manufacturing and assembly tolerances, unmodelled elastic deformation, trajectory or state-estimation uncertainty, and any mandated operational buffer. The present case adopts dreq = 0 only as a first-contact criterion; it is not a conservative certification allowance. The groove height H = 40 mm is used as dref because it is the documented local release-stroke scale available in the retained geometry, not because it is a physical safety threshold. Changing a positive dref changes ηd and may change ranking among positive margins, but it cannot change the sign of gd or the clearance pass/fail decision.

Table 4 and Figure 7 are reinterpreted through the trajectory component. With dreq = 0 and dref = 40 mm, ηd at 3°, 4°, 5°, 6°, and 7° is 0.310, 0.370, 0.130, −0.038, and −0.053, respectively. The sampled results therefore show a sign change between 5° and 6°; they do not resolve a unique critical angle inside this interval. Because only ηd is numerically reconstructable for all five cases, the retained-data implementation is reported as a clearance-decision template rather than a numerically complete unified safety index.

Table 5. Clearance-decision application of the constraint-normalized framework using the retained deterministic records (dreq = 0; dref = 40 mm).

AOA	ησ	ηM	ηd	Ssep	Active Constraint	Status
3°	>0 †	>0 †	0.310	>0 †	Not resolved †	Nominal pass †
4°	>0 †	>0 †	0.370	>0 †	Not resolved †	Nominal pass †
5°	>0 †	>0 †	0.130	>0 †	Not resolved †	Nominal pass †
6°	>0 †	>0 †	−0.038	−0.038	Clearance	Collision risk
7°	>0 †	>0 †	−0.053	−0.053	Clearance	Collision risk

† The retained record establishes only the positive sign reported for the structural and control checks in the nominal cases; it does not contain independently reproducible numerical σallow and Mctrl values for every trajectory case. Accordingly, no numerical ησ or ηM is fabricated, and the active constraint among positive margins is not assigned. For 6° and 7°, the negative ηd necessarily governs because the other two reported checks remain positive.

records the decision sign and identifies clearance where it necessarily governs. For the 3–5° passing cases, the exact value of Ssep and the active constraint remain undefined until numerical σallow and Mctrl are supplied externally.

Two post-processing sensitivity checks were performed using only the reported dmin values. First,

Table 6. Required-clearance sensitivity of ηd using the retained clearance values (dref = 40 mm).

dreq (mm)	ηd at 3°	ηd at 4°	ηd at 5°	ηd at 6°	ηd at 7°	Sampled Transition Interval
0	0.310	0.370	0.130	−0.038	−0.053	5–6°
5	0.185	0.245	0.005	−0.163	−0.178	5–6° (5° marginal)
6	0.160	0.220	−0.020	−0.188	−0.203	4–5°

summarizes the required-clearance sensitivity. For dreq = 5 mm, ηd becomes 0.185, 0.245, 0.005, −0.163, and −0.178 for 3–7°, respectively; the 5° case retains only 0.2 mm of absolute clearance above the requirement. For dreq = 6 mm, the 5° margin becomes −0.020 and the sampled pass/fail transition shifts to the 4–5° interval. Second, dref was varied from 20 to 80 mm while dreq remained zero. As summarized in

Table 7. Normalization-scale sensitivity of ηd using the retained clearance values (dreq = 0).

dref (mm)	ηd at 3°	ηd at 4°	ηd at 5°	ηd at 6°	ηd at 7°
20	0.620	0.740	0.260	−0.075	−0.105
40	0.310	0.370	0.130	−0.038	−0.053
80	0.155	0.185	0.065	−0.019	−0.026

, this rescaling changes the numerical ηd values but leaves their signs unchanged; therefore, clearance remains the necessarily active failed constraint at 6° and 7°. For the positive-margin 3–5° cases, the active constraint remains unresolved because numerical ησ and ηM are unavailable. These scenarios are sensitivity illustrations, not proposed certification values.

3.4. Experimental Evidence for the Low-Impact Response Channel

Ground experiments provide component-level evidence for the dynamic-response channel. A six-degree-of-freedom force/torque sensor and laser displacement measurement record the release transient at 1000 Hz. The present analysis examines response smoothness and consistency with the prescribed mechanism motion. A ground test at one aerodynamic-equivalent condition does not independently validate the complete airborne safe/unsafe classification.

For compact description, the stacked-mean acceleration is represented by the sinusoidal regression in Equation (7). The form is motivated qualitatively by the prescribed smooth periodic groove motion, but the fitted time history is not treated as a causal dynamic model. Its goodness of fit is reported only as a descriptive response indicator:

afit(t) = A sin(ωt + φ) + b

(7)

The parameters A, ω, φ, and b denote fitted amplitude, angular frequency, phase, and bias. The coefficient R² = 0.96 indicates that the stacked mean is represented closely by this empirical form over the measured record. No acceptance threshold is assigned to this R² value, and it is not used as a safety criterion. It does not establish a physical impact threshold or prove structural, control, or clearance safety. Because averaging can attenuate isolated run-specific peaks, the individual Tests 1–4 remain visible in Figure 5. The final constraint decision must be based on the separately defined limit states and independently specified requirement values.

The experimental setup described in this work is shown in Figure 8. The experiment therefore supports only the observed response-smoothness component. It is not presented as independent validation of Ssep or of the airborne operational boundary.

3.5. Verification Scope, Uncertainty, and Limitations

Calculation verification was performed by direct substitution of the reported dmin values into ηd = (dmin − dreq)/dref and by checking the logical equivalence between Ssep > 0 and simultaneous satisfaction of all three individual limit states. This verifies the algebraic implementation and reproduces the reported contact-risk labels at 6° and 7°. It does not constitute independent physical validation of the aerodynamic, structural, or control models.

The complete safe/unsafe classifier has not been independently validated with a separate full-state airborne dataset. The ground experiment addresses response smoothness only, while the trajectory record is reused both to calculate and to interpret clearance. The present results should therefore be read as a retrospective deterministic case application rather than a validated certification envelope.

Aerodynamic loads, friction, contact parameters, material properties, control authority, geometric tolerances, and measurement error may all perturb the margins, particularly near Ssep = 0. For example, with dreq = 5 mm, the reported 5° case has only 0.2 mm of absolute excess clearance. The archived record provides neither validated probability distributions nor error bounds for these inputs, and it also lacks CFD and finite-element mesh-convergence series. A Monte Carlo or first-order reliability calculation would therefore be driven by unverified assumed distributions and is not introduced in this revision. No robustness probability or confidence interval is claimed; probabilistic uncertainty propagation and independent full-state validation remain future work.

4. Conclusions

A deterministic, constraint-based assessment framework has been applied to the existing low-impact separation platform of a combined UAV. Its contribution is an application-specific, traceable organization of structural, control, and clearance requirements; it is not a new general theory of series-system reliability and it does not repeat the mechanism design reported in Ref. [26]. With the retained records, the demonstrated implementation is specifically a clearance-decision template because numerical structural and control capacities are unavailable for complete cross-channel ranking.

The main findings are as follows:

(1) Strength, roll-control authority, and collision clearance are written as separate limit-state margins and linked using a non-compensatory minimum function. This eliminates empirical weighted averaging but does not eliminate the influence of selected normalization scales on positive-margin ranking. A numerically complete three-channel index and active-constraint ranking require externally specified σallow, Mctrl, and dreq.

(2) Consistent with Figure 4 and Table 2, parametric refinement reduces the maximum equivalent von Mises stress from 17.2 MPa to 10.4 MPa, corresponding to a 39.5% decrease. This is a component-level structural result.

(3) Reanalysis gives clearance margins of 0.310, 0.370, 0.130, −0.038, and −0.053 from the sampled 3–7° cases at dreq = 0 and dref = 40 mm. The sign change brackets the contact transition between 5° and 6° but does not determine a more precise critical angle. Table 6 shows that increasing dreq to 5 mm makes the 5° case marginal, whereas dreq = 6 mm shifts the sampled transition to 4–5°. Table 7 shows that changing dref from 20 to 80 mm rescales ηd without changing the clearance decision.

(4) The ground-test stacked mean yields R² = 0.96 for a descriptive sinusoidal regression. This indicates a smooth fitted trend, but it is neither a causal proof of low impact nor independent validation of the complete classifier.

The formulation can be reused as a numerically complete three-channel assessment when traceable allowable stress, control authority, and clearance requirements are supplied. Until then, the present retained-data implementation should be interpreted as a clearance-decision template. The evidence is deterministic and partly reanalyzes Ref. [26]; therefore, full independent validation and probabilistic robustness are outside the demonstrated scope. Future work should quantify input uncertainties and estimate P(Ssep > 0) using independent full-state observations.