Energy-Efficient Trochoidal Path Planning for Unmanned Aircraft Under Wind and Performance Constraints

Highlights

What are the main findings?

• The developed bank-limited trochoidal lane-change strategies can yield wind-aware optimal maneuvers for fixed-wing UAVs under stall-margin and roll-rate limits.
• The new path optimization workflow can identify energy-efficient aerial-mapping mission plans that account for aerodynamic loads and enable data-acquisition quality assessment.

What are the implications of the main findings?

• Aerial mapping can be executed with reduced reliance on conventional overshoot points and conservative buffer accommodations, improving area coverage and data-acquisition quality while lowering energy demand and respecting aircraft limitations.
• Simulation case studies demonstrate that, compared to conventional strategies, the optimized paths can reduce energy by up to 17% and improve image coverage uniformity by up to 10%.

Abstract

Fixed-wing unmanned aircraft are widely used for aerial mapping because they can acquire high-resolution data at relatively low cost, but maintaining both energy efficiency and image quality in the presence of wind and flight-performance limits remains challenging. In practice, operators introduce buffer regions and extended waypoints outside the area of interest to cope with deviations during turning, which increases flight distance and energy use; yet, this approach can still degrade image overlap near the boundary. This paper presents a path-planning framework that designs turning maneuvers compatible with bank-angle, stall-margin, and roll-rate constraints while aligning mapping lanes directly with the area of interest. The framework combines analytically structured turn patterns, an energy-based metric that accounts for increased aerodynamic load in banked flight, and a two-stage path-angle selection procedure that uses a fast, simplified model to guide a more detailed optimization. Simulation studies on both idealized and real survey geometries indicate that, within the considered maneuver families and assumptions, the proposed method can reduce the integrated aerodynamic energy metric and improve coverage compliance relative to a conventional path-following approach that relies on overshoot points.

1. Introduction

The rapid advancement of unmanned aerial vehicle (UAV) technology has notably improved cost-effectiveness, flexibility, and high-resolution data acquisition in aerial mapping [1]. Yet, executing mapping missions efficiently with small fixed-wing UAVs remains challenging, especially under uncertain wind conditions. Adding buffer areas during tight lane-change turns is common during operations, increasing flight distance and degrading overlap uniformity near area of interest (AOI) boundaries [2]. In addition, wind alters ground speed and turn profile (curvature), which may violate the bank and stall constraints of a vehicle; hence, lane transition must also respect aircraft performance limits in addition to ensuring a good alignment of the vehicle path.

Starting from Dubins’ curvature-constrained shortest-path maneuvering [3], a wide array of time-optimal paths have been established, as summarized by Shkel and Lumelsky [4] for the no-wind case. Under steady wind, time-minimal “trochoidal” paths were established by McGee et al. [5] and Techy and Woolsey [6]. Incorporating wind in the path planning enabled applications such as reachable forced-landing design [7] and performance-aware planning in uniform wind fields [8]. Achieving better alignments between lane transitions and desired ground tracks could be attained by integrating trochoidal turns with boustrophedon patterns [9] or by incorporating the AOI polygon directly into the optimization [10]. Field investigations revealed discrepancies between planned and executed paths [11], highlighting a gap in method implementation. Benders et al. [12] mitigated this issue by adaptively re-planning the paths under wind and model mismatch via in-flight parameter estimation. Recent works put more emphasis on energy efficiency, environmental, and computational constraints, especially in the context of coverage path planning (CPP). For example, Duan et al. [13] considered energy-optimized fixed-wing planning in non-uniform wind fields, while Porcelli et al. [14] incorporated weather, residual energy, and battery-swapping logistics into context-aware coverage planning. Pang et al. [15] developed a wind-resilient cruise-phase energy-estimation framework for multirotor UAVs, in which the measured wind vector is transformed into body-frame axial and lateral components to improve power-consumption prediction under varying wind conditions. Lian et al. [16] derived an energy-aware path-planning algorithm for UAVs in dynamic wind environments, coupling wind-field information with energy-aware planning decisions. Other recent studies addressed complex-area decomposition and route optimization for UAV coverage [17], multi-fixed-wing collaborative 3D coverage planning [18], and learning-based continuous coverage planning for fixed-wing vehicles [19]. The CPP methods covered in these works mainly focus on global coverage decomposition, sweep ordering, obstacle handling, multi-UAV allocation, or route-level planning. One less-studied topic is the local lane-transition arising in fixed-wing aerial mapping, which we aim to address in this work. In particular, we develop a wind-aware path-planning strategy that optimizes an aerodynamic energy metric and explicitly considers bank- and roll-limited turns between closely spaced mapping lanes. The method development is focused on the lane-transition problem between adjacent mapping lanes, which defines the scope of this paper.

We focus on two limitations pertaining to wind-aware fixed-wing mapping. First, classical Bang–Singular–Bang (BSB) lane-change maneuvers tend to become infeasible in tight sidelap-driven lane spacing under bank and stall constraints. Prior approaches often assume sufficient bank authority or restore feasibility through conservative turn-rate limits and added safety distances [8,9,10], thereby reintroducing buffers and reducing efficiency near AOI boundaries. Second, performance is commonly evaluated using path length or flight-time-in-wind (FTIW) [10], which does not reflect the additional energetic burden of banked turns under wind-varying ground speed. An increased load factor, in turn, raises induced drag and power, so time-efficient choices are not necessarily energy-efficient for battery-limited UAVs.

We introduce three key contributions to address these limitations. First, we extend wind-aware trochoidal lane-change formulations to insufficient-bank regimes by deriving a practical infeasibility test for BSB under wind and bank constraints and by constructing lane-aligned alternatives when BSB is infeasible: Singular–Bang–Bang (SBB), Bang–Bang–Singular (BBS), and Bang–Bang–Bang (BBB). These alternative maneuvers minimize manual overshoot tuning while respecting aircraft limits. Second, we introduce the Integrated Energy Metric (IEM) that sums aerodynamic power over the entire maneuver, accounting for load-factor effects. IEM quantifies how bank-angle history and wind effects impact total aerodynamic energy consumption throughout the maneuver. Third, we introduce a two-stage path-angle selection procedure. By initializing a Bayesian optimization procedure with a simplified $L_1$ path-following model, we eliminate the need for exhaustive angle sweeps [20]. The method assumes steady wind and constant airspeed, though extensions to time-varying environments are discussed in our concluding remarks.

The paper is organized as follows. Section 2 reviews challenges in fixed-wing mapping, wind-aware trochoidal maneuvering, boustrophedon coverage and footprint geometry, and prevailing performance metrics. Section 3 presents the extended maneuver synthesis for insufficient-bank regimes, the IEM formulation, and the accelerated path-angle selection procedure. Section 4 evaluates the approach on a single-lane alternation scenario, an axisymmetric AOI, and a complex real-world AOI at the Hong Kong University of Science and Technology (HKUST) campus area. Finally, conclusions and future directions are summarized in Section 5.

2. Background

In this section, we review challenges in aerial mapping using fixed-wing UAVs, trochoidal path formulations under wind, boustrophedon path planning and aerial footprints in polygons, and current performance evaluation metrics. In addition, we also discuss how these limitations motivate and shape the current work.

2.1. Challenges in Aerial Mapping Using Fixed-Wing UAVs

Aerial mapping supports applications across urban planning, agriculture, and environmental monitoring. Using UAVs has improved the cost-effectiveness and resolution of mapping [1,21]. Small fixed-wing UAVs, however, face tight performance envelopes and reduced control authority compared to multirotors [22]. Lane-entry deviations degrade overlap and uniformity, prompting operational practices such as buffer regions around the AOI and overshoot/lead-in points [23]. Standards vary: for example, guidelines cap entry deviation at 2% of lane length [24] or at 5% of image size along-track (South African Department of Agriculture, Land Reform and Rural Development, Standard for the Acquisition of Digital Aerial Imagery. Available at: https://ngi.dlrrd.gov.za/index.php/technical-information/standards-menu?download=2:standard-for-the-acquisition-of-digital-aerial-imagery (last accessed on 25 May 2026)).

As illustrated in Figure 1, the planned straight lanes and the realized trajectory—even under ideal conditions—can still leave end effects near the AOI (shown by the dashed blue lines), reducing the effective coverage region (shown by the green zone in Figure 1a) and resulting in a region where the data acquisition quality might be compromised (red zone). Extending outer waypoints with overshoot/lead-in points enlarges the maneuvering zone and better aligns usable coverage with the AOI, but at the cost of additional distance, as shown in Figure 1b. Under wind, lane-entry curvature and ground-speed variation shift the effective coverage relative to the boundary; overshoot sizing that works under a no-wind condition can become insufficient or overly conservative when wind magnitude or bearing changes (Figure 1c).

Because fixed-wing platforms operate at modest airspeeds (often <80 km/h), typical winds of 20–30 kt can be quite substantial [25], making lane-entry alignment sensitive to wind. In practice, buffer and overshoot parameters are frequently smaller than needed (coverage loss near the AOI boundary) or larger than needed (reduced efficiency due to extra maneuvering) [26]. Field evidence echoes these effects: Tziavou et al. [27] observed compromised overlap uniformity in structural geological mapping when using a fixed-wing platform despite large turning areas.

The above observations motivate planning methods that synthesize lane alternations consistent with aircraft limits and wind, reducing reliance on manual overshoot tuning and improving data quality near AOI boundaries.

2.2. Maneuvering into the Wind: Trochoidal Path Under Sufficient Bank Angle

Dubins paths can reduce deviation along intended lane centerlines and, therefore, have been commonly used for maneuver planning [4]. The Dubins formulation yields the shortest planar curve connecting two states with specified initial and final headings, subject to bounded curvature [3]. Under ideal, no-wind conditions, following a Dubins path aligns the vehicle with the next lane without overshoot points or buffer regions, improving flight efficiency and eliminating unnecessary segments in aerial mapping missions.

Applied to mapping, alternating lanes require sharp turns that typically switch between right- and left-hand directions, with total heading changes of ${180}^{°}$ or less, depending on the profile geometry. The feasibility of Dubins connections is fundamentally constrained by the minimum turn radius, which, in fixed-wing aircraft, is limited by the allowable bank angle and stall margin. Figure 2 illustrates a 180-degree lane change in ideal conditions: a continuous feasible path is formed by tangents between two circles of a radius set by the bank angle and speed.

In real-world operations, the presence of wind introduces additional challenges. The aircraft’s ground-relative motion is the vector sum of its true airspeed and the wind; as the aircraft turns, the relative bearing between the airspeed vector and the wind changes, which alters both the instantaneous ground speed and the curvature achieved on the ground track, as depicted in Figure 3. Without compensation, this induces drift off the intended lane and degrades alignment at lane entry and exit.

Throughout, headings $\psi$ and ground-track courses $\chi$ are measured clockwise from north ($0^{°}$ north, ${90}^{°}$ east) in an inertial frame with the positive x and y axes pointing east and north, respectively. Wind is assumed to be steady and horizontal, with magnitude $V_{\mathrm{WS}}$ and meteorological bearing (coming from) angle ${\theta }_{\mathrm{wa}}$. Under this convention, the inertial wind components in the x and y directions are

$$w_x=-V_{\mathrm{WS}}sin{\theta }_{\mathrm{wa}},w_y=-V_{\mathrm{WS}}cos{\theta }_{\mathrm{wa}}.$$

(1)

To manage these effects, fixed-wing aircraft are flown in coordinated flight, maintaining near-zero sideslip ($\beta \approx 0$) so that the lift vector remains aligned with the body lateral axis, and the turn is generated by banking rather than yawing or skidding. Coordinated flight minimizes lateral aerodynamic inefficiencies and preserves predictable turn dynamics. Achieving coordination in wind requires applying a wind correction angle ${\theta }_{\mathrm{w}}$—crabbing the nose into the wind—so that the ground track aligns with the desired course despite the lateral wind component. We define the wind correction angle as the difference between heading $\psi$ and ground-track course $\chi$,

$${\theta }_{\mathrm{w}}=\psi -\chi .$$

(2)

Assuming a steady wind of magnitude $V_{\mathrm{WS}}$ with wind bearing (coming from) angle ${\theta }_{\mathrm{wa}}$, the wind correction angle ${\theta }_{\mathrm{w}}$ required to hold course $\chi$ at true airspeed $V_{\mathrm{TAS}}$ is

$${\theta }_{\mathrm{w}}=arcsin\left({\displaystyle \frac{V_{\mathrm{WS}}}{V_{\mathrm{TAS}}}}sin({\theta }_{\mathrm{wa}}-\chi )\right).$$

(3)

The corresponding ground-speed magnitude becomes

$$V_{\mathrm{GS}}=\sqrt{V_{\mathrm{TAS}}^2+V_{\mathrm{WS}}^2-2V_{\mathrm{TAS}}{V}_{\mathrm{WS}}cos(\psi -{\theta }_{\mathrm{wa}})},$$

(4)

which shows that $V_{\mathrm{GS}}$ varies with heading (and, thus, with course, when holding course via ${\theta }_{\mathrm{w}}$), wind bearing, $V_{\mathrm{TAS}}$, and wind speed $V_{\mathrm{WS}}$.

With coordinated flight at constant $V_{\mathrm{TAS}}$, the changing $V_{\mathrm{GS}}$ during a constant-bank turn alters the instantaneous turn rate and radius. Over a nominal full-circle turn, the endpoint does not coincide with the starting point, producing a roulette curve—classically termed a trochoidal path [28]. Figure 4 shows examples for a full right-hand turn at $V_{\mathrm{TAS}}=17$ m/s, $\varphi ={30}^{°}$, and varying wind bearings.

Extending Dubins concepts to wind leads to trochoidal analogs of the classic families. Under sufficient bank angle, the shortest lane-change remains a turn–straight–turn—often referred to as BSB—maneuver [5,6,29]. In a wind field, the straight segment is characterized by a constant ground-track course ${\chi }_{\tan }$, whereas the aircraft flies a heading ${\psi }_{\tan }={\chi }_{\tan }+{\theta }_{\mathrm{w}}\left({\chi }_{\tan }\right)$ that incorporates the wind correction angle. The two turn segments are flown as coordinated, constant-bank turns at the bank limit. The BSB construction is defined by two tangency conditions: the first turn must end at a state whose course is tangent to the straight segment, and the second turn must begin from a state that is tangent to the same straight segment while still reaching the desired terminal state.

In this work, we evaluate such turn segments by numerically integrating the coordinated-turn model

$$\begin{array}{cc}\hfill \dot{x}\left(t\right)& =V_{\mathrm{TAS}}sin\psi \left(t\right)+w_x,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \dot{y}\left(t\right)& =V_{\mathrm{TAS}}cos\psi \left(t\right)+w_y,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \dot{\psi }\left(t\right)& ={\displaystyle \frac{gtan\varphi }{V_{\mathrm{TAS}}}},\hfill \end{array}$$

(7)

where the instantaneous ground speed is $V_{\mathrm{GS}}\left(t\right)=\sqrt{\dot{x}{\left(t\right)}^2+\dot{y}{\left(t\right)}^2}$, and the ground-track course is $\chi \left(t\right)=arctan2\left(\dot{x}\left(t\right),\dot{y}\left(t\right)\right)$. Under constant-$V_{\mathrm{TAS}}$ coordinated turns and steady wind, the motion in the air-relative frame is circular at fixed bank, but because $V_{\mathrm{GS}}\left(t\right)$ and the drift angle vary, the resulting ground track is trochoidal.

For missions with tight lane spacing (e.g., for high-resolution imagery), the BSB straight segment can shrink to zero, driving the signed straight-segment length $s_{\Vert }$ to become nonpositive and making BSB infeasible, as depicted in the right-hand side figure of Figure 2. Wind exacerbates this by altering $V_{\mathrm{GS}}$ and the effective turn curvature, thereby increasing the demand for tighter turns. A practical remedy is to raise speed to enable larger bank angles and a smaller radius; however, this may reduce endurance efficiency and increase structural loads. Operating away from best-endurance conditions increases power consumption, and a higher bank raises the load factor. Swatton [30] noted that the minimum-drag speed $V_{\mathrm{IMD}}$ for propeller aircraft was approximately $1.3V_{\mathrm{s}}$, limiting how aggressively turns could be tightened while preserving stall margins.

These limitations motivate enhanced trochoidal formulations that respect fixed-wing performance constraints without relying on steeper banks or higher speeds. The method developed in this work extends the maneuver set beyond BSB to recover feasible lane alternations in many insufficient-bank cases and introduces an energy-aware evaluation to prefer energetically efficient choices under wind.

3. Methodology

This section describes the methods used in the developed energy-efficient path planning framework for aerial mapping tasks. Building upon classical trochoidal and Dubins-style constructions for constant-airspeed vehicles in steady wind [5,6,10], the framework adds three key elements tailored to small fixed-wing UAV mapping: (1) bank-limited trochoidal lane-change maneuvers that remain feasible even under insufficient bank condition and can be refined to include the finite roll rate; (2) an energy-based performance metric IEM that acts as an aerodynamic proxy and evaluates the maneuver time history, including load-factor effects in turns; and (3) a two-stage path-angle optimization that combines a simplified $L_1$-based exploration with a maneuver-resolved, Bayesian-optimization refinement. Each of these will be described below.

3.1. Generalized Wind- and Bank-Constrained Lane-Change Path Formulation

To ensure an efficient lane-change turn even when BSB is infeasible, we establish a hierarchical bank-limited maneuver-selection procedure. This procedure starts with the BSB feasibility test (Section 3.1.1). When BSB is infeasible, the planner evaluates the alternative maneuvers (BBB and SBB/BBS—which are described in Section 3.1.2 and Section 3.1.3, respectively) and selects the one with the lowest IEM for the lane transition.

We model planar motion in an inertial frame $(x,y)$ with heading $\psi$ measured clockwise from north. The UAV flies at constant true airspeed $V_{\mathrm{TAS}}$ in steady horizontal wind $(w_x,w_y)$. Under coordinated flight with a bank angle $\varphi$ and negligible sideslip, the yaw dynamics and inertial kinematics are

$${\displaystyle \frac{\mathrm{d}\psi }{\mathrm{d}t}}={\displaystyle \frac{gtan\varphi }{V_{\mathrm{TAS}}}},$$

(8)

$${\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=V_{\mathrm{TAS}}sin\psi +w_x,{\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}=V_{\mathrm{TAS}}cos\psi +w_y,$$

(9)

where g is gravitational acceleration, and the other variables are as previously defined. These equations should be interpreted as an air-mass-relative coordinated turn model embedded in a steady inertial wind field. Wind affects the ground-relative trajectory, ground speed, ground track curvature, maneuver duration, and lane-entry/exit geometry through the vector sum in Equation (9). The aerodynamic loads used in the energy metric, however, are evaluated in the air-relative frame at the prescribed true airspeed. Therefore, under the assumptions of steady uniform wind, constant $V_{\mathrm{TAS}}$, and negligible sideslip, wind does not directly modify the drag polar coefficients. Gusts, wind shear, turbulence, sideslip, and unsteady aerodynamic effects are not modeled in the present study and are planned for future work.

Recent works on time-optimal path planning for curvature-bounded vehicles [20] classify extremal trajectories into BSB and BBB families, where a Bang segment uses maximum bank ($\varphi =\pm {\varphi }_b$) and a Straight (Singular) segment uses $\varphi =0$. (In classical Dubins notation, these correspond to turn–straight–turn and turn–opposite turn–turn patterns, respectively; we use the B/S notation throughout.) In the present paper, we extend the feasible maneuver set beyond classical BSB in the insufficient-bank regimes.

Figure 5 shows several right-turn lane-change maneuvers using different bank angles, comprising feasible and infeasible BSB regimes. The nominal solution is a BSB maneuver flown at the maximum allowable bank magnitude $|\varphi |={\varphi }_b$ (blue dotted line), corresponding to the minimum achievable turn radius ($R_3$). At turning radius $R_1$ (green dotted line), the straight segment vanishes (the computed straight-segment length becomes nonpositive), beyond which the BSB maneuver is infeasible, and we need to switch to alternate saturated-bank structures, such as BBB (purple dashed line) or SBB/BBS (orange dashed line). These maneuvers preserve lane alignment while satisfying the same bank limit; these alternatives produce a wider maneuvering envelope, denoted by the effective turn radius $R_0$.

3.1.1. BSB Feasibility Test

BSB feasibility under wind can be tested by checking the signed straight-segment length $s_{\Vert }$ upon solving for a straight-segment course ${\chi }_{\tan }$ that yields consistent tangency for both turns. If no tangency solution exists or if the converged straight length satisfies $s_{\Vert }\le 0$, the BSB structure is infeasible, and alternative saturated-bank maneuver families are required. These detailed tangency solutions and feasibility tests are presented in Appendix B.

This feasibility test is deterministic and is evaluated using the nominal wind estimate. In practice, the boundary between BSB-feasible and BSB-infeasible regimes can be sensitive to wind-estimation error, especially when the computed straight-segment length $s_{\Vert }$ is close to zero. A conservative implementation can therefore replace the nominal condition $s_{\Vert }>0$ with a margin condition

$$s_{\Vert }>\Delta s_{\mathrm{rob}},$$

(10)

where $\Delta s_{\mathrm{rob}}$ is a robustness margin selected from wind-speed and wind-bearing uncertainty, worst-case perturbation analysis, or Monte Carlo sampling over likely wind estimates. Equivalently, the planner can evaluate the BSB construction over an uncertainty set $\mathcal{W}$ and impose

$$\underset{\mathbf{w}\in \mathcal{W}}{min}{s}_{\Vert }\left(\mathbf{w}\right)>0.$$

(11)

If this robust condition is not satisfied, the planner should switch to BBB/SBB/BBS alternatives to ensure operability. In the present paper, the reported simulations use the nominal deterministic criterion; robust switching under uncertain and time-varying wind is identified as future work.

3.1.2. BBB Maneuvers

When a standard BSB lane change is infeasible, a BBB maneuver (purple dashed line in Figure 5) offers a curvature-feasible alternative that preserves continuous heading and bounded curvature. Each arc is flown at a constant-magnitude bank $|{\varphi }_b|$ with the sign chosen by turn direction, under the dynamics expressed in Equations (8) and (9).

Let $t_0$ denote the maneuver start time, and let $t_1$ and $t_2$ denote the switching times between the first/second and second/third Bang arcs, respectively. We parameterize the first two arcs by their signed heading changes:

$$\Delta {\psi }_1=\psi \left(t_1\right)-\psi \left(t_0\right),\Delta {\psi }_2=\psi \left(t_2\right)-\psi \left(t_1\right).$$

(12)

For a given pair $(\Delta {\psi }_1,\Delta {\psi }_2)$, arcs $B_1$ and $B_2$ are integrated until the corresponding heading changes are achieved. The third arc $B_3$ is then integrated at a saturated bank until the terminal heading constraint $\psi ={\psi }_2$ is met. Thus, $(\Delta {\psi }_1,\Delta {\psi }_2)$ determine how the net heading change is distributed across the first two arcs, while the third arc is determined implicitly by the requirement to align with the final heading.

We solve for $(\Delta {\psi }_1,\Delta {\psi }_2)$ so that the composite BBB trajectory connects the initial state $(x_1,y_1,{\psi }_1)$ to the final state $(x_2,y_2,{\psi }_2)$ while respecting the bank-angle constraint. Specifically, we integrate the three arcs forward in time and use a two-dimensional root finder to drive the terminal position error to zero in a lane-aligned coordinate frame. Initialization is obtained from simple geometric heuristics (or a no-wind Dubins BBB guess). The full derivation and implementation are provided in Appendix C (Algorithm A1).

3.1.3. SBB/BBS Maneuvers

SBB and BBS maneuvers provide additional structures to recover feasibility in insufficient-bank regimes while remaining close to the geometric shortest path. In SBB, a short straight segment precedes a pair of opposite-sense turns; in BBS, the straight segment follows the two consecutive turns.

A straight segment is flown at a desired ground course $\chi$ with a wind correction angle (${\theta }_{\mathrm{w}}$),

$$sin\left({\theta }_{\mathrm{w}}\right)=-{\displaystyle \frac{w_{\perp }}{V_{\mathrm{TAS}}}},$$

(13)

so that the along-track ground speed is

$$V_{\mathrm{GS},\mathrm{along}}=V_{\mathrm{TAS}}cos\left({\theta }_{\mathrm{w}}\right)+w_{\Vert },$$

(14)

with $(w_{\Vert },w_{\perp })$ defined relative to $\chi$; here, $w_{\Vert }$ and $w_{\perp }$ denote the parallel and perpendicular components of wind velocity, respectively. The SBB/BBS geometries are parameterized by an intermediate heading change $\Delta {\psi }_1$ and a straight length $d_{\mathrm{straight}}$. A one-dimensional bisection on $\Delta {\psi }_1$ in a lane-aligned frame is used to match the required cross-track displacement (i.e., to end on the target lane). The straight-segment length is then chosen to remove the remaining along-track mismatch. The full scheme and implementation (laid out in Algorithm A2) are given in Appendix D.

3.1.4. Finite-Roll Refinement

The bank-limited maneuvers described so far assume instantaneous transitions between 0 and $\pm {\varphi }_b$. Real fixed-wing UAVs, however, cannot change bank angle instantaneously because roll rate is bounded by $|\dot{\varphi }|\le p_{max}$. Each turn is, therefore, decomposed into: (1) roll-in from ${\varphi }_{\mathrm{in}}$ to ${\varphi }_{\mathrm{cmd}}$ at $|\dot{\varphi }|=p_{max}$; (2) a steady-bank hold at ${\varphi }_{\mathrm{cmd}}$; and (3) roll-out from ${\varphi }_{\mathrm{cmd}}$ to ${\varphi }_{\mathrm{out}}$ at $|\dot{\varphi }|=p_{max}$. During roll, the heading change can be expressed analytically as

$$\Delta {\psi }_{\mathrm{roll}}({\varphi }_{\mathrm{a}}\to {\varphi }_{\mathrm{b}})={\displaystyle \frac{g}{V_{\mathrm{TAS}}{p}_{max}}}\left[-lncos\left({\varphi }_{\mathrm{b}}\right)+lncos\left({\varphi }_{\mathrm{a}}\right)\right],$$

(15)

and the remaining heading budget for the steady-bank segment is

$$\Delta {\psi }_{\mathrm{steady}}=\Delta {\psi }_{\mathrm{req}}-\left(\Delta {\psi }_{\mathrm{in}}+\Delta {\psi }_{\mathrm{out}}\right),$$

(16)

with duration

$$T_{\mathrm{steady}}={\displaystyle \frac{\Delta {\psi }_{\mathrm{steady}}}{gtan{\varphi }_{\mathrm{cmd}}/V_{\mathrm{TAS}}}}.$$

(17)

The translational motion during roll-in and roll-out is evaluated numerically with time-varying bank angle and the kinematics in Equation (9), while the steady-bank part is still handled analytically with constant ${\varphi }_{\mathrm{cmd}}$ and duration $T_{\mathrm{steady}}$.

To illustrate the impact of roll-rate limits, we consider a simple lane-alternation maneuver between two parallel lanes. The UAV starts at $(x_1,y_1)=(0,0)$ and must align with the next lane at $(x_2,y_2)=(100,0)$ under steady wind. The true airspeed is $V_{\mathrm{TAS}}=17$ m/s, and the commanded maximum bank is set to ${\varphi }_b={60}^{°}$ to emphasize the geometric and energetic impact of roll-rate limits; in the mission-level test cases in Section 4, we use a more conservative operational bank-angle limit (approximately ${44}^{°}$) derived from the stall-margin constraint. For this geometry and wind condition, we synthesize energy-optimal BSB-type maneuvers under three different roll-rate limits: “instantaneous” roll (idealized case, $p_{max}\to \infty$), a moderate roll rate representative of small fixed-wing UAVs ($p_{max}={60}^{°}/\mathrm{s}$), and a low roll rate representative of general aviation/transport-like aircraft ($p_{max}={20}^{°}/\mathrm{s}$). These three cases are illustrated in Figure 6, superimposed on wind direction indicated by red arrows (this wind-direction indication applies to subsequent flight-path figures). All three cases share the same boundary conditions, wind, and bank limit; only $p_{max}$ is changed. With instantaneous roll, each turn switches immediately from 0 to $\pm {\varphi }_b$, holds that bank, and then switches back to zero. The resulting BSB path closely follows the idealized trochoidal construction: two sharp, nearly circular-arc turns separated by a straight segment. When realistic roll-rate limits are imposed, the optimal ground tracks notably change, as shown in Figure 6: roll-in and roll-out occupy a non-negligible fraction of each turn, the effective centers and separation of the turns shift, and the length of the “straight” segment differs from that predicted by the instantaneous-bank model. With $p_{max}={20}^{°}/\mathrm{s}$ (i.e., the most limiting case), these distortions are even stronger, and the overall maneuver shape departs substantially from the ideal BSB pattern. Under these cases, the variation in the bank angles is shown in Figure 7, which shows that as $p_{max}$ decreases, a growing fraction of the maneuver is spent at intermediate banks rather than at $|\varphi |={\varphi }_b$, which directly affects both the achievable ground track and the energetic cost.

Classical minimum-time trochoidal formulations [5,6] idealize turns as instantaneous switches between straight and maximum-turn-rate arcs. The finite-roll refinement introduced here ensures that the synthesized maneuvers remain compatible with realistic roll-rate limits and with the resulting heading and load-factor histories of small fixed-wing UAVs. A more detailed derivation and discussion of roll-rate impact are provided in Appendix E.

3.2. Energy-Based Performance Evaluation

Maneuver efficiency is typically assessed using distance- and time-based metrics, which cannot fully quantify the energetic cost associated with bank-limited turns and wind-compensated trajectories. We therefore introduce a new metric, namely IEM, that explicitly accounts for aircraft aerodynamic characteristics and the increased load factor caused by turns, and that is evaluated along the actual finite-roll-refined maneuver trajectories. The derivation of IEM is described below.

In a steady, level cruise at airspeed $V_{\mathrm{TAS}}$, the required power is

$$P_{\mathrm{cruise}}={\displaystyle \frac{1}{2}}\rho V_{\mathrm{TAS}}^{3}S\left(C_{D_0}+K{C}_L^2\right),$$

(18)

with

$$C_L={\displaystyle \frac{2W}{\rho V_{\mathrm{TAS}}^{2}S}},$$

(19)

where $\rho$ is air density, W is weight, S is wing area, and $(C_{D_0},K)$ are drag-polar parameters. In a coordinated turn at bank angle $\varphi$, the load factor is $n=1/cos\varphi$, and the lift coefficient becomes

$$C_L\left(\varphi \right)={\displaystyle \frac{C_{L,\mathrm{cruise}}}{cos\varphi }},$$

(20)

leading to

$$P_{\mathrm{turn}}\left(\varphi \right)={\displaystyle \frac{1}{2}}\rho V_{\mathrm{TAS}}^{3}S\left[C_{D_0}+K{\left({\displaystyle \frac{2W}{\rho V_{\mathrm{TAS}}^{2}Scos\varphi }}\right)}^2\right].$$

(21)

Equation (20) follows from the quasi-steady force balance for a coordinated, level turn: the load factor is $n=1/cos\varphi$, the required lift is $L=nW$, and the dynamic pressure is fixed at constant $V_{\mathrm{TAS}}$ [30,31]. Thus, for the same aircraft weight, density, speed, and wing area, the required lift coefficient increases by the load factor. The drag model used here is a parabolic drag polar with constant $C_{D_0}$ and K, so the bank-angle effect enters through the induced-drag term. This is an intentionally compact aerodynamic model deemed sufficient for the scope of the present study, without considering changes in $C_{D_0}$ or K due to Reynolds-number variation, control-surface deflection, sideslip, aeroelastic effects, or nonlinear behavior near stall.

For a maneuver with time-varying bank angle $\varphi \left(t\right)$, the total energy consumption is obtained by integrating the required power along the maneuver time history. We define IEM as this total energy, reported in watt-hours (Wh):

$$\mathrm{IEM}\triangleq E_{\mathrm{Wh}}={\displaystyle \frac{1}{3600}}{\int }_{t_0}^{t_f}P\left(\varphi \left(t\right)\right)\mathrm{d}t,$$

(22)

where $P\left(\varphi \right(t\left)\right)$ is evaluated from the power model using the instantaneous bank angle $\varphi \left(t\right)$ along the finite-roll-refined trajectory, and the factor $1/3600$ converts joules to watt-hours $(1\mathrm{Wh}=3600\mathrm{J}$) when P is in watts.

In the numerical implementation, this is approximated consistently with the trajectory integration scheme:

$$\mathrm{IEM}\approx {\displaystyle \frac{1}{3600}}{\displaystyle \sum _{k=1}^N}P\left({\varphi }_k\right)\Delta t_k.$$

(23)

When comparing heterogeneous metrics (e.g., IEM, flight time, distance) on a common scale, we use a normalized IEM for visualization only. For a set of candidate designs $\mathcal{X}$ (e.g., a sweep of path angles), let

$${\mathrm{IEM}}_{max}=\underset{x\in \mathcal{X}}{max}\mathrm{IEM}\left(x\right).$$

(24)

We define the capped normalized IEM as

$$\widetilde{\mathrm{IEM}}\left(x\right)=min\left({\displaystyle \frac{\mathrm{IEM}\left(x\right)}{{\mathrm{IEM}}_{max}}},1\right),$$

(25)

so that $\widetilde{\mathrm{IEM}}\in [0,1]$. This capped normalization is used only for metric-to-metric comparisons and does not affect the optimization, which is performed on the physical IEM.

In this study, IEM is used as an aerodynamic proxy for relative onboard energy demand rather than as a full battery-electrical model. It captures the relative effect of bank-angle history, load factor, maneuver duration, and wind-induced trajectory changes under a fixed aircraft and operating condition. It does not explicitly model motor/propeller efficiency maps, avionics load, battery voltage sag, state-of-charge dependence, thermal effects, or discharge-rate limitations. This simplified energy metric is deemed sufficient for the present study because IEM is primarily used for comparative evaluation of candidate trajectories. If the purpose is to validate absolute energy consumption, experimental current/voltage measurements would be required.

3.3. Flight Path Optimization

The flight path optimization combines the maneuver-level models with a global, sample-efficient search. Each high-fidelity evaluation requires integrating wind-influenced, bank- and roll-limited dynamics and computing the IEM, which renders exhaustive grids over path angles or maneuver parameters not practical. We address this challenge by adopting two key strategies, as described briefly below.

3.3.1. Using Both Low- and High-Resolution Models

Using this strategy, we combine a low-resolution model for initial screening and surrogate fitting and a high-resolution model for finding promising candidates. In particular, the low-resolution model is based on an $L_1$ waypoint controller, which generates approximate trajectories quickly and is used to explore a wide range of path angles under the prescribed vehicle-performance constraints. The high-resolution model, on the other hand, is based on the BSB/BBB/SBB/BBS maneuver set with finite-roll refinement, which produces dynamically consistent trajectories and accurate IEM values but at a higher computational cost.

3.3.2. Implementing Bayesian Optimization to Minimize IEM

Here, we implement Bayesian optimization (BO) with an expected-improvement (EI) acquisition function to find the path corresponding to the minimum IEM value. The optimization iteratively selects the next high-resolution evaluation that best balances exploration of uncertain regions and exploitation near the current best IEM [32]. In this work, we use Gaussian process (GP) regression [33] as the BO surrogate model. GP provides predictive uncertainty bands, which we use to guide sampling and assess model reliability. The procedure is summarized in Algorithm 1, which consists of four main steps: (1) generate a low-resolution dataset over the path-angle design space; (2) fit a surrogate and use predictive uncertainty to identify a promising sub-region; (3) initialize a high-resolution surrogate from a subset of that region; and (4) run BO with a limited high-fidelity budget to locate the minimum IEM.

Algorithm 1 Flight Path Optimization

Require: Low-resolution model $y_{\mathrm{LR}}\left(x\right)$, high-resolution model $y_{\mathrm{HR}}\left(x\right)$, total high-resolution evaluation budget $B_{\mathrm{tot}}$, number of initial high-resolution samples $N_{\mathrm{init}}$ Ensure: Best high-resolution design $x_{\mathrm{best}}$   1: Generate initial low-resolution dataset $D_{\mathrm{LR}}=\left\{(x_i,y_{\mathrm{LR}}\left(x_i\right))\right\}$   2: Train low-resolution surrogate ${\widehat{y}}_{\mathrm{LR}}\left(x\right)$ on $D_{\mathrm{LR}}$   3: Compute prediction intervals (PIs) for candidate points using ${\widehat{y}}_{\mathrm{LR}}\left(x\right)$   4: Identify promising region ${\mathcal{R}}_{\mathrm{cand}}$ based on PIs   5: Select $N_{\mathrm{init}}$ initial high-resolution points from ${\mathcal{R}}_{\mathrm{cand}}$ and evaluate $y_{\mathrm{HR}}\left(x\right)$   6: Initialize high-resolution dataset $D_{\mathrm{HR}}$ and surrogate ${\widehat{y}}_{\mathrm{HR}}\left(x\right)$   7: Set $(x_{\mathrm{best}},y_{\mathrm{best}})$ as the pair $(x,y)\in D_{\mathrm{HR}}$ with the smallest value of y   8: while $|D_{\mathrm{HR}}| <B_{\mathrm{tot}}$ do   9:      Build the expected-improvement acquisition function $\alpha \left(x\right)$ from ${\widehat{y}}_{\mathrm{HR}}\left(x\right)$ 10:      Select $x_{\mathrm{next}}=arg{max}_x\alpha \left(x\right)$ 11:      if $\alpha \left(x_{\mathrm{next}}\right)<{ϵ}_{\mathrm{EI}}$ then 12:            break 13:      end if 14:      Evaluate $y_{\mathrm{next}}=y_{\mathrm{HR}}\left(x_{\mathrm{next}}\right)$ 15:      Update dataset $D_{\mathrm{HR}}\leftarrow D_{\mathrm{HR}}\cup \left\{(x_{\mathrm{next}},y_{\mathrm{next}})\right\}$ 16:      if $y_{\mathrm{next}}<y_{\mathrm{best}}$ then 17:            Set $x_{\mathrm{best}}\leftarrow x_{\mathrm{next}}$ 18:            Set $y_{\mathrm{best}}\leftarrow y_{\mathrm{next}}$ 19:      end if 20:      Update the high-resolution surrogate ${\widehat{y}}_{\mathrm{HR}}\left(x\right)$ 21: end while 22: return $x_{\mathrm{best}}$

The design variable in this optimization is the path angle, and the constraints include the true airspeed and maximum allowable bank angle. This choice reflects typical aerial-mapping practice: airspeed is constrained by endurance, image exposure, motion blur, sensor triggering, and desired ground sample distance (GSD), while the bank-angle limit is constrained by stall margin and platform handling qualities. The maneuver generator may use bank angles up to this limit to satisfy lane-transition constraints, but the mission-level optimizer selects the mapping direction. This reduces the dimensionality of the optimization and isolates the effect of wind-relative path orientation. A natural future work extension is to include airspeed scheduling, bank-angle scheduling, altitude, and camera-trigger timing as additional design variables.

Algorithm 1 is a budgeted surrogate-optimization procedure rather than a finite-time globally convergent optimizer. The total number of expensive high-resolution evaluations is limited by the prescribed budget $B_{\mathrm{tot}}$. After $N_{\mathrm{init}}$ initial high-resolution samples, the BO loop performs at most $B_{\mathrm{tot}}-N_{\mathrm{init}}$ additional high-resolution evaluations. The loop can also terminate earlier when the maximum expected improvement falls below ${ϵ}_{\mathrm{EI}}$. While global convergence is not guaranteed in this finite-budget optimization, the optimality of the results can still be qualitatively tested by comparing them against exhaustive high-fidelity sweeps over the considered path-angle grid, as presented in Section 4.

The computational benefit of employing BO can be measured by comparing the number of required expensive high-fidelity evaluations in BO $N_{\mathrm{HR},\mathrm{BO}}$ against that of an exhaustive high-fidelity sweep $N_{\mathrm{HR},\mathrm{exh}}=|{\Theta }_{\mathrm{grid}}|$, expressed as

$${\eta }_{\mathrm{HR}}=1-{\displaystyle \frac{N_{\mathrm{HR},\mathrm{BO}}}{N_{\mathrm{HR},\mathrm{exh}}}}.$$

(26)

Using the number of expensive high-fidelity evaluations, instead of wall-clock time, to quantify the computational benefits eliminates the assessment’s dependence on implementation, processor, integration tolerance, and plotting/diagnostic overhead. The computational cost saving pertaining to our test cases is presented in Section 4.5.

3.4. Acquisition Quality Metric

To complement the energy-based evaluation, a uniformity-based acquisition quality metric U is introduced that accounts for both image overlap and local GSD variations. First, the AOI is discretized into grid cells, and for each cell, an ideal overlap $c_i^{\mathrm{ideal}}$ and an actual overlap $c_i^{\mathrm{act}}$ are computed, with $c_i^{\mathrm{act}}$ corrected by the ratio of nominal to local GSD. This metric U quantifies the fraction of overlap deficit relative to the ideal specification and takes values in $[0,1]$,

$$U=1-{\displaystyle \frac{{\sum }_{i=1}^N{A}_{i}max\left(0,c_i^{\mathrm{ideal}}-c_i^{\mathrm{act}}\right)}{{\sum }_{i=1}^N{A}_i{c}_i^{\mathrm{ideal}}}},$$

(27)

where $A_i$ is the area of grid cell i. A higher U indicates more uniform, specification-compliant coverage. This metric is evaluated for each candidate path to characterize trade-offs between energy efficiency (via IEM) and acquisition quality.

It should be noted that U primarily penalizes underlap relative to the specification and does not symmetrically penalize excessive overlap. Accordingly, in this paper, U is interpreted as a coverage-compliance/underlap-uniformity score rather than as a fully symmetric dispersion metric. This choice is appropriate for mapping missions in which under-coverage is the primary operational concern. Additional discussion and alternative formulations for other applications (spraying, seeding, search and rescue) are provided in Appendix F.

4. Results and Discussion

In this section, we demonstrate the developed methodologies for optimal path planning in aerial mapping missions. Various aspects of the methodology are evaluated via three test cases, from fundamental maneuvers to complex real-world missions. All simulations were performed using Raybe (a fixed-wing UAV developed by BETA-UAS, Bandung, West Java, Indonesia), a small UAV typical for aerial mapping; its specifications are summarized in

Table 1. Specifications of the BETA Raybe used in the simulations.

General Parameter	Value
Maximum takeoff weight	5400 g
Wingspan	1830 mm
Length	1270 mm
Wing area	0.49 ${\mathrm{m}}^2$
Aerodynamic Parameter	Value
$C_{D_0}$	0.0210
K	0.055
$C_{L_{max}}$	1.25
${{\mathrm{V}}_{\mathrm{EAS}}}_{{(L/D)}_{\max }}$	16.8 m/s

. The numerical simulations and optimization routines were implemented in Python 3.11.1 using NumPy 1.24.1, SciPy 1.14.0, scikit-learn 1.3.0, scikit-optimize 0.10.2, Matplotlib 3.6.3, and Statsmodels 0.14.1. The numerical values used in the test cases are intended to represent a realistic, small, fixed-wing mapping scenario rather than a universal operating standard. The mapping speed is expressed in equivalent airspeed (EAS) and converted to true airspeed (TAS) using the density ratio, and the allowable bank angle is derived from the coordinated-turn stall-speed increase, together with a 20% stall-speed margin. The comparisons presented herein are made against fixed-wing-relevant maneuver assumptions and operational baselines, namely instantaneous-roll/instantaneous-bank trochoidal transitions and conventional waypoint following with overshoot points, rather than against global CPP algorithms directly.

In the simulations, the environmental conditions were set to reflect the realistic scenarios often encountered in aerial mapping missions, particularly in tropical and humid environments. Specifically, the takeoff altitude is 150 m above sea level, with an ambient temperature of 30 °C and a relative humidity above 90%. Under these conditions, the ambient air density is lower than the standard sea-level value. Approximating the static pressure at 150 m using the standard-atmosphere lapse-rate model [34] gives $p\approx 99.5$ kPa. With $T=303.15$ K and a relative humidity (RH) of 90%, the water-vapor partial pressure is $e\approx 3.82$ kPa. Following the standard moist-air density formulation based on the dry-air and water-vapor gas constants [35], the moist-air density is estimated as

$$\rho ={\displaystyle \frac{p-e}{R_{d}T}}+{\displaystyle \frac{e}{R_{v}T}}\approx 1.13{\mathrm{kg}/\mathrm{m}}^3,$$

(28)

where $R_d=287.05$ J/(kg·K) and $R_v=461.50$ J/(kg·K). The simulations and performance constraints are formulated using equivalent airspeed (EAS). Therefore, the mapping speed ${V_{\mathrm{EAS}}}_{{(L/D)}_{\max }}=16.8$ m/s corresponds to a true airspeed

$$V_{\mathrm{TAS}}=V_{\mathrm{EAS}}\sqrt{{\displaystyle \frac{{\rho }_0}{\rho }}}\approx 16.8\sqrt{{\displaystyle \frac{1.225}{1.13}}}\approx 17.5\mathrm{m}/\mathrm{s},$$

(29)

with ${\rho }_0=1.225$ kg/m³.

We compute an aerodynamic upper bound on the steady, coordinated-turn bank angle based on a standard stall-speed margin requirement. In a coordinated level turn, the load factor increases with bank angle, and the stall speed increases with the square root of the load factor [30,31]. The wing-level stall speed in EAS is estimated by

$$V_{s,\mathrm{EAS}}=\sqrt{{\displaystyle \frac{2W}{{\rho }_{0}S{C}_{L_{max}}}}},$$

(30)

where $C_{L_{max}}$ is the maximum lift coefficient and other variables are as previously defined. Using values from Table 1 ($m=5.4$ kg, $S=0.49$ m², $C_{L_{max}}=1.25$) gives

$$V_{s,\mathrm{EAS}}\approx 11.9\mathrm{m}/\mathrm{s}.$$

(31)

In a coordinated, steady, level turn, the load factor is $n=1/cos\varphi$, and the stall speed increases as

$$V_{s,\varphi }=V_s\sqrt{n}={\displaystyle \frac{V_s}{\sqrt{cos\varphi }}}.$$

(32)

To maintain a conservative margin against stalling during maneuvering, we impose the operational constraint

$$V_{\mathrm{EAS}}\ge 1.2V_{s,\varphi }.$$

(33)

Combining the above relations yields an upper bound on the allowable load factor at the selected mapping speed:

$$n\le n_{max}\triangleq {\left({\displaystyle \frac{V_{\mathrm{EAS}}}{1.2V_{s,\mathrm{EAS}}}}\right)}^2={\left({\displaystyle \frac{16.8}{1.2\times 11.9}}\right)}^2\approx 1.39.$$

(34)

Therefore, the corresponding bank-angle bound is

$${\varphi }_{max}=arccos\left({\displaystyle \frac{1}{n_{max}}}\right)=arccos\left({\displaystyle \frac{1}{1.39}}\right)\approx {44}^{°}.$$

(35)

Accordingly, we capped the operational bank angle at $\varphi \le {44}^{°}$ in the simulation to preserve at least a 20% margin against a stall at the selected mapping airspeed.

Next, we describe the three test cases used in this study. Each test case highlights distinct aspects of the methodology and its performance under various conditions:

• Single-lane alternation: The UAV is tasked with performing a single-lane alternation maneuver, which involves alternating between two parallel flight lanes. The evaluation considers variations in lane distance, bank angle, and wind conditions. This test is designed to examine the sensitivity and feasibility of the methodology under controlled and simple conditions.
• Basic axisymmetric polygon: The second test case involves a more complex geometry, specifically an axisymmetric polygon that approximates a circular flight pattern. This configuration enables a controlled assessment of the IEM and the path optimization algorithms. The symmetry of the polygon ensures consistent performance across all headings, allowing a detailed evaluation of energy efficiency and maneuverability.
• Complex polygon mission: The final test case simulates a more realistic mapping mission at the Hong Kong University of Science and Technology (HKUST) campus area. This mission involves navigating an irregular polygon and performing data acquisition over a non-uniform AOI. The complex geometry of the AOI, combined with varying environmental conditions, tests the methodology’s adaptability and performance in realistic scenarios.

For the second and third test cases, additional simulations were performed on a variety of AOIs. Each AOI represents the region where data acquisition is conducted. To ensure complete and uniform coverage of the AOI, a buffer is added to the initial region, as shown in Figure 8. This approach guarantees equal and consistent footprints of the data acquisition. The same aircraft model, with the previously defined parameters, was used across all test cases, ensuring consistency and comparability of the results.

Through these test cases, we can gain insights into several component effects, although they do not constitute a complete factorial sensitivity study over all model parameters. Test Case 1 isolates the effects of bank angle, wind bearing, crosswind condition, and lane spacing on maneuver feasibility and IEM. Then, Test Cases 2 and 3 isolate the role of the path-angle optimization workflow by comparing low-fidelity screening, BO-based high-fidelity refinement, and exhaustive high-fidelity validation. A full component-wise sensitivity analysis over all maneuver families, roll-rate limits, wind uncertainties, vehicle parameters, and sensor settings would require a much larger design of experiments beyond the scope of the current paper and is left for future work.

4.1. Trajectory-Level Comparison with Real Flight Logs

Before evaluating the proposed path-angle optimization framework, we first qualitatively compare the simulation environment against a real fixed-wing platform under comparable mission conditions. Two archived Raybe flight logs were reconstructed by extracting their waypoint missions and evaluating the same missions in the simulator with the same aircraft configuration and $L_1$ guidance controller. The purpose of this comparison is not to provide a complete flight-test validation of the proposed optimizer, but to verify that the baseline simulation and guidance dynamics are reasonably consistent with actual flight behavior, which is imperative in the simulation-based study presented in this paper.

Table 2. Summary of real-flight cases used for trajectory-level simulator comparison. Wind speed and direction were obtained from the aircraft extended Kalman filter (EKF) estimate.

Case	EAS	TAS/EAS	TAS	Mean Wind	Wind Dir.	Wind/TAS
	(m/s)	(−)	(m/s)	(m/s)	(° from)	(−)
Light wind	17.66	1.06	18.64	$3.02\pm 0.80$	320.6	0.16
Strong wind	17.76	1.09	19.39	$7.95\pm 0.92$	79.4	0.41

summarizes the two flight cases representing light-wind and stronger-wind conditions, with onboard extended Kalman filter (EKF)-estimated mean wind speeds of approximately $3.02\mathrm{m}/\mathrm{s}$ and $7.95\mathrm{m}/\mathrm{s}$, respectively. The two flights used the same Raybe aircraft but were conducted at marginally different equivalent airspeeds and atmospheric conditions, resulting in different TASs. The second case represents a more demanding condition, both because of the larger wind magnitude and because the mean wind speed corresponds to approximately $41\%$ of the TAS, compared with approximately $16\%$ in the light-wind case.

Figure 9a shows the comparison for the light-wind case. Under this condition, the simulated trajectory closely follows the logged flight path. The lane-following segments, turn locations, and overall maneuver geometry are reproduced well, indicating that the simulator, aircraft model, and $L_1$ guidance implementation provide a reasonable representation of the real aircraft behavior when the wind disturbance is moderate.

Figure 9b shows the comparison for the stronger-wind case. As expected, the discrepancy between the simulated and logged trajectories becomes more pronounced. This is reasonable because the EKF-estimated wind is only an onboard estimate and does not fully capture spatially varying wind, turbulence, gusts, actuator effects, sensor delays, or possible differences between the real and simulated aircraft states. Nevertheless, the simulator still captures the main maneuver structure, including the direction of turn deformation, the relative overshoot tendency, and the general path shape. This suggests that the simulator is suitable for the comparative evaluation performed in this work, while also motivating the use of wind-aware planning methods for more demanding atmospheric conditions.

Overall, this qualitative comparison supports the use of the simulator as a baseline evaluation environment for the proposed planning framework. However, the results should be interpreted as a trajectory-level consistency check rather than a complete validation of the proposed method under all real-flight conditions.

4.2. Test Case 1: Single-Lane Alternation

The first test case evaluates the energy consumption and optimal maneuver strategies for performing a full right turn under varying wind conditions. The objective is to analyze how wind direction and speed affect the required bank angle and energy cost during the maneuver. Additionally, the influence of lane distance on maneuver performance is explored to assess its impact on high-resolution mapping missions.

4.2.1. Effect of Wind on Bank Angle and Maneuver Cost

The initial simulations involve performing a full right turn with a lane distance of 100 m under the following conditions: without wind disturbance, with a wind speed of 5 m/s from ${60}^{°}$, and with a wind speed of 5 m/s from ${300}^{°}$. For each condition, the bank angle was varied, and the corresponding energy cost (in Wh) was recorded. Figure 10 shows the relationship between bank angle and energy consumption for these three scenarios.

From the results, two distinct zones of energy consumption are observed, separated by a threshold bank angle ${\varphi }_{\mathrm{th}}$ (marked in the plots):

• Sufficient bank-angle zone (green-shaded): For $\varphi \ge {\varphi }_{\mathrm{th}}$, the aircraft performs the BSB maneuver. The energy cost is relatively low and exhibits a convex trend (see Figure 11), indicating an optimal bank angle that minimizes energy consumption within the BSB regime.
• Insufficient bank-angle zone (red-hatched): For $\varphi <{\varphi }_{\mathrm{th}}$, the bank angle is insufficient for BSB, and the aircraft executes more complex maneuvers (BBB, BBS/SBB, or their inverses). In this zone, energy cost increases sharply as $\varphi$ decreases further away from ${\varphi }_{\mathrm{th}}$, highlighting the benefit of operating as close as possible to the allowable bank-angle limit.

The effect of wind on the minimum bank angle required for the BSB maneuver is significant. When the wind comes from ${60}^{°}$, the aircraft predominantly experiences upwind conditions during the right turn, reducing the required bank angle from ${32}^{°}$ to ${22}^{°}$, shown in Figure 11a and Figure 11b, respectively. Conversely, when the wind comes from ${300}^{°}$, the aircraft faces downwind conditions during the turn, increasing the ground speed and shifting the boundary between the sufficient and insufficient bank-angle zones to a higher bank angle, as seen in Figure 11c.

The two-zone behavior is also evident in the resulting ground-track geometries. Figure 12 shows representative maneuver paths for selected bank angles in each wind scenario. For insufficient bank angles, the aircraft cannot complete the turn with a single BSB segment and instead follows larger and/or multi-arc trajectories (e.g., BBB or BBS/SBB), which increases path length and, therefore, energy consumption. Once $\varphi \ge {\varphi }_{\mathrm{th}}$, the trajectory collapses to a compact BSB turn, consistent with the lower energy-cost curves in Figure 10.

To further illustrate the impact of wind, an additional simulation was performed with wind coming from ${270}^{°}$ (perpendicular to the lane). As shown in Figure 13, the crosswind case increases the penalty in the insufficient bank-angle zone: for $\varphi <{\varphi }_{\mathrm{th}}$, the energy cost grows more steeply with decreasing $\varphi$ compared to the no-wind and oblique-wind cases (cf. Figure 10). Moreover, the transition to the low-cost BSB regime occurs at ${43}^{°}$, a larger bank angle (higher ${\varphi }_{\mathrm{th}}$), indicating that crosswind reduces the margin for completing the turn with a single BSB maneuver.

4.2.2. Effect of Lane Distance on Minimum Bank Angle

The second set of simulations investigates the influence of lane distance on the energy cost and the minimum bank angle required to perform the BSB maneuver. Lane distances ranging from 25 m to 200 m were tested, with and without wind disturbance, where the former considers a wind speed of 5 m/s from ${270}^{°}$ (crosswind). In these simulations, the maximum allowable bank angle was fixed at ${44}^{°}$, as previously described. The results, presented in Figure 14, show that as the lane distance decreases, the energy cost initially decreases due to shorter travel paths, but it increases sharply when the BSB maneuver is no longer feasible. This transition occurs because the aircraft cannot meet the required turning dynamics at smaller lane distances and must switch to less efficient maneuvers.

Under calm conditions (no wind), the energy cost decreases gradually as the lane distance decreases due to shorter travel distances. However, as the lane distance becomes very small, the required bank angle approaches the maximum allowable limit, beyond which the BSB maneuver is no longer feasible. Once this limit is reached, the aircraft must transition to more complex and less efficient maneuvers, leading to a sharp increase in energy cost for smaller lane distances.

When crosswind conditions are introduced (wind from ${270}^{°}$), the maneuver becomes more challenging because the wind distorts the ground-track turn geometry and effectively tightens the feasibility constraints for remaining within the lane while completing the turn. As a result, the threshold lane distance at which BSB remains feasible shifts to larger values compared to the no-wind case (see the rightward shift of the zone boundary in Figure 14b relative to Figure 14a). For smaller lane distances, the energy cost increases more rapidly than in the no-wind scenario because the solution switches from BSB to less efficient multi-arc maneuver types (BBB or SBB/BBS variants), as indicated by the red-hatched region in Figure 14. This effect is particularly significant in high-resolution mapping missions, where narrow lane distances are often required to achieve finer detail.

In summary, the results show that while narrow lane distances may be desirable for high-resolution mapping, they introduce significant challenges, which may force a switch away from BSB to higher-cost maneuver types, especially under crosswind conditions. These findings highlight the need for careful consideration of lane distance, wind conditions, and bank-angle limits when optimizing flight parameters to minimize energy consumption and maintain maneuver feasibility.

4.2.3. Key Observations for Test Case 1

The following key observations summarize the insights gained from this test case. First, the energy cost of a maneuver is strongly influenced by bank angle. In the sufficient bank-angle zone, energy cost is minimal and exhibits a convex trend, with an optimal bank angle minimizing energy expenditure. In the insufficient bank-angle zone, energy consumption rises drastically, emphasizing the importance of maximizing the allowable bank angle in this region. Second, wind direction significantly impacts the required bank angle and energy cost. Upwind conditions reduce the required bank angle, while downwind and crosswind conditions increase it. Crosswind conditions impose the greatest challenges, particularly for maintaining the trajectory during the turn. Lastly, lane distance plays a critical role in determining the feasibility of a maneuver. Smaller lane distances, as required in high-resolution mapping, increase the minimum bank angle needed for the BSB maneuver. This effect is exacerbated by crosswind, making it essential to carefully plan flight paths to efficiently execute the mission.

4.3. Test Case 2: Basic Axisymmetric Polygon

The second test case is designed to test the basic maneuver strategies, the effectiveness of IEM, and the path optimization methodology in a controlled environment by using an axisymmetric polygon configuration that approximates a circular flight pattern. We use FTIW, integrated energy, and footprint metrics to assess the interplay between energy consumption, time efficiency, and data acquisition quality during aerial mapping operations. The selected shape ensures consistent behavior across all headings from $0^{°}$ to ${360}^{°}$, as depicted in Figure 15. The symmetry of the polygon allows us to isolate the fundamental aspects of the methodology without introducing external errors or inconsistencies. The specific setup conditions for this test case are summarized in

Table 3. Setup conditions for Test Case 2: Basic axisymmetric polygon.

Parameter	Value
AOI shape	Axisymmetric polygon (circle-like)
AOI radius	1000 m
Wind speed	5 m/s
Wind direction	$0^{°}$ (from the north)
Ground sampling distance (GSD)	5 cm/pixel

.

4.3.1. Initial Evaluation with Straight Paths

We first evaluate straight boustrophedon paths, where the aircraft follows straight lines between waypoints without accounting for maneuver dynamics. As shown in Figure 16, the distance-based cost is mostly constant across all flight path angles due to the axisymmetricity. However, the time-based evaluation reveals variations based on the wind’s effect on the aircraft. Specifically, the time-based evaluation highlights the best and worst flight paths for completing the mission, based on the FTIW metric. While this approach provides a quick insight into the optimal flight path, it still neglects the aircraft’s need to perform turns and other maneuvers, making it an incomplete representation of real-world performance.

4.3.2. Incorporating Maneuver Dynamics

To achieve a more realistic evaluation, we incorporated the classical $L_1$ control algorithm into the simulation. This enables the aircraft to execute turns and, therefore, account for maneuver dynamics, as shown in Figure 17. To realistically mimic the actual operation, we added the overshoot points at the end of each lane, before the aircraft turns, to improve the alignment. However, despite the additional overshoot points, deviations from the intended flight paths (orange lines) relative to straight lines between waypoints (black dashed lines) are still observed. These deviations can reduce data acquisition quality near waypoint entries, often requiring additional buffer zones to ensure complete AOI coverage.

Incorporating maneuver dynamics allows us to apply the IEM, which evaluates energy consumption based on the aerodynamic loads experienced during flight. As shown in Figure 18a, the time-based cost now varies due to the wind’s effect on the realized flight path. Additionally, the energy-based evaluation using IEM differs from the time-based evaluation because of the aerodynamic efficiency and loads during maneuvers. For example, the time-based evaluation identifies the optimal flight path angle at ${90}^{°}$, consistent with the straight-path evaluation. However, the energy-based evaluation identifies the optimal angle at ${270}^{°}$, with ${90}^{°}$ being a close second. This difference highlights the importance of using the IEM for more accurate predictions in optimization tasks.

4.3.3. Path-Angle Optimization via Prediction-Interval Screening and BO

The optimization process follows the workflow described in Section 3.3 and summarized in Algorithm 1. In brief, we use prediction intervals (PIs) from the low-fidelity surrogate to identify a candidate region of path angles. An example of this PI-based screening using low-fidelity data is shown in Figure 18b, where candidate values are selected based on a threshold on the predictive upper bound (here, the 75% upper-bound cut). We then refine the solution using BO on the high-fidelity IEM evaluations.

We fit a GP surrogate to the low-fidelity IEM data and use its predictive mean and confidence intervals to quantify model uncertainty over path angle. Figure 19 illustrates the GP posterior before and after refinement: the initial surrogate (left) provides a coarse trend with relatively wide uncertainty, whereas the refined surrogate (right) yields a smoother mean prediction with tighter confidence bands consistent with the observed low-fidelity points. This uncertainty quantification is subsequently exploited by the BO acquisition function to balance exploration and exploitation during the high-fidelity refinement stage.

The BO process iteratively refines the solution. Figure 20 shows the posterior distribution and acquisition function after the second BO iteration, illustrating how the acquisition function guides sampling toward promising path angles. Within the selected high-fidelity evaluation budget, the best sampled solution and the GP posterior stabilize near the optimum. For Test Case 2, the BO workflow uses 17 total high-fidelity evaluations, consisting of eight initial high-fidelity samples and nine BO-selected samples. The computational cost will be discussed further in Section 4.5.

To verify the optimization result within the simulated high-fidelity model, we compared it with exhaustive high-fidelity evaluations over flight path angle values spanning the design space, which is the alternative search procedure when the optimization is not in place [20]. As shown in Figure 21, the optimization results accurately predict the global minimum in the region of interest; the “true function” (green line) is constructed from the high-fidelity evaluation sweep mentioned earlier. Overall, the surrogate model can match the true function well, demonstrating the effectiveness of the developed methodology.

The conventional baseline used here is an $L_1$ path-following strategy with manually added overshoot points, representative of common fixed-wing mapping practice to improve lane alignment. The overshoot time was set to 4 s in this study, representing an expected (average) value instead of an optimum one. The comparison presented herein is meant to compare our method’s performance against a representative operational baseline, instead of the best possible overshoot-tuned method under given conditions. Optimizing the overshoot duration could reduce the performance gap, but it would also introduce an additional tuning problem that depends on wind and vehicle dynamics. The proposed method instead constructs lane-aligned maneuver transitions directly, removing the need for ad hoc overshoot tuning within the considered maneuver families.

To further evaluate the performance of the proposed optimal maneuver strategy, we compared the normalized IEM across three methods: the optimal maneuver strategy, the conventional path-following technique, and the ideal condition obtained from the shortest-distance evaluation. The normalized IEM values are shown in Figure 22a. The ideal condition represents the theoretical minimum energy consumption required, without considering realistic maneuvering requirements. As expected, its IEM curve is the lowest in Figure 22a, setting the baseline for comparison. From the graph, it is evident that the optimal maneuver strategy achieves notably lower energy consumption compared to the conventional path-following technique. Across different flight path angle values, the optimal maneuver strategy achieves an 8–13% reduction in energy consumption compared to the conventional approach, as shown in Figure 22b. These reductions highlight the importance of incorporating maneuver dynamics and aerodynamic efficiency into the optimization process.

4.3.4. Acquisition Quality Evaluation for Test Case 2

We analyzed the acquisition quality using two key metrics: (1) the footprint-overlap count and (2) the uniformity metric U defined in Section 3.4, which summarizes how closely the achieved overlap distribution matches the specified overlap after accounting for local GSD variations. These metrics are evaluated across three flight path angles: $0^{°}$, ${45}^{°}$, and ${90}^{°}$, representing different aerodynamic and wind-alignment conditions. The results are visualized in individual footprint maps (Figure 23, Figure 24, Figure 25 and Figure 26), which are elaborated below, and are consolidated in Figure 27.

In Figure 23, Figure 24, Figure 25 and Figure 26, the background color encodes the footprint-overlap count, i.e., the number of captured-image footprints covering each grid cell (see the color bar). Darker colors indicate higher overlap counts, whereas lighter colors indicate fewer footprints and potential under-coverage. A desirable coverage pattern exhibits (1) no low-overlap pockets inside the AOI, especially near lane entry/exit regions and AOI boundaries, and (2) a spatially uniform color field. Very high overlap is not necessarily beneficial (it can indicate redundancy and reduced efficiency), so we interpret the overlap maps together with the uniformity metric U, which penalizes overlap deficits relative to the specification.

To provide a baseline for comparison, we first evaluated the conventional path-following approach on the original AOI without additional overshoot points and without applying the buffer used in the subsequent footprint evaluation. As seen in Figure 23, turning dynamics cause lane-entry misalignment and a non-uniform overlap distribution, particularly near lane transitions and boundaries. For the conventional strategy used in the following comparisons, we added overshoot points corresponding to 4 s of flight time to improve lane alignment; in practice, such overshoot and buffer settings are often determined by trial and error and can be sensitive to wind and aircraft dynamics. Note that Figure 23 is plotted on the original (unbuffered) AOI to illustrate end effects, whereas Figure 24, Figure 25 and Figure 26 use the buffered AOI employed in the footprint evaluation.

At a $0^{°}$ path angle, flight lanes are aligned parallel to the wind flow. As shown in Figure 24, the optimal maneuver strategy achieves near-ideal acquisition quality, with highly uniform footprints and consistent overlap across the AOI. The magnified region A shows that the optimized trajectory remains aligned with the lane direction, resulting in uniform overlap. In contrast, the conventional path-following technique shows lane-entry misalignment, highlighted in region B, which produces local non-uniform overlap and reduced coverage near the boundary.

At a ${45}^{°}$ path angle, the aircraft heading is no longer parallel to the lane due to wind correction. This introduces “crabbing,” where the aircraft flies at an angle relative to the desired track, causing the footprints to yaw across the lane (not parallel with the flight path). As shown in Figure 25, the optimal maneuver strategy partially mitigates this effect and maintains a more uniform footprint distribution. The highlighted region A shows more consistent overlap under the optimized maneuver, whereas region B shows a compromised boundary region with lower uniformity and shifted/overlapping footprints in the conventional strategy.

At a ${90}^{°}$ path angle, the lane is perpendicular to the wind flow, representing the most challenging condition for acquisition quality. The crabbing effect is more pronounced, causing significant yawing of the footprints across the lane, as shown in Figure 26. The highlighted region A illustrates the footprint yawing caused by wind correction, while region B shows that the optimized trajectory still conforms more closely to the intended lane-transition direction under crosswind. Despite these challenges, the optimal maneuver strategy maintains higher overlap and better uniformity compared to the conventional method.

For easier comparison, we superimpose the overlapping footprints and uniformity for different cases in Figure 27. Overall, the results show that the conventional maneuver strategy with additional turnaround points and buffer effectively increases the acquisition data by around 4%. However, the optimal maneuver strategy consistently achieves higher overlap and uniformity compared to the conventional method across all path angles. For the $0^{°}$ path angle case, the optimal strategy achieves approximately 5% more overlapped footprints and significantly higher uniformity compared to the conventional approach. Furthermore, the uniformity metric effectively captures footprint distortions caused by wind-induced crabbing effects, particularly at ${45}^{°}$ and ${90}^{°}$, and confirms that the proposed strategy is more robust in all conditions. The optimal maneuver is also able to provide a 7–10% increase in both overlapped footprints and the uniformity value over the baseline value, while also offering an 8–13% reduction in energy consumption.

4.3.5. Key Observations for Test Case 2

The results from Test Case 2 highlight the effectiveness of the proposed optimal maneuver strategy and optimization methodology across multiple evaluation metrics. The main findings are as follows:

• Optimization solution quality: The methodology accurately identifies the best path angle within the considered search space, as validated by comparisons with exhaustive high-fidelity evaluations.
• Energy consumption: The optimal maneuver strategy reduces energy consumption by 8–13% on average compared to the conventional $L_1$ path-following technique and approaches ideal (shortest-distance) performance while remaining feasible under dynamic constraints.
• Acquisition quality: The optimal strategy consistently achieves higher overlap and better uniformity than the conventional method, particularly at $0^{°}$, where acquisition quality is near-ideal. Even at ${45}^{°}$ and ${90}^{°}$, it mitigates wind-induced crabbing effects better than the baseline.
• Trade-offs: The most energy-efficient and most acquisition-friendly path angles are not identical, illustrating that users may need to balance energy minimization and acquisition quality based on mission priorities.

4.4. Test Case 3: Complex Mapping Mission at HKUST

To evaluate the proposed methodology in a real-world scenario, Test Case 3 involves a complex mapping mission over the HKUST campus area, as shown in Figure 28. Compared with the axisymmetric AOI, this case introduces additional challenges, including an irregular AOI polygon, a higher-resolution GSD of 2 cm/pixel, and a wind direction of ${150}^{°}$ (southeast). The procedure follows the same workflow as Test Case 2, and the setup conditions for this test case are summarized in

Table 4. Setup conditions for Test Case 3: Complex mapping mission at HKUST.

Parameter	Value
AOI shape	Irregular polygon (HKUST campus)
Ground sampling distance (GSD)	2 cm/pixel
Wind speed	5 m/s
Wind direction	${150}^{°}$ (southeast)

.

4.4.1. Initial Cost Evaluation: Straight Path vs. Low-Fidelity Simulation

We evaluated the cost using two evaluation methods: (1) straight-path evaluation, where the aircraft follows a boustrophedon pattern without accounting for maneuver dynamics, and (2) low-fidelity simulation, which incorporates flight dynamics, including wind-induced effects and maneuvering constraints.

In the straight-path evaluation, the aircraft follows straight boustrophedon paths between waypoints. Due to the irregular AOI shape, the distance covered varies significantly with path angle, affecting distance and time-based metrics, as shown in Figure 29a. The straight-path analysis identifies a potential optimal region between ${180}^{°}$ and ${270}^{°}$, but does not account for maneuvering or wind compensation.

When the low-fidelity simulation is used with an $L_1$ controller, the potential optimal region shifts. As shown in Figure 29b, the most promising region now lies between ${330}^{°}$ and ${30}^{°}$ (i.e., within $\pm {30}^{°}$ of $0^{°}$), with $0^{°}$ identified as a strong candidate for both time and energy metrics. This shift highlights the limitation of relying solely on straight-path evaluations. This discrepancy is expected because the straight-path evaluation neglects turn dynamics, wind-correction (crab) effects, and the additional distance, time, and energy incurred during repeated lane-change maneuvers. The low-fidelity simulation incorporates these effects through closed-loop path following and, thus, provides a more representative trend for screening, while the final optimum is determined using the high-fidelity maneuver-resolved model and BO.

4.4.2. Optimization via Prediction and Confidence Intervals

Similarly to the previous test case, this optimization process also uses prediction intervals to identify promising regions from the low-fidelity model, followed by high-fidelity evaluations and BO. This is particularly useful for the irregular AOI, where the underlying cost function can be jagged.

Figure 30 shows the evolution of the BO process. Initially, the acquisition function promotes broad exploration; by the fourth iteration, sampling focuses more on angles around $0^{°}$ to ${45}^{°}$. Within the selected high-fidelity evaluation budget, the surrogate and best sampled design stabilize around the optimum near $22.5^{°}$. For Test Case 3, the BO workflow uses 16 total high-fidelity evaluations, consisting of six initial high-fidelity samples and 10 BO-selected samples.

The optimization result was validated against exhaustive high-fidelity evaluations over a sweep of flight path angles, as shown in Figure 31. The comparison confirms that the BO-based optimization accurately finds the best solution within the considered search space at approximately $22.5^{°}$, despite the irregular cost function.

4.4.3. Energy Consumption Evaluation

The energy consumption for Test Case 3 was evaluated using the normalized IEM across all path angles. The results shown in Figure 32 highlight the effectiveness of the optimal maneuver strategy under an irregular AOI and wind. The optimal maneuver strategy consistently achieves lower energy consumption across all path angles, with a reduction of 2–17% compared to the conventional method. The energy-optimal path angle is given by the minimum of the normalized IEM curve (here approximately $22.5^{°}$), whereas the peak percentage reduction can occur at a different angle because it reflects the gap between two methods rather than the absolute minimum of IEM. The results also show that the optimal strategy remains robust in a broad region around this angle, which is valuable for real missions where exact adherence to a single path angle may be difficult.

4.4.4. Acquisition Quality Evaluation for Test Case 3

The acquisition quality in Test Case 3 was, again, assessed using overlap and uniformity metrics. As shown in Figure 33, the optimal maneuver strategy outperforms the conventional method across all path angles.

The optimal strategy achieves higher overlap values overall, with peak performance near the optimal path angle. The uniformity metric confirms that footprint distribution is more consistent for the optimal strategy, even as the AOI geometry induces jagged behavior in overlap. Figure 34 illustrates the improved coverage patterns of the optimal maneuver (left-hand side figure) compared to the conventional method (right-hand side figure) for a representative footprint map.

A notable observation in this test case is that energy efficiency and acquisition quality exhibit better alignment than in Test Case 2: both are favorable in the neighborhood of $22.5^{°}$. In this HKUST case study, the path-angle region that minimizes IEM also yields high acquisition-quality scores, suggesting that the energy and coverage objectives can be compatible under certain AOI geometries and wind conditions.

4.4.5. Key Observations for Test Case 3

We now summarize our key observations from this test case’s results. First, the BO-based optimization, guided by prediction and confidence intervals, successfully identifies the optimum solution within the considered search space in a jagged cost landscape induced by an irregular AOI and wind. We observe consistent energy savings relative to the conventional baseline (approximately 2–17%, depending on path angle), with robust performance around the optimum. Lastly, the overlap and uniformity metrics confirm that the optimal strategy yields better acquisition quality across all path angles and indicate that, in this HKUST case study, the path-angle region that minimizes IEM also yields high acquisition-quality scores.

4.5. Computational Cost Reduction with BO

We now discuss the computational benefits of performing BO on Test Cases 2 and 3, using Equation (26) presented in Section 3.3.2. The resulting reductions in the required number of expensive high-fidelity evaluations are summarized in 

Table 5. Reduction in expensive high-fidelity evaluations achieved by the BO-based workflow relative to exhaustive high-fidelity path-angle sweeps.

Case	${\mathit{N}}_{\mathbf{HR},\mathbf{BO}}$	${\mathit{N}}_{\mathbf{init}}$	${\mathit{N}}_{\mathbf{BO}}$	${\mathit{\eta }}_{\mathbf{HR}}$ vs. 36	${\mathit{\eta }}_{\mathbf{HR}}$ vs. 72
Test Case 2	17	8	9	52.8%	76.4%
Test Case 3	16	6	10	55.6%	77.8%

. Here, $N_{\mathrm{init}}$ is the number of initial high-fidelity samples, $N_{\mathrm{BO}}$ is the number of BO-selected additional high-fidelity samples, and $N_{\mathrm{HR},\mathrm{BO}}=N_{\mathrm{init}}+N_{\mathrm{BO}}$ is the total number of high-fidelity evaluations used by the optimization workflow. The 36- and 72-point exhaustive sweeps correspond to ${10}^{°}$ and $5^{°}$ path-angle resolutions, respectively.

As shown in Table 5, even relative to a coarse ${10}^{°}$ exhaustive sweep, our workflow reduces expensive high-fidelity calls by more than 50%. Relative to a finer $5^{°}$ exhaustive sweep, the reduction increases to approximately 76–78%. For the two mission-level cases, the best solution stabilizes within 16–17 total high-fidelity evaluations. The low-fidelity $L_1$ evaluations are retained because they are substantially cheaper and are used only to identify promising regions and initialize the high-fidelity surrogate. In both reported case studies, the best high-fidelity solution within the selected budget matched the optimum identified by the exhaustive high-fidelity evaluation sweep.

4.6. Limitations, Generalizability, and Validation Plan

The results reported in this study should be interpreted within the assumptions and scope of the developed framework. First, the primary validation remains simulation-based. The trajectory-level comparison with real flight logs in Section 4.1 provides a useful consistency check for the baseline simulator, aircraft model, and $L_1$ guidance behavior, but it does not constitute a full flight-test validation of the proposed optimized maneuver planner. The finite-roll, wind-aware trajectories are dynamically more realistic than instantaneous-bank geometric paths, but additional flight-test data are still required to confirm tracking accuracy, wind-estimation robustness, and actual onboard energy savings. Second, IEM is an aerodynamic energy proxy rather than a complete electrical-energy model. It captures the relative influence of bank angle, load factor, maneuver duration, and wind-induced trajectory changes, but it does not include propulsion-efficiency maps, avionics load, battery voltage sag, state-of-charge effects, or thermal limitations. Third, the quantitative savings are reported relative to a representative conventional $L_1$ baseline with fixed overshoot time, not relative to every possible optimized baseline or global CPP algorithm. The derived maneuver generator should, therefore, be viewed as a fixed-wing transition-feasibility and transition-cost module that could be embedded within broader CPP frameworks.

The method is nevertheless transferable to other fixed-wing UAVs if the relevant vehicle and mission parameters are supplied. Important nondimensional quantities include the wind ratio $V_{\mathrm{WS}}/V_{\mathrm{TAS}}$, the ratio between lane spacing and minimum turn radius, the allowable load factor or stall-margin constraint, the roll-rate limit relative to maneuver duration, and the sensor footprint-to-lane-spacing ratio. Different airframes may, therefore, produce different percentage savings, even though the same planning procedure applies.

A practical validation framework would comprise three stages. First, software-in-the-loop or hardware-in-the-loop tests can verify that the generated waypoints and maneuver commands are compatible with the autopilot guidance implementation. Second, controlled flight tests can be conducted for repeated lane-change maneuvers under calm and moderate wind, recording Global Positioning System (GPS) and Inertial Measurement Unit (IMU) states, commanded and measured bank angles, wind estimates, and current/voltage data. Third, full mapping missions can compare measured electrical energy, IEM, footprint overlap, and reconstruction quality across the proposed and conventional strategies. The relationship between IEM and measured energy should be assessed using both absolute error and rank correlation, since IEM is primarily intended to compare candidate trajectories. If needed, propulsion-efficiency and battery-discharge correction factors can be added to calibrate IEM into a more complete onboard energy model.

Finally, the acquisition-quality metric U used in this paper is intentionally formulated as an underlap-compliance metric. This is appropriate for photogrammetric mapping, where insufficient overlap near AOI boundaries is a primary failure mode. For other applications, such as spraying, seeding, or search and rescue, excessive overlap may also be undesirable. In such cases, U can be extended to include symmetric penalties for both under- and over-coverage, nonlinear penalties for severe gaps, or task-specific definitions based on deposition rate, sensor dwell time, or detection probability.

5. Conclusions

This study presents a framework for optimizing the maneuver strategies and flight paths of fixed-wing UAVs, specifically in aerial mapping missions, addressing challenges posed by wind, aircraft dynamics, and performance constraints. By integrating extended trochoidal path formulations, an energy-based performance metric, and a two-stage optimization scheme, the methodology improves both energy efficiency and data acquisition quality compared with a conventional path-planning baseline.

A key contribution is the development of bank-limited trochoidal lane-change maneuvers that accommodate both sufficient and insufficient bank-angle conditions. The classical BSB structure is extended with BBB, BBS, and SBB maneuvers, which remain feasible under tighter lane spacing and bank constraints while preserving lane alignment under wind. These maneuvers are further refined with a finite-roll model that enforces bounded roll rate, ensuring compatibility with the roll- and load-factor limits of small fixed-wing UAVs. The second contribution is the introduction of the new IEM, which acts as an aerodynamic proxy and evaluates energy consumption by integrating aerodynamic power along the maneuver, including load-factor effects in turns. Unlike distance- or FTIW-based metrics, the IEM captures the energetic impact of bank angle, roll-rate limits, and wind effects through the resulting maneuver cost history, enabling explicit aerodynamic energy optimization. Lastly, the third contribution is the two-stage path-angle optimization strategy that combines a low-fidelity $L_1$ controller with prediction intervals and a high-fidelity maneuver-based model embedded in a BO framework. This reduces the number of expensive high-fidelity evaluations needed to locate energy-efficient path angles, even in irregular AOIs and under wind.

Simulations on a single-lane alternation, an axisymmetric AOI, and a complex real-world AOI at HKUST campus area demonstrated that the developed framework reduced energy consumption by approximately 8–13% (and up to 17% in the HKUST case) compared to a conventional $L_1$ path-following approach with fixed overshoot points, improved overlap uniformity by up to 10%, particularly near AOI boundaries and under challenging wind conditions, and reduced reliance on ad hoc overshoot-point tuning by generating lane-change maneuvers that were dynamically feasible and aligned with the AOI geometry. These improvements are especially important for high-resolution mapping missions, where both energy efficiency and data quality are critical. The ability to trade between energy minimization and acquisition quality, and in some cases to improve both simultaneously, underscores the practical value of the framework.

The present study remains subject to several limitations. First, although a trajectory-level comparison with real flight logs supports the baseline simulator behavior, the proposed optimized planner has not yet been validated through dedicated flight tests. Second, IEM is an aerodynamic proxy for relative energy demand and not a full battery-electrical model. Third, the reported improvements are relative to a representative conventional $L_1$ path-following baseline with fixed overshoot time; an optimized overshoot baseline or other advanced fixed-wing planners can potentially further reduce the observed gap. Fourth, the deterministic BSB switching rule is evaluated using nominal wind and should be augmented with robustness margins for uncertain or time-varying wind. Future work will, therefore, focus on experimental flight-test validation, time-varying winds, airspeed scheduling, more detailed propulsion-energy models, and real-time or onboard implementation. Extensions to other applications—such as precision agriculture, environmental monitoring, and search and rescue—constitute promising directions, where the same ideas of bank-limited, roll-limited trajectory synthesis and energy-aware evaluation can be applied.