WA-LPA*: An Energy-Aware Path-Planning Algorithm for UAVs in Dynamic Wind Environments

Highlights

What are the main findings?

• Developed a wind-adaptive lifelong planning A* (WA-LPA*) algorithm that couples UAV energy modeling with dynamic wind-field perception to achieve energy-aware path optimization.
• Introduced a composite heuristic integrating wind-alignment and altitude-layer optimization, together with an adaptive replanning mechanism responsive to environ- mental changes.

What are the implications of the main findings?

• The proposed approach enables UAVs to maintain energy-efficient and stable flight performance under complex, time-varying wind conditions.
• This framework offers a practical foundation for real-world UAV deployment and provides methodological guidance for intelligent navigation in energy-constrained aerial systems.

Abstract

Energy optimization is crucial for unmanned aerial vehicle (UAV) path planning, particularly in complex wind-field environments. Most existing path-planning algorithms rely on simplified energy consumption models, which often fail to adequately capture the effects of wind fields. To address this limitation, a wind-adaptive lifelong planning A* algorithm (WA-LPA*) is proposed for energy-aware path planning in dynamic wind environments. WA-LPA* constructs a composite heuristic function incorporating wind-field alignment factors and integrates a hierarchical height-aware optimization strategy. Meanwhile, an adaptive replanning mechanism is designed based on the change characteristics of the wind field. Simulation experiments conducted across representative scenarios demonstrate that, compared to conventional algorithms that neglect wind-field effects, WA-LPA* achieves energy efficiency improvements of 5.9–29.4%.

1. Introduction

In recent years, unmanned aerial vehicles (UAVs) have been widely applied in fields such as logistics distribution, environmental monitoring, and infrastructure inspection [1,2,3,4,5]. This has driven a shift in path-planning technology from traditional objectives focused on safety and feasibility to the optimization of low power consumption and low risk [6,7]. Wind fields, with velocities reaching 50% or more of UAV airspeed, significantly affect path execution, energy consumption, and flight safety, making wind-aware path planning a critical research priority [8,9,10,11].

Researchers have developed various computational methods to solve path-planning problems [12,13,14]. Search algorithms such as A* [15] and Dijkstra [16] guarantee solution optimality through systematic exploration of the state space. Sampling-based methods such as rapidly exploring random trees [17] and probabilistic road map [18] efficiently handle high-dimensional configuration spaces through random sampling strategies. Heuristic optimization methods, such as evolutionary algorithms [19,20], approach global optima through population-based iterative evolution and possess multi-objective optimization capabilities. Learning-based methods such as reinforcement learning [21,22] learn optimal path-planning strategies through environment interaction and demonstrate environmental adaptability [23,24,25].

However, when applied to dynamic environments where wind conditions evolve continuously during mission execution, these methods face significant computational efficiency challenges [26,27]. Traditional search and sampling-based methods require full solution recomputation when environments change, resulting in prohibitive computational overhead for replanning. Evolutionary algorithms can enhance dynamic adaptability via prior knowledge integration [28] or parallel computing acceleration [29], but their iterative nature still restricts real-time replanning. Learning-based methods demand hours to days of offline training on high-performance computing resources, with limited generalization to wind-field distributions outside the training dataset. Additionally, both evolutionary and learning-based methods provide limited interpretability and performance guarantees, which are critical concerns for safety-critical UAV operations requiring deterministic, verifiable solutions.

Incremental search algorithms, as an important branch of search algorithms, demonstrate computational advantages in dynamic scenarios by preserving and reusing historical computational results. Unlike traditional search methods, incremental search algorithms update only the affected portions of the search space when environments change [30,31]. Foundational work in this field includes the D* algorithm proposed by Stentz [32] and the Lifelong Planning A* (LPA*) algorithm developed by Koenig and Likhachev [33], which, respectively, established backward and forward incremental search frameworks. Building upon these foundational algorithms, researchers have continuously improved incremental search methods. In terms of algorithmic improvements, Ren et al. enhanced the D* Lite method using pruning strategies [30]. Lim et al. combined LPA* with guided local search to optimize replanning efficiency [34]. In application extensions, Gao et al. applied D* Lite to dynamic wind-field environments [35]. Xu et al. improved the D* Lite algorithm for Arctic ice condition changes [36]. In addition, Karlsson et al. made D* series methods more suitable for complex navigation scenarios with imperfect maps and dynamic changes through innovations in risk awareness, unknown space modeling, and dynamic map updates [37].

Although incremental search algorithms perform well in dynamic environments, deficiencies remain when applying them to UAV path planning in wind-field environments. First, the energy consumption of UAVs in wind fields involves complex coupling of induced power, profile power, and parasitic power [38,39,40]. The cost functions and heuristic functions of existing algorithms are primarily based on geometric distance, preventing UAVs from both utilizing wind-field information for path optimization and accurately reflecting the impact of wind fields on flight energy consumption. Second, existing incremental search algorithms have a limited search scope, constraining UAVs to focus only on local environmental information during path planning, making it difficult to achieve global wind-field awareness and long-term energy consumption optimization.

To address these issues, this paper proposes a wind-adaptive improved LPA* algorithm (WA-LPA*) for energy-optimal path planning of UAVs in dynamic wind fields with dynamic obstacles. The contributions of this work are as follows:

• Establishing a detailed energy consumption model for rotorcraft UAVs, which accurately describes the interaction mechanisms among power components under wind-field influences.
• Integrating physically accurate energy models into edge cost functions and designing composite heuristic functions that incorporate wind-field information, combined with a hierarchical height-aware optimization strategy for improved energy efficiency.
• Proposing a dynamic replanning decision algorithm based on wind-field change characteristics, which adaptively selects between global and local adjustment strategies to improve response efficiency.

The paper is organized as follows: Section 2 establishes the mathematical problem formulation, dynamic wind-field modeling, and energy consumption modeling. Section 3 presents the WA-LPA* algorithm methodology. Section 4 validates algorithm performance through various simulations. Finally, Section 5 concludes the paper and outlines future research directions.

2. Problem Formulation and System Modeling

2.1. Problem Definition

UAVs operating in complex environments encounter two primary challenges: time-varying wind conditions that significantly affect energy consumption, and dynamically appearing obstacles that compromise flight safety. This paper formulates these real-world complexities as an energy-optimal path-planning problem on a time-varying graph (Figure 1).

Let $\mathcal{S}:=[0,L_x]\times [0,L_y]\times [0,L_z]$ represent the three-dimensional continuous flight space, discretized into an $N_x\times N_y\times N_z$ regular cubic grid. The grid node set is defined as $\mathcal{V}=\{(i,j,k):0\le i<N_x,0\le j<N_y,0\le k<N_z\}$, with the mapping function between grid coordinates $(i,j,k)$ and physical coordinates $(x,y,z)$ given by:

$$\mathit{\varphi }(i,j,k)={(i·{\displaystyle \frac{L_x}{N_x}},j·{\displaystyle \frac{L_y}{N_y}},k·{\displaystyle \frac{L_z}{N_z}})}^{\top }$$

(1)

For any time instant t, the time-varying graph is defined as $\mathcal{G}\left(t\right)=(\mathcal{V},\mathcal{E}(t\left)\right)$, where the edge set $\mathcal{E}\left(t\right)$ represents the connectivity relationships between nodes at time t.

Given initial state ${\mathit{v}}_0\in \mathcal{V}$ and goal state ${\mathit{v}}_{\mathrm{n}}\in \mathcal{V}$, the energy-optimal path planning problem aims to find the path ${\mathit{\pi }}^{*}=\{{\mathit{v}}_0,{\mathit{v}}_1,\dots ,{\mathit{v}}_{\mathrm{n}}\}$ that minimizes total energy consumption:

$$\begin{array}{cc}\hfill {\mathit{\pi }}^{*}& =arg\underset{\mathit{\pi }\in \mathsf{\Phi }({\mathit{v}}_0,{\mathit{v}}_{\mathrm{n}})}{min}\mathcal{J}\left(\mathit{\pi }\right)=arg\underset{\mathit{\pi }\in \mathsf{\Phi }({\mathit{v}}_0,{\mathit{v}}_{\mathrm{n}})}{min}\sum _{k=0}^{n-1}{E}_{\mathrm{tot}}({\mathit{v}}_k,{\mathit{v}}_{k+1},\mathit{w}({\mathit{v}}_k,t_k))\hfill \\ \hfill \mathrm{subject}\mathrm{to}& \left\{\begin{array}{cc}{\mathit{v}}_k\notin \mathcal{O}\left(t_k\right),\hfill & \forall k\in \{0,1,\dots ,n\}\hfill \\ \parallel {\mathit{v}}_{\mathrm{a}}({\mathit{v}}_k,t_k)\parallel \le v_{max},\hfill & \forall k\in \{0,1,\dots ,n\}\hfill \\ ({\mathit{v}}_k,{\mathit{v}}_{k+1})\in \mathcal{E}\left(t_k\right),\hfill & \forall k\in \{0,1,\dots ,n-1\}\hfill \end{array}\right.\hfill \end{array}$$

(2)

where $\mathsf{\Phi }({\mathit{v}}_0,{\mathit{v}}_{\mathrm{n}})$ represents the set of all feasible paths from ${\mathit{v}}_0$ to ${\mathit{v}}_{\mathrm{n}}$, $E_{\mathrm{tot}}({\mathit{v}}_k,{\mathit{v}}_{k+1},\mathit{w}({\mathit{v}}_k,t_k))$ denotes the energy consumption for traversing from node ${\mathit{v}}_k$ to ${\mathit{v}}_{k+1}$ under wind field $\mathit{w}({\mathit{v}}_k,t_k)$, and $t_k$ represents the time when the UAV reaches node ${\mathit{v}}_k$. The constraints ensure collision avoidance, dynamic feasibility, and graph connectivity, respectively.

2.2. Dynamic Wind-Field Modeling

Wind fields, as fundamental elements of atmospheric boundary layer dynamics, exert a decisive influence on UAV path planning, energy consumption, and mission execution efficiency. To accurately characterize the spatiotemporal evolution of wind fields in realistic atmospheric environments, this study develops a time-varying wind-field mathematical model that integrates basic flow patterns, localized vortex perturbations, and logarithmic wind profile theory (Figure 2).

Assumption 1. 

Given that vertical wind speeds are significantly smaller in magnitude compared to horizontal components in low-altitude flight environments, this paper ignores the existence of vertical wind.

This model employs a layered modeling strategy. The three-dimensional space $\mathcal{S}$ is equally discretized into $N_z$ horizontal layers along the vertical direction. Through establishing a two-dimensional wind-field distribution at a reference height of $z_{\mathrm{ref}}$ and subsequently utilizing logarithmic wind profile theory to establish inter-layer wind speed relationships, the three-dimensional wind field is constructed.

Within the reference height plane, the baseline wind field ${\mathit{w}}_0={[u_0,v_0]}^{\top }$ is constructed through linear superposition of uniform flow and stochastic turbulent perturbations. For grid node $(i,j)$, the horizontal components of the baseline wind field are expressed as:

$$\left\{\begin{array}{cc}\hfill u_0(i,j)& =V_0{c}_{{\theta }_w}+{\epsilon }_x(i,j)\hfill \\ \hfill v_0(i,j)& =V_0{s}_{{\theta }_w}+{\epsilon }_y(i,j)\hfill \end{array}\right.$$

(3)

where $c_{.}=cos\left(·\right)$ and $s_{.}=sin\left(·\right)$. $V_0\in [V_{min},V_{max}]$ denotes the baseline wind speed intensity, ${\theta }_w\in [0,2\mathit{\pi }]$ represents the prevailing wind direction angle, and ${\epsilon }_x(i,j),{\epsilon }_y(i,j)\sim \mathcal{N}(0,{\sigma }^2)$ are independent Gaussian white noise turbulent perturbation terms that simulate the stochastic characteristics of small-scale atmospheric turbulence.

To simulate local circulation in the atmosphere, this model introduces $N_{\mathrm{v}}$ independent vortex structures based on a simplified form of Rankine vortex theory. The n-th vortex is centered at grid coordinates $(c_{x,n},c_{y,n})$, and its spatial influence function adopts a Gaussian distribution form:

$${\mathcal{G}}_n\left(r\right)=exp\left(-{\displaystyle \frac{r^2}{2r_{0,n}^2}}\right)$$

(4)

where $r_{0,n}$ denotes the radius of influence of the n-th vortex. In addition, r represents the radial distance from the physical coordinates $(x,y)$ to the vortex center, where $r=\sqrt{{(x-c_{x,n})}^2+{(y-c_{y,n})}^2}$.

The vortex wind field ${\mathit{w}}_{\mathrm{v}}={[u_{\mathrm{v}},v_{\mathrm{v}}]}^{\top }$ simulates the effects of local cyclonic and anticyclonic circulations. For a counterclockwise rotating cyclone vortex, the tangential velocity components are:

$$\left\{\begin{array}{cc}\hfill u_{\mathrm{v},n}(i,j,t)& =-{\displaystyle \frac{y-c_{y,n}\left(t\right)}{r_{\mathrm{s}}}}{I}_n{\mathcal{G}}_n\left(r\right)\hfill \\ \hfill v_{\mathrm{v},n}(i,j,t)& ={\displaystyle \frac{x-c_{x,n}\left(t\right)}{r_{\mathrm{s}}}}{I}_n{\mathcal{G}}_n\left(r\right)\hfill \end{array}\right.$$

(5)

where $I_n$ represents the vortex intensity parameter, and $r_{\mathrm{s}}=max(r,0.5)$ is the regularization distance that effectively avoids numerical singularities at vortex centers.

For the clockwise rotating anticyclone vortex, the corresponding velocity component has the opposite sign. The total contribution of the multi-vortex system is obtained through the linear superposition principle:

$${\mathit{w}}_{\mathrm{v}}(i,j,t)=\sum _{n=1}^{N_{\mathrm{v}}}{\mathit{w}}_{\mathrm{v},n}(i,j,t)$$

(6)

The vector superposition of the baseline flow and vortex fields constitutes the composite wind field at the reference height:

$${\mathit{w}}_{\mathrm{ref}}(i,j,t)={\mathit{w}}_0(i,j)+{\mathit{w}}_{\mathrm{v}}(i,j,t)$$

(7)

Although this paper assumes that vertical wind effects are negligible, the horizontal wind field varies with altitude. Therefore, based on Monin-Obukhov similarity theory [41], the logarithmic wind profile theory [42] is introduced to modify the wind speed distribution:

$$u\left(z\right)={\displaystyle \frac{u^{*}}{\kappa }}ln\left({\displaystyle \frac{z}{z_0}}\right)$$

(8)

where $u^{*}=0.5 \mathrm{m}/\mathrm{s}$ is the surface friction velocity, $\kappa =0.4$ is the Karman constant, and $z_0=0.1 \mathrm{m}$ represents the aerodynamic roughness length.

The height correction factor is defined as the relationship between the reference height and the target height wind speeds:

$$\alpha \left(z\right)={\displaystyle \frac{ln(z_{\mathrm{s}}/z_0)}{ln(z_{\mathrm{ref}}/z_0)}}$$

(9)

where $z_{\mathrm{s}}=max(z,z_0+0.1)$ ensures numerical stability of the logarithmic function. Therefore, the wind field $\mathit{w}(i,j,k,t)$ at an arbitrary height z can be obtained through vertical extrapolation:

$$\left\{\begin{array}{cc}\hfill u(i,j,k,t)& =u_{\mathrm{ref}}(i,j,t)·\alpha \left(z\right)\hfill \\ \hfill v(i,j,k,t)& =v_{\mathrm{ref}}(i,j,t)·\alpha \left(z\right)\hfill \\ \hfill h(i,j,k,t)& =0\hfill \end{array}\right.$$

(10)

where $i,j,k$, respectively, represent the grid nodes corresponding to the length, width, and height of the three-dimensional space, and z is the physical height corresponding to grid index k. The wind fields at different heights are shown in Figure 3.

To simulate the time-varying characteristics of atmospheric turbulence, vortex center positions undergo random updates at prescribed time intervals $\Delta T$. The process of wind-field changes is shown in Figure 4. At time $t+\Delta T$, the new position of the n-th vortex center is determined through the following equations:

$$\left\{\begin{array}{cc}\hfill c_{x,n}(t+\Delta T)& =c_{x,n}\left(t\right)+\delta x_n\hfill \\ \hfill c_{y,n}(t+\Delta T)& =c_{y,n}\left(t\right)+\delta y_n\hfill \end{array}\right.$$

(11)

where $\delta x_n,\delta y_n$ are random displacement increments following uniform distributions.

2.3. Energy Consumption Modeling for Rotorcraft UAVs Under Wind Disturbances

Three-dimensional path planning for UAVs operating in complex wind environments necessitates accurate kinematic and dynamic models that precisely quantify energy consumption. This work establishes comprehensive energy models using both the North-East-Down (NED) inertial reference ${\mathcal{F}}_{\mathrm{G}}$ and body frame ${\mathcal{F}}_{\mathrm{B}}$ to characterize UAV motion dynamics.

As illustrated in Figure 5, the inertial frame ${\mathcal{F}}_{\mathrm{G}}$ is centered at the mission starting point $O_{\mathrm{g}}$ with the $X_{\mathrm{g}}$-axis directed eastward, $Y_{\mathrm{g}}$-axis northward, and $Z_{\mathrm{g}}$-axis pointing downward, describing the UAV’s global position and velocity. The body frame ${\mathcal{F}}_{\mathrm{B}}$ is centered at the UAV’s center of mass $O_{\mathrm{b}}$, with the $X_{\mathrm{b}}$-axis representing the forward direction of the UAV, $Z_{\mathrm{b}}$-axis points downward relative to the fuselage, and $Y_{\mathrm{b}}$-axis is determined based on the right-hand coordinate system.

The UAV attitude is parameterized by three Euler angles: roll angle $\varphi$ about the $X_{\mathrm{b}}$-axis, pitch angle $\theta$ about the $Y_{\mathrm{b}}$-axis, and yaw angle $\psi$ about the $Z_{\mathrm{b}}$-axis. The transformation between coordinate frames is achieved through the rotation matrix $\mathbf{R}\in SO\left(3\right)$:

$$\mathbf{R}=\left[\begin{array}{ccc}{c}_{\theta }{c}_{\psi }& -c_{\varphi }{s}_{\psi }+s_{\varphi }{s}_{\theta }{c}_{\psi }& s_{\varphi }{s}_{\psi }+c_{\varphi }{s}_{\theta }{c}_{\psi }\\ c_{\theta }{s}_{\psi }& c_{\varphi }{c}_{\psi }+s_{\varphi }{s}_{\theta }{s}_{\psi }& -s_{\varphi }{c}_{\psi }+c_{\varphi }{s}_{\theta }{s}_{\psi }\\ -s_{\theta }& s_{\varphi }{c}_{\theta }& c_{\varphi }{c}_{\theta }\end{array}\right]$$

(12)

In ${\mathcal{F}}_{\mathrm{G}}$, the UAV’s position and ground velocity are represented by the vectors:

$$\mathit{p}={[x,y,z]}^{\top }\in {\mathbb{R}}^3$$

(13)

$${\mathit{v}}_{\mathrm{g}}=\dot{\mathit{p}}$$

(14)

Considering a three-dimensional time-varying wind field $\mathit{w}(x,y,z,t)\in {\mathbb{R}}^3$, the UAV’s airspeed vector ${\mathit{v}}_{\mathrm{a}}$ is:

$${\mathit{v}}_{\mathrm{a}}={\mathit{v}}_{\mathrm{g}}-\mathit{w}$$

(15)

Based on Newton–Euler equations, the translational dynamics are governed by:

$$m{\dot{\mathit{v}}}_{\mathrm{g}}=mg{\mathbf{e}}_z+\mathbf{R}{\mathit{f}}_{\mathrm{B}}+{\mathit{f}}_{\mathrm{d}}$$

(16)

where m denotes the UAV mass, g represents gravitational acceleration, and ${\mathbf{e}}_z={[0,0,1]}^{\top }$ is the vertical unit vector. The thrust vector in body frame is ${\mathit{f}}_{\mathrm{B}}={[0,0,-T]}^{\top }$, where the total thrust $T={\sum }_{i=1}^4{T}_i={\sum }_{i=1}^4\rho S{\left({\omega }_{i}r\right)}^2{C}_T$ is generated by the four rotors with air density $\rho$, rotor area S, rotor speed ${\omega }_i$, rotor radius r, and thrust coefficient $C_T$. The parasitic drag ${\mathit{f}}_{\mathrm{d}}=-{\displaystyle \frac{1}{2}}\rho C_{\mathrm{D}}A\left|{\mathit{v}}_{\mathrm{a}}\right|{\mathit{v}}_{\mathrm{a}}$ depends on the drag coefficient $C_{\mathrm{D}}$ and equivalent frontal area A.

In rotational dynamics, the torque generated by the rotor and the external disturbance torque act together to determine the attitude change in the UAV:

$$J\dot{\mathit{\omega }}+\mathit{\omega }\times J\mathit{\omega }={\mathit{\tau }}_{\mathrm{b}}+{\mathit{\tau }}_{\mathrm{d}}$$

(17)

where J is the inertia matrix, $\mathit{\omega }$ is the angular velocity vector, ${\mathit{\tau }}_{\mathrm{b}}={[l(T_2-T_4),l(T_3-T_1),{\sum }_{i=1}^4{(-1)}^{i+1}{Q}_i]}^{\top }$ is the control torque with rotor arm length l and rotor torque $Q_i=\rho Sr{\left({\omega }_{i}r\right)}^2{C}_Q$, where $C_Q$ is the torque coefficient. The term ${\mathit{\tau }}_{\mathrm{d}}$ represents wind-induced aerodynamic torque, which is typically negligible.

Assumption 2. 

The UAV maintains a constant ground speed ${\mathit{v}}_{\mathrm{g}}$, and its attitude remains basically stable.

Under this assumption, $|{\mathit{v}}_{\mathrm{g}}|$ is constant with ${\dot{\mathit{v}}}_{\mathrm{g}}=0$. The translational dynamic equation reduces to force equilibrium:

$$\mathbf{R}{\mathit{f}}_{\mathrm{B}}+{\mathit{f}}_{\mathrm{d}}=mg{\mathbf{e}}_z$$

(18)

For nearly horizontal flight with small angle approximations $cos\theta \approx 1$ and $sin\theta \approx \theta$, the vertical and horizontal force balances become $Tcos\theta \approx mg$ and $Tsin\theta \approx |{\mathit{f}}_{\mathrm{d}}|$, yielding the tilt angle $\theta$ and required thrust T:

$$\theta \approx {\displaystyle \frac{|{\mathit{f}}_{\mathrm{d}}|}{mg}}={\displaystyle \frac{\rho C_{\mathrm{D}}A{\left|{\mathit{v}}_{\mathrm{a}}\right|}^2}{2mg}}$$

(19)

$$T={\displaystyle \frac{mg}{cos\theta }}$$

(20)

In a stable flight state, the rotating dynamic is mainly used to maintain the stable attitude of the UAV, and its influence on the subsequent energy consumption calculation can be ignored. Therefore, when deriving the energy consumption model, this paper focuses on the impact of translational dynamics on energy consumption.

The power consumption of rotorcraft UAVs comprises four primary components. According to momentum theory, induced power generates thrust:

$$P_{\mathrm{i}}={\displaystyle \frac{T{v}_{\mathrm{ie}}}{{\eta }_{\mathrm{p}}}}$$

(21)

where ${\eta }_{\mathrm{p}}$ is the propeller efficiency and $v_{\mathrm{ie}}$ is the effective induced velocity: $v_{\mathrm{ie}}=v_i+v_z/2$ for climb ($v_z>0$) and $v_{\mathrm{ie}}=max(v_i-v_z/2,0)$ for descent ($v_z<0$), with hover induced velocity $v_i=\sqrt{T/\left(2\rho A_r\right)}$ and total rotor disk area $A_r=N_r\mathit{\pi }{r}^2$.

The profile power from rotor blade resistance is:

$$P_{\mathrm{p}}={\displaystyle \frac{\rho N_r{C}_{\mathrm{p}}{A}_r{\left(\omega r\right)}^3}{8}}$$

(22)

where the angular velocity $\omega =\sqrt{T/\left(N_r{C}_T\rho A_r{r}^2\right)}$ is determined by thrust coefficient $C_T$.

The parasitic power results from fuselage drag:

$$P_{\mathrm{d}}={\displaystyle \frac{1}{2}}\rho C_{\mathrm{D}}A{\left|{\mathit{v}}_{\mathrm{a}}\right|}^3$$

(23)

The climb power represents additional power to overcome gravitational potential energy:

$$P_{\mathrm{c}}=max(0,mg{v}_z)$$

(24)

Considering motor efficiency ${\eta }_{\mathrm{m}}$, electronic speed controller efficiency ${\eta }_{\mathrm{e}}$, and onboard electronics power $P_{\mathrm{e}}$, the total energy consumption from position ${\mathit{p}}_1$ to ${\mathit{p}}_2$ is:

$$E_{\mathrm{tot}}=\left[{\displaystyle \frac{P_{\mathrm{i}}+P_{\mathrm{p}}+P_{\mathrm{d}}+P_{\mathrm{c}}}{{\eta }_{\mathrm{m}}{\eta }_{\mathrm{e}}}}+P_{\mathrm{e}}\right]·{\displaystyle \frac{|{\mathit{p}}_2-{\mathit{p}}_1|}{|{\mathit{v}}_{\mathrm{g}}|}}$$

(25)

Substituting the power components yields the energy consumption model, considering wind-field effects:

$$E_{\mathrm{tot}}\approx \left[{\displaystyle \frac{mg\sqrt{mg/\left(2\rho A_r\right)}+P_{\mathrm{p}}+\frac{1}{2}\rho C_{\mathrm{D}}A{|{\mathit{v}}_{\mathrm{g}}-\mathit{w}|}^3}{{\eta }_{\mathrm{t}}}}+P_{\mathrm{e}}\right]·{\displaystyle \frac{|{\mathit{p}}_2-{\mathit{p}}_1|}{v_{\mathrm{g}}}}$$

(26)

where ${\eta }_{\mathrm{t}}={\eta }_{\mathrm{p}}{\eta }_{\mathrm{m}}{\eta }_{\mathrm{e}}$ is the total efficiency.

3. Methodology

3.1. Wind-Adaptive LPA* Algorithm Framework

Classical path-planning algorithms exhibit fundamental limitations in dynamic environments. While A* guarantees optimality in static settings, it requires complete re-search upon environmental changes.

The LPA* algorithm addresses this through incremental search capabilities. The core of LPA* lies in maintaining two key values to describe node state. One is the cost function $g\left(\mathit{v}\right)$ that represents the current best-known cost from start to node $\mathit{v}$. The other is the right-hand-side (rhs) value $\mathrm{rhs}$ that denotes a one-step lookahead estimate based on predecessors, defined as:

$$\mathrm{rhs}\left(\mathit{v}\right)=\left\{\begin{array}{cc}0\hfill & \mathrm{if}\mathit{v}={\mathit{v}}_{\mathrm{n}}\hfill \\ \underset{\mathit{u}\in \mathcal{P}\left(\mathit{v}\right)}{min}\{g\left(\mathit{u}\right)+c(\mathit{u},\mathit{v})\}\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(27)

where $\mathcal{P}\left(\mathit{v}\right)$ is the set of predecessors of node $\mathit{v}$, and $c(\mathit{u},\mathit{v})$ is the edge cost function.

A node $\mathit{v}$ is locally consistent if $g\left(\mathit{v}\right)=\mathrm{rhs}\left(\mathit{v}\right)$. Otherwise, it is locally inconsistent and must be refined through search. All locally inconsistent nodes are stored in a priority queue, ordered by two-component keys:

$$\kappa \left(\mathit{v}\right)=[min(g\left(\mathit{v}\right),\mathrm{rhs}\left(\mathit{v}\right))+h(\mathit{v}),min(g\left(\mathit{v}\right),\mathrm{rhs}\left(\mathit{v}\right)\left)\right]$$

(28)

However, the LPA* algorithm adopts a geometric heuristic method that ignores the impact of wind fields on energy consumption, making it unsuitable for energy optimization planning in wind-field environments. Thus, this paper proposes the WA-LPA*, which is improved from four aspects: edge cost, heuristic function, hierarchical height-aware optimization strategy, and replanning strategy. The overall process of the algorithm is shown in Algorithm 1.

Algorithm 1 Wind-Adaptive LPA* Algorithm

Require: Start ${\mathit{v}}_0$, Goal ${\mathit{v}}_{\mathrm{n}}$, Environment $\mathcal{E}$ Ensure: Optimal energy path ${\mathit{\pi }}^{*}$   1: Initialize $g\leftarrow \varnothing$, $\mathrm{rhs}\leftarrow \varnothing$, $\mathcal{Q}\leftarrow \varnothing$   2: Set $\mathrm{rhs}\left[{\mathit{v}}_{\mathrm{n}}\right]\leftarrow 0$, insert $(\kappa \left({\mathit{v}}_{\mathrm{n}}\right),{\mathit{v}}_{\mathrm{n}})$ into $\mathcal{Q}$   3: Execute LPA* search to obtain initial path $\mathit{\pi }$, set $k\leftarrow 0$   4: while $k<\left|\mathit{\pi }\right|$ and $\mathit{\pi }\left[k\right]\ne {\mathit{v}}_{\mathrm{n}}$ do   5:       Advance time step, detect wind-field changes   6:       if wind field changed then   7:             $L_{\mathrm{r}}\leftarrow \left|\mathit{\pi }\right|-k$, $\Delta w_{max}\leftarrow$ maximum wind change   8:             if $L_{\mathrm{r}}\le L_{\mathrm{h}}$ or $\Delta w_{max}>\Delta w_{\mathrm{sig}}$ then   9:             Clear wind cache, reinitialize LPA* from $\mathit{\pi }\left[k\right]$ to ${\mathit{v}}_{\mathrm{n}}$ 10:             Global replanning to obtain new path ${\mathit{\pi }}^{\prime }$ 11:             else 12:             Define $\mathcal{R}\leftarrow {\bigcup }_{\mathit{v}\in \mathcal{P}}{\mathcal{N}}^{\left(r\right)}\left(\mathit{v}\right)$ 13:             Locally update $\mathrm{rhs}$ values and $\mathcal{Q}$ within $\mathcal{R}$ 14:             end if 15:             Extract updated path ${\mathit{\pi }}^{\prime }$, adjust k 16:       end if 17:       ${\mathit{v}}_{\mathrm{c}}\leftarrow \mathit{\pi }\left[k\right]$ 18:       Determine next optimal movement ${\mathit{v}}_{\mathrm{next}}$ based on cost function 19:       if ${\mathit{v}}_{\mathrm{next}}$ represents horizontal movement then 20:             Find candidate height $z_{\mathrm{c}}$ and calculate $E_{\mathrm{alt}}$ 21:             Compute net benefit $B=E_{\mathrm{d}}-E_{\mathrm{alt}}$ 22:             if $B>{\beta }_{\mathrm{h}}$ then 23:             Execute vertical movement to optimal height 24:             else 25:             Execute original horizontal movement ${\mathit{v}}_{\mathrm{next}}$ 26:             end if 27:       else 28:             Execute vertical movement ${\mathit{v}}_{\mathrm{next}}$ directly 29:       end if 30:       $k\leftarrow k+1$ 31: end while         return Executed path ${\mathit{\pi }}_{\mathrm{exe}}$

3.2. Physically Accurate Edge Cost Function

The accuracy of the edge cost function directly determines the physical rationality and practical applicability of the path-planning results. In complex wind-field environments, simplified geometric metrics cannot accurately reflect the actual flight costs of UAVs. The edge cost function in this paper is directly based on the detailed aerodynamic model established in Equation (25):

$$c({\mathit{v}}_i,{\mathit{v}}_j)=E_{\mathrm{tot}}({\mathit{v}}_i,{\mathit{v}}_j,\mathit{w}({\mathit{v}}_i,t))$$

(29)

Compared with simplified geometric metrics, this method can correctly identify the optimal flight path under different wind-field conditions. Especially in strong wind environments, it can effectively avoid flying against the wind and prioritize downwind or crosswind paths, thereby significantly reducing overall energy consumption.

3.3. Wind-Aware Heuristic Function Design

The design of the heuristic function is a key determinant of the performance of path-planning algorithms. It not only affects the search efficiency but also directly determines whether the algorithm can find the true optimal solution. In dynamic wind-field environments, traditional geometric heuristic functions, which fail to account for the impact of wind fields on flight energy consumption, often mislead the search direction, resulting in the algorithm wasting computational resources on suboptimal energy paths.

To address this issue, this paper designs a composite heuristic function that effectively incorporates the effects of wind fields into the heuristic evaluation process while maintaining computational efficiency. The improved heuristic function is:

$$h\left(\mathit{v}\right)=h_0\left(\mathit{v}\right)·(1-\alpha ·\xi \left(\mathit{v}\right))$$

(30)

where $h_0\left(\mathit{v}\right)$ is the base geometric heuristic, $\alpha$ is the wind-field influence weight coefficient, and $\xi \left(\mathit{v}\right)$ is the normalized wind-field alignment factor.

The base geometric heuristic employs a computationally efficient physical distance metric:

$$h_0\left(\mathit{v}\right)={\parallel \mathit{\varphi }\left(\mathit{v}\right)-\mathit{\varphi }\left({\mathit{v}}_{\mathrm{n}}\right)\parallel }_2·{\displaystyle \frac{P_{\mathrm{h}}}{v_{\mathrm{c}}}}$$

(31)

where $\mathit{\varphi }\left(\mathit{v}\right)$ represents the physical coordinates of node $\mathit{v}$, $P_{\mathrm{h}}$ is the hover power, and $v_{\mathrm{c}}$ is the cruise velocity.

The wind-field alignment factor quantifies the compatibility between the current wind field and flight direction:

$$\xi \left(\mathit{v}\right)={\displaystyle \frac{\mathit{d}·\mathit{w}\left(\mathit{v}\right)}{\left|\mathit{d}\right|·\left|\mathit{w}\right(\mathit{v}\left)\right|+ϵ}}$$

(32)

where $\mathit{d}=\mathit{\varphi }\left({\mathit{v}}_{\mathrm{n}}\right)-\mathit{\varphi }\left(\mathit{v}\right)$ is the flight direction vector from the current node to the goal, $\mathit{w}\left(\mathit{v}\right)$ is the wind velocity vector at the current position, and $ϵ=1\times {10}^{-6}$ is a numerical stability term. The wind alignment $\xi \in [-1,1]$, with positive values indicating tailwind and negative values indicating headwind.

Through the wind correction term $(1-\alpha ·\xi )$, the heuristic function dynamically adjusts estimates: reducing estimated costs under tailwind conditions to encourage exploration, and increasing estimated costs under headwind conditions to avoid unfavorable paths. The weight coefficient $\alpha$ controls the intensity of wind-field influence.

Consistent with the standard LPA* framework, the WA-LPA* algorithm describes node states using two key values: the cost function $g\left(\mathit{v}\right)$ and the right-hand-side value $\mathrm{rhs}\left(\mathit{v}\right)$ defined in Equation (27). The algorithm operates on the principle of node consistency. A node $\mathit{v}$ is locally consistent when $g\left(\mathit{v}\right)=\mathrm{rhs}\left(\mathit{v}\right)$, indicating accurate cost estimates under the current graph state. Inconsistent nodes are maintained in a priority queue ordered by the key function $\kappa \left(\mathit{v}\right)$. During execution, the algorithm iteratively processes the node with the smallest key. For overconsistent nodes which $g\left(\mathit{v}\right)>\mathrm{rhs}\left(\mathit{v}\right)$, it sets $g\left(\mathit{v}\right)=\mathrm{rhs}\left(\mathit{v}\right)$ and updates all successors. For underconsistent nodes which $g\left(\mathit{v}\right)<\mathrm{rhs}\left(\mathit{v}\right)$, it sets $g\left(\mathit{v}\right)=\infty$ and recomputes $\mathrm{rhs}$ values for the node and its successors.

3.4. Hierarchical Height-Aware Optimization Strategy

Classical three-dimensional LPA* algorithms usually consider six basic movement directions at each search node. However, this direct six-connectivity search strategy exposes fundamental myopic limitations in dynamic wind-field environments. The algorithm makes greedy decisions based solely on current local information, unable to anticipate potentially more favorable wind-field conditions in adjacent height layers, thereby missing energy optimization opportunities.

To overcome the limitations of traditional approaches, this paper proposes a hierarchical height-aware optimization strategy that decomposes the three-dimensional path-planning problem into collaborative optimization of vertical height selection and horizontal path-planning.

The core of the hierarchical height-aware strategy lies in the intelligent action selection mechanism. Let the UAV’s current position be ${\mathit{v}}_{\mathrm{c}}$, and the LPA* algorithm determines the next optimal movement ${\mathit{v}}_{\mathrm{next}}$ based on the cost function at the current height layer.

When ${\mathit{v}}_{\mathrm{next}}$ represents vertical movement, the algorithm directly executes this action. This ensures that LPA*’s original vertical path-planning capabilities are preserved. When ${\mathit{v}}_{\mathrm{next}}$ represents horizontal movement, the algorithm initiates the height optimization evaluation process.

Based on insights from logarithmic wind profile theory, the paper establishes an adaptive height search mechanism. The algorithm evaluates both upward and downward adjacent height layers to identify potentially more favorable wind conditions. Under headwind conditions, lower altitude layers typically exhibit reduced wind speeds due to boundary layer effects, making downward search a priority. Under tailwind conditions, higher altitude layers can provide additional propulsive force through stronger winds, prioritizing upward search. However, the final decision is made through rigorous energy consumption evaluation rather than directional assumptions, ensuring adaptability to diverse wind profiles.

For each candidate height $z_{\mathrm{c}}$, the algorithm calculates the total alternative strategy energy consumption:

$$E_{\mathrm{alt}}=E_{\mathrm{v}}({\mathit{v}}_{\mathrm{c}},{\mathit{v}}_{\mathrm{c}}^{z_{\mathrm{c}}})+E_{\mathrm{h}}({\mathit{v}}_{\mathrm{c}}^{z_{\mathrm{c}}},{\mathit{v}}_{\mathrm{next}}^{z_{\mathrm{c}}})$$

(33)

where $E_{\mathrm{v}}$ represents the vertical movement energy consumption from current position to the candidate height layer, $E_{\mathrm{h}}$ denotes the horizontal movement energy consumption at the candidate height layer, ${\mathit{v}}_{\mathrm{c}}^{z_{\mathrm{c}}}$ is the position at candidate height corresponding to current position, and ${\mathit{v}}_{\mathrm{next}}^{z_{\mathrm{c}}}$ is the target position at candidate height corresponding to the next planned position.

Decision optimization achieves optimal selection through the evaluation of net benefits. The net benefit B is defined as the energy consumption difference between direct and alternative strategies:

$$B=E_{\mathrm{d}}-E_{\mathrm{alt}}$$

(34)

The vertical movement is executed only when net benefit exceeds ${\beta }_{\mathrm{h}}$. Otherwise, the original horizontal movement strategy is maintained.

3.5. Dynamic Environment Adaptation Mechanisms

Environmental changes in dynamic settings manifest as two distinct perturbation types: wind-field evolution and obstacle transitions. These differ fundamentally in their impact mechanisms: wind-field changes alter edge costs without affecting graph connectivity, while obstacle transitions constitute topological modifications that directly affect path feasibility.

When wind conditions evolve, the algorithm employs an intelligent replanning strategy that dynamically selects between global replanning and local adjustment based on wind variation amplitude and remaining path characteristics. The decision logic is:

$$\mathcal{S}=\left\{\begin{array}{cc}\mathrm{Global}\mathrm{replanning}\hfill & \mathrm{when}{\mathcal{C}}_{\mathrm{s}}\vee {\mathcal{C}}_{\mathrm{d}}\vee {\mathcal{C}}_{\mathrm{m}}\hfill \\ \mathrm{Local}\mathrm{adjustment}\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(35)

where the three triggering conditions are defined as:

$$\begin{array}{cc}\hfill {\mathcal{C}}_{\mathrm{s}}& :L_{\mathrm{r}}\le L_{\mathrm{h}}\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {\mathcal{C}}_{\mathrm{m}}& :\underset{(x,y,z)\in \mathcal{L}}{max}{\parallel \mathit{w}(x,y,z,t+1)-\mathit{w}(x,y,z,t)\parallel }_2>\Delta w_{\mathrm{sig}}\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {\mathcal{C}}_{\mathrm{d}}& :\underset{(x,y,z)\in \mathcal{L}}{max}min\left(\right|{\theta }_{t+1}-{\theta }_t|,2\mathit{\pi }-|{\theta }_{t+1}-{\theta }_t\left|\right)>\Delta {\theta }_{\mathrm{sig}}\hfill \end{array}$$

(38)

Global replanning is triggered when any one of these conditions is satisfied. Here, $L_{\mathrm{r}}$ is the remaining path length, $L_{\mathrm{h}}$ is the short path threshold, $\mathcal{L}$ represents the spatial domain covered by the current path, and $\Delta w_{\mathrm{sig}}$, $\Delta {\theta }_{\mathrm{sig}}$ are significance thresholds for wind magnitude and direction changes. When $L_{\mathrm{r}}$ falls below $L_{\mathrm{h}}$, global replanning becomes computationally feasible. When wind changes exceed thresholds, comprehensive re-optimization maintains energy efficiency.

During global replanning, all wind-related cached costs are invalidated, and a complete path re-optimization is executed from the current position. When wind changes are localized or moderate, local adjustment is performed instead. Local adjustment defines an affected region:

$$\mathcal{R}=\bigcup _{\mathit{v}\in \mathcal{P}}{\mathcal{N}}^{\left(r\right)}\left(\mathit{v}\right)$$

(39)

where ${\mathcal{N}}^{\left(r\right)}\left(\mathit{v}\right)$ denotes the r-layer neighborhood around node $\mathit{v}$ and $\mathcal{P}$ represents the path segment requiring update (Figure 6).

The local adjustment execution follows an orchestrated sequence: cache invalidation removes edge cost entries from the affected region, vertex updates recalculate $\mathrm{rhs}$ values for all nodes within $\mathcal{R}$, and incremental search optimizes the path from the current position while preserving the traversed portion. The resulting path combines the fixed segment with the locally re-optimized remainder, maintaining continuity while adapting to wind-field changes.

Obstacle changes constitute topological modifications that directly affect graph connectivity, potentially rendering existing path segments unreachable. For newly appeared obstacles ${\mathcal{O}}_{\mathrm{new}}\left(t\right)$, the affected region is defined as the 6-connected neighborhood of all obstacle boundary nodes:

$$\mathcal{A}=\bigcup _{\mathit{o}\in {\mathcal{O}}_{\mathrm{b}}}{\{\mathit{v}\in \mathcal{G}:\parallel \mathit{v}-\mathit{o}\parallel }_1=1,\mathit{v}\notin \mathcal{O}\left(t\right)\}$$

(40)

where ${\mathcal{O}}_{\mathrm{b}}$ represents the boundary nodes of newly appeared obstacles and $\mathcal{O}\left(t\right)$ denotes all obstacles at time t.

Once the affected region is determined, the algorithm performs selective cache invalidation, removing cached edge costs for affected nodes. If a node becomes an obstacle, its g and $\mathrm{rhs}$ values are set to infinity and marked as impassable. The $\mathrm{rhs}$ values are then recalculated for each affected node and its neighbors, with inconsistent nodes added to the priority queue for subsequent processing.

4. Experimental Evaluation

4.1. Simulation Setup

This paper conducts simulation experiments on a Windows 11 system (Huawei, Shenzhen, China) equipped with an Intel Core i7-10510U processor (1.8 GHz, 4 cores, 8 threads) and 16 GB RAM. The UAV platform is modeled based on the DJI Mavic 3T (DJI Innovation Technology Co., Ltd., Shenzhen, China) parameters shown in

Table 1. Parameter settings related to UAV.

Parameter	Symbol	Typical Value
Mass	m	0.92 kg
Ground speed	$v_g$	15 m/s
Gravitational acceleration	g	9.81 m/s2
Air density	$\rho$	1.225 kg/m3
Number of rotors	$N_r$	4
Rotor radius	r	0.12 m
Equivalent frontal area	A	0.06 m2
Thrust coefficient	$C_T$	0.0180
Torque coefficient	$C_Q$	0.0016
Profile drag coefficient	$C_{\mathrm{D}}$	0.015
Average blade drag coefficient	$C_{\mathrm{p}}$	0.5
Propeller efficiency	${\eta }_{\mathrm{p}}$	0.78
Motor efficiency	${\eta }_{\mathrm{m}}$	0.82
Controller efficiency	${\eta }_{\mathrm{e}}$	0.92
Electronic power	$P_{\mathrm{e}}$	6 W

.

The simulation environment constructs a 1000 m × 1000 m × 20 m cubic flight space using equidistant grid discretization. Grid resolution is set to $20\times 20\times 20$, corresponding to cell dimensions of 50 m × 50 m × 1 m. The wind-field model employs the layered modeling strategy from Section 2.2, and the update interval is set to 25 time steps. Obstacle configuration includes static obstacles and dynamic obstacles that appear after predetermined time steps to simulate evolving environmental conditions. The UAV is modeled as a point mass, and collision detection is performed by checking grid cell occupancy—a path is considered collision-free if its nodes do not overlap with obstacle cells.

The WA-LPA* algorithm requires several key parameters to control its adaptive behavior. The wind-field influence weight $\alpha =0.3$ balances wind information contribution to the heuristic function without overwhelming geometric distance. The energy benefit threshold ${\beta }_{\mathrm{h}}=10.0$ J prevents frequent vertical movements by requiring significant energy savings.

4.2. Parameter Sensitivity Analysis

To systematically evaluate the impact of key parameters on algorithm performance, a comprehensive sensitivity analysis was conducted for four core parameters in the wind-aware replanning mechanism: the wind speed variation threshold $\Delta w_{\mathrm{sig}}$, the wind direction variation threshold $\Delta {\theta }_{\mathrm{sig}}$, the remaining path length threshold $L_{\mathrm{h}}$, and the affected neighbor range r. The baseline parameter configuration was established as $\Delta w_{\mathrm{sig}}=10.0 \mathrm{m}/s$, $\Delta {\theta }_{\mathrm{sig}}={180}^{\circ }$, $L_{\mathrm{h}}=15$ steps, and $r=1$ step, serving as the reference for comparative analysis. For each parameter, representative value ranges were selected based on theoretical considerations and practical constraints: $\Delta w_{\mathrm{sig}}$ was varied across $[4.0,6.0,8.0,10.0,12.0]$ m/s to capture different environmental responsiveness levels; $\Delta {\theta }_{\mathrm{sig}}$ spanned $[{45}^{\circ },{90}^{\circ },{135}^{\circ },{180}^{\circ }]$ to examine various angular sensitivity ranges; $L_{\mathrm{h}}$ explored $[5,10,15,20]$ steps to investigate the trade-off between local efficiency and global optimality; and r examined $[1,2,3,4]$ steps to assess the spatial scope of local updates. Each parameter configuration was evaluated across $N=10$ distinct wind-field environments with randomized vortex configurations while maintaining identical obstacle layouts. ${\mathrm{CV}}_E=({\overline{\sigma }}_E/{\overline{\mu }}_E)\times 100\%$ quantifies the coefficient of variation across environments, reflecting parameter robustness to environmental uncertainty.

Experimental results in

Table 2. Experimental results under different parameter configurations. Results represent mean ± standard deviation across 10 randomized wind-field environments with identical obstacle configurations.

Parameter	Value	Planning Time (s)	Energy Consumption (J)
$\Delta {\theta }_{\mathrm{sig}}$ (∘)	45	14.85 ± 2.42	328.6 ± 48.2
	90	12.73 ± 2.18	335.4 ± 40.1
	135	11.54 ± 2.05	361.8 ± 42.5
	180 †	10.67 ± 1.92	415.3 ± 33.7
$\Delta w_{\mathrm{sig}}$ (m/s)	4.0	14.85 ± 2.42	331.5 ± 49.8
	6.0	12.73 ± 2.18	357.2 ± 45.4
	9.0	11.54 ± 2.05	362.1 ± 42.8
	10.0 †	10.67 ± 2.92	415.3 ± 33.7
	12.0	9.24 ± 2.86	428.7 ± 29.1
$L_{\mathrm{h}}$ (steps)	5	9.52 ± 1.58	430.9 ± 51.6
	10	10.08 ± 2.21	422.7 ± 44.2
	15 †	10.67 ± 2.92	415.3 ± 33.7
	20	13.83 ± 3.01	363.4 ± 31.5
	25	15.51 ± 3.08	342.8 ± 28.9
r (steps)	1 †	10.67 ± 1.92	415.3 ± 33.7
	2	12.24 ± 2.12	362.8 ± 41.1
	3	14.81 ± 2.36	359.5 ± 42.8
	4	17.02 ± 2.65	351.2 ± 41.7
	5	19.15 ± 2.73	347.9 ± 41.3

^† Baseline configuration.

reveal that smaller thresholds for $\Delta {\theta }_{\mathrm{sig}}$ and $\Delta w_{\mathrm{sig}}$, coupled with larger $L_{\mathrm{h}}$, enhance wind-field responsiveness and achieve substantial energy reduction, albeit with increased computation time. While increasing $r_{\mathrm{neighbor}}$ expands the local optimization scope, marginal energy benefits diminish beyond $r>3$ steps, accompanied surge in computation time. Figure 7 presents the parameter robustness ranking based on the ${\mathrm{CV}}_E$. Based on the ranking of the parameters’ coefficient of variation and by balancing energy consumption optimization with computational efficiency, subsequent experiments adopt the configuration: $\Delta {\theta }_{\mathrm{sig}}={45}^{\circ }$, $\Delta w_{\mathrm{sig}}=4.0 \mathrm{m}/s$, $L_{\mathrm{h}}=10$ steps, and $r=2$ steps.

4.3. Wind-Field Adaptability Testing

To comprehensively evaluate the algorithm’s adaptivity across different wind-field regimes, this section presents experiments under both steady wind fields and dynamic wind fields. All experiments adopt unified task settings with starting point $(2,2,5)$ and target point $(18,18,18)$.

The steady wind-field experiment verifies the algorithm baseline performance in static environments. Figure 8, Figure 9, Figure 10 and Figure 11 show three-dimensional path-planning results under four steady wind conditions.

Table 3. Performance comparison under steady wind at different angles.

Scenario	Path Length (m)	Energy Consumption (J)	Energy Efficiency (m/J)
${\theta }_w={45}^{\circ }$	1613.0	28,306.9	17.549
${\theta }_w={135}^{\circ }$	1623.0	28,306.9	22.751
${\theta }_w={225}^{\circ }$	1623.0	44,444.5	27.384
${\theta }_w={315}^{\circ }$	1623.0	36,994.5	22.794

provides performance comparisons for these scenarios. Results indicate significant wind-field impact on UAV path planning and energy consumption, demonstrating the proposed method’s ability to utilize wind information for energy-efficient planning.

Simulation experiments under complex dynamic wind fields verify the proposed method’s ability to respond to dynamic environmental changes and the effectiveness of the replanning strategy. Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16 show path-planning results in five different scenarios, with performance indicators presented in

Table 4. Performance comparison in time-varying complex wind fields.

Scenario	Initial Path Length (m)	Initial Path Energy (J)	Replanning Times	Final Path Length (m)	Final Path Energy (J)	Energy Savings (J)
1	1613	26,269.9	1	1613	25,621.2	648.7
2	1613	28,317.8	1	1613	28,197.3	120.5
3	1613	32,368.6	1	1613	32,220.9	147.7
4	1629	42,059.9	2	1629	39,624.8	2385.2
5	1637	41,261.3	2	1623	37,523.1	3738.2

.

The initial path is planned based on wind-field conditions at $t=0$, exhibiting wind-field adaptation characteristics similar to steady experiments. WA-LPA* fully utilizes favorable wind conditions at different altitudes while avoiding strong headwind areas. As wind fields evolve, the algorithm triggers appropriate replanning mechanisms based on changing characteristics. The replanned paths effectively utilize favorable conditions under new wind-field states, demonstrating the algorithm’s capability to handle dynamic wind environments.

4.4. Performance Comparison Experiments

To comprehensively validate the performance advantages of the proposed WA-LPA* algorithm, this section designs three comparison experimental scenarios with increasing complexity, systematically comparing WA-LPA* with two classical path-planning algorithms: traditional A* and standard LPA*. All three algorithms operate within a 6-connected grid framework, where the action space is restricted to six cardinal directions $(x,y,z)$. Traditional A* employs the evaluation function $f\left(n\right)=g\left(n\right)+h\left(n\right)$, where $g\left(n\right)$ is the step count from the start node and $h\left(n\right)$ is the Euclidean distance to the goal.

Figure 17 shows path-planning comparisons under a steady wind field and static building conditions.

Table 5. Algorithm performance indicators under a steady wind field with static buildings.

Algorithm	Planning Time (s)	Path Steps	Path Length (m)	Energy Consumption (J)
Traditional A*	$0.37$	46	$1613.0$	34,299.0
Standard LPA*	$8.62$	46	$1613.0$	33,633.3
WA-LPA*	$9.01$	56	$1623.0$	31,282.7

presents key performance indicators. The A* algorithm selects geometrically shortest paths without considering wind-field effects. Standard LPA* optimizes path length but lacks wind-field adaptivity. WA-LPA* effectively utilizes favorable wind fields through the height-aware strategy, achieving energy consumption optimization.

Under dynamically complex wind fields with changes at 25 time step intervals, Figure 18 illustrates the path evolution process of the three algorithms, while

Table 6. Algorithm performance indicators under dynamic wind field with static buildings.

Algorithm	Planning Time (s)	Path Steps	Path Length (m)	Energy Consumption (J)
Traditional A*	$0.37$	46	$1613.0$	51,276.2
Standard LPA*	$8.62$	46	$1613.0$	43,943.6
WA-LPA*	$15.81$	70	$1637.0$	36,181.5

summarizes corresponding performance comparisons.

In dynamic environments, WA-LPA* advantages are more pronounced. Traditional A* and standard LPA* lack dynamic wind-field adaptation capability, whereas WA-LPA* achieves energy-optimal path planning through intelligent replanning strategies.

The most challenging experimental scenario involves both wind-field changes and dynamic obstacle emergence. In addition to wind-field changes at predetermined intervals, a new obstacle is introduced at the 10th time step. Figure 19 presents path-planning results of the three algorithms in this environment. The A* algorithm fails when encountering dynamic obstacles and cannot reach the destination.

Table 7. Algorithm performance indicators under the dynamic wind field with dynamic obstacles.

Algorithm	Planning Time (s)	Path Steps	Path Length (m)	Energy Consumption (J)
Traditional A*	$0.37$	21	$755.00$	Failed
Standard LPA*	$8.62$	46	$1613.00$	35,847.1
WA-LPA*	$19.78$	56	$1623.00$	33,724.9

shows performance comparison results under this scenario. Even under the most complex environmental conditions, WA-LPA* maintains significant performance advantages.

The experimental results demonstrate that WA-LPA* exhibits longer computational times due to its integrated wind-field replanning mechanism and height optimization strategy. Across the three experimental scenarios, WA-LPA* requires 9.01 s, 15.81 s, and 19.78 s, respectively, significantly higher than traditional A* and standard LPA*. This computational overhead stems from the algorithm’s need to continuously analyze wind-field characteristics, execute height optimization decisions, and perform incremental replanning when adapting to dynamic environmental changes.

However, from the energy consumption perspective, WA-LPA* demonstrates remarkable performance advantages across all testing scenarios. Under steady wind-field conditions, WA-LPA* achieves 8.8% energy savings compared to traditional A* and 7.0% improvement over standard LPA*. In dynamic wind-field environments, this advantage becomes more pronounced: 29.4% energy reduction compared to traditional A* and 17.7% improvement over standard LPA*. Under the most complex scenario involving both dynamic wind fields and dynamic obstacles, while traditional A* fails to adapt to dynamic changes, WA-LPA* still achieves 5.9% energy efficiency improvement compared to standard LPA*. These results indicate that although WA-LPA* requires additional computational time for intelligent path planning, the energy optimization benefits achieved through wind-field adaptivity and height-aware strategies are significant, demonstrating stronger practical value in complex dynamic environments.

5. Conclusions

This paper addresses the energy-optimal path-planning problem for UAVs in three-dimensional dynamic wind-field environments by proposing the WA-LPA*. By integrating energy consumption models with wind-aware heuristic functions, hierarchical height-aware optimization, and dynamic replanning mechanisms, WA-LPA* achieves physics-based energy optimization. Experimental results demonstrate that the WA-LPA* algorithm can generate energy-efficient paths under various wind-field conditions, showing improvements of 5.9–29.4% over A* and standard LPA* algorithms in terms of energy efficiency.

However, this work has limitations that warrant future research. First, the current obstacle avoidance strategy lacks explicit safety margins, allowing paths to be planned adjacent to obstacle boundaries, which may pose risks in practical deployment when wind disturbances cause trajectory deviations. Second, the research focuses on kinematic path planning without fully integrating UAV dynamic constraints, creating a gap between planned paths and executable trajectories. Third, while the planning time is acceptable for medium to long-range missions, the computational overhead may limit applicability for time-sensitive tasks. Future work will focus on four aspects: addressing safety margin design, integrating dynamic constraints to improve algorithm practicality, optimizing computational efficiency for time-sensitive applications, and conducting flight experiments to validate the proposed algorithm’s performance in real dynamic wind environments.