A Lie group-based hybrid optimization framework for multi-objective UAV path planning using L-VGWO

Abstract

To address the challenges of high-precision pose alignment, dynamic path smoothness, and efficient multi-objective optimization in three-dimensional UAV path planning under complex environments, this study proposes the Lie Group-based Griffon Vulture Grey Wolf Hybrid Optimizer (L-VGWO) framework. First, the framework employs the rotation-vector representation of the Lie group $\text{SO}(3)$ to parameterize the path. Through the use of exponential and logarithmic mappings, the proposed model inherently preserves geometric continuity and avoids singularities in rotational motion. Second, to overcome the difficulty of efficient optimization in the high-dimensional Lie algebra space, we design the core solver of L-VGWO, which integrates the local refinement capability of the Grey Wolf Optimizer (GWO) with the global exploration ability of the Griffon Vulture Optimizer (GVOA). This cooperative mechanism substantially improves the balance between exploration and exploitation when optimizing the multi-objective cost function. Under a comprehensive objective formulation that includes obstacle avoidance, pose accuracy terms, and angular and linear velocity smoothness constraints, L-VGWO is validated through multi-scenario simulations and statistical analyses.

Introduction

Unmanned Aerial Vehicles (UAVs), as flexible and efficient autonomous platforms, are now widely used in military reconnaissance, logistics delivery, infrastructure inspection, and high-risk disaster response^1,2,3. In fire detection and rescue missions, UAVs must generate safe paths in real time within complex and uncertain three-dimensional environments^4,5,6. Such tasks impose stringent requirements on path planning: the planner must guarantee collision avoidance and short path length, while maintaining continuous vehicle attitude and accurate terminal alignment to ensure stable sensing and reliable mission execution.

For 3D path planning, classical graph-search algorithms such as A* and Dijkstra are commonly used^7,8, together with heuristic approaches based on intelligent optimization. Among these, metaheuristic algorithms are widely adopted for their global search ability. Zhan et al.⁹ proposed a 3D path planning method for Autonomous Underwater Vehicles (AUVs) using standard Particle Swarm Optimization (PSO); P. Priya et al.¹⁰ employed a variant of the Grey Wolf Optimizer (GWO) for UAV trajectory tracking; and Wang et al.¹¹ introduced an improved Whale Optimization Algorithm (WOA) for 3D path planning. Emerging swarm intelligence algorithms, including the Snow Leopard Optimization Algorithm (SLOA)¹², the Candidates Cooperative Competitive Algorithm (CCCA)¹³, and the Griffon Vulture Optimization Algorithm (GVOA)¹⁴, also show promise in complex spaces. However, when applied to high-dimensional continuous optimization, these methods still face three main challenges: (i) convergence speed is difficult to guarantee in multi-constrained scenarios⁹; (ii) achieving a good balance between exploration and exploitation remains nontrivial; and (iii) the dynamic smoothness of the resulting path is often inadequate, causing abrupt changes in angular or linear velocity that degrade motion stability.

It is worth noting that metaheuristic algorithms have also been extensively applied to the Vehicle Routing Problem (VRP), which primarily focuses on the discrete combinatorial optimization of vehicle–customer visiting sequences and scheduling costs, with mature applications in cold-chain logistics and hazardous materials transportation^15,16. In contrast to VRP, UAV path planning typically involves continuous-space trajectory generation subject to geometric smoothness, motion feasibility, and kinematic or attitude-related constraints, requiring joint optimization of path continuity and motion stability^17,18.

To better handle path and attitude constraints, some studies incorporate attitude limits directly into the planning model. Early work mainly used Euler angles to describe attitude. Jin et al.¹⁹ introduced rotation and heading constraints into an improved A* algorithm; Yang et al.²⁰ proposed a lexicographic A* method that integrates attitude constraints and turning-radius limits for mobile robots; and Funk et al.²¹ combined attitude control with turning-radius constraints for UAV trajectory planning. However, Euler angle representations inherently suffer from gimbal lock, which can cause rotational discontinuities and singularities under large-angle manoeuvres, severely limiting their robustness in highly dynamic 3D missions²².

As an alternative, Lie group theory, particularly the Special Euclidean group SE(3) and the Special Orthogonal group SO(3), provides a unified framework for rigid-body translational and rotational motion on manifolds^23,24. Dhullipalla et al.²⁵ formulated path planning on SE(3) by incorporating pointing-direction constraints; Teng et al.²⁶ addressed control problems on Lie groups using cost functions defined in the associated Lie algebras; and Zheng et al.²⁷ utilized SE(3) to represent spacecraft attitude and trajectory²⁸. With tools such as rotation vectors and exponential/logarithmic maps^29,30, Lie group-based modeling ensures geometric continuity and differentiable optimization of pose trajectories, providing a rigorous mathematical foundation for trajectory generation under complex attitude constraints²³. Despite these geometric advantages, most existing studies still rely on traditional optimization methods such as gradient descent, variational techniques, or optimal control²⁹.

Despite the geometric advantages of Lie-group-based modeling over Euler-angle parameterizations, it fundamentally reshapes the optimization landscape. Unlike Euclidean formulations where decision variables directly correspond to waypoint coordinates, Lie-group-based path planning typically parameterizes trajectories using Lie algebra increments propagated through exponential mappings and group compositions. This induces a more indirect and strongly coupled relationship between decision variables and objective functions, such that small perturbations in early increments may be amplified through recursive group operations. Moreover, the intrinsic curvature and non-commutativity of SO(3) and SE(3), together with logarithmic-map-based error metrics, lead to highly non-convex and multi-modal fitness landscapes, thereby increasing convergence difficulty compared with traditional Euclidean search spaces^31,32. Although recent manifold-aware optimization frameworks aim to preserve feasibility on general manifolds, they do not explicitly address the high-dimensional coupling effects inherent in Lie-group-parameterized trajectory generation, leaving exploration–exploitation balance challenging for both gradient-based and metaheuristic methods³³.

Consequently, a critical gap remains: the lack of an effective optimization framework that integrates the strengths of Lie-group-based modeling for pose continuity with the global search capability and balanced exploration–exploitation behavior of hybrid metaheuristics in SE(3)-based path planning problems.

To bridge this gap, this paper proposes a Lie Group-based Griffon Vulture Grey Wolf Hybrid Optimizer (L-VGWO) framework for 3D path planning. This framework parameterizes paths on the Lie group manifold and incorporates multiple objectives to ensure dynamic feasibility and pose consistency, aiming to provide a robust and efficient path optimization solution for demanding tasks such as rotational attitude alignment. The main contributions are summarized as follows:

• Path poses are parameterized on the Lie group SE(3) using pose increments between waypoints, thereby casting 3D path planning as an optimization problem on a continuous manifold.

• A unified multi-objective cost function is constructed that jointly accounts for path smoothness, velocity constraints, obstacle avoidance, and consistency of terminal pose and heading.

• A hybrid optimization algorithm, VGWO, is developed by combining the Griffon Vulture Optimization Algorithm (GVOA) and the Grey Wolf Optimizer (GWO), aiming to improve global search capability and local convergence efficiency in high-dimensional Lie-algebra-based path optimization.

The remainder of this paper is organized as follows. “Problem formulation” section introduces the Lie group-based SE(3) path parameterization and the multi-objective cost function. “Methodology” section presents the proposed L-VGWO framework and its hybrid optimization mechanism. “Experiments and discussions” section reports the simulation setup, path comparison results, convergence behavior, and statistical analysis. “Conclusion and future work” section summarizes the main findings and outlines future research directions.

Problem formulation

Path parameterization based on Lie group

On the basis of the Lie group SE(3), this section presents a path-parameterization mechanism that provides a unified description of both rotational and translational motion of rigid bodies, thereby effectively overcoming the singularity and discontinuity issues inherent in traditional Euler angle representations.

SE(3) pose representation and mapping

A typical element g of the Lie group SE(3) comprises a rotation matrix R and a translation vector b, and can be written as

$$\begin{aligned} g = \begin{bmatrix} R & b \\ 0 & 1 \end{bmatrix} \in SE(3), \end{aligned}$$

(1)

where $R \in SO(3)$ denotes the rotation matrix, which describes the attitude of the UAV, and $b \in \mathbb {R}^3$ represents the translation vector, describing its position in three-dimensional space. Within the SE(3) framework, the rotation matrix R and its corresponding rotation vector $\textbf{r} \in \mathbb {R}^3$ (with $\hat{\textbf{r}} \in \mathfrak {so}(3)$) are related via the exponential map $R = \exp (\hat{\textbf{r}})$ and the logarithmic map $\hat{\textbf{r}} = \log (R)$. The Lie group SE(3) thus provides a unified representation of rigid-body rotation and translation, enabling continuous, singularity-free pose parameterization in path planning and offering a stable mathematical foundation for three-dimensional UAV path optimization.

Rotation-vector-based path recursive generation

Lie groups serve as a standard modeling tool for rigid-body motion in three-dimensional space, enabling the natural generation of pose sequences along path nodes within a unified framework that integrates both rotation and translation. Building upon this theory, we present a complete path parameterization approach.

Given an initial pose:

$$\begin{aligned} T_0= \begin{bmatrix} R_0 & p_0 \\ 0 & 1 \end{bmatrix} \end{aligned}$$

(2)

The pose of the i-th node along the path is generated incrementally on the Lie group manifold as

$$\begin{aligned} T_i = \begin{bmatrix} R_i & \textbf{p}_i \\ \textbf{0}^\top & 1 \end{bmatrix}, \quad R_i = R_{i-1}\exp \!\big (\hat{\textbf{r}}_i\big ), \quad i=1,2,\ldots ,N-1, \end{aligned}$$

(3)

where $\exp (\hat{\textbf{r}}_i)\in \text{SO}(3)$ denotes the incremental rotation generated from the Lie algebra $\mathfrak {so}(3)$, and $\hat{\textbf{r}}_i$ is the skew-symmetric matrix corresponding to the rotation vector $\textbf{r}_i\in \mathbb {R}^3$. The translation vector $\textbf{p}_i$ is not directly optimized but is obtained by propagating a fixed displacement direction in the current body frame, ensuring geometric continuity of the trajectory. The overall shape of the path is generated by combining the rotation vector with a fixed displacement direction, such as along the x-axis, as illustrated in Fig. 1a.

Fig. 1

Illustrations of path propagation and point generation mechanisms: (a) 3D pose propagation based on Lie group increments (b) path point generation mechanism based on Lie algebra increment mapping.

As shown in Fig. 1b, let the norm of the rotation increment vector $\textbf{r}_i$, denoted as $\Vert \textbf{r}_i \Vert$, serve as the step length of the current path segment, while $\exp (\hat{\textbf{r}}_i)\textbf{e}_x$ indicates the forward direction of the displacement along this segment. Then we have

$$\begin{aligned} p_i = p_{i-1} + \Vert \textbf{r}_i \Vert \bigl ( \exp (\hat{\textbf{r}}_i)\,\textbf{e}_x \bigr ) \end{aligned}$$

(4)

which means that the i-th waypoint of the path is obtained by advancing from the previous point $p_{i-1}$ in the direction specified by the Lie group increment $\textbf{r}_i$, with a step size equal to $\Vert \textbf{r}_i \Vert$.

Construction of the objective cost function

To achieve comprehensive optimization of high-precision attitude alignment, dynamic smooth flight, pose consistency, and obstacle-avoidance safety, a unified cost function is constructed in the SE(3) Lie group space, covering four major categories of sub-objectives. These objectives are combined into a single-objective optimization problem via a weighted-sum method, where the weighting coefficients are tuned to balance the priorities among different goals^34,35.

Task pose accuracy objectives

• (1) Terminal Attitude Error Cost The terminal attitude error is measured using a right-invariant rotation error metric, where the rotation between the terminal attitude $R_t \in SO(3)$ of the path and the target attitude $R_T \in SO(3)$ is mapped to the Lie algebra. The squared norm of this mapping is taken as the cost:

  $$\begin{aligned} J_{\text {post}} = \left\| \log (R_t^{-1} R_T) \right\| ^2 \end{aligned}$$

  (5)

  where $\log (\cdot )$ denotes the logarithmic map from SO(3) to $\mathfrak {so}(3)$, and $\Vert \cdot \Vert$ represents the vector norm.

• (2) Terminal Position Error Cost This paper introduces the terminal position error cost based on the Euclidean distance between the terminal path point and the target point in the spatial coordinate system. The position of the target point is denoted as $p_T \in \mathbb {R}^3$, and the position of the terminal path point is denoted as $p_t \in \mathbb {R}^3$. The Euclidean distance between the two points is expressed as:

  $$\begin{aligned} J_{\text {pos}} = \Vert p_t - p_T \Vert ^2 \end{aligned}$$

  (6)

  where $p_t$ and $p_T$ represent the positions of the terminal path point and target point, respectively.

Dynamic smoothness objectives

• (1) Angular Velocity Smoothness Cost Considering the geometric differences between consecutive attitude changes along the path, the logarithmic mapping on the Lie group is used to measure the difference vector between adjacent rotations:

  $$\begin{aligned} J_{\text {ang}} = \sum _{i=2}^{N} \left\| \log \!\big (R_{i-1}^{-1} R_i\big ) \right\| ^2. \end{aligned}$$

  (7)

• (2) Linear Velocity Smoothness Cost Let the position of the i-th point along the path be denoted as $\textbf{p}_i$. The discrete linear velocity is then defined by:

  $$\begin{aligned} v_i = \Vert \textbf{p}_i - \textbf{p}_{i-1} \Vert \end{aligned}$$

  (8)

  The linear velocity variation cost term is formulated as:

  $$\begin{aligned} J_{\text {lin}} = \sum _{i=2}^{N} \left( v_i - v_{i-1} \right) ^2 \end{aligned}$$

  (9)

Pose consistency objectives

Fig. 2

Conceptual diagram of orientation consistency indicator: (a) clustered orientations (b) dispersed orientations.

To better capture the overall coordination and orientation alignment capability of the UAV’s attitude in 3D path planning³⁰, this paper introduces a pose consistency objective as follows:

$$\begin{aligned} J_{\text {ori}} = \sum _{i=1}^{N} \left\| \log (R_i^{-1} R_T) \right\| ^2 \end{aligned}$$

(10)

where $\log (\cdot )$ denotes the logarithmic map on the Lie group SO(3), which measures the minimal rotation distance between the current attitude $R_i$ and the target attitude $R_T$. As shown in Fig. 2, a smaller value of this metric indicates stable pose distribution along the path, while a larger value suggests abrupt or irregular rotational behaviors.

Obstacle avoidance and path length penalty objectives

• (1) Obstacle Avoidance Penalty Cost In disaster environments such as fire scenes, an obstacle-avoidance penalty term $J_{\text {obs}}$ is introduced to encode the distance relationship between each discrete waypoint and the obstacle set. Let the i-th waypoint be $\textbf{p}_i$ and the obstacle set be $\mathcal {O}$. An exponentially decaying soft penalty function $\phi (\textbf{p}_i,\mathcal {O})$ is defined as

  $$\begin{aligned} J_{\text {obs}} = \sum _{i=1}^{N} \phi (\textbf{p}_i,\mathcal {O}), \end{aligned}$$

  (11)

  where

  $$\begin{aligned} \phi (\textbf{p}_i,\mathcal {O}) = \sum _{j=1}^{M} \exp \bigl (-\alpha \,\Vert \textbf{p}_i - \textbf{o}_j\Vert ^{2}\bigr ), \end{aligned}$$

  (12)

  $\textbf{o}_j \in \mathcal {O}$ denotes the position of the j-th obstacle point, and $\alpha >0$ is an exponential decay coefficient that controls the penalty intensity (as shown in Fig. 3). This formulation imposes large penalties on waypoints close to obstacles while keeping distant segments weakly penalized.

• (2) Path Length Cost To avoid unnecessary detours in the generated path during the optimization process, a path length cost term is defined as:

  $$\begin{aligned} J_{\text {disp}} = \sum _{i=1}^{N-1} \Vert \textbf{p}_{i+1} - \textbf{p}_i \Vert _2, \end{aligned}$$

  (13)

  where $\textbf{p}_i \in \mathbb {R}^3$ denotes the position vector of the i-th pose point along the path.

Fig. 3

Illustration of obstacle avoidance via soft constraint mechanism.

Integrated cost function formulation

To systematically integrate the construction methods of the sub-objective functions described above, this paper formalizes the 3D path optimization task as the following weighted multi-objective minimization problem based on the Lie group structure:

$$\begin{aligned} J_{\text {total}} =\lambda _{\text {post}} J_{\text {post}} + \lambda _{\text {pos}} J_{\text {pos}} + \lambda _{\text {ang}} J_{\text {ang}} + \lambda _{\text {lin}} J_{\text {lin}} + \lambda _{\text {ori}} J_{\text {ori}} + \lambda _{\text {obs}} J_{\text {obs}} + \lambda _{\text {disp}} J_{\text {disp}}, \end{aligned}$$

(14)

where $J_{\text {total}}$ represents the overall path cost function. Each $J_i$ term specifically characterizes the path in terms of terminal accuracy, pose continuity, velocity smoothness, path consistency, obstacle avoidance capability, and total displacement.

To achieve path optimization, this study adopts a weighted-sum formulation to integrate multiple objective cost functions into a single aggregate cost function^36,37. In the integrated cost function, each coefficient $\lambda _i$ denotes the weight of the corresponding sub-objective and is set according to its importance in the mission³⁸. In this work, relatively high weights are assigned to terminal accuracy and obstacle avoidance, with $\lambda _{\text {post}} = 0.15$, $\lambda _{\text {pos}} = 0.15$, $\lambda _{\text {ori}} = 0.15$, and $\lambda _{\text {obs}} = 0.25$, in order to ensure path success and safety. Lower weights are used for dynamic smoothness and path length, with $\lambda _{\text {ang}} = 0.15$, $\lambda _{\text {lin}} = 0.10$, and $\lambda _{\text {disp}} = 0.05$, so as to improve path efficiency and smoothness without over-penalizing trajectory length. All weight values given here are after normalization. For clarity in subsequent experiments, they are scaled up by a factor of 1000. These choices follow empirical weighting rules reported in the literature^36,37,38. Proper weight allocation thus helps balance the various objectives during the optimization process and leads to high-quality path solutions.

Methodology

To address the challenges of preserving pose continuity and achieving efficient multi-objective optimization in complex 3D environments, this paper proposes a Lie Group-based Griffon Vulture Grey Wolf Hybrid Optimizer (L-VGWO). The proposed framework integrates SE(3)-based geometric path parameterization with an improved VGWO hybrid metaheuristic, providing a unified solution that simultaneously respects rigid-body motion constraints and attains high-quality global path optimization.

Grey wolf optimizer

The Grey Wolf Optimizer (GWO), proposed by Mirjalili et al. in 2014 ³⁹, is a metaheuristic algorithm inspired by the social hierarchy and cooperative hunting behavior of grey wolf packs. GWO divides the population into four hierarchical levels: the best solution $\varvec{X}_\alpha$ (leader), the second-best solution $\varvec{X}_\beta$, the third-best solution $\varvec{X}_\delta$, and the remaining wolves, which follow these three leaders. The core mechanism of GWO is to simulate the wolves’ encircling and hunting behaviors around the prey.

For an individual wolf with position $\varvec{X}(t)$ in iteration t, the distance vectors to the three leading wolves are computed as

$$\begin{aligned} \varvec{D}_\alpha = \bigl |\varvec{C}_1 \cdot \varvec{X}_\alpha - \varvec{X}(t)\bigr |,\quad \varvec{D}_\beta = \bigl |\varvec{C}_2 \cdot \varvec{X}_\beta - \varvec{X}(t)\bigr |,\quad \varvec{D}_\delta = \bigl |\varvec{C}_3 \cdot \varvec{X}_\delta - \varvec{X}(t)\bigr | \end{aligned}$$

(15)

where $\varvec{C}_i$ $(i=1,2,3)$ are random coefficient vectors.

The candidate positions obtained by encircling the three leaders are then given by

$$\begin{aligned} \varvec{X}_1 = \varvec{X}_\alpha - \varvec{A}_1 \cdot \varvec{D}_\alpha ,\quad \varvec{X}_2 = \varvec{X}_\beta - \varvec{A}_2 \cdot \varvec{D}_\beta ,\quad \varvec{X}_3 = \varvec{X}_\delta - \varvec{A}_3 \cdot \varvec{D}_\delta \end{aligned}$$

(16)

where the coefficient vectors $\varvec{A}_i$ and $\varvec{C}_i$ are defined as

$$\begin{aligned} \varvec{A}_i = 2 a \varvec{r}_{1,i} - a,\qquad \varvec{C}_i = 2 \varvec{r}_{2,i},\qquad i=1,2,3 \end{aligned}$$

(17)

where $\varvec{r}_{1,i}$ and $\varvec{r}_{2,i}$ are random vectors whose elements are uniformly distributed in [0, 1], and a decreases linearly from 2 to 0 over the course of iterations to gradually shift the search from global exploration to local exploitation.

Finally, the position of the individual is updated as the average of the three candidate positions:

$$\begin{aligned} \varvec{X}(t+1) = \frac{\varvec{X}_1 + \varvec{X}_2 + \varvec{X}_3}{3} \end{aligned}$$

(18)

This update rule enables GWO to balance exploration and exploitation by following the three best solutions, thereby providing a simple yet effective mechanism for continuous optimization problems.

Griffon vulture optimization algorithm

The Griffon Vulture Optimization Algorithm (GVOA), proposed by Hasan et al. in 2025 ¹⁴, is a swarm intelligence optimizer inspired by the group foraging behavior of griffon vultures in nature. The algorithm adopts a four-phase search strategy consisting of Following Behavior, Group Foraging, Independent Search, and Near-Carcass Scouting. Among these phases, the Following Behavior plays a primary role in global exploration, where less-informed vultures follow an informed individual $\varvec{X}_{\text {SelGV}}$ to locate potential food sources.

The position update rule for this behavior is defined as:

$$\begin{aligned} \varvec{X}_{\text {new}}(t+1) = \varvec{X}(t) + b \cdot \Bigl (\varvec{X}_{\text {SelGV}} - r_{\text {num}} \cdot \varvec{X}(t)\Bigr ), \end{aligned}$$

(19)

where $\varvec{X}(t)$ denotes the current individual position, and $r_{\text {num}}\in [0,1]$ is a uniformly distributed random scalar. The dynamic parameter b controls the step size and regulates the movement strength toward the selected leader, serving as a key factor for balancing exploration and exploitation. As iterations proceed, a larger b enhances global exploration, whereas a smaller b promotes localized refinement.

Compared with GWO, however, GVOA does not employ a hierarchical leadership structure such as $\varvec{X}_\alpha$, $\varvec{X}_\beta$, and $\varvec{X}_\delta$. This limitation reduces the algorithm’s exploitation capability and may lead to slower convergence near the global optimum, motivating the development of hybrid strategies to improve performance.

L-VGWO hybrid strategy and mechanism

To enhance path planning in the SE(3) Lie group space, this paper proposes a L-VGWO. The core idea is to encode the path as rotation increments in the Lie algebra $\mathfrak {so}(3)$ and to combine the complementary advantages of GVOA and GWO in a sequential manner. In each iteration, GVOA is first used to perform global exploratory perturbations, after which GWO refines the solutions locally. A greedy selection rule then preserves the better candidate, yielding a balanced exploration–exploitation behavior on the Lie group manifold. The procedure can be summarized as follows:

• (1) Lie algebraic encoding. The optimization variables are the rotation vectors in the Lie–algebra space associated with SO(3). For the i-th individual, the decision vector $\textbf{X}_i$ collects the $N\!-\!1$ pose increments along the path:

  $$\begin{aligned} \textbf{X}_i = \begin{bmatrix} \textbf{r}_1^{T},\ \textbf{r}_2^{T},\ \ldots ,\ \textbf{r}_{N-1}^{T} \end{bmatrix}^{T} \end{aligned}$$

  (20)

  where $\textbf{r}_k \in \mathbb {R}^3$ is the rotation vector of the k-th path segment. During cost evaluation, each $\textbf{r}_k$ is mapped to a rotation matrix

  $$\begin{aligned} R_k = \exp (\widehat{\textbf{r}}_k) \in SO(3) \end{aligned}$$

  (21)

  and the relative rotation between consecutive poses is computed as $\Delta R_k = R_{k+1} R_k^{-1}$. The corresponding Lie–algebra increment is then obtained via the logarithmic map:

  $$\begin{aligned} \Delta \textbf{r}_k = \log (\Delta R_k) \end{aligned}$$

  (22)

  which guarantees geometric continuity and smooth variation of the path on the Lie–group manifold.

• (2) GVOA-based global exploration. Given the current individual $\textbf{X}_i(t)$ in iteration t, GVOA generates an exploratory candidate in the Lie algebra as

  $$\begin{aligned} \textbf{X}_i^{\text {GVOA}}(t+1) = \textbf{X}_i(t) + b \bigl (\textbf{X}_{\text {SelGV}} - r_{\text {num}}\,\textbf{X}_i(t)\bigr ), \end{aligned}$$

  (23)

  where $\textbf{X}_{\text {SelGV}}$ is the selected informed vulture, $r_{\text {num}}\in [0,1]$ is a random scalar, and b is the dynamic exploration parameter controlling the step size. The composite objective is then evaluated as

  $$\begin{aligned} F_i^{\text {GVOA}} = J_{\text {total}}\bigl (\textbf{X}_i^{\text {GVOA}}(t+1)\bigr ). \end{aligned}$$

  (24)

• (3) GWO-based local refinement. Using the population after Step (2) as the new search points, GWO identifies the three leaders $X_\alpha$, $X_\beta$ and $X_\delta$ and computes three candidate positions $X_1$, $X_2$ and $X_3$ according to the standard GWO exploitation rule (see equation (18)). The refined position of the i-th individual is denoted by $\textbf{X}_i^{\text {GWO}}(t+1)$, and its fitness is computed as

  $$\begin{aligned} F_i^{\text {GWO}} = J_{\text {total}}\bigl (\textbf{X}_i^{\text {GWO}}(t+1)\bigr ). \end{aligned}$$

  (25)

• (4) Optimal cluster–guided search strategy. To further enhance exploitation capability in the later optimization stage while avoiding significant computational overhead, a lightweight optimal cluster–guided search strategy is incorporated into the L-VGWO hybrid framework. Specifically, at iteration t, the current population is first ranked according to fitness values, and the top K individuals are selected to form an elite subset, which is regarded as the optimal cluster at the current iteration. The center of this cluster is defined as the centroid of elite individuals in the Lie algebra space:

  $$\begin{aligned} \textbf{X}_{\text {center}}(t) = \frac{1}{K}\sum _{j=1}^{K} \textbf{X}_{j}(t), \end{aligned}$$

  (26)

  where $\textbf{X}_{j}(t)$ denotes the decision vector of the j-th elite individual. Based on the cluster center, a small number of low-fitness individuals are selected and guided toward the optimal cluster in the Lie algebra space according to

  $$\begin{aligned} \textbf{X}_{i}^{\text {OC}}(t+1) = \textbf{X}_{i}(t) + \eta \bigl (\textbf{X}_{\text {center}}(t) - \textbf{X}_{i}(t)\bigr ), \end{aligned}$$

  (27)

  where $\eta \in (0,1)$ is a guidance step size that controls the influence strength of the cluster information. This cluster-guided update is triggered only at predefined iteration intervals and is subject to a greedy acceptance criterion. As a result, the proposed strategy introduces directed exploitation based on population structural information while preserving the original GVOA–GWO search dynamics and maintaining a low additional computational cost.

• (5) Greedy selection and update. Finally, a simple greedy rule retains the better of the exploratory and refined candidates:

  $$\begin{aligned} \textbf{X}_i(t+1) = {\left\{ \begin{array}{ll} \textbf{X}_i^{\text {GWO}}(t+1), & F_i^{\text {GWO}} \le F_i^{\text {GVOA}},\\ \textbf{X}_i^{\text {GVOA}}(t+1), & \text {otherwise}. \end{array}\right. } \end{aligned}$$

  (28)

In this way, GVOA provides large-scale exploration in the Lie algebra space, while GWO strengthens local exploitation, and their cooperative interaction gradually drives the population toward high-quality Lie group paths. The overall procedure of L-VGWO is summarized in Algorithm 1.

Algorithm 1

L-VGWO for SE(3) Path Planning

Experiments and discussions

Experiment setup

Fig. 4

Illustration of start/target points and terminal attitude settings.

• Start and Goal Points Setting The starting point of the UAV is set to (1, 1, 1), and the goal point is set to (11, 11, 9). The target goal orientation is set in three directions: $0^\circ$ (toward the x-axis), $45^\circ$ (toward the xy-direction), and $90^\circ$ (toward the y-axis), to test the adaptability of the path endpoint to different orientation constraints, as shown in Fig. 4.

• Path Modeling Parameters Number of path points: $N=10$. Initial path: All path points are initialized to coincide with the start position. The variables to be optimized are the sequence of rotation vectors $\{ \textbf{r}_1, \dots , \textbf{r}_{N-1} \}$. The objective is to achieve alignment using a rotation-matrix representation: $R_t \rightarrow R_T$.

• Obstacle Configuration To simulate realistic and complex obstacle-avoidance challenges, the experiment introduces diverse three-dimensional obstacles—including spheres, cylinders, and cuboids—with a total of 14 obstacles placed within the constructed three-dimensional environment.

• Method Comparison Strategy To validate the proposed L-VGWO, a comparative study is conducted between Lie group rotation-vector modeling and traditional Euler angle modeling. In addition, the following algorithms are selected for comparison: the Grey Wolf Optimizer (GWO)³⁹, the Griffon Vulture Optimization Algorithm (GVOA)¹⁴, the Snow Leopard Optimization Algorithm (SLOA)¹², the Particle Swarm Optimization (PSO)⁹, and the Candidates Cooperative Competitive Algorithm (CCCA)¹³. To mitigate the influence of randomness, each experiment is independently executed 30 times. The final results will be quantitatively compared based on the mean (AVG) and standard deviation (STD) obtained from the 30 independent runs, and statistical methods such as the Friedman test and the Wilcoxon rank-sum test will be employed to verify the statistically significant superiority of the algorithm. The main parameters are listed in Table 1.

Table 1 Algorithm parameter settings.

Sensitivity analysis of weighting coefficients

To evaluate the robustness of the proposed method under different mission priorities, a sensitivity analysis is conducted. In the baseline configuration, the following weight values are used: $\lambda _{\text {post}} = 0.15$, $\lambda _{\text {pos}} = 0.15$, $\lambda _{\text {ori}} = 0.15$, $\lambda _{\text {obs}} = 0.25$, $\lambda _{\text {ang}} = 0.15$, $\lambda _{\text {lin}} = 0.10$, and $\lambda _{\text {disp}} = 0.05$. The remaining coefficients are kept unchanged while the following three task-biased configurations are considered: (i) safety-prioritized: $\lambda _{\text {obs}} = 0.75$; (ii) smoothness-prioritized: $\lambda _{\text {ang}} = 0.75$ and $\lambda _{\text {lin}} = 0.50$; (iii) accuracy-prioritized: $\lambda _{\text {pos}} = 1.50$, $\lambda _{\text {post}} = 1.50$, and $\lambda _{\text {ori}} = 1.50$. These configurations are referred to as VGWO_Safety, VGWO_Smooth, and VGWO_Accuracy, respectively, while the baseline configuration is denoted as VGWO.

Fig. 5

Sensitivity analysis of L-VGWO, a 3D trajectory comparison, b Convergence curves, c Angular velocity profiles, d Linear velocity profiles, e Relative performance radar chart.

Figure 5e presents a normalized radar comparison of safety, smoothness, and accuracy metrics under different weight settings. The safety-prioritized configuration exhibits a clear improvement in obstacle clearance, accompanied by moderately increased velocity variations, reflecting the expected trade-off between collision avoidance and motion smoothness. Conversely, the smoothness-prioritized configuration significantly suppresses angular and linear velocity fluctuations, at the expense of slightly reduced obstacle margins. The accuracy-prioritized configuration achieves the lowest terminal position error, while accepting higher curvature and obstacle proximity in certain trajectory segments. Overall, the radar visualization demonstrates that the proposed framework responds consistently and predictably to changes in task priorities.

Further insights are provided by the trajectory plots, convergence curves, and velocity profiles shown in Fig. 5a–d. When safety is emphasized, the optimized paths exhibit more conservative detours around obstacles, whereas smoothness-oriented optimization yields gentler curvature and reduced angular velocity peaks. In contrast, accuracy-oriented optimization produces more direct trajectories toward the target, with faster convergence but increased dynamic activity. Importantly, in all cases, the proposed L-VGWO maintains stable convergence behavior and generates physically feasible paths without abrupt divergence or numerical instability.

These results indicate that the proposed framework is not overly sensitive to specific empirical weight selections. Instead, L-VGWO adapts smoothly to different weighting distributions and produces interpretable solutions aligned with the intended mission objectives. The sensitivity analysis therefore validates the robustness of the integrated cost formulation and confirms that the baseline weighting configuration represents a reasonable compromise rather than a fragile, over-tuned choice.

Visual analysis

This section presents a visual analysis of the simulation results. All visual representations, including trajectory plots and convergence curves, are generated from the run (among 30 independent trials) whose total cost $J_{\text {total}}$ is closest to the sample average (AVG).

Three-dimensional path convergence

Fig. 6

3D trajectory comparison under Lie Group and Euler angle models at rotation targets of $0^\circ$, $45^\circ$, and $90^\circ$ (a–c Lie group; d–f Euler angles).

Figure 6 compares Lie group modeling with Euler angle modeling under three target orientations: $0^\circ$, $45^\circ$, and $90^\circ$. For clarity, all trajectories are visualized using B-spline interpolation applied to the optimized discrete waypoints.

The comparison highlights the geometric advantage of Lie group modeling in large-angle rotation scenarios. As the target orientation increases to $45^\circ$ and $90^\circ$, trajectories obtained under the Euler angle representation exhibit noticeable folding and local discontinuities, which are indicative of the singularity-related limitations of this parameterization. In contrast, the trajectories generated using Lie group modeling remain smooth and continuous across all three orientation settings, demonstrating improved geometric consistency.

Within the Lie group framework, L-VGWO produces the smoothest and most regular trajectories among all compared algorithms. In obstacle-dense regions, the paths generated by L-VGWO follow reasonable detours while avoiding redundant oscillations. Compared with GWO and GVOA, which show sharper turns or local path irregularities in some cases, L-VGWO maintains a more stable trajectory shape, reflecting an improved balance between path smoothness and feasibility.

Objective function convergence analysis

Fig. 7

Convergence curves under Lie group and Euler angles models at rotation targets of $0^\circ$, $45^\circ$, and $90^\circ$ (a–c Lie group; d–f Euler angles).

Collectively examining Fig. 7, the convergence behavior under Lie group modeling is markedly superior to that under the Euler angle representation. Under the Lie group formulation, most algorithms steadily converge to lower objective values, with L-VGWO consistently reaching the lowest final cost levels (typically around $3.4\times 10^{3}$–$4.2\times 10^{3}$ across different orientation scenarios). In contrast, under the Euler angle model, the convergence trajectories remain at noticeably higher cost plateaus, generally above $4.5\times 10^{3}$, reflecting the adverse impact of geometric discontinuities on optimization efficiency.

Among all compared methods, L-VGWO exhibits the most favorable optimization behavior in terms of both convergence speed and final solution quality. The VGWO curve shows the fastest initial descent in all three orientation tasks and stabilizes after approximately 150–250 iterations at the lowest cost level. By comparison, GWO demonstrates slower convergence and higher residual costs, indicating inferior accuracy and stability. GVOA, which places stronger emphasis on exploration, converges more gradually and yields slightly higher final objective values. These results underscore the effectiveness of the hybrid design in L-VGWO, which achieves a balanced trade-off between exploration and exploitation within the Lie group search space.

Angular and linear velocity profiles

Fig. 8

Angular and linear velocity profiles under Lie group and Euler angle models at rotation targets of $0^\circ$, $45^\circ$, and $90^\circ$ (a–f angular velocity; g–l linear velocity).

Figure 8 presents the angular (a–f) and linear (g–l) velocity profiles under Lie group and Euler angle modeling. A clear advantage of the Lie group approach is observed in both cases. Under the Euler angle model, all algorithms exhibit pronounced angular velocity fluctuations, with several trajectories showing peaks exceeding $2.5\ \text {rad/s}$ in the $45^\circ$ and $90^\circ$ tasks. In contrast, when Lie group modeling is employed, angular velocity oscillations are substantially reduced and remain within a relatively steady range of approximately $0.5\ \text {rad/s}$ to $2.0\ \text {rad/s}$ for most time steps. This behavior confirms that the Lie group parameterization helps to maintain rotational continuity and effectively suppress undesired attitude jitter. The proposed L-VGWO consistently achieves the smoothest angular velocity profiles among all methods, exhibiting lower fluctuation amplitudes and fewer sharp peaks across all three orientation tasks.

A similar advantage is observed in the linear velocity profiles. Under the Euler angle model, the linear velocities of all algorithms exhibit pronounced oscillations and occasional sharp peaks, particularly in the $45^\circ$ and $90^\circ$ tasks. By contrast, when Lie group modeling is adopted, the overall fluctuation amplitude is notably reduced, and the majority of velocity trajectories remain within a comparatively bounded range, with significantly fewer abrupt jumps. This behavior highlights the effectiveness of combining the Lie group parameterization with the linear-velocity cost term $J_{\text {lin}}$ in suppressing excessive velocity variations. Moreover, in both the $45^\circ$ and $90^\circ$ scenarios, L-VGWO consistently maintains the smallest variation between local peaks and troughs and demonstrates the most compact and stable velocity distribution among all compared methods. These characteristics indicate that the proposed L-VGWO framework effectively mitigates abrupt changes in both angular and linear velocities, thereby improving the dynamic feasibility and smoothness of the planned trajectories.

Computational efficiency analysis

To further evaluate the practical computational efficiency of different algorithms under realistic time constraints, we analyze their convergence behaviors from the perspective of wall-clock time. As illustrated in Fig. 9, although all methods exhibit rapid objective reduction during the early optimization stage, their time-normalized convergence efficiencies differ significantly. Compared with conventional GWO, PSO, and GVOA, the proposed L-VGWO consistently achieves lower objective values within the same time budget. For instance, within the first 15 s and 25 s, L-VGWO reduces the cost to approximately $3.6\times 10^{3}$ and $3.5\times 10^{3}$, respectively, whereas the competing algorithms remain trapped at substantially higher cost levels. This indicates that L-VGWO is able to exploit computational time more effectively and deliver higher-quality solutions under limited execution time.

When jointly interpreted with the iteration-based convergence curves shown in Fig. 7a, it can be observed that the fast convergence behavior of L-VGWO in the iteration domain translates into tangible advantages in wall-clock time. Although L-VGWO introduces a sequential–parallel exploration and refinement mechanism that incurs a moderately higher per-iteration computational cost, this overhead is compensated by more informative population updates and accelerated convergence toward high-quality solutions. As a result, L-VGWO attains superior solution quality within the same or even shorter execution time compared to the baseline methods. These results indicate that the proposed approach improves not only convergence speed in terms of iteration counts, but also overall computational efficiency in time-critical mission planning scenarios.

Fig. 9

Objective cost convergence with respect to wall-clock time in Scenario $0^\circ$.

Statistical results

This section reports a non-parametric statistical evaluation of the overall optimization performance of six algorithms under the Lie-group-based modeling framework, using the total cost $J_{\text {total}}$ as the performance metric. For each orientation-constrained task ($0^\circ$, $45^\circ$, and $90^\circ$), all algorithms were independently executed 30 times to obtain reliable performance distributions. As the normality assumption could not be guaranteed, the Friedman rank-sum test was first employed to compare the overall performance of all algorithms across different scenarios. Subsequently, Wilcoxon rank-sum tests were conducted for pairwise comparisons between L-VGWO and each baseline algorithm, in order to assess whether the observed performance differences were statistically significant.

The Friedman ranking results summarized in Table 2 demonstrate that L-VGWO consistently achieves the best overall performance under all terminal orientation scenarios. Specifically, L-VGWO ranks first in each individual task ($0^\circ$, $45^\circ$, and $90^\circ$), and attains the lowest aggregated average rank of 2.07 in the overall evaluation. Among the baseline methods, GVOA and GWO follow with aggregated average ranks of 3.20 and 3.33, respectively, indicating competitive but clearly inferior performance compared with L-VGWO. In contrast, PSO, SLOA, and CCCA exhibit substantially higher average ranks (greater than 4.0), reflecting weaker optimization capability and robustness across different orientation constraints. These results confirm that L-VGWO maintains superior solution quality and stability under varying terminal orientation conditions. Moreover, the Friedman ranking outcomes are consistent with the convergence trends and trajectory characteristics observed in the previous analyses, further validating the effectiveness of the proposed L-VGWO framework.

Table 2 Friedman ranking results under different terminal orientation scenarios.

Table 3 presents the Wilcoxon rank-sum test results for pairwise comparisons between L-VGWO and the baseline algorithms under different terminal orientation scenarios. For the $0^\circ$ task, L-VGWO achieves statistically significant performance improvements over all baseline methods, with all p-values below 0.01. In the $45^\circ$ scenario, L-VGWO significantly outperforms GWO, PSO, SLOA, and CCCA, while the difference with GVOA is not statistically significant ($p = 5.652\times 10^{-2}$). Under the $90^\circ$ orientation constraint, statistically significant advantages are observed over PSO, SLOA, and CCCA, whereas no significant differences are found with respect to GWO and GVOA. Importantly, when aggregating results across all scenarios, L-VGWO exhibits statistically significant superiority over all baseline algorithms, with all corresponding p-values below 0.05 and most well below $10^{-3}$, confirming the overall robustness and effectiveness of the proposed method.

Table 3 Wilcoxon rank-sum test p-values for pairwise comparisons between L-VGWO and baseline algorithms under different terminal orientation scenarios ($N=30$ independent runs per algorithm).

Taken together, these results indicate that L-VGWO demonstrates statistically robust overall superiority, particularly in challenging multi-scenario settings, by effectively balancing global exploration and local exploitation in the Lie-group-based optimization space, thereby achieving improved optimization accuracy, stability, and overall performance compared with the baseline methods.

Conclusion and future work

Aiming at the stringent requirements on pose continuity and efficient multi-objective optimization in 3D UAV path planning under complex scenarios, this paper proposes a Lie group-based Griffon Vulture–Grey Wolf Hybrid Optimizer (L-VGWO) framework. L-VGWO integrates the Lie group $\\text{SO}(3)$ rotation-vector parameterization with a sequential–parallel cooperative optimization mechanism of VGWO, thereby enabling the search for high-quality solutions under rigid-body motion constraints. Experimental results clearly demonstrate the effectiveness of the framework. In terms of geometric modeling, the total path-planning cost under Lie group modeling is reduced by approximately 40–70% on average compared with Euler angle modeling, substantially enhancing optimization feasibility. In terms of algorithmic performance, L-VGWO attains the lowest average total cost across all orientation tasks. The Friedman test shows that L-VGWO consistently ranks first and achieves the lowest overall average rank of 2.07, confirming its superior comprehensive performance. Moreover, Wilcoxon rank-sum tests indicate that L-VGWO significantly outperforms PSO, SLOA, and CCCA across all evaluated scenarios (all $p < 0.01$), demonstrating that the proposed hybrid mechanism provides statistically significant advantages in avoiding premature convergence and mitigating local-optimum entrapment. Overall, the L-VGWO framework is capable of generating high-precision trajectories that satisfy stringent dynamic constraints while maintaining high path quality, thus offering a reliable solution for autonomous mission planning in complex environments.

Although the proposed L-VGWO algorithm demonstrates strong performance under statically constrained environments, there remains room for further improvement when addressing more complex and dynamic tasks, such as real-world rescue missions. Future research will be pursued along the following three directions. First, a temporal dimension will be incorporated to extend the current path planning formulation toward full trajectory optimization, enabling the framework to effectively handle dynamic obstacles with time-varying characteristics and environmental uncertainties. Second, the proposed framework will be extended to multi-UAV cooperative planning scenarios, with particular emphasis on adapting L-VGWO to multi-agent systems for addressing path allocation, formation maintenance, and cooperative obstacle avoidance in collaborative UAV missions. Third, it should be noted that the proposed method has so far been evaluated exclusively in simulated environments. Its practical feasibility, robustness, and performance on real UAV platforms remain to be further validated through real-world flight experiments under realistic operating conditions.