A two-stage manifold optimization method for low-altitude route planning of unmanned aerial vehicles

Abstract

This invention discloses a two-stage manifold optimization-based method for low-altitude flight path planning for unmanned aerial vehicles (UAVs). The method involves acquiring the UAV's low-altitude operational mission, identifying virtual obstacle zones (VORs) containing obstacles and complex building clusters on a 3D mission map, and identifying any undefined intersection points around these VORs. Initial path planning is performed for the UAV to avoid obstacles and VORs. This initial path undergoes two-stage manifold optimization, comprising a first-stage optimization based on constraints related to obstacle avoidance, energy consumption, smoothness, and proximity to intersection points, establishing a first cost function, and a second-stage optimization based on the optimized path from the first stage. The optimal or suboptimal path trajectory is then selected according to the UAV mission requirements. This method facilitates the development of airspace density control strategies, avoidance of hotspot airspace congestion and conflicts, and deployment of mission priority scheduling schemes. The two-stage path planning scheme based on manifold optimization provides key technical support for safe, smooth, and efficient flight under complex constraints.

Description

Technical Field

[0001] This invention relates to unmanned aerial vehicle (UAV) control, specifically to a two-stage manifold optimization method for UAV low-altitude flight path planning. Background Technology

[0002] Extremely low altitude refers to the airspace where flight altitude is 0-120 meters (or 0-400 feet) above the ground. Current research has discussed autonomous navigation for unmanned aerial vehicles (UAVs), safety management of urban low-altitude areas, and low-altitude sensing and communication. However, safety regulations and efficient flight planning methods for extremely low-altitude traffic planning in urban areas at the kilometer level still need improvement. The core functions of existing UAV traffic management systems include flight plan application and approval, dynamic airspace allocation, conflict warning and obstacle avoidance, and real-time monitoring and command issuance.

[0003] However, for tasks such as formation flying, logistics delivery, or emergency response, existing low-altitude flight safety regulations do not quantify the horizontal dimension (the plane formed by longitude and latitude) and vertical dimension (altitude) of dense airspace below 120 meters in urban areas. This discretization and quantification process can often reduce the computational complexity required to find safe paths for UAVs, because path searching only traverses a limited number of discrete latitude, longitude, and altitude variables. The aforementioned tasks also involve a large number of UAV flight plans or flight scheduling based on time, and such scheduling is related to low-altitude system parameters such as airspace classification, flight altitude stratification, isolation requirements, and flight control areas.

[0004] Using 120 meters as a controlled airspace boundary, finding safe routes for a large number of drones in uncontrolled airspace involves more than just airspace management regulations and primary flight missions for the low-altitude economy; drone path planning also needs to consider various safety constraints. First, the quantification of vertical dimension (altitude) needs to be influenced by flight altitude stratification, but existing low-altitude regulations lack fine-grained definitions for this. For example, the division of uncontrolled airspace by altitude is often influenced by city level and specific city areas. The following definitions exist for uncontrolled airspace division: building height and density: ground level to 60 meters: recommended as a restricted flight zone, allowing only specific purposes (such as police or emergency); 60 meters to 120 meters: designated as a buffer zone, requiring flight permits. According to existing airspace management regulations, high-density business districts with an estimated 80-120 high-rise buildings per square kilometer, small building spacing (often less than 20 meters between buildings) are unsuitable for low-altitude flights. How to enable drones with mission requirements such as formation flying, logistics delivery, or emergency response to reach the vicinity of such business districts, and how to optimize safe routes for a large number of drones, are all current challenges. Summary of the Invention

[0005] Purpose of the invention: To address the above-mentioned shortcomings, this invention provides a two-stage manifold optimization method for UAV low-altitude flight path planning, enabling ultra-low-altitude flight missions and optimizing safe routes. It incorporates a problem-decomposition approach, transforming vague requirements into clear steps. For urban low-altitude airspace up to 120 meters, this solution does not simply treat it as a no-fly zone or uncontrolled area, but rather redefines safe traffic planning.

[0006] Technical solution: To solve the above problems, this invention adopts a two-stage manifold optimization method for UAV low-altitude flight path planning, the steps of which are as follows:

[0007] Step 1: Obtain the UAV low-altitude operation mission, including the mission start point, mission end point, and whether the mission passes through the intersection point; obtain the mission 3D map based on the UAV low-altitude operation mission, and confirm the obstacles, virtual obstacle areas, and the intersection points of UAV flight paths around the virtual obstacle areas in the mission 3D map. The virtual obstacle areas are the areas where complex building clusters are located.

[0008] Step 2: Based on the drone's mission start and end points, perform initial path planning for the drone to avoid obstacles and virtual obstacle zones, and obtain the initial path;

[0009] Step 3: Perform two-stage manifold optimization on the initial path. The two-stage manifold optimization includes a first-stage optimization and a second-stage optimization. In the first-stage optimization, a first cost function is established based on the constraints of obstacle avoidance, energy consumption, smoothness, and proximity to the intersection point of the UAV path. The Riemannian manifold optimization tool is used to obtain the optimal path trajectory with the goal of minimizing the first cost function. In the second-stage optimization, a second cost function is established based on the constraints of the safe distance between the UAV path, the optimal path trajectory, and obstacles. The Riemannian manifold optimization tool is used to obtain the suboptimal path trajectory with the goal of minimizing the second cost function.

[0010] Step 4: Determine the path trajectory of the UAV based on the intersection point of the UAV low-altitude operation mission. If the UAV passes through the intersection point, the optimal path trajectory is adopted; if the UAV does not pass through the intersection point, the suboptimal path trajectory is adopted.

[0011] Furthermore, the channel where the intersection point is located includes horizontal channels and vertical channels, and the UAV low-altitude operation task includes whether the vertical channel of the intersection point is required.

[0012] Furthermore, the initial path planning for the UAV to avoid obstacles and virtual obstacle zones can be performed using graph search-based algorithms, random sampling-based algorithms, or biomimetic-based algorithms.

[0013] Furthermore, the first cost function for:

[0014] ;

[0015] in, For the first The total length of the drone's path, For the first Energy consumption of the drone's flight path For the first Severe weather penalties for drone flight paths. For the first Penalty for the intersection of drone paths. For the first Obstacles in the drone's path are penalized. For the first Spatial boundary penalty for drone flight paths, For the first Penalty for directional consistency of drone path. For the first Speed ​​penalty for the drone's path For the first Penalty for consistency between speed and displacement of the drone's path. , , , , , , , , These are the weighting coefficients. This represents the total number of drones.

[0016] Furthermore, the first Total length of the drone's path The calculation formula is:

[0017] ;

[0018] in, For the first A drone in time step Location, For the first A drone in time step Location, This represents the total number of steps in time.

[0019] The first Energy consumption of drone flight path The calculation formula is:

[0020] ;

[0021] in, For hovering power consumption, The air drag coefficient, Sampling time, The speed of the drone relative to the air;

[0022] The first Penalty for the intersection of drone paths The calculation formula is:

[0023] ;

[0024] in, Set of intersection points Time interval The location of the internal drone and the first Intersection The basis functions for the distance relationship between them. Set of intersection points The total number of intersections;

[0025] No. Obstacles in the drone's path One calculation formula is:

[0026] ;

[0027] ;

[0028] in, For the first The weighting coefficient of each obstacle For the first An obstacle, For the first Safe distance radius of each obstacle, single-point penalty It is a quadratic penalty function with a threshold. For position To the The shortest Euclidean distance to each obstacle;

[0029] The first Spatial boundary penalty for drone path The calculation formula is:

[0030] ;

[0031] in, For position x-coordinate For position The ordinate, For position The vertical coordinate, These are the boundary values ​​for the x-axis. The boundary value of the ordinate. This is the minimum height boundary value. Maximum height boundary value;

[0032] The first Directional consistency penalty for drone paths The calculation formula is: ;

[0033] in, In time step The angle between the trajectory segment direction and the ideal direction;

[0034] The first Speed ​​penalty for drone path The calculation formula is:

[0035] ;

[0036] in, For the first A drone in time step of, This represents the minimum speed of the drone. This represents the maximum speed of the drone.

[0037] The first Consistency penalty for speed and displacement of drone path The calculation formula is:

[0038] .

[0039] Furthermore, the second cost function for:

[0040] ;

[0041] in, The solution required by the second cost function is the first... A drone in time step Location, To solve the problem using the first cost function, the second... A drone in time step Location, This represents the minimum safe distance between the suboptimal and optimal path trajectories. The minimum safe distance between the suboptimal path trajectory and the obstacle. For position With the Safe distance between obstacles.

[0042] Furthermore, the Riemannian manifold optimization tool uses the Pymanopt tool.

[0043] Furthermore, based on the optimal and suboptimal path trajectories of several UAVs obtained from steps 1 to 4, different takeoff delays are added to several UAVs to achieve interval takeoff of several UAVs.

[0044] The present invention employs a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the above method.

[0045] The present invention employs a computer-readable storage medium on which a computer program is stored, and when the computer program is executed by a processor, it implements the steps of the above method.

[0046] Beneficial Effects: Compared to existing technologies, the significant advantages of this invention are: through initial path search and two-stage optimization, it effectively supports UAV path optimization in cities with specific high-rise building distributions. The initial path, with low computational complexity, primarily considers avoiding sparse obstacles and virtual obstacle zones, but it is not smooth enough and does not consider how to approach points of interest or intersections. The first-stage optimization rationally utilizes the structural characteristics of Riemannian manifold optimization to achieve a flattened optimization and safe temporal and spatial scheduling. The second-stage optimization does not consider crossing points of interest but has lower computational complexity and safety, as it can quickly calculate the UAV trajectory at nearby altitudes or along the flight path, reducing the computation time to one-tenth of the first stage. In short: the inclusion of initial path logic, two-stage manifold optimization, and flattened temporal and spatial scheduling logic helps save computational resources, improve the efficiency of UAV control, and enhance the safety and backtracking of UAV UAV ultra-low-altitude flight path planning. In addition, it lays the groundwork for redefining the construction of the urban ultra-low airspace system below 120 meters and the flight plan management of its dense airspace, because the proposed scheme does not simply regard the urban ultra-low airspace as a no-fly zone or an uncontrolled zone, but rather as a redefined safe traffic plan. Attached Figure Description

[0047] Figure 1 This is a schematic diagram of the path optimization for ultra-low altitude UAVs in this invention. The environment contains sparse obstacles and a cluster of low-altitude high-rise buildings.

[0048] Figure 2 This is a flowchart illustrating the waterway planning method of the present invention.

[0049] Figure 3 This is a schematic diagram illustrating the principle of environmental analysis involving stratification and waterways in this invention. The uncontrolled area includes an uncontrolled layer. Detailed Implementation

[0050] like Figure 1 and Figure 2 As shown in this embodiment, a two-stage manifold optimization method for UAV low-altitude flight path planning includes the following steps:

[0051] Step 1: Obtain the UAV low-altitude operation task, including the task start point, task end point, and whether the task passes through the intersection point; obtain the task 3D map based on the UAV low-altitude operation task, and confirm the obstacles, virtual obstacle areas, and the intersection points of UAV flight paths around the virtual obstacle areas in the task 3D map. The virtual obstacle areas are the areas where complex building clusters are located.

[0052] Introducing the concept of a "virtual obstacle zone" in complex obstacle environments is not simply about creating artificial no-fly zones. Instead, it abstracts complex spatial structures such as high-rise buildings and drone (traffic) intersections into virtual obstacles, achieving decoupled modeling of low-altitude airspace. This means that the region of interest is equivalent to a virtual obstacle with corresponding geometric parameters such as latitude, longitude, and altitude. Furthermore, these virtual obstacles contain intersections, so drone low-altitude flight path planning can be simplified to: guiding drones through intersections, which are defined as intermediate logistics delivery points or points of interest. This completely avoids drones crossing no-fly zones. By introducing a decoupled model of part of the airspace in a dense 3D obstacle environment, complex building clusters and low-altitude obstacles are transformed into computable spatial constraints, thereby reducing the complexity of high-dimensional searches in path planning and improving trajectory smoothness and optimization efficiency. This decoupling model effectively alleviates the computational complexity problem caused by the increasing number and spacing of obstacles in high-dimensional space search of the Riemannian manifold optimization algorithm, enabling the two-stage manifold optimization to maintain efficiency and scalability in both local and global path planning. Simulations have verified its applicability in formation flying, logistics distribution, and emergency response tasks.

[0053] This paper provides three points regarding layering and flight paths, further decoupling the aforementioned spatial structure. The first point introduces a layering and flight pathing system for extremely low-altitude airspace, encompassing the extremely low-altitude environment and a semi-structured UAV flight path. The extremely low-altitude environment includes urban high-rise clusters, UAV meeting points, weather conditions, and UAV density. Urban high-rises specifically refer to buildings 6 stories or taller, starting at approximately 18 meters, and the terms "urban high-rise" and "obstacle" are interchangeable, as are "high-density business district" and "area of ​​interest." Next, two types of UAV flight paths are defined: horizontal and vertical. Depending on the UAV mission type, UAVs can fly on horizontal, vertical, or a combination of these paths. The concept of virtual obstacles can also be used to meet the mission requirements of deployment at the edge of no-fly zones and to quickly obtain initial paths.

[0054] The second point to clarify regarding stratification and navigation is that stratification and navigation in the very low altitude requires city-level 3D models (such as CityGML, 3D Tiles, BIM, or digital twin data with buildings), which typically include the geometric location (longitude and latitude) and shape (length, width, height, and base polygons) of the buildings.

[0055] The third point regarding stratification and airways is that, for scenarios spanning several kilometers in cities, the divide-and-conquer approach should be introduced. Low-altitude stratification and airway designation can be defined in blocks according to different geofences. Low-altitude stratification and airways are related to low-altitude traffic flow (busyness), obstacle density, city-level 3D model data, and multiple geofences.

[0056] The modeling of the mission map includes:

[0057] Two-dimensional map: The city's extremely low altitude is divided into several geographic fences, each fence representing a location state, and each fence contains several obstacles at a certain height.

[0058] 3D waterway model: Each grid layer is represented by a 2D matrix. Considering the varying average heights of buildings within the area of ​​interest, a semi-structured, non-uniform height hierarchy is used, with each grid layer (equivalent to a geofence) represented by a 2D matrix. "Semi-structured" represents undefined horizontal waterways below a specific height (e.g., a specific height of 60 meters), and "non-uniform" represents that the spacing or safety distances between multiple horizontal waterways are not uniform.

[0059] The application module for low-altitude drone operations typically relies on mission prompts to clarify mission intent, enhance mission safety and compliance, and ensure mission reusability and traceability. For example, the terms for Mission 1 are shown in Table 1.

[0060] Table 1 Task Application Form

[0061] Task Request A A1 Time: Busy Area A2: Dense A3 Risk: High A4 Intersection Point: True A5 requires a perpendicular passage at the intersection point: False A6 Starting Point: Latitude, Longitude, and Altitude A7 endpoint: latitude, longitude and altitude

[0062] The value of A4 in Task 1 is "True", which means that the UAV path planning has a high safety level and will adopt the popular optimization from the first stage. Its output is defined as oriented towards the main UAV flight path.

[0063] For example, in the terms of Mission 2, A4 and A5 are both set to "True". This means that the difference between Mission 2 and Mission 1 is that the drone first travels along the horizontal path to the rendezvous point, and then switches to the vertical path and performs a descent.

[0064] For example, in the terms of Task 3, the value of A4 is "False", which means that the safety level of the UAV path planning is low and the popular optimization from the second stage will be adopted, and its output is defined as oriented towards the UAV flight path.

[0065] The approval module for low-altitude drone operations typically includes mission registration, airspace inspection, risk assessment, safety configuration, communication support, and approval results. For example, see Table 2 for the terms and conditions for flight plan approval.

[0066] Table 2 Flight Plan Approval

[0067] Approval B B1 Import City-Level 3D Model B2 UAV speed, UAV time interval: 10m / s, 5s B3 Obstacles, Virtual Obstacles Count and Geometry List: 4, 1, (omitted) B4 Horizontal Channel Number, Horizontal Channel Attributes, Altitude, Anomaly Area Parameters, Wind Speed: 1, Main Path, Primary, 75m, (details omitted), 10m / s B5 recommends the following interaction point count and interaction point geographic parameters: 2 Horizontal and vertical channel indices for the B6 junction: (omitted) The B7 horizontal navigation channel list contains horizontal navigation channel indices, starting points, junction indices, and ending points: (omitted)

[0068] Step 2: Based on the UAV's mission start and end points, perform initial path planning to avoid obstacles and virtual obstacle zones, obtaining the initial path. This mainly focuses on environmental analysis within a single geofence and the generation of its initial path. Considering the complex airspace of geofences and the high computational complexity and easy failure of optimization-based path planning, a tentative obstacle avoidance initial path solution is selected. The aforementioned initial path is not suboptimal, and the pathfinding method in this solution is based on uncontrolled hierarchical scheduling logic. Virtual obstacles help the UAV effectively avoid densely populated urban high-rise areas and also particularly help the UAV pass through certain UAV rendezvous points and regions of interest. After reaching the rendezvous point, the UAV may follow a vertical flight path to a lower high-rise platform to unload its cargo. Note: The rendezvous point can be a high-rise platform or a virtual low-altitude 3D geographic coordinate.

[0069] This section explains the intersection point function and segmented pathfinding. Conventionally, the most direct method for finding an initial path in uncontrolled airspace is to find a straight line between the starting and ending points. However, when there are dense urban high-rise buildings along this line, simple "point-to-point" straight-line flight is clearly impractical. As a compromise, practical path planning methods typically abandon a single linear trajectory and instead attempt to search for an initial path in segments, with each segment potentially employing a quadratic curve or other nonlinear function. Considering the complex environment typically faced by the UAV flight plan in this study, with several sparse obstacles and a single region of interest between the starting and ending points, obtaining a suboptimal initial path that satisfies the task of this embodiment is difficult under limited computational resources and strict time constraints. Commonly used path planning algorithms include graph search-based algorithms, random sampling-based algorithms, and biomimetic-based algorithms. Referring to the safety and various constraints of the UAV low-altitude operation task in this embodiment, a tentative obstacle avoidance initial path solution is selected. Note that although A*, RRT, etc. are more general path search schemes, in order to adapt to the requirements of the scenario and optimization algorithm considered in this paper, we only consider this low-complexity initial path method. The premise is that an environmental analysis is performed on a specific city 3D model that simultaneously contains several sparsely distributed high-rise buildings and a low-altitude high-rise cluster to obtain virtual obstacle parameters.

[0070] See Table 3 for the terms regarding the generation of the initial path.

[0071] Table 3 Environmental Analysis and Initial Path Generation Process

[0072] Environmental Analysis C Initial path generation D C1 implements a low-altitude airway stratification strategy D1 associates the primary channel and horizontal channel indexes (high security level when the horizontal channel attribute is set to Primary; low security level when it is set to Subordinate). C2 Statistical Geographic Distribution of Obstacles Within Geographic Fences and Virtual Obstacles D2 Associates the main channel and vertical channel index (the vertical channel is assigned only if the intersection vertical channel is not empty). C3 Statistical Geographical Fence Wind Speed D3 generates the initial path for the main channel using a strategy of laterally bypassing virtual obstacles. C4 predicts dynamic obstacle distribution (optional) Sampling of D4 horizontal waterway (latitude, longitude, and altitude) C5 Predicted Channel Traffic Flow (Optional) Sampling of D5 interaction points (latitude, longitude, and altitude)

[0073] C1 contains the distribution of high-rise buildings at specific heights within the region of interest, derived from a city-level 3D model. The initial path planning process requires using the high-rise building distribution data to calculate the geometric parameters of virtual obstacles. Here, the initial path planning is a segmented planning approach that only considers physical and virtual obstacles.

[0074] Step 3: Perform two-stage manifold optimization on the initial path. This includes a first-stage optimization and a second-stage optimization. In the first-stage optimization, a first cost function is established based on constraints related to obstacle avoidance, energy consumption, smoothness, and proximity to intersection points of the UAV path. The optimal path trajectory is obtained using a Riemannian manifold optimization tool, with the goal of minimizing the first cost function. In the second-stage optimization, a second cost function is established based on constraints on the safe distance between the UAV path, the optimal path trajectory, and obstacles. The suboptimal path trajectory is obtained using a Riemannian manifold optimization tool, with the goal of minimizing the second cost function. The first and second stages of optimization each have a first and a second cost function, respectively, to progressively discover the primary and secondary UAV paths. Here, primary and secondary UAVs represent high and low path safety, respectively, which also implies a greater and lesser number of constraints, and higher and lower time and space computational complexity.

[0075] A manifold is a geometric concept that can be understood as a "low-dimensional surface / curve embedded in a high-dimensional space," whose local structure resembles that of a low-dimensional Euclidean space (such as a line or plane). Popular optimization of geometrically constrained models refers to using current mainstream optimization methods (such as projected gradient method, Lagrange multiplier method, ADMM, etc.) to guide the spatial distribution of solutions through geometric constraints (such as distance, angle, shape regularity, etc.) while optimizing the objective function, ensuring that the result meets geometric requirements. The core is to efficiently solve for solutions that are both geometrically feasible and optimal by combining constraints with gradient information (such as projecting onto the feasible region, introducing penalty terms or multipliers), and this method is widely used in computer graphics, robot path planning, and other fields.

[0076] This embodiment proposes a combination of an initial path solution based on tentative obstacle avoidance and two-stage manifold learning to effectively avoid the problem of the manifold learning optimizer getting stuck in high-dimensional space due to numerous obstacles. Specifically, in the studied UAV global path planning problem, obstacles exhibit a repulsive force on the UAV, while (traffic) intersections exhibit an attractive force. The results of the verification process in this embodiment demonstrate the feasibility of using a manifold learning optimizer in the later stages of global path planning.

[0077] The initial path planning output is suboptimal because it is not smooth enough and does not directly consider the operation of switching to a vertical channel at the junction, nor does it consider constraints such as areas with abnormal wind speeds. This embodiment specifically relies on a Pymanopt-based manifold optimization method to obtain a suboptimal trajectory, and specifically proposes a two-stage Pymanopt strategy including first and second stages to address different needs such as passing through junctions and not passing through junctions. The loss function design of the first stage integrates multiple indicators such as path penalty, directionality penalty, obstacle avoidance penalty, wind speed penalty, and interest point cost, and also considers path smoothness, making the path both reasonable and safe.

[0078] The following a)-f) outline one implementation scheme for initial path planning:

[0079] a) Functionality of the class: Generate an initial safe trajectory for each drone for subsequent optimization, diverting drones by base / route and avoiding obstacles near the destination. Figure 1 The bases are represented by start_points and end_points, and the base number is Base_number.

[0080] b) Constructor parameters (__init__):

[0081] The variables are defined as follows: start_points, end_points, obstacles_np, obstacle_sizes, safe_margin.

[0082] The class also uses global configurations such as T, L, Base_number, n, y_up, and y_down.

[0083] c) Main methods: interpolate_trajectory(waypoints): generates a smooth trajectory by interpolating with cubic splines over T time steps; generate(): constructs waypoints for each UAV (start point → intermediate waypoints → end point, adding detour points if necessary), calls the interpolation and verifies safety, and returns all trajectories.

[0084] d) Key behaviors: Assign up / down routes based on Base_number or initial y-coordinate; add waypoints around real obstacles near the endpoint; check the minimum distance to obstacles for each trajectory using validation_safe_box_distance and print the status. See also Figure 3 The terms "upper" and "lower" only indicate different horizontal routes.

[0085] e) Return value: generate() returns X0, which is the initial trajectory array after stacking (T rows per drone, with 3D coordinates in each column, ordered by drone).

[0086] f) Key points for parameter tuning: safe_margin controls the distance between the initial trajectory and obstacles; if the initial trajectory is "dangerous", it will be further corrected during the optimization phase; the detour offset and the waygate position can be adjusted to improve initial feasibility.

[0087] In terms of handling constraints, the first and second stages are similar; both stages use penalty terms instead of constraint terms. This can be explained by the fact that Pymanopt's support for complex constraints is incomplete. The most practical and reliable method is to not directly use constraint parameters, but instead treat distance constraints as penalty terms and add them to the cost function. In this embodiment, the second stage did not impose any constraints or penalties on the speed of the subordinate UAVs, while the first stage included speed range and speed consistency constraints. Therefore, the reliability of the second stage in the verification process is significantly lower than that of the first stage. Nevertheless, the second stage is not only used for quickly constructing horizontal channels but can also be extended to cases with speed constraints.

[0088] The first cost function of the first-stage optimization will not only output the main flight path that passes through the point of interest and is close to optimization, but also take into account obstacle avoidance, energy consumption, and smoothness. The second-stage optimization requires the main UAV trajectory optimized in the first stage as input to quickly search for the trajectory of UAVs in other flight paths, so as to realize a flight path manifold constraint optimization for slave UAVs using Pymanopt. The second-stage optimization does not consider crossing the point of interest but has the characteristics of lower computational complexity and lower safety. It can quickly calculate the UAV trajectory at nearby altitudes or on the flight path, such as reducing the computation time to one-tenth of the first stage.

[0089] By using manifold optimization to naturally embed constraints into the variable space, not only can the geometric structure be preserved, but the solution efficiency can also be improved. The number of steps in the scheduling time is defined. , , , and wait; Indicates the first The path of the drone ( (step and x, y, z coordinates), belonging to trajectory manifolds or fixed-rank matrix manifolds; Indicates the first The cruise speed sequence of the drone belongs to a positive real number manifold; Indicates the first The energy consumption sequence of the drone belongs to a positive real number manifold; Indicates the first The airspace state of a drone (intersection, weather, obstacles) belongs to a discrete manifold or an embedded manifold.

[0090] First cost function in main channel manifold modeling (phase one) for:

[0091] ;

[0092] in, For the first The total length of the drone's path, For the first Energy consumption of the drone's flight path For the first Severe weather penalties for drone flight paths. For the first Penalty for the intersection of drone paths. For the first Obstacles in the drone's path are penalized. For the first Spatial boundary penalty for drone flight paths, For the first Penalty for directional consistency of drone path. For the first Speed ​​penalty for the drone's path For the first Penalty for consistency between speed and displacement of the drone's path. , , , , , , , , These are weighting coefficients, which can be adjusted according to task requirements. The total number of main drones.

[0093] Use Pymanopt to optimize the trajectory X_opt of the master unmanned aerial vehicles (UAVs) (n vehicles), including:

[0094] Constraints and penalties include path length, energy consumption, wind speed, obstacles, weather, intersection points, and safe distance. Output: X_opt, with shape (n * T, 3), representing the position of each UAV at T time steps. The output of the first cost function can also be written as X_all = [X1, X2, ..., Xn], with a total of n × T × 3 variables.

[0095] The terms of the first cost function are explained below:

[0096] No. Total length of the drone's path The calculation formula is:

[0097] ;

[0098] in, For the first A drone in time step Location, For the first A drone in time step Location, This represents the total number of time steps. By calculating the positional differences between consecutive time points, it encourages simple and short paths.

[0099] No. Energy consumption of drone flight path The calculation formula is:

[0100] ;

[0101] in, For hovering power consumption, The air drag coefficient, Sampling time, Let be the airspeed of the drone, that is, the speed of the drone relative to the air. The energy consumption formula penalizes deviations from expected speed and energy consumption under adverse weather conditions.

[0102] Based on the above variable definitions, the matrix is ​​first calculated within the focal time period. For each t, ​​take the minimum value of the column and then sum them up, thus the th... Penalty for the intersection of drone paths The calculation formula is:

[0103] ;

[0104] No. Obstacles in the drone's path The two calculation formulas are:

[0105] ;

[0106] ;

[0107] in, For the first The weighting coefficient of each obstacle For the first An obstacle, A collection of obstacles. For the first Safe distance radius of each obstacle, single-point penalty It is a quadratic penalty function with a threshold. For position To the The shortest Euclidean distance to each obstacle will be expressed using an expression for the center and size of the obstacle. Instead of the former; for obstacles Apply a safety distance penalty, forcing a path deviation to avoid approaching obstacles. If the first The UAV was too close to the first When there is an obstacle, it generates a positive penalty (the closer to the center of the obstacle, the larger the penalty). For a detailed definition of an obstacle center and size expression, see a)-c) below.

[0108] a) Let the center and size of the j-th obstacle be aligned. ,Right now .

[0109] b) Define a component-wise truncated box projection and distance:

[0110] , ;

[0111] ;

[0112] ;

[0113] c) Two definitions of penalty items:

[0114] ;

[0115] When considering exponential penalties, first define the penetration amount. The penalty intensity coefficient α, the steepness of the exponential penalty κ, and the scalar multiplication factor. Multiply by this factor as you approach the finish line. The exponential penalty expression is:

[0116] ;

[0117] No. Spatial boundary penalty for drone path The calculation formula is:

[0118] ;

[0119] in, For position x-coordinate For position The ordinate, For position The vertical coordinate, These are the boundary values ​​for the x-axis. The boundary value of the ordinate. This is the minimum height boundary value. Maximum height boundary value.

[0120] No. Directional consistency penalty for drone paths The calculation formula is: ;

[0121] in, In time step The angle between the trajectory segment direction and the ideal direction. Here, the starting point is defined. and the end point , For the first The equivalent form of the trajectory point sequence of a drone, with the penalty term for consistent direction, is:

[0122]

[0123] No. Speed ​​penalty for drone path The calculation formula is:

[0124] ;

[0125] in, For the first A drone in time step of, This represents the minimum speed of the drone. This represents the maximum speed of the drone.

[0126] No. Consistency penalty for speed and displacement of drone path The calculation formula is:

[0127] .

[0128] Use Pymanopt to optimize the trajectory X_opt of the m slave UAVs. Let the trajectory of the i-th slave UAV be... The main drone's trajectory is Then the second cost function for:

[0129] ;

[0130] in, The solution required by the second cost function is the first... A drone in time step Location, To solve the problem using the first cost function, the second... A drone in time step Location, This represents the minimum safe distance between the suboptimal and optimal path trajectories. This is the minimum safe distance between the suboptimal path trajectory and obstacles (e.g., 25.0 meters), but it needs to be optimized based on the scenario. For position With the The safe distance between obstacles. The UAV can also be viewed as a companion drone. Secondly, the second cost function here assumes parameter configuration. However, as mentioned above, it can also be extended to other configurations, such as... In this process, the second-stage optimization process will involve calculations using two different safety distance parameters to search for path trajectories from the UAV. Each trajectory corresponds to a channel supporting m subordinate UAVs, and the two paths (or channels) can be in the same uncontrolled layer (e.g., uncontrolled layer [i+1]), but belong to different horizontal lateral channels. Figure 3 The diagram illustrates different horizontal lateral flight paths, where the uncontrolled layer represents the extremely low altitude of urban areas below 120 meters. The minimum distance between flight paths for vertical formation flight and horizontal lateral formation flight is defined as the safe distance D_safe. However, the distance between two flight paths in different uncontrolled layers is usually greater than D_safe, depending on the geometric relationship between the flight paths. For example, in a certain verification experiment, the main UAV flight path or main UAV flight path is located in the [i]th uncontrolled layer, and the secondary UAV flight path is located in the [i+1]th uncontrolled layer, meaning the main UAV flight path is directly below the secondary UAV flight path. Here, the [x]th uncontrolled layer represents the low-altitude state, and the flight path represents the configuration of the flight plan.

[0131] Step 4: Determine the UAV's path trajectory based on the intersection point of the UAV's low-altitude operation tasks. If the UAV passes through the intersection point, the optimal path trajectory is adopted; otherwise, the suboptimal path trajectory is adopted. Construct a "flattened" joint manifold for time-space scheduling. The considered application scenario is one where all UAVs depart from their respective bases simultaneously and arrive at their respective destinations simultaneously. In this embodiment, taking advantage of pymanopt's lack of support for nested Cartesian product manifolds, the joint optimization space variables of multiple UAVs are transformed into a fixed, flattened vector. Furthermore, a takeoff delay is added to the two-stage manifold optimization output during post-processing to preserve the effectiveness of the current optimization framework.

[0132] The two-stage flow optimization and time-space scheduling process is shown in Table 4.

[0133] Table 4 Two-stage population optimization and time-space scheduling process

[0134] Main channel manifold modeling E Optimization of F from channel manifold constraints Time air conditioning temperature G E1 implements Pymanopt based on cost function 1 (optimization objectives such as safe distance, energy consumption, etc.). F1 implements Pymanopt based on cost function 2 (checking the safe distance between the optimized channel and the main channel). G1 implements scheduling based on spatiotemporal trajectory matrices (main channel and secondary channel). E2 searches for the suboptimal main route passing through the interaction point (obstacle avoidance, energy consumption, smoothness, etc.). F2 searches for suboptimal main routes that do not pass through interaction points (obstacle avoidance, energy consumption, smoothness, etc.). E3 Output Phase 1: High-Safety Main Channel Path Sequence The F3 output phase 2 has relatively low safety from the channel path sequence.

[0135] In the verification process, both the first and second cost functions employ instantiated Pymanopt's Riemannian Trust-Region optimizer. Trust-Regions are specifically designed for optimization on manifolds (such as Euclidean, Positive, and Product). In this embodiment, the problem's variables (position and velocity) are constrained by the manifold, making ordinary Euclidean space optimizers (such as L-BFGS and SGD) unsuitable or ineffective. Preliminary results from the verification process show that Trust-Regions can efficiently and stably handle complex constraints and multi-objective cost functions within the manifold space, exhibiting better convergence and robustness than simple first-order methods.

[0136] The functional division of the main function in the verification program is as follows:

[0137] 1) Generate the start point, end point, and obstacles;

[0138] 2) Initial path and optimizer;

[0139] 3) Define the cost function and run the optimizer;

[0140] Note that since the total number of variables involved in scheduling is n × T × 3, but pymanopt does not support nested Product manifolds, multiple variables are concatenated into a large manifold, that is, by concatenating the variables of all drones together, a "flattened" joint manifold is constructed.

[0141] This validation program simulates a scenario where all UAVs simultaneously depart from their respective bases and arrive at their respective destinations, outputting the main UAV flight path and (cooperative) secondary UAV flight paths. Its optimization goal is to find cooperative, safe, and efficient flight paths for all UAVs within this fixed time window. First, it optimizes the ideal parallel trajectories for all UAVs, then adds takeoff delays to them in the post-processing stage. This preserves the effectiveness of the current optimization framework while demonstrating the effect of spaced takeoffs in the final visualization and analysis. Furthermore, the validation program sets the initial energy of each UAV to a fixed constant, so initial energy is not an optimization variable, thus reducing the dimensionality of the simulation / optimization.

Claims

1. A two-stage manifold optimization method for low-altitude flight path planning of unmanned aerial vehicles (UAVs), characterized in that, Includes the following steps: Step 1: Obtain the UAV low-altitude operation mission, including the mission start point, mission end point, and whether the mission passes through the intersection point; obtain the mission 3D map based on the UAV low-altitude operation mission, and confirm the obstacles, virtual obstacle areas, and the intersection points of UAV flight paths around the virtual obstacle areas in the mission 3D map. The virtual obstacle areas are the areas where complex building clusters are located. Step 2: Based on the drone's mission start and end points, perform initial path planning for the drone to avoid obstacles and virtual obstacle zones, and obtain the initial path; Step 3: Perform two-stage manifold optimization on the initial path, which includes a first-stage optimization and a second-stage optimization. In the first-stage optimization, a first cost function is established based on constraints such as obstacle avoidance, energy consumption, smoothness, and proximity to intersection points of the UAV path. The optimal path trajectory is obtained using a Riemannian manifold optimization tool with the goal of minimizing the first cost function. In the second-stage optimization, a second cost function is established based on constraints such as the safe distance between the UAV path, the optimal path trajectory, and obstacles. The suboptimal path trajectory is obtained using a Riemannian manifold optimization tool with the goal of minimizing the second cost function. The first cost function... for: ， in, For the first The total length of the drone's path, For the first Energy consumption of the drone's flight path For the first Severe weather penalties for drone flight paths. For the first Penalty for the intersection of drone paths. For the first Obstacles in the drone's path are penalized. For the first Spatial boundary penalty for drone flight paths, For the first Penalty for directional consistency of drone path. For the first Speed ​​penalty for the drone's path For the first Penalty for consistency between speed and displacement of the drone's path. , , , , , , , , These are the weighting coefficients. The total number of drones; The second cost function for: ； in, The solution required by the second cost function is the first... A drone in time step Location, To solve the problem using the first cost function, the second... A drone in time step Location, This represents the minimum safe distance between the suboptimal and optimal path trajectories. The minimum safe distance between the suboptimal path trajectory and the obstacle. For position With the Safe distance between obstacles; Step 4: Determine the path trajectory of the UAV based on the intersection point of the UAV low-altitude operation mission. If the UAV passes through the intersection point, the optimal path trajectory is adopted; if the UAV does not pass through the intersection point, the suboptimal path trajectory is adopted.

2. The UAV low-altitude flight path planning method based on two-stage manifold optimization according to claim 1, characterized in that, The channel where the intersection point is located includes horizontal and vertical channels, and the UAV low-altitude operation mission includes whether or not the vertical channel of the intersection point is required.

3. The UAV low-altitude flight path planning method based on two-stage manifold optimization according to claim 1, characterized in that, The initial path planning for drones to avoid obstacles and virtual obstacle zones can be performed using graph search-based algorithms, random sampling-based algorithms, or biomimetic-based algorithms.

4. The UAV low-altitude flight path planning method based on two-stage manifold optimization according to claim 1, characterized in that, The first Total length of the drone's path The calculation formula is: ； in, For the first A drone in time step Location, For the first A drone in time step Location, This represents the total number of steps in time. The first Energy consumption of drone flight path The calculation formula is: ； in, For hovering power consumption, The air drag coefficient, Sampling time, The speed of the drone relative to the air; The first Penalty for the intersection of drone paths The calculation formula is: ； in, Set of intersection points Time interval The location of the internal drone and the first Intersection The basis functions for the distance relationship between them. Set of intersection points The total number of intersections; No. Obstacles in the drone's path One calculation formula is: ； ； in, For the first The weighting coefficient of each obstacle For the first An obstacle, For the first Safe distance radius of each obstacle, single-point penalty It is a quadratic penalty function with a threshold. For position To the The shortest Euclidean distance to each obstacle; The first Spatial boundary penalty for drone path The calculation formula is: ； in, For position x-coordinate For position The ordinate, For position The vertical coordinate, These are the boundary values ​​for the x-axis. The boundary value of the ordinate. This is the minimum height boundary value. Maximum height boundary value; The first Directional consistency penalty for drone paths The calculation formula is: ; in, In time step The angle between the trajectory segment direction and the ideal direction; The first Speed ​​penalty for drone path The calculation formula is: ； in, For the first A drone in time step of, This represents the minimum speed of the drone. This represents the maximum speed of the drone. The first Consistency penalty for speed and displacement of drone path The calculation formula is: 。 5. The UAV low-altitude flight path planning method based on two-stage manifold optimization according to claim 1, characterized in that, The Riemannian manifold optimization tool used is Pymanopt.

6. The UAV low-altitude flight path planning method based on two-stage manifold optimization according to claim 1, characterized in that, Based on the optimal and suboptimal path trajectories of several UAVs obtained from steps 1 to 4, different takeoff delays are added to several UAVs to achieve interval takeoff of several UAVs.

7. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the method according to any one of claims 1 to 6.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the steps of the method according to any one of claims 1 to 6.

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