Time optimal trajectory planning method suitable for deformable four-rotor aircraft

Abstract

The invention provides a time optimal trajectory planning method suitable for a deformable four-rotor aircraft, and relates to the technical field of trajectory planning methods.The time optimal trajectory planning method comprises the following steps that deformation factors are considered, and a kinetic model of the deformable four-rotor aircraft is established; constructing a time optimal objective function; simplifying a time optimal objective function; solving an unconstrained optimization problem; an L-BFGS algorithm is used to solve the unconstrained optimization problem; calculating optimal time distribution and waypoint positions; and calculating the trajectory of the deformed unmanned aerial vehicle according to an optimization result. Simulation results show that the method can realize 10-second-level rapid crossing of a seven-level variable-diameter obstacle of the four transformable rotors.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of trajectory planning methods, in particular, especially relates to a time optimal trajectory planning method suitable for a morphing quadrotor. BACKGROUND

[0002] Quadrotor unmanned aerial vehicle (UAV) has a wide application in military, agriculture, fire fighting, logistics and other fields. The traditional fixed quadrotor unmanned aerial vehicle often has limitations due to large space requirement for take-off and landing, and difficulty in passing through narrow and obstacle-dense environment. The morphing unmanned aerial vehicle enhances its flexibility and reliability in narrow space by changing the shape parameters, optimizes the maneuvering performance and environmental adaptability, thereby better meeting the multi-task requirements.

[0003] Trajectory planning, as a key link for unmanned aerial vehicle to realize autonomous flight, is the basis for ensuring the unmanned aerial vehicle to safely, stably and quickly reach the predetermined position. The morphing factor of the unmanned aerial vehicle is rarely considered in the trajectory planning process in the past work. The conventional trajectory planning of quadrotor unmanned aerial vehicle mainly uses the following three methods.

[0004] 1. The sampling-based planning algorithm constructs a search tree or graph to find a feasible path by randomly or strategically sampling in the environment space. In a complex environment, the sample points may be unevenly distributed, which makes it difficult to find a feasible path in a complex environment (such as obstacle-dense or narrow channel), and the generated path is generally not smooth and optimized, which needs additional smoothing processing.

[0005] 2. The discretization-based method discretizes the environment into a graph structure, and then uses a graph search algorithm to find the optimal path. A* algorithm is a representative, which uses heuristic function to guide the search direction, and can efficiently find the optimal path from the starting point to the ending point. Although the trajectory generated by this method may be optimal, in a high-dimensional environment, the size of the graph will increase dramatically, resulting in increased computation time.

[0006] 3. The polynomial-based method takes certain performance indicators of the path as the optimization target, and represents the trajectory as a sequence of quadrotor states and control inputs with respect to time. The quadrotor dynamics and input constraints are considered in the solution, and the smooth trajectory that meets the physical characteristics of the unmanned aerial vehicle and environmental constraints is obtained by converting these constraint conditions into constraint terms of the optimization problem. However, when the problem is complex and the solution time is long, it may not be able to solve the global optimal solution. SUMMARY

[0007] According to the technical problems mentioned in the background art, a time optimal trajectory planning method suitable for a morphing quadrotor aircraft is provided. The purpose of the present application is to consider the influence of morphing in trajectory planning in view of the shortcomings of existing quadrotor trajectory planning, and to propose a trajectory optimization method based on the dynamics model of the morphing quadrotor aircraft.

[0008] The present application first establishes the dynamics model of the quadrotor, then constructs the time optimal objective function combined with the morphing factor of the unmanned aerial vehicle, then simplifies the constraint conditions in the function, and finally solves the time, position and unmanned aerial vehicle arm length value by means of the solver, thereby generating the optimal trajectory of the morphing quadrotor aircraft satisfying the constraint conditions.

[0009] The technical means adopted by the present application are as follows: A time optimal trajectory planning method suitable for a morphing quadrotor aircraft, comprising the following steps: Step 1, considering the morphing factor, establishing the dynamics model of the morphing quadrotor aircraft; Step 2, constructing a time optimal objective function; defining an optimization problem in which the aircraft starts from the starting point, passes through a plurality of waypoints in turn and finally stops at the end point; introducing the constraint conditions of the waypoints, constructing an optimization function with the minimum total flight time as the target; the constraint conditions include: position, radius and aircraft arm length; Step 3, simplifying the time optimal objective function; dividing the trajectory into multiple segments, introducing a functional expression, converting the dynamic equation, state and control input into constraint conditions; using differential flat mapping to convert the state and control trajectory into polynomial form; introducing variable substitution to eliminate the constraints of waypoints and time, and converting the problem into an unconstrained optimization problem; Step 4, solving the unconstrained optimization problem; using L-BFGS algorithm to solve the unconstrained optimization problem; calculating the optimal time allocation and waypoint position; calculating the trajectory of the morphing unmanned aerial vehicle according to the optimization result.

[0010] Further, the dynamics model of the morphing quadrotor aircraft is established by defining the mass, acceleration, position, rotation matrix, angular velocity and moment of inertia of the aircraft.

[0011] Further, the dynamics model of the morphing quadrotor aircraft is: ; ; ; ; wherein, m represents the mass of the quadrotor aircraft, g represents the acceleration of gravity, represents the position of the quadrotor, , represents the rotation unit quaternion from the world frame to the body frame, represents the rotation matrix parameterized by ; , and represent the angular velocity, velocity and acceleration in the body frame, angular velocity, represents the moment of inertia of the quadrotor, represents the moment generated by the propeller on the body axis, the relationship between thrust and moment is: ; where, represents the rotor moment constant, represents the total thrust generated by the motor.

[0012] Further, in step 2, assuming a total of waypoints, a starting point, an ending point, the time optimal problem can be described as the UAV starting from the starting point, passing through the waypoints in turn, and finally stopping at the ending point, and satisfying the state constraints and input constraints throughout the flight; define as the th circle in the trajectory planning, as the radius of the circle, and the center point is , that is, the position of the th waypoint, is the position of the deformed quadrotor when passing through the th waypoint, is the arm length of the deformed quadrotor, in order to ensure the safety of the deformed quadrotor aircraft passing through the waypoint, it is necessary to satisfy: ; define to represent that the deformed quadrotor has successfully passed through this circle; then the optimization function can be expressed in the form of the total time spent by the UAV flying as the optimization objective:

[0013] ,

[0014]

[0015]

[0016]

[0017] where the quadrotor state is defined as , the control input , all dynamic equations are encapsulated as , all state input constraints are encapsulated as ; represents the position component of the quadrotor state when passing the th waypoint, represents the time when passing the th lap, represents the initial state, represents the final state.

[0018] Further, in step 3, the whole trajectory is divided into segments by waypoints, define as the set of time spent on each segment, total time , cumulative time of the first segments , define as the set of waypoint coordinates; introduce a functional expression to express the above time optimal function as:

[0019]

[0020] where is the time integral penalty functional, when satisfies conditions, is zero, if violates any constraints in , grows rapidly, with the help of can guarantee that the solution meets all conditions; define the spatial trajectory , where represents the trajectory of the quadrotor's center of gravity, represents the yaw angle of the quadrotor; use the differential flatness mapping and to convert the state and control trajectories and into : ); ; Similarly, the relevant constraints​ is expressed as: ; At this time, is the only constraint left, then, By the minimum control trajectory functional expression is: ; Where, denotes the sampling frequency, ; Introducing the variable , any satisfy the following smooth mapping, so that satisfies , thus the in the constraint condition is eliminated: ; Since implicit upper and lower bound conditions, the optimization calculation needs to add constraints, so the variable s is introduced, let: ; Where, and respectively represent the minimum and maximum value of , the final mapping is obtained as: ); Let the optimization function , the gradient of two variables is as follows: ; ; In order to eliminate the addition of constraints, then:

[0021] By using the new variable and , it is ensured that the quadrotor unmanned aerial vehicle always remains within the corresponding circle, and is used to ensure the nonnegativity of time; Now, the waypoint position and time are obtained from and , and the time optimal function is converted into an unconstrained form: .

[0022] Further, the functional expression is:

[0023]

[0024]

[0025]

[0026]

[0027] The functional expression contains a position vector satisfying initial and final states, state equations, path constraint conditions .

[0028] Compared with the prior art, the present application has the following advantages: The present application aims at the technical defects of ignoring the influence of body deformation in the existing four-rotor trajectory planning, and proposes a trajectory optimization method based on the dynamics model of the deformed four-rotor aircraft. The method first establishes the dynamics model of the four-rotor aircraft, then combines the deformation factors of the unmanned aerial vehicle to construct a time-optimal objective function, then simplifies the constraint conditions in the function, and finally solves the time, position and unmanned aerial vehicle arm length value by means of a solver, thereby generating the optimal trajectory of the deformed four-rotor aircraft satisfying the constraint conditions. The simulation results show that the present application can realize the 10-second rapid crossing of the seven-level diameter-changing obstacles by the deformed four-rotor aircraft.BRIEF DESCRIPTION OF DRAWINGS BRIEF DESCRIPTION OF DRAWINGS

[0029] In order to more clearly illustrate the technical solutions in the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or the prior art description. Obviously, the drawings in the following description are some embodiments of the present application, and other drawings can also be obtained by those skilled in the art without creative labor.

[0030] Figure 1 It is a flowchart of the time-optimal trajectory planning method of the deformed four-rotor aircraft of the present application.

[0031] Figure 2 It is a 3D trajectory generation diagram of the deformed four-rotor aircraft of the present application.

[0032] Figure 3 It is a specific diagram of the 3D trajectory of the present application.

[0033] Figure 4 It is a diagram showing the change of the arm length l of the unmanned aerial vehicle of the present application.

[0034] Figure 5 It is a diagram showing the change of the arm length l of the unmanned aerial vehicle of the present application.

[0035] It is a diagram showing the change of the arm length l of the unmanned aerial vehicle of the present application. Figure 6 Acceleration variation diagram of direction. DETAILED DESCRIPTION

[0036] In order to make the personnel in the technical field better understand the present application scheme, the technical scheme in the embodiments of the present application will be clearly and completely described below in combination with the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, not all. Based on the embodiments in the present application, all other embodiments obtained by the person skilled in the art without creative labor should belong to the scope of protection of the present application.

[0037] It should be noted that the terms "first", "second", and the like in the specification and claims of the present application and the above-described drawings are used to distinguish similar objects, and do not necessarily have to describe a specific order or sequence. It should be understood that the data thus used can be interchanged under appropriate circumstances, so that the embodiments of the present application described herein can be implemented in an order other than those illustrated or described herein. In addition, the terms "include" and "have" and any variations thereof are intended to cover non-exclusive inclusion, for example, a process, method, system, product or device including a series of steps or units does not have to be limited to those steps or units clearly listed, but can include other steps or units not clearly listed or inherent to these processes, methods, products or devices.

[0038] As Figures 1-6 shown, the present application provides a time optimal trajectory planning method suitable for a morphing quadrotor aircraft, comprising the following steps: Step 1, considering the morphing factor, a dynamic model of the morphing quadrotor aircraft is established; by defining the mass, acceleration, position, rotation matrix, angular velocity, moment of inertia of the aircraft, a dynamic model of the morphing quadrotor aircraft is established. The dynamic model of the morphing quadrotor aircraft is: ; ; ; ; wherein, m represents the mass of the quadrotor aircraft, g represents the acceleration of gravity, p represents the position of the quadrotor, , q represents the rotation unit quaternion from the world coordinate system to the body coordinate system, R represents the rotation matrix parameterized by ; , and denotes the angular velocity, velocity and acceleration in the body frame, angular velocity, denotes the moment of inertia of the quadrotor, denotes the moment generated by the propeller on the body axis, the relationship between thrust and moment is: ; where, denotes the rotor moment constant, denotes the total thrust generated by the motor.

[0039] Step 2, construct the time optimal objective function; define the optimization problem of the aircraft starting from the starting point, passing through a plurality of waypoints in turn and finally stopping at the end point; introduce the constraint condition of the waypoint, construct the optimization function with the minimum total flight time as the objective; the constraint conditions include: position, radius and aircraft arm length; in step 2, assuming that there are waypoints, a starting point, an end point, the time optimal problem can be described as that the unmanned aerial vehicle starts from the starting point, passes through the waypoints in turn, and finally stops at the end point, and needs to meet the state constraint and input constraint in the whole flight; define as the th circle in the trajectory planning, as the radius of the circle, and the center point is , that is, the position of the th waypoint, is the position of the morphing quadrotor when passing through the th waypoint, is the arm length of the morphing quadrotor, in order to ensure that the morphing quadrotor aircraft safely passes through the waypoint, it is necessary to meet: ; define to indicate that the morphing quadrotor has successfully passed through this circle; then the optimization function can be expressed in the form of taking the total flight time as the optimization objective:

[0040] ,

[0041]

[0042]

[0043]

[0044] wherein, define the quadrotor state , control input , encapsulate all dynamic equations as , encapsulate all state input constraints as ; Indicates that the quadcopter is passing through Waypoint status The position component of Indicates that after A circle of time, Indicates the initial state, Indicates the end state.

[0045] Step 3: Simplify the time optimal objective function; divide the trajectory into multiple segments, introduce functional expressions, transform the dynamic equations, state and control input into constraints; use differential flat mapping to transform the state and control trajectory into polynomial form; introduce variable substitution, eliminate the waypoint and time constraints, and transform the problem into an unconstrained optimization problem; in step 3, use The waypoints divide the entire trajectory into Segment, definition is the collection of time spent in each section, the total time ,forward Cumulative time of the segment ,definition is a set of waypoint coordinates; A functional expression is introduced, which is:

[0046]

[0047]

[0048]

[0049]

[0050] The functional expression contains the position vector that satisfies the initial and final states, state equations, and path constraints. .

[0051] The above time optimal function is expressed as:

[0052]

[0053] in is the time-integrated penalty functional, when satisfy When the condition is zero if violate any constraints in , grow rapidly, by means of can guarantee that the solution satisfies all conditions; define the spatial trajectory where denotes the trajectory of the center of mass of the quadrotor, denotes the yaw angle of the quadrotor; using the differential flatness mapping and convert the state and control trajectories and into : ); ; Similarly, the relevant constraints are expressed as: ; At this point, is the only constraint left in , then, ; where denotes the sampling frequency, ; Introduce the variable , any satisfies the following smooth mapping, so that satisfies , thus in the constraint condition is eliminated: ; Since implies the upper and lower bound conditions, additional constraints need to be added in the optimization calculation, so introduce the variable s, let: ; where and denote the minimum and maximum values of , respectively, and the final mapping is: ); Let the optimization function , the gradient of the two variables is as follows: ; ; To eliminate the addition of constraints, we have:

[0054] By using new variables and , it is ensured that the quadrotor always stays within the corresponding circle, and ensures the non-negativity of time; Now, the waypoint positions and times are obtained from and , and the time-optimal function is converted to an unconstrained form: .

[0055] Step 4, solve the unconstrained optimization problem; use the L-BFGS algorithm to solve the unconstrained optimization problem; calculate the optimal time allocation and waypoint positions; calculate the trajectory of the morphing UAV according to the optimization results. The above simplified problem is solved using the L-BFGS algorithm. After solving , the is calculated by the conversion function, thereby generating the trajectory.

[0056] Embodiment 1 The present application provides a trajectory planning method based on the dynamic model of a morphing quadrotor, comprising the following steps: S1, establish the dynamic model of the morphing quadrotor.

[0057]

[0058]

[0059]

[0060]

[0061] wherein, is the mass of the quadrotor, is the acceleration of gravity, is the position of the quadrotor, , represents the rotation unit quaternion from the world coordinate system to the body coordinate system, is a rotation matrix parameterized by . , and are the angular velocity, velocity and acceleration in the body coordinate system, in this example , , The moment of inertia of the quadrotor for rotation, represents the moment of force generated by the propeller on the body axis; ; S2, construct the time optimal objective function. There are circles in this example, and the coordinates of the center points are , , , , , , , the coordinates of the starting point are , and the coordinates of the ending point are . The time optimal problem can be represented as the UAV starting from the starting point, passing through these waypoints in order, and finally stopping at the ending point, while satisfying the state constraints and input constraints in flight; for these circles, the radii are set to ; Let be the circle in the trajectory planning, be the radius of the circle, and the center point be , i.e., the position of the waypoint, be the position of the morphing quadrotor when passing through the waypoint, be the arm length of the morphing quadrotor. To make the morphing quadrotor vehicle pass through this waypoint safely, the following conditions should be met:

[0062] Here, let represent that the morphing quadrotor has successfully traversed the circle; Then the optimization function can be represented in the following form with the total time spent in flight as the optimization objective:

[0063] ,

[0064]

[0065]

[0066]

[0067] where the quadrotor state is defined, and the control input , encapsulate all dynamic equations as , encapsulate all state input constraints as . Indicates that after The state of the quadrotor drone when there are waypoints The position component of Indicates that after The time of a circle, is the initial state, is the end state; S3, simplified time optimal objective function. The entire trajectory is divided into Segment, definition is the collection of time spent in each section, so the total time , after period of time ,definition is a set of waypoint coordinates; A functional expression is introduced as follows:

[0068]

[0069]

[0070]

[0071]

[0072] This functional includes the position vector that satisfies the initial and final states, state equations, and path constraints. , so the above time optimal function can be expressed as:

[0073]

[0074] in is the time-integrated penalty functional, when satisfy When the condition is zero if Violation For any constraint in , the penalty functional Rapid growth through It can be guaranteed that the solution will meet all conditions; The trajectory in the definition space is ,in is the trajectory of the quadrotor drone's center of gravity, is the yaw angle of the quadrotor; Then, the differential flat map is used to convert the state and control trajectories and to the flat trajectories and as follows: )

[0075] Then similarly, the associated constraints are expressed as:

[0076] At this point, is the only constraint left in , then, can be expressed by the MINCO trajectory functional

[0077] where is the sampling frequency,

[0078] Introduce a variable , any satisfies the following smooth map, so that satisfies , thus the in the constraint condition is eliminated:

[0079] Since implies the upper and lower bound conditions, the value of each is also different, which adds constraints in the optimization calculation process, so introduce the variable s, let:

[0080] where, and represent the minimum and maximum values of at this time, in this example the range of is , get the final mapping: Optimization and need to get the gradient of the optimization function respectively, let the optimization function , the gradients of the two variables are as follows:

[0081]

[0082] Similarly, time also contains non-negative implicit conditions, which adds constraints in the optimization calculation process, which are eliminated using the following formula:

[0083] By using new variables and , it is ensured that these waypoints always remain within the corresponding circles, and ensures the non-negativity of time; Now, the waypoint positions and times are obtained from and , and the time-optimal function is converted to the following unconstrained form:

[0084] S4, solve the unconstrained optimization problem. The above simplified problem is solved using the L-BFGS algorithm. The optimal time is , and is obtained. After that, the is calculated by the conversion function, so that and are obtained. Regarding the time segmented trajectory, the trajectory length is .

[0085] Example two: The simulation gives the trajectory planning results with seven circles to illustrate that the morphing quadrotor aircraft can pass through the target circles by stretching the arms under the condition of meeting the shortest time; Figure 1 and Figure 2 are the trajectory planning results, both of which are 3D graphs, the black line represents the generated trajectory, and the red part is the circle that needs to be passed through, the circles passed through by the unmanned aerial vehicle and the order are marked, it can be seen that the size of the circles from to decreases in turn; Figure 2 is a local enlarged view of the Figure 1 point; From the simulation results, it can be seen that the morphing unmanned aerial vehicle starts from the first waypoint, safely passes through seven circles that decrease in size, and reaches the end point; Through the simulation, it is shown that the present application can realize the morphing quadrotor to pass through 7 levels of variable diameter obstacles in 10 seconds.

[0086] ​The above-mentioned embodiment numbers of the present application are only for description, and do not represent the advantages and disadvantages of the embodiments.

[0087] In the above-mentioned embodiments of the present application, the description of each embodiment has its own focus, and the parts not described in detail in a certain embodiment can be referred to the relevant description of other embodiments. In the several embodiments provided in the present application, it should be understood that the disclosed technical contents can be implemented by other ways.

[0088] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present application, and not to limit them; although the present application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that: it can still modify the technical solutions recorded in the foregoing embodiments, or make equivalent replacement for part or all of the technical features; and these modifications or replacements do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present application.

Claims

1. A time-optimal trajectory planning method for a morphing quadrotor aircraft, characterized in that: The following steps are involved: Step 1: Considering the deformation factor, a dynamic model of the deformable quadrotor is established; Step 2: Construct a time-optimal objective function. Define an optimization problem where a vehicle starts from a starting point, passes through multiple waypoints in sequence, and finally stops at a destination. Introduce constraints on the waypoints and construct an optimization function with the goal of minimizing the total flight time. The constraints include position, radius, and vehicle arm length. Step 3: Simplify the time-optimal objective function; divide the trajectory into multiple segments, introduce functional expressions, and transform the dynamic equations, states, and control inputs into constraints; use differential flat mapping to transform the states and control trajectories into polynomial forms; introduce variable substitution to eliminate the waypoint and time constraints and transform the problem into an unconstrained optimization problem; Step 4: Solve the unconstrained optimization problem; use the L-BFGS algorithm to solve the unconstrained optimization problem; calculate the optimal time allocation and waypoint positions; and calculate the trajectory of the deformed UAV based on the optimization results.

2. The time-optimal trajectory planning method for a deformable quadrotor aircraft according to claim 1, characterized in that: The dynamic model of the deformable quadrotor is established by defining the mass, acceleration, position, rotation matrix, angular velocity, and moment of inertia of the aircraft.

3. The time-optimal trajectory planning method for a morphing quadrotor aircraft according to claim 1, characterized in that: The dynamic model of the deformable quadrotor aircraft is as follows: ； ； ； ； in, represents the mass of the quadrotor, represents the acceleration due to gravity, Indicates the position of the quadrotor, , Represents the rotation unit quaternion from the world coordinate system to the body coordinate system, Indicated by parameterized rotation matrices; , and Represents the angular velocity, velocity and acceleration in the body coordinate system, angular velocity, represents the moment of inertia of the quadrotor, It represents the torque generated by the propeller on the fuselage shaft. The relationship between thrust and torque is: ； in, represents the rotor torque constant, Indicates the total thrust generated by the motor.

4. The time-optimal trajectory planning method for a morphing quadrotor aircraft according to claim 1, characterized in that: In step 2, it is assumed that there are There are waypoints, a starting point, and an end point. The time optimization problem can be described as the drone starting from the starting point and passing through these waypoints in sequence. The flight must reach the destination point and meet the state and input constraints throughout the flight. definition For trajectory planning A circle, is the radius of the circle and the center point is , that is, The location of the waypoints, For the deformation of the quadrotor Position at the time of the waypoint, is the arm length of the deformable quadrotor. To ensure that the deformable quadrotor aircraft passes this waypoint safely, the following conditions must be met: ； definition Indicates that the deformed quadrotor has successfully passed this circle; then the optimization function can be expressed as follows based on the total flight time The optimization objective is of the form: , Among them, the definition of the quadrotor state , control input , encapsulate all dynamic equations as , encapsulate all state input constraints as ; Indicates that the quadcopter is passing through Waypoint status The position component of Indicates that after A circle of time, Indicates the initial state, Indicates the end state.

5. The time-optimal trajectory planning method for a morphing quadrotor aircraft according to claim 1, characterized in that: In step 3, use The waypoints divide the entire trajectory into Segment, definition is the collection of time spent in each section, the total time ,forward Cumulative time of the segment ,definition is a set of waypoint coordinates; By introducing functional expressions, the above time optimal function can be expressed as: in is the time-integrated penalty functional, when satisfy When the condition is zero if Violation Any constraints in Rapid growth, with It can be guaranteed that the solution meets all conditions; Defining spatial trajectories ,in represents the trajectory of the quadrotor drone's center of gravity, Indicates the yaw angle of the quadrotor drone; Using differentially flat mapping and The state and control trajectory and Convert to : )； ； Similarly, related constraints Expressed as: ； at this time, yes The only constraint left in , then, The minimum control trajectory functional is expressed as: ； in, represents the sampling frequency, ; Introducing variables , arbitrary All satisfy the following smooth mapping, so that satisfy , so the constraints Eliminate: ； because Implicit upper and lower bound conditions, which need to be added during optimization calculations constraints, so we introduce the variable s and let: ； in, and Respectively The minimum and maximum values ​​of , the final mapping is: )； Let the optimization function , the gradients of the two variables are as follows: ； ； To eliminate add constraints, then: By using the new variable and , ensuring that the quadcopter always stays in the corresponding circle, using Ensure the non-negativity of time; Now, the waypoint position and time are respectively changed from and Obtain, transform the time optimal function into an unconstrained form: 。 6. The time-optimal trajectory planning method for a morphing quadrotor aircraft according to claim 5, characterized in that: The functional expression is: The functional expression contains the position vector that satisfies the initial and final states, state equations, and path constraints. .

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