An affine formation maneuver control method and system for an underactuated quadrotor unmanned aerial vehicle
By constructing an underactuated quadrotor UAV model and a fixed-time, predefined-time controller, combined with a leader-follower strategy, the problems of inaccurate quadrotor UAV models and the inability of formation controllers to meet complex requirements were solved, and efficient formation maneuver control was achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-06-30
- Publication Date
- 2026-04-07
AI Technical Summary
Existing quadcopter drone models are not accurate enough, formation controllers cannot meet complex requirements, and the convergence time of formation control systems is not ideal.
A mathematical model of an underactuated quadrotor UAV is constructed, the desired formation is designed and the stress matrix is obtained. By combining fixed time and predefined time controllers and adopting a leader-follower strategy, translational and rotational tracking controllers are constructed to achieve control of the desired formation.
It improves the accuracy of formation control and the controllability of convergence time, expands the applicability of the control scheme, and realizes formation maneuverability in complex environments.
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Figure CN116755471B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of quadrotor unmanned aerial vehicle (UAV) technology, and relates to a method and system for controlling the affine formation maneuver of an underactuated quadrotor UAV. Background Technology
[0002] Quadrotor drones adjust their attitude and speed by controlling motor speed, thus smoothly flying to their target location. They offer advantages such as simple and flexible structure, vertical takeoff and landing capability, and ease of operation. A quadcopter is a typical six-degree-of-freedom underactuated system, and its model can be divided into a three-input, three-output fully driven rotational part and a single-input, three-output underactuated translational part. Current research on quadcopter formation control still faces the following three main problems:
[0003] The models of quadcopter drones are not accurate enough. A quadcopter drone is a typical highly coupled underactuated system with four inputs and six outputs. However, in many studies, quadcopter drones are often simplified to low-order linear or nonlinear models, thus ignoring the drone's rotational motion. Furthermore, some studies assume that quadcopter drone models are decoupled or fully actuated, such as first- or second-order integrator models, or Euler-Lagrange models, all of which ignore the underactuated and highly coupled characteristics of the drone.
[0004] Formation controllers are insufficient to meet the increasingly complex requirements of formation control. In the formation control of quadrotor UAVs, the formation maneuverability of quadrotor UAVs largely depends on the constraints imposed on the desired formation. Based on the different constraints imposed on the desired formation, common formation controllers can be divided into three categories: relative position-based formation controllers, which use connected graphs as their theoretical tool; absolute position-based formation controllers, which use rigid graphs as their theoretical tool; and relative azimuth-based formation controllers, which use azimuth Laplace matrices as their theoretical tool. However, in practical formation control, all three types of formation controllers have different drawbacks. For example, relative position-based formation controllers can only be applied to tracking formations with time-varying translations. In contrast, absolute position-based controllers are suitable for tasks requiring time-varying translations and turns, but not for tasks requiring time-varying scaling, while relative azimuth-based controllers are suitable for tasks with time-varying translations and scaling, but not for tasks with time-varying turns. In practical engineering, we usually need drone formations to be able to perform translation, turning, scaling and other maneuvers simultaneously in order to better cope with complex and unknown mission environments and requirements.
[0005] The convergence time of formation control systems is not ideal. Convergence time, a metric describing the convergence rate of a closed-loop system, is widely considered a performance specification in control system design. In practical formation control applications, fast convergence is typically pursued, meaning a sufficiently short convergence time to achieve better control performance and system robustness. In practical formation control systems, achieving finite-time formation control to meet specific system requirements is particularly important. In recent years, finite-time cooperative control has received widespread attention. However, although system convergence can be achieved within a finite time, the calculation of the convergence time depends on the initial conditions of the formation system. When the initial state information of a multi-quadrotor UAV formation system is unknown or unknowable beforehand, this greatly limits the application scope of existing finite-time cooperative control. Summary of the Invention
[0006] The purpose of this invention is to solve the problems in the prior art that the model of the quadrotor UAV is not accurate enough, the formation controller of the quadrotor UAV cannot meet the increasingly complex formation control requirements, and the convergence time of the formation control system is not ideal. The invention provides a method and system for affine formation maneuver control of underactuated quadrotor UAVs.
[0007] To achieve the above objectives, the present invention employs the following technical solution:
[0008] A method for controlling the affine formation maneuvers of an underactuated quadrotor UAV, comprising:
[0009] Construct a mathematical model for an underactuated quadrotor unmanned aerial vehicle;
[0010] Based on the mathematical model of an underactuated quadrotor UAV, the desired formation is designed and the stress matrix is obtained.
[0011] Based on the designed formation and stress matrix, the desired navigation trajectory and time-varying formation are determined.
[0012] Based on the time estimator, obtain the current actual position and actual velocity of the affine formation;
[0013] Construct a fixed-time translational tracking controller to make the actual position approach the desired position, thereby completing the control of the desired formation;
[0014] The design of a translational tracking controller is constructed, intermediate auxiliary inputs are obtained, attitude calculation is performed based on the intermediate auxiliary inputs, and the desired attitude and desired angular velocity are obtained.
[0015] Based on the obtained desired attitude and desired angular velocity, a time-rotation tracking controller is constructed to control the rotational motion of the underactuated quadcopter.
[0016] A further improvement of the present invention is that:
[0017] Furthermore, the mathematical model for the underactuated quadrotor UAV includes a translational model and a rotational model; the translational model is as follows:
[0018]
[0019] in The intermediate auxiliary input is used for attitude calculation to obtain the desired attitude and angular velocity; p and v are the position and velocity of the underactuated quadcopter, respectively; g is the acceleration due to gravity, m is the mass of the underactuated quadcopter, and T is the thrust of the underactuated quadcopter; Q = [q T ,η] T R(Q) is a quaternion used to describe the attitude of an underactuated quadrotor UAV. R(Q) is the rotation matrix, expressed as R(Q) = (ηQ / Q). 2 -q T q)I3+2q T q-2ηS(q), where I3 is a 3×3 identity matrix and S(·) is a skew-symmetric matrix.
[0020] The rotation model is as follows:
[0021]
[0022] Where τ is the control torque, ω is the angular velocity, J is the moment of inertia, and G(Q) = [ηI³ + S(q), -q] T ] T Among them, torque τ, thrust T, and thrust [f1,f2,f3,f4] generated by the four motors. T The relationship between them is represented as follows:
[0023]
[0024] Where C is the reaction torque coefficient, and d is the distance between the center of mass of the motor and the underactuated quadcopter.
[0025] Furthermore, the desired formation includes a first formation and a second formation; the spatial coordinates of the first formation are:
[0026]
[0027] The spatial coordinates of the second formation are:
[0028]
[0029] The stress matrix is obtained as follows: based on the desired formation, an undirected graph G is used to describe the relationships between the various underactuated quadcopter UAVs, and the correlation matrix H,h of the undirected graph G is obtained. i(i = 1, 2, 3, ..., n) represent the column vectors of the incidence matrix H;
[0030] To obtain the matrix, add a unit column vector after the last column of the spatial coordinate matrix r of the desired formation. Then, singular value decomposition is performed on it to obtain matrix U, where the matrix formed by the first 4 columns is the first matrix U1, and the matrix formed by the remaining columns is the second matrix U2;
[0031] Based on matrix The transpose of the incidence matrix H of an undirected graph G and the diagonal matrix diag(h) of the column vectors of the incidence matrix H. i Construct matrix E, and solve for the null space z of matrix E using Matlab. i ;
[0032] Based on the second matrix U2, the incidence matrix H of the undirected graph G, and the null space z i diagonal matrix diag(z) i ), obtain matrix M i ,in Solving LMI using the Matlab toolbox Thus, we obtain c i ;
[0033] Based on C i and zero space z i Obtain angular velocity The stress matrix Ω is obtained based on the correlation matrix H of the undirected graph G and the diagonal matrix diag(w) of the angular velocity ω.
[0034] Furthermore, based on the designed formation and stress matrix, the desired navigation trajectory and time-varying formation are determined, specifically: the desired navigation trajectory and time-varying formation are respectively determined by p0(t) and δ. ij (t) is used to represent the expected flight trajectory; for the first formation, with a simulation time of t = 210s, the expected flight trajectory p0(t) for the first formation is:
[0035]
[0036] The first formation's time-varying formation δ ij (t) is:
[0037]
[0038] For the second formation, the simulation time is t = 240s, and the expected flight trajectory for the second formation is:
[0039]
[0040] The time-varying formation of the second formation is:
[0041]
[0042] Furthermore, based on the time estimator, the current actual position and actual velocity of the affine formation are obtained, specifically:
[0043] For the first formation, the current actual position and actual speed of the underactuated quadrotor affine formation are obtained by using a fixed-time estimator;
[0044] For the second formation, the current actual position and actual speed of the underactuated quadrotor affine formation are obtained through a predefined time estimator.
[0045] Furthermore, for the first formation, the current actual position and actual velocity of the underactuated quadcopter affine formation are obtained through a fixed-time estimator, specifically:
[0046] The fixed-time estimator for the navigator in the first formation is:
[0047]
[0048] in Where N is the number of navigators, ε and σ d It is a constant greater than zero; λ max (·) and λ min (·) represent the maximum and minimum eigenvalues of the matrix, respectively; Ω ll This is the stress matrix between the navigator and the navigator obtained by partitioning the stress matrix Ω; sig α (·)=|·| α sgn(·), where sgn(·) is the sign function;
[0049] The estimation error of the expected speed is in After convergence, the estimation error of the desired position is within Later convergence; when Then, the desired position and desired speed of the navigator are estimated;
[0050] The fixed-time estimator for the followers in the first formation is:
[0051]
[0052] in, M represents the number of navigators; in Then, the expected position and expected speed of the followers are estimated;
[0053] For the second formation, the current actual position and actual velocity of the underactuated quadcopter affine formation are obtained through a predefined time estimator, specifically:
[0054] The fixed-time estimator for the navigator in the second formation is:
[0055]
[0056] For the navigator in the second formation:
[0057]
[0058] For the followers in the second formation:
[0059]
[0060] Where α, β, ρ, ζ, k, c1 and c2 > 0, 0 <kρ<1,kζ> 1; For the navigator in the second formation, we have:
[0061]
[0062] For the followers in the second formation, we have:
[0063]
[0064] in, and For a predefined time parameter, Γ(·) is the Gamma function.
[0065] For the navigator in the second formation, the expected speed estimation error At a predefined time Convergence; estimation error of the desired position At a predefined time Convergence; that is, when At that time, the estimation errors of the navigator in the second formation regarding the desired velocity and desired position converge within a predefined time; for the followers in the second formation, the estimation error of the desired velocity... Can be done at a predefined time Convergence; similarly, the estimation error of the expected position. Can be done at a predefined time Convergence.
[0066] Furthermore, a fixed-time translational tracking controller is constructed to make the actual position approach the desired position, thereby achieving control over the desired formation. Specifically:
[0067] Select the first sliding surface as:
[0068]
[0069] in and Let be the tracking errors of the actual position versus the desired position and the actual velocity versus the desired velocity of the underactuated quadcopter UAV, respectively. Let k1 be the positive control gain. The sliding mode face is differentiated over time, and the sliding mode reaching law is chosen as follows: Where μ1 and ν1 are positive constants, 0 <a1<1,b1> 1. We can obtain:
[0070]
[0071] Thus, intermediate auxiliary inputs are obtained.
[0072]
[0073] To verify the stability of the fixed-time translational tracking controller, a Lyapunov function was selected. Differentiating it with respect to time yields:
[0074]
[0075]
[0076] when When the tracking error between the actual position and velocity and the desired position and velocity is converged, the tracking error can be reduced.
[0077] Furthermore, the design of a translational tracking controller is constructed, intermediate auxiliary inputs are obtained, and attitude calculation is performed based on the intermediate auxiliary inputs to obtain the desired attitude and desired angular velocity, specifically as follows:
[0078] Based on the design of the fixed-time translational tracking controller, an intermediate auxiliary input U is obtained. Attitude calculation is then performed on the intermediate auxiliary input U to obtain the desired attitude and desired angular velocity. The attitude calculation method is as follows:
[0079]
[0080] The desired attitude and desired angular velocity are obtained by performing attitude calculation on intermediate auxiliary inputs.
[0081] Furthermore, based on the acquired desired attitude and desired angular velocity, a time-rotation tracking controller is constructed to control the rotational motion of the underactuated quadcopter UAV, specifically as follows:
[0082] Select the second sliding surface as:
[0083]
[0084] in and Let be the tracking error between the actual position and the desired position of the underactuated quadcopter UAV, and k2 be the positive control gain. The sliding mode plane is differentiated over time, and the sliding mode reaching law is chosen as follows: Where μ2 and ν2 are positive constants, 0 <a2<1,b2> 1. We can obtain:
[0085]
[0086] The control torque can be obtained as follows:
[0087]
[0088] The method for proving the stability of the fixed-time rotational tracking controller is similar to that of the fixed-time translational tracking controller; the attitude tracking error and angular velocity tracking error converge within a fixed time.
[0089] A control system for affine formation maneuvers of an underactuated quadrotor unmanned aerial vehicle includes:
[0090] A construction module that constructs a mathematical model of an underactuated quadcopter unmanned aerial vehicle;
[0091] The first acquisition module, based on the mathematical model of the underactuated quadcopter UAV, designs the desired formation and acquires the stress matrix;
[0092] The determination module determines the desired navigation trajectory and time-varying formation based on the designed formation and stress matrix;
[0093] The second acquisition module, based on a time estimator, acquires the current actual position and actual speed of the affine formation.
[0094] The first control module constructs a fixed-time translational tracking controller to make the actual position approach the desired position, thereby completing the control of the desired formation.
[0095] The solution module constructs the design of the translational tracking controller, obtains intermediate auxiliary input, performs attitude calculation based on the intermediate auxiliary input, and obtains the desired attitude and desired angular velocity.
[0096] The second control module constructs a time-rotation tracking controller based on the acquired desired attitude and desired angular velocity to control the rotational motion of the underactuated quadcopter.
[0097] Compared with the prior art, the present invention has the following beneficial effects:
[0098] This invention employs an underactuated quadrotor UAV model and combines it with radial formation maneuver control based on fixed-time and predefined-time control. This approach considers both the complex characteristics of quadrotor UAVs and the practical requirements for formation maneuverability in engineering, while also constraining the convergence time of the control system.
[0099] Furthermore, this invention employs a leader-follower strategy, whereby the navigator accurately tracks the desired formation, allowing the entire formation to track the characteristics of the target formation, thus achieving the goal of controlling the entire formation system with a small number of quadcopter drones.
[0100] Furthermore, in the fixed-time control scheme of this invention, the convergence time of the control system is only related to the design parameters and the algebraic connectivity of the Laplace matrix, and the upper bound of the convergence time can be calculated, which greatly improves the convergence and performance of the control system and expands the applicability of the control scheme. Simultaneously, in the design of predefined-time control, the convergence time can be directly obtained from the design parameters, and large initial control inputs can also be addressed by adjusting the design parameters. In this invention, compared to the more conservative finite-time control in terms of performance and applicability, the fixed-time and predefined-time control schemes are adopted and have achieved good results. Attached Figure Description
[0101] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.
[0102] Figure 1 This is a schematic diagram of the affine formation maneuver control method for underactuated quadrotor UAVs according to the present invention;
[0103] Figure 2 This is a schematic diagram of the first queue structure of the present invention;
[0104] Figure 3 This is a schematic diagram of the second queue structure of the present invention;
[0105] Figure 4 This is a schematic diagram of the affine formation maneuver control system for an underactuated quadcopter UAV according to the present invention.
[0106] Figure 5 A schematic diagram of the formation simulation trajectory for the first formation;
[0107] Figure 6 This is a schematic diagram of the formation simulation results for the first formation;
[0108] Figure 7 This is a schematic diagram of the formation simulation trajectory for the second formation.
[0109] Figure 8 This is a schematic diagram of the formation simulation results for the second formation. Detailed Implementation
[0110] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.
[0111] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.
[0112] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.
[0113] In the description of the embodiments of the present invention, it should be noted that if terms such as "upper," "lower," "horizontal," or "inner" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings, or the orientation or positional relationship commonly used when the product of the invention is in use, they are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of the present invention. Furthermore, terms such as "first" and "second" are only used to distinguish descriptions and should not be construed as indicating or implying relative importance.
[0114] Furthermore, the use of the term "horizontal" does not imply that the component must be absolutely horizontal, but rather that it can be slightly tilted. For example, "horizontal" simply means that its direction is more horizontal than "vertical," and does not mean that the structure must be completely horizontal, but can be slightly tilted.
[0115] In the description of the embodiments of the present invention, it should also be noted that, unless otherwise explicitly specified and limited, the terms "set," "install," "connect," and "link" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in the present invention according to the specific circumstances.
[0116] The present invention will now be described in further detail with reference to the accompanying drawings:
[0117] See Figure 1 This invention discloses a method for controlling the affine formation maneuver of an underactuated quadrotor unmanned aerial vehicle, comprising:
[0118] Step 1: Construct a mathematical model of an underactuated quadcopter drone.
[0119] A quadcopter drone is a typical underactuated, highly coupled control system. We will build an underactuated quadcopter drone model from both translational and rotational perspectives, starting with the translational model:
[0120]
[0121] in, This is an intermediate auxiliary input used for attitude calculation to obtain the desired attitude and angular velocity. p and v represent the position and velocity of the underactuated quadcopter, respectively. g is the acceleration due to gravity, m is the mass of the underactuated quadcopter, and T is the thrust of the underactuated quadcopter. Q = [q T ,η] T Let q be a quaternion used to describe the attitude of an underactuated quadrotor UAV, and η be the vector part of the quaternion. R(Q) is the rotation matrix, which can be represented as R(Q) = (η) 2 -q T q)I3+2q T q-2ηS(q), where I3 is a 3×3 identity matrix and S(·) is a skew-symmetric matrix.
[0122] The rotational model of the underactuated quadcopter UAV is as follows:
[0123]
[0124] Where τ is the control torque, ω is the angular velocity, J is the moment of inertia, and G(Q) = [ηI³ + S(q), -q] T ] T Among them, the torque τ, the thrust T, and the thrust [f1, f2, f3, f4] generated by the four motors. T The relationship between them can be represented as:
[0125]
[0126] Where C is the reaction torque coefficient, and d is the distance between the center of mass of the motor and the underactuated quadcopter.
[0127] Step 2: Based on the mathematical model of the underactuated quadcopter UAV, the desired formation is designed, and the stress matrix is calculated. To avoid potential randomness, this invention employs two different formations to better verify the effectiveness and universality of the invention; these formations are the first formation and the second formation. The first formation is as follows: Figure 2 As shown.
[0128] In this formation, hexagons numbered 1, 2, 3, and 4 represent the navigator of the first formation, while squares numbered 5, 6, and 7 represent the followers. The lines depict the communication connections between the various underactuated quadcopter drones. The spatial coordinates of the formation can be represented as:
[0129]
[0130] The second formation is as follows Figure 3 As shown in the diagram. Hexagons numbered 1, 2, and 3 represent the navigators of the second formation, while squares numbered 4, 5, 6, and 7 represent the followers. The lines depict the communication connections between the various underactuated quadcopter drones. The spatial coordinates of the formation can be represented as:
[0131]
[0132] The stress matrix is obtained as follows: Based on the desired formation, an undirected graph G is used to describe the relationships between the various underactuated quadcopter UAVs, thus yielding the correlation matrix H,h of the undirected graph G. i (i = 1, 2, 3, ..., n) represent the column vectors of the incidence matrix H;
[0133] To obtain the matrix, add a unit column vector after the last column of the spatial coordinate matrix r of the desired formation. Then, singular value decomposition is performed on it to obtain matrix U, where the matrix formed by the first 4 columns is the first matrix U1, and the matrix formed by the remaining columns is the second matrix U2;
[0134] Based on matrix The transpose of the incidence matrix H of an undirected graph G and the diagonal matrix diag(h) of the column vectors of the incidence matrix H. i Construct matrix E, And solve the null space z of matrix E using Matlab. i ;
[0135] Based on the second matrix U2, the incidence matrix H of the undirected graph G, and the null space z i diagonal matrix diag(z) i ), obtain matrix M i ,in Solving LMI using the Matlab toolbox Thus, we obtain c i ;
[0136] Based on C i and zero space z i Obtain angular velocity Based on the incidence matrix H of the undirected graph G and the diagonal matrix diag(w) of the angular velocity ω, the stress matrix Ω is obtained; Ω = H T diag(ω)H.
[0137] The stress matrix for the first formation is:
[0138]
[0139] The stress matrix for the second formation is:
[0140]
[0141] Step 3: Determine the desired flight path and the desired time-varying formation. Based on the formation designed in Step 2 and the calculated stress matrix, design the desired flight path and time-varying formation, respectively defined by p0(t) and δ. ij (t) is used to represent this.
[0142] For the first formation, with a simulation time of t = 210s, the expected flight trajectory is:
[0143]
[0144] The time-varying formation of the first formation is:
[0145]
[0146] For the second formation, with a simulation time of t = 240s, the expected flight trajectory is:
[0147]
[0148] The time-varying formation of the second formation is:
[0149]
[0150] Step 4: For the first formation, obtain the current actual position and actual velocity of the underactuated quadrotor affine formation using a fixed time estimator; for the second formation, obtain the current actual position and actual velocity of the underactuated quadrotor affine formation using a predefined time estimator.
[0151] For the first formation, this invention designs different fixed-time estimators for the leader and followers. The fixed-time estimator for the leader in the first formation is:
[0152]
[0153] in Where N is the number of navigators, ε and σ d It is a constant greater than zero. λ max (·) and λ min (·) represent the maximum and minimum eigenvalues of the matrix, respectively. Ω ll This is the stress matrix between the navigator and the navigator obtained by partitioning the stress matrix Ω. sig α (·)=|·| α sgn(·), where sgn(·) is the sign function. To verify the stability of the fixed-time estimator described above, the Lyapunov stability criterion needs to be used. For the Lyapunov principal stability theorem or other theorems of the Lyapunov second method, the conditions stated by the theorem are only sufficient conditions to guarantee system stability. That is, for a specific Lyapunov function V, if it fails to satisfy the stability criterion, the stability of the system cannot be directly denied. In this case, we have to choose another Lyapunov function to re-verify. To verify the stability of the system, the above selection of the Lyapunov function may need to be repeated multiple times, so the form of the Lyapunov function is not unique. Here, the Lyapunov function chosen is:
[0154]
[0155] Differentiating with respect to time, we can derive:
[0156]
[0157] According to Lemma 1 in reference
[18] , the estimation error of the desired velocity can be... After convergence, similarly, the estimation error of the desired position can be obtained within... It converges later. Then, when... Then, the navigator can accurately estimate the desired position and desired speed.
[0158] The fixed-time estimator for the followers in the first formation is:
[0159]
[0160] in, M represents the number of navigators. Similarly, it can be seen that under the action of the fixed-time estimator, the follower drones can... Then, the desired position and desired velocity are accurately estimated.
[0161] For the second formation, this invention designs predefined time estimators for both the leader and the follower. The fixed time estimator for the leader in the second formation is:
[0162]
[0163] For the navigator in the second formation:
[0164]
[0165] For the followers in the second formation:
[0166]
[0167] Where α, β, ρ, ζ, k, c1 and c2 > 0, 0 <kρ<1,kζ> 1. For the navigator, there are:
[0168]
[0169] For the followers in the second formation, we have:
[0170]
[0171] in, and For a predefined time parameter, Γ(·) is the Gamma function. To verify the stability of the predefined time estimator mentioned above, taking the Navigator predefined time estimator as an example, the Lyapunov function is chosen as follows:
[0172]
[0173] By differentiating the Lyapunov function with respect to time and deriving the result, we can obtain:
[0174]
[0175] For the navigator, the estimation error of the expected speed Can be done at a predefined time Convergence. Similarly, the estimation error of the desired position can be obtained. Can be done at a predefined time Convergence. That is, when At this time, the navigator's estimation errors of the desired velocity and desired position converge within a predefined time. For the follower's predefined time estimator, the Lyapunov function is chosen as:
[0176]
[0177] By differentiating the Lyapunov function with respect to time and deriving the result, we can obtain:
[0178]
[0179] For followers, the estimation error of expected speed Can be done at a predefined time Convergence. Similarly, the estimation error of the desired position can be obtained. Can be done at a predefined time Convergence. That is, when At that time, the navigator's estimation errors of the desired speed and desired position can converge within a predefined time.
[0180] Step 5: Design and employ a fixed-time translational tracking controller to bring the actual position close to the desired position, thereby achieving control of the desired formation. The first sliding surface is selected as:
[0181]
[0182] in and Let be the tracking errors of the actual position versus the desired position and the actual velocity versus the desired velocity of the underactuated quadcopter UAV, respectively. Let k1 be the positive control gain. The sliding mode face is differentiated over time, and the sliding mode reaching law is chosen as follows: Where μ1 and ν1 are positive constants, 0 <a1<1,b1> 1. We can obtain:
[0183]
[0184] Thus, intermediate auxiliary inputs are obtained.
[0185]
[0186] To verify the stability of the fixed-time translational tracking controller, a Lyapunov function was chosen. Differentiating it with respect to time yields:
[0187]
[0188] when When the tracking error between the actual position and velocity and the desired position and velocity is converged, the tracking error can be reduced.
[0189] Step Six: By designing the translational tracking controller, an intermediate auxiliary input can be obtained. This intermediate auxiliary input is then used for attitude calculation to obtain the desired attitude and angular velocity. Based on the design of the fixed-time translational tracking controller in Step Five, an intermediate auxiliary input U can be obtained. Subsequently, attitude calculation can be performed on this intermediate auxiliary input to further obtain the desired attitude and angular velocity. The attitude calculation method is as follows:
[0190]
[0191] It is evident that the desired attitude and desired angular velocity can be obtained by attitude calculation of the intermediate auxiliary input.
[0192] Step 7: Based on the desired attitude and desired angular velocity obtained from the attitude calculation, design and employ a fixed-time rotation tracking controller to control the rotational motion of the underactuated quadrotor UAV. The second sliding surface is selected as:
[0193]
[0194] in and Let be the tracking error between the actual position and the desired position of the underactuated quadcopter UAV, and k2 be the positive control gain. The sliding mode plane is differentiated over time, and the sliding mode reaching law is chosen as follows: Where μ2 and ν2 are positive constants, 0 <a2<1,b2> 1. We can obtain:
[0195]
[0196] The control torque can be obtained as follows:
[0197]
[0198] The proof of the stability of the fixed-time rotational tracking controller is similar to that of the fixed-time translational tracking controller. It can be seen that the attitude tracking error and angular velocity tracking error converge within a fixed time.
[0199] See Figure 4 This invention discloses a control system for affine formation maneuvers of an underactuated quadcopter unmanned aerial vehicle, characterized by comprising:
[0200] A construction module that constructs a mathematical model of an underactuated quadcopter unmanned aerial vehicle;
[0201] The first acquisition module, based on the mathematical model of the underactuated quadcopter UAV, designs the desired formation and acquires the stress matrix;
[0202] The determination module determines the desired navigation trajectory and time-varying formation based on the designed formation and stress matrix;
[0203] The second acquisition module, based on a time estimator, acquires the current actual position and actual speed of the affine formation.
[0204] The first control module constructs a fixed-time translational tracking controller to make the actual position approach the desired position, thereby completing the control of the desired formation.
[0205] The solution module constructs the design of the translational tracking controller, obtains intermediate auxiliary input, performs attitude calculation based on the intermediate auxiliary input, and obtains the desired attitude and desired angular velocity.
[0206] The second control module constructs a time-rotation tracking controller based on the acquired desired attitude and desired angular velocity to control the rotational motion of the underactuated quadcopter.
[0207] See Figure 5 and Figure 6 , Figure 6 The left side shows the position tracking error. Figure 6 The right side shows the attitude tracking error. It can be seen that the control scheme in this invention can achieve affine formation maneuvers for underactuated quadcopter UAVs.
[0208] See Figure 7 and Figure 8 , Figure 8 The left side shows the position tracking error. Figure 8 The right side shows the attitude tracking error. It can be seen that the control scheme in this invention can realize affine formation maneuvers for underactuated quadcopter UAVs.
[0209] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A method for controlling the affine formation maneuvers of an underactuated quadrotor unmanned aerial vehicle, characterized in that, include: Construct a mathematical model for an underactuated quadrotor unmanned aerial vehicle; Based on the mathematical model of an underactuated quadrotor UAV, the desired formation is designed and the stress matrix is obtained. Based on the designed formation and stress matrix, the desired navigation trajectory and time-varying formation are determined; specifically, the desired navigation trajectory and time-varying formation are respectively determined by... and The simulation time is as follows: For the first formation, the simulation time is... The expected flight path of the first formation for: The first formation's time-varying formation for: For the second formation, the simulation time is... The expected flight path for the second formation is: The time-varying formation of the second formation is: Based on a time estimator, the current actual position and velocity of the affine formation are obtained; specifically: For the first formation, the current actual position and velocity of the underactuated quadrotor affine formation are obtained using a fixed-time estimator; specifically: The fixed-time estimator for the navigator in the first formation is: in , , ,in For the number of navigators, and It is a constant greater than zero; and These represent the maximum and minimum values of the matrix's eigenvalues, respectively. For stress matrix The stress matrix between the navigator and the navigator is obtained by partitioning the system. ,in It is a symbolic function; The estimation error of the expected speed is in After convergence, the estimation error of the desired position is within Later convergence; when Then, the desired position and desired speed of the navigator are estimated; The fixed-time estimator for the followers in the first formation is: in, , , , For the number of navigators; in Then, the expected position and expected speed of the followers are estimated; For the second formation, the current actual position and velocity of the underactuated quadrotor affine formation are obtained using a predefined time estimator; specifically: The fixed-time estimator for the navigator in the second formation is: For the navigator in the second formation: For the followers in the second formation: in, , , , , , and , , For the navigator in the second formation, we have: For the followers in the second formation, we have: in, and For predefined time parameters, It is the Gamma function. ; For the navigator in the second formation, the expected speed estimation error At a predefined time Convergence; estimation error of the desired location At a predefined time Convergence; that is, when At that time, the estimation errors of the navigator in the second formation regarding the desired velocity and desired position converge within a predefined time; for the followers in the second formation, the estimation error of the desired velocity... Can be done at a predefined time Convergence; similarly, the estimation error of the expected position. Can be done at a predefined time Convergence; Construct a fixed-time translational tracking controller to make the actual position approach the desired position, thereby completing the control of the desired formation; The design of a translational tracking controller is constructed, intermediate auxiliary inputs are obtained, attitude calculation is performed based on the intermediate auxiliary inputs, and the desired attitude and desired angular velocity are obtained. Based on the obtained desired attitude and desired angular velocity, a time-rotation tracking controller is constructed to control the rotational motion of the underactuated quadcopter.
2. The affine formation maneuver control method for underactuated quadrotor UAVs according to claim 1, characterized in that, The mathematical model for constructing the underactuated quadrotor UAV includes a translational model and a rotational model; the translational model is as follows: in It serves as an intermediate auxiliary input for attitude calculation, thereby obtaining the desired attitude and angular velocity; and The positions and velocities of the underactuated quadcopter drone are shown below. It is gravitational acceleration. It is the mass of the underactuated quadcopter drone. It is the thrust of an underactuated quadcopter drone; It is a quaternion used to describe the attitude of an underactuated quadcopter drone. It is a rotation matrix, represented as ,in for The identity matrix, It is an oblique symmetric matrix. ; The rotational model is as follows: in, To control the torque, Angular velocity, For rotational inertia, Among them, torque ,thrust and the thrust generated by the four motors The relationship between them is represented as follows: in This is the reaction moment coefficient. It is the distance between the motor and the center of mass of the underactuated quadcopter drone.
3. The affine formation maneuver control method for underactuated quadrotor UAVs according to claim 2, characterized in that, The desired formation includes a first formation and a second formation; the spatial coordinates of the first formation are: The spatial coordinates of the second formation are: The stress matrix is obtained by: based on the desired formation, through an undirected graph. To describe the relationships between various underactuated quadcopter drones and obtain an undirected graph. Correlation matrix , Represents the correlation matrix Column vectors; The spatial coordinate matrix of the desired formation Add a unit column vector after the last column to obtain the matrix. And perform singular value decomposition on it to obtain the matrix , among which the former The matrix formed by the columns is the first matrix. The matrix formed by the remaining columns is the second matrix. ; Based on matrix Undirected graph Correlation matrix transpose matrix and incidence matrix diagonal matrix of column vectors Constructing a matrix And solve the matrix based on Matlab. zero space ; Based on the second matrix Undirected graph Correlation matrix Zero space diagonal matrix Obtain the matrix ,in Solving LMI using the Matlab toolbox Thus obtain ; based on and zero space Obtain angular velocity Based on undirected graphs Correlation matrix and angular velocity diagonal matrix Obtain the stress matrix .
4. The affine formation maneuver control method for underactuated quadrotor UAVs according to claim 1, characterized in that, The construction of a fixed-time translational tracking controller, which makes the actual position approach the desired position and completes the control of the desired formation, specifically involves: Select the first sliding surface as: in and These represent the tracking errors of the underactuated quadcopter UAV, specifically the actual position versus the desired position and the actual velocity versus the desired velocity. To achieve positive control gain, the sliding mode face is differentiated over time, and the sliding mode reaching law is chosen as follows: ,in and For positive integers, , We can obtain: Thus, intermediate auxiliary inputs are obtained. To verify the stability of the fixed-time translational tracking controller, a Lyapunov function was selected. Differentiating it with respect to time yields: when When the tracking error between the actual position and velocity and the desired position and velocity is converged, the tracking error can be reduced.
5. The affine formation maneuver control method for underactuated quadrotor UAVs according to claim 4, characterized in that, The design of the translational tracking controller involves obtaining intermediate auxiliary inputs, performing attitude calculations based on these inputs, and obtaining the desired attitude and angular velocity. Specifically: Based on the design of the fixed-time translational tracking controller, the intermediate auxiliary input is obtained. For intermediate auxiliary input The desired attitude and desired angular velocity are obtained by attitude calculation. The attitude calculation method is as follows: The desired attitude and desired angular velocity are obtained by performing attitude calculation on intermediate auxiliary inputs.
6. The affine formation maneuver control method for underactuated quadrotor UAVs according to claim 5, characterized in that, Based on the acquired desired attitude and desired angular velocity, a time-rotation tracking controller is constructed to control the rotational motion of the underactuated quadcopter UAV, specifically as follows: Select the second sliding surface as: in and These represent the tracking error between the actual position and the desired position of the underactuated quadcopter UAV. To obtain a positive control gain, differentiate the sliding mode face in time and choose the sliding mode reaching law as follows: ,in and For positive integers, , We can obtain: The control torque can be obtained as follows: The method for proving the stability of a fixed-time rotational tracking controller is similar to that for a fixed-time translational tracking controller; The attitude tracking error and angular velocity tracking error converge within a fixed time.
7. A control system for affine formation maneuvers of an underactuated quadrotor unmanned aerial vehicle, characterized in that, include: A construction module that constructs a mathematical model of an underactuated quadcopter unmanned aerial vehicle; The first acquisition module, based on the mathematical model of the underactuated quadcopter UAV, designs the desired formation and acquires the stress matrix; The determination module determines the desired navigation trajectory and time-varying formation based on the designed formation and stress matrix; Specifically, the desired flight trajectory and time-varying formation are respectively determined by: and The simulation time is as follows: For the first formation, the simulation time is... The expected flight path of the first formation for: The first formation's time-varying formation for: For the second formation, the simulation time is... The expected flight path for the second formation is: The time-varying formation of the second formation is: The second acquisition module, based on a time estimator, acquires the current actual position and actual velocity of the affine formation; specifically: For the first formation, the current actual position and velocity of the underactuated quadrotor affine formation are obtained using a fixed-time estimator; specifically: The fixed-time estimator for the navigator in the first formation is: in , , ,in For the number of navigators, and It is a constant greater than zero; and These represent the maximum and minimum values of the matrix's eigenvalues, respectively. For stress matrix The stress matrix between the navigator and the navigator is obtained by partitioning the system. ,in It is a symbolic function; The estimation error of the expected speed is in After convergence, the estimation error of the desired position is within Later convergence; when Then, the desired position and desired speed of the navigator are estimated; The fixed-time estimator for the followers in the first formation is: in, , , , For the number of navigators; in Then, the expected position and expected speed of the followers are estimated; For the second formation, the current actual position and velocity of the underactuated quadrotor affine formation are obtained using a predefined time estimator; specifically: The fixed-time estimator for the navigator in the second formation is: For the navigator in the second formation: For the followers in the second formation: in, , , , , , and , , For the navigator in the second formation, we have: For the followers in the second formation, we have: in, and For predefined time parameters, It is the Gamma function. ; For the navigator in the second formation, the expected speed estimation error At a predefined time Convergence; estimation error of the desired location At a predefined time Convergence; that is, when At that time, the estimation errors of the navigator in the second formation regarding the desired velocity and desired position converge within a predefined time; for the followers in the second formation, the estimation error of the desired velocity... Can be done at a predefined time Convergence; similarly, the estimation error of the expected position. Can be done at a predefined time Convergence; The first control module constructs a fixed-time translational tracking controller to make the actual position approach the desired position, thereby completing the control of the desired formation. The solution module constructs the design of the translational tracking controller, obtains intermediate auxiliary input, performs attitude calculation based on the intermediate auxiliary input, and obtains the desired attitude and desired angular velocity. The second control module constructs a time-rotation tracking controller based on the acquired desired attitude and desired angular velocity to control the rotational motion of the underactuated quadcopter.