Air-ground cooperative multi-task constraint following control method based on Udwadiia-Kalaaba equation

Abstract

The invention relates to the field of robot control, in particular to an air-ground cooperative multi-task constraint following control method based on a Udwadiia-Kalaaba equation, and the method comprises the following steps: building a kinetic model of an air-ground cooperative system composed of unmanned control equipment, and building system constraints of the air-ground cooperative system according to a U-K method; based on the analysis of the control problem, establishing a multi-domain behavior control constraint between the unmanned control devices; the constraint following error is expressed as a control following object, and a control problem is converted into a solvable constraint following problem; multi-domain behavior control tasks among unmanned control devices are integrated into a problem framework, constraint force in the problem framework is obtained, and an adaptive robust controller is designed. By adopting the multi-task processing framework, a plurality of tasks can be integrated into a unified constraint, the complexity of the system is effectively reduced, and the conciseness and reliability are improved.

Description

Technical Field

[0001] The present invention relates to the field of robot control, and in particular, to an air-ground collaborative multi-task constrained following control method based on the Udwadia-Kalaba equation. Background Art

[0002] In order to apply unmanned devices to more complex scenarios, more and more researchers are committed to the research of an air-ground collaborative unmanned system composed of multiple drones and unmanned vehicles. This system has the ability to combine the reconnaissance capabilities of drones with the heavy lifting capabilities of unmanned vehicles, and plays an important role in intelligence collection, autonomous target strikes, collaborative search and rescue, etc.

[0003] Regarding the control of the air-ground collaborative unmanned system, a multi-task processing framework for the air-ground collaborative unmanned system is proposed, and the fundamental purpose is to achieve the effective coordination and management of multiple targets. In the collaborative operation of drones and unmanned ground vehicles, especially in complex environments, it is usually necessary to handle multiple targets simultaneously, including trajectory tracking, collision avoidance, and uncertainty suppression. Traditional control designs often focus on a single target, resulting in low efficiency of the system in dealing with diversity and dynamic changes. Summary of the Invention

[0004] In view of the deficiencies of the existing technology, the present invention provides the following technical solutions:

[0005] An air-ground collaborative multi-task constrained following control method based on the Udwadia-Kalaba equation, comprising the following steps:

[0006] S10: Establish a dynamic model of an air-ground collaborative system composed of unmanned control devices, and establish system constraints of the air-ground collaborative system according to the Udwadia-Kalaba method.

[0007] S20: Based on the analysis of the control problem, establish multi-domain behavior control constraints between each unmanned control device;

[0008] S30: Represent the constrained following error as a control following object, and transform the control problem into a solvable constrained following problem.

[0009] S40: Integrate the multi-domain behavior control tasks between each unmanned control device into a problem framework, obtain the binding force within the problem framework, and design an adaptive law and an adaptive robust controller according to the binding force.

[0010] As an improvement of the above technical solution, the dynamic model in step S10 is as follows:

[0011]

[0012] Where Yi = [x i , y i , z i T represents the position of the drone or the unmanned vehicle in the inertial coordinate system with the ground as the zero potential energy. Among them, x i , y i , z i respectively represent the coordinates of the drone or the unmanned vehicle in the three directions x, y, and z. represents the rate of change of the position of the drone in the inertial coordinate system over time. represents the rate of change of the velocity of the drone in the inertial coordinate system over time; τ i = [τ xi , τ yi , τ zi T represents the control input. Among them, τ xi , τ yi , τ zi respectively represent the control inputs in the three directions x, y, and z. is the set of serial numbers of all unmanned control devices; t ∈ R represents the time variable. represents the uncertainty in the unmanned cluster system. represents the potential boundary of σ i ; M i (Y i , σ i , t) > 0 represents the inertia matrix. represents the Coriolis force, g i (Y i (t), σ i (t), t) represents the gravity. represents the external disturbance.

[0013] The system constraints of the air-ground cooperation system include the following formula:

[0014]

[0015] Among them, 1 ≤ m ≤ n, m represents the number of constraint conditions, n represents the total number of state variables of the unmanned vehicle or the drone in the system, and r represents the index variable of the constraint conditions. represents the rate of change of the velocity of the drone in the inertial coordinate system over time. is the u-th component of the velocity vector ; S iru (·) represents the weight coefficient when each state component satisfies the constraint, c ir (·) represents the target value of the constraint that the system motion needs to satisfy given the state Y i and t, S i (Y​​i , t) represents the constraint matrix, b i is the dynamic compensation term of the constraint, is the acceleration of the system under unconstrained conditions, is S i (Y i , t) of the generalized inverse, is the inverse matrix of the mass matrix, S i =[S iru m×n , b i =[b i1 , b i2 , …, b im T respectively represent the weight coefficients when each state component satisfies the constraint and the target value of the quadratic constraint required to be satisfied under the given state Y i , and t.

[0016] As an improvement of the above technical solution, the step S20 includes the following steps:

[0017] S21: Adopt the method of spatial measurement to construct the collision avoidance constraint between the cooperative flight space-time domain unmanned control devices and the collision avoidance constraint between the airspace unmanned control device and the ground during flight.

[0018] S23: Construct the agile formation constraint of the airspace unmanned control device group by introducing the platform activation function method.

[0019] S24: Establish the air-ground cooperation constraint between the airspace unmanned control device group and the regional unmanned control device according to the distance between the airspace unmanned control device and the regional unmanned control device.

[0020] S25: Construct the trajectory tracking constraint of the unmanned vehicle according to the trajectory tracking error of the regional unmanned control device.

[0021] As an improvement of the above technical solution, the step S21 includes the following steps:

[0022] S211: Obtain the distance representation between any two airspace unmanned control devices and the distance representation between any airspace unmanned control device and the ground.

[0023] S212: Define the corresponding spatial measure according to the distance representation, and construct the collision avoidance constraint between the cooperative flight space-time domain unmanned control devices and the collision avoidance constraint between the airspace unmanned control device and the ground during flight by this spatial measure.

[0024] As an improvement of the above technical solution, converting the control problem into a solvable constraint following problem in the step S30 includes the following steps: ​​

[0025] S31: Convert the multi-domain behavior control constraints between the unmanned control devices into a multi-domain behavior control constraint matrix.

[0026] As an improvement to the above technical solution, the step S40 includes the following steps:

[0027] S41: Integrate the multi-domain behavior control constraint matrix and define the constraint following error according to the integration result.

[0028] S42: Use the Udwadia-Kalaba method to represent the control objective as a minimized norm.

[0029] S43: Decompose the parameters of the dynamic model and define the dynamic model of the nominal system.

[0030] S44: Based on the dynamic model of the nominal system and the system constraints of the air-ground cooperation system, obtain the binding force that makes the dynamic model of the nominal system satisfy the system constraints and has the minimum norm according to the D'Alembert principle in Lagrangian form.

[0031] S45: Introduce a control torque that limits the uncertainty of the system, dynamically compensates for the uncertainty and interference of the system to construct a continuously differentiable and concave uncertainty boundary function, and construct an adaptive law for the online uncertainty vector according to the constraint following error.

[0032] S46: And construct an adaptive robust controller according to this adaptive law.

[0033] As an improvement to the above technical solution, the constraint following error defined in the step S41 is as follows:

[0034]

[0035] where χ i represents the constraint following error, represents the velocity vector of the unmanned aerial vehicle or unmanned vehicle, where S i represents the constraint matrix, and c i represents the constraint target value.

[0036] The minimized norm representation in the step S42 is as follows:

[0037]

[0038] where, represents the rate of change of the velocity of the unmanned aerial vehicle with time in the inertial coordinate system, is the acceleration of the system under unconstrained conditions, is the constraint matrix S i (Y i, the generalized inverse of t), is the inverse matrix of the mass matrix, b i is the dynamic compensation term of the constraint, represents the time derivative of the constraint following error, that is, its rate of change, represents the time derivative of the constraint target value.

[0039] The dynamic model of the nominal system in step S43 is shown as follows:

[0040]

[0041] Where, respectively represent the nominal parts of the inertia matrix, Coriolis force matrix, gravity vector and external disturbance force, represents the nominal control input relative to the system control input τ i That is, the theoretical control input of the system under the condition of no uncertainty, represents the rate of change of the velocity of the UAV changing with time in the inertial coordinate system, represents the velocity vector of the system.

[0042] The binding force that makes the dynamic model of the nominal system satisfy the system constraints and has the minimum norm in step S44, that is, the control input that can make the nominal system satisfy the system constraints, and its acquisition method depends on the following formula:

[0043]

[0044] Where, τ i represents the control input that makes the nominal system satisfy the system constraints, represents the process of calculating the control input based on the state of UAV i, is the square root of the inertia matrix of the nominal system, is the inverse matrix of the square root of the inertia matrix of the nominal system, is the inverse matrix of the inertia matrix of the nominal system; S i (Y i , t), are the constraint matrix and the dynamic compensation term of the constraint of the system respectively; is the Coriolis force matrix, gravity vector and external disturbance force of the nominal system, represents the velocity vector of the system.

[0045] As an improvement of the above technical solution, the matrix function of the uncertainty of the system is as follows:

[0046]

[0047] Where, P iDenote the weight matrix, which is a symmetric positive definite matrix designed by the designer according to the actual system stability requirements, and is used to adjust the weight and stability of the system. is the inverse matrix of the weight matrix; S i (Y i , t) is the constraint matrix of the system. is the transpose of the system constraint matrix; is the intermediate variable of the system dynamics matrix transformation; is the nominal part of the system inertia matrix.

[0048] The control torque for dynamically compensating system uncertainties and disturbances:

[0049]

[0050] Among them, is the inertia matrix of the nominal system, is the square root of the nominal system inertia matrix, is the inverse matrix of the square root of the nominal system inertia matrix, is the inverse matrix of the nominal system inertia matrix; S i (Y i , t) is the constraint matrix of the system, is the dynamic compensation term of the constraint, c i (Y i , t) represents the target value of the constraint that the system motion needs to satisfy under the given state Y i and t, is the transpose of the constraint matrix; is the Coriolis force matrix, gravitational vector and external disturbance force of the nominal system; is the inverse matrix of the weight matrix; κ i is a positive design constant used to control the adjustment of the input, represents the velocity vector of the system.

[0051] The adaptive law of the on-line uncertainty vector in the step S45 includes the following formula:

[0052]

[0053] Among them, is the j-th component of the uncertainty vector , represents the time derivative of the uncertainty vector , describing the update rate of this estimated value over time; is the constraint following error, is the Euclidean norm of the constraint following error, represents the uncertainty boundary function with respect to αi Derivative of transpose, k i1 , k i2 ∈R, k i1 , k i2 >0 is a design parameter adjusted by the designer himself, used to adjust the update rate and the magnitude of the control correction.

[0054] As an improvement of the above technical solution, the adaptive robust controller includes the following formula:

[0055]

[0056] Where p i1 and p i2 are the control torques for dynamically compensating system uncertainties and disturbances, and p i3 is the adaptive robust control torque based on the adaptive parameter .

[0057] Advantages of the present invention:

[0058] By adopting the multi-task processing framework, multiple tasks can be integrated into a unified constraint, effectively reducing the complexity of the system, thereby enhancing the simplicity and reliability of the design. This framework also introduces an adaptive robust control scheme, enabling the system to maintain the stability of its performance in the face of uncertainties and external disturbances. This design not only enhances the adaptability of the system but also improves its performance under dynamic operating conditions. Brief description of the drawings

[0059] Figure 1 is a flow block diagram of the working principle of the present invention. Specific embodiments

[0060] The following specific examples illustrate the implementation manners of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific implementation manners, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention.

[0061] An air-ground collaborative multi-task constrained following control method based on the Udwadia-Kalaba equation, characterized by comprising the following steps:

[0062] S10: Establish the dynamic model of the air-ground collaborative system composed of unmanned control devices, and establish the system constraints of the air-ground collaborative system according to the Udwadia-Kalaba method.

[0063] The dynamic model in step S10 is as follows:

[0064]

[0065] where Y i =[x i , y i , z i T represents the position of the UAV or UGV in the inertial coordinate system with the ground as the zero potential energy, where x i , y i , z i represent the coordinates of the UAV or UGV in the three directions x, y, and z, respectively, represents the rate of change of the position of the UAV with respect to time in the inertial coordinate system, represents the rate of change of the velocity of the UAV with respect to time in the inertial coordinate system; τ i =[τ xi , τ yi , τ zi T represents the control input, where τ xi , τ yi , τ zi represent the control inputs in the three directions x, y, and z, respectively, is the set of serial numbers of all UAVs and UGVs, where N is a constant, represents the set of serial numbers assigned to the UAVs, and Γ := N + 1 represents the set of serial numbers assigned to the UGVs; t ∈ R represents the time variable, represents the uncertainty in the multi - UAV system, represents the potential boundary of σ i ; M i (Y i , σ i , t)>0 represents the inertia matrix, represents the Coriolis force, and g i (Y i (t), σ i (t), t) represents the gravity, represents the external disturbance;

[0066] where the external disturbance usually refers to frictional force, air resistance in the system, or other external disturbances. Based on the above model, the acceleration in the unconstrained case can be determined as:

[0067]

[0068] where, is the inverse matrix of the mass matrix, and τ i =[τ xi ​​, τ yi , τ zi T represents the control input, represents the Coriolis force, g i (Y i (t), σ i (t), t) represents the gravity, represents the external disturbance; represents the rate of change of the position of the UAV with respect to time in the inertial coordinate system, represents the rate of change of the velocity of the UAV with respect to time in the inertial coordinate system.

[0069] Based on the acceleration in the unconstrained case, the constraint conditions that the system needs to follow can be described in first-order form as follows:

[0070]

[0071] where 1 ≤ m ≤ n, m represents the number of constraint conditions, n represents the total number of state variables of the unmanned vehicle or UAV in the system, and r represents the index variable of the constraint conditions, represents the rate of change of the velocity of the UAV with respect to time in the inertial coordinate system, is the u-th component of the velocity vector S iru (·) represents the weight coefficient when each state component satisfies the constraint, c ir (·) represents the target value of the constraint that the system motion needs to satisfy given the state Y i and t, and S iru (·) and c ir (·) are both C i with respect to Y 1 and t, that is, continuously differentiable. The first-order form constraint is represented in matrix form as:

[0072]

[0073] where S i = [S iru m×n , b i = [b i1 , b i2 , …, b im T respectively represent the weight coefficient when each state component satisfies the constraint and the target value of the quadratic constraint that needs to be satisfied given the state Y i , and t.

[0074] Differentiating both sides of the first-order form constraint with respect to t gives the second-order form constraint of the system: ​​​

[0075]

[0076] wherein, is the u-th component of the acceleration vector of, is the u-th component of the velocity vector and:

[0077]

[0078] wherein, Y ik represents the k-th component of the system state vector Y i where k is an index variable, represents the time rate of change of Y ik of, and respectively represent the weight coefficients S iru (Y i , t) with respect to Y ik and time partial derivatives, and respectively represent the weight coefficients c ir (Y i , t) when the state components satisfy the constraints c ir (Y i , t) with respect to Y ik and time partial derivatives.

[0079] Rewrite the second-order form constraint as:

[0080]

[0081] wherein, is the u-th component of the acceleration vector of, is the u-th component of the velocity vector of, represents the defined constraint term, S iru (·) and c ir (·) respectively represent the weight coefficients when the state components satisfy the constraints and the target value of the constraints that the system motion needs to satisfy under the given state Y i and t.

[0082] Represent the second-order form constraint in matrix form as:

[0083]

[0084] wherein, S i = [S iru m×n , b i = [b​i1 , b i2 , …, b im T represents the objective value of the quadratic constraint that the system motion needs to satisfy given the state Y i , and t, and represents the objective value of the quadratic constraint that the system motion needs to satisfy given the state Y represents the objective value of the quadratic constraint for each state component, where 1 ≤ r ≤ m. S iru (·) represents the weight coefficient when each state component satisfies the constraint.

[0085] According to the Udwadia-Kalaba method, the actually constrained acceleration can be obtained by adding a correction term of minimum intervention to the unconstrained acceleration, as shown in the following formula:

[0086]

[0087] where 1 ≤ m ≤ n, is the u-th component of the velocity vector of S iru (·) represents the weight coefficient when each state component satisfies the constraint, c ir (·) represents the objective value of the constraint that the system motion needs to satisfy given the state Y i and t, is the acceleration of the system under unconstrained conditions, is S i (Y i , t) of the generalized inverse, is the inverse matrix of the mass matrix, S i = [S iru m×n , b i = [b i1 , b i2 , …, b im T respectively represent the weight coefficient when each state component satisfies the constraint and the objective value of the quadratic constraint that needs to be satisfied given the state Y i , and t.

[0088] S20: Based on the analysis of the control problem, establish the multi-domain behavior control constraints between unmanned control devices.

[0089] Specifically, the multi-domain behavior control constraints between unmanned control devices usually include five types, namely the collision avoidance constraint between unmanned aerial vehicles, the collision avoidance constraint between unmanned aerial vehicles and the ground, the agile formation constraint of unmanned aerial vehicle swarms, the air-ground cooperation constraint, and the trajectory tracking constraint of unmanned vehicles. Based on this, it is also necessary to specifically describe step S20, which specifically includes the following steps:

[0090] ​​​S21: Construct the collision avoidance constraints between the unmanned control devices in the collaborative flight space-time domain and the collision avoidance constraints between the unmanned control devices in the airspace and the ground during flight by using the method of spatial measurement.

[0091] Among them, the step S21 includes the following steps:

[0092] S211: Obtain the distance representation between any two unmanned control devices in the airspace and the distance representation between any unmanned control device in the airspace and the ground.

[0093] The distance between two unmanned aerial vehicles k and j is expressed as:

[0094] Δd kj (t) = ||Y k (t) - Y j (t)|| - 2s

[0095] Among them, Y k = [x k , y k , z k T and Y j = [x j , y j , z j T respectively represent the positions of the unmanned aerial vehicles k and j in the inertial coordinate system with the ground as the zero potential energy, is the index variable, represents the set of number indices of the unmanned aerial vehicles, s > 0 represents the maximum radius of the unmanned aerial vehicle, and Δd kj (t) represents the distance between the unmanned aerial vehicles k and j.

[0096] The distance between the unmanned aerial vehicle and the ground is expressed as:

[0097] Δd kh (t) = z k (t)

[0098] Δd kh (t) represents the distance between the unmanned aerial vehicle k and the ground, and z k (t) represents the coordinate value of the unmanned aerial vehicle k on the vertical axis.

[0099] S212: And define the corresponding spatial measurement according to the distance representation, and construct the collision avoidance constraints between the unmanned control devices in the collaborative flight space-time domain and the collision avoidance constraints between the unmanned control devices in the airspace and the ground during flight by using this spatial measurement.

[0100] Based on the representation of the distance in step S211, the definition of the spatial measurement is as follows:

[0101] ​​Define the spatial measure corresponding to UAV k and UAV j as:

[0102] d kj (t) = ||Y k (t) - Y j (t)|| 2 -4s 2

[0103] Among them, d kj (t) represents the spatial measure used in the UAV collision avoidance problem, Y k = [x k , y k , z k T and Y j = [x j , y j , z j T respectively represent the positions of UAV k and UAV j in the inertial coordinate system with the ground as the zero potential energy, and s > 0 represents the maximum radius of the UAV.

[0104] Define the spatial measure between the UAV and the ground as:

[0105] d kh (t) = z k (t) - h k

[0106] Among them, d kh (t) represents the spatial measure used in the UAV and ground collision avoidance problem, z k (t) represents the coordinate value of UAV k on the vertical axis, and h k > 0 represents the minimum safe distance between the UAV and the ground. h c > 0 represents the distance from the highest point of the unmanned vehicle to the ground, and h k > h c .

[0107] And construct the collision avoidance constraint from the above spatial measure, specifically as follows:

[0108] The collision avoidance constraint between UAVs is: For d kj (t) > 0.

[0109] Among them, t > t0 and

[0110] The collision avoidance constraint between the UAV and the ground is: For and d kh (t) > 0. ​​

[0111] where \(t > t_0, d\) kh \(\rho(t)\) represents the spatial measure used in the collision avoidance problem between the UAV and the ground.

[0112] S23: Construct the agile formation constraint of the unmanned control device group in the airspace by introducing the platform activation function method.

[0113] In this step, the platform activation function \(\varphi(t)\) is as shown in the following formula:

[0114]

[0115] where \(k\) 1，2，3，4，5 \(> 0\) is the constant parameter of the platform activation function, which controls the amplitude, baseline, decay rate, and conversion steepness of the activation function respectively, and is selected and adjusted by the designer according to the actual system situation. \(T > 0\) represents the central moment of the platform interval of the activation function, and \(d(t)=(t - T)\) 2 represents the square of the time difference between the current moment \(t\) and the reference moment \(T\), which is used to regulate the output change of the activation function when approaching or departing from the reference moment \(T\), so as to achieve smooth switching of the formation.

[0116] From the above form of the activation function, we have that is, when \(t\) is within a certain range \(T\), \(\varphi(t)\) remains at near; when \(t\) is far from this range, \(\varphi(t)\) will drop sharply to near 0. The parameter \(k\) 3，4，5 \(> 0\) can adjust the decay rate of this trend, etc.

[0117] Based on the above reasons, this platform activation function is further generalized as follows:

[0118]

[0119] where \(\varPhi(t)\) represents the activation function, \(R\) h \(> 0\), \(h = 1, 2, \cdots, l\) represents the index of the platform activation function, and \(l\) represents the total number of platform activation functions that activate different formation states or time periods. Among them, \(\varphi\) h \((t)\) represents the platform activation function of different formation states or time periods, as shown in the following formula:

[0120]

[0121] where \(k\) 1h to \(k\) 5h are the constant parameters used to adjust the shape of the \(h\)-th platform activation function, which control the amplitude, baseline, decay rate, and conversion steepness of the activation function respectively, and are selected by the designer according to the actual situation. \(d\) h(t) represents the square of the time difference between the current moment and the central moment, which is used to adjust the change of the activation function output over time, as shown in the following formula:

[0122] d n (t) = (t - T h ) 2

[0123] where T h represents the central moment of the platform interval of the h-th platform activation function, and the values of k 1h to k 5h and T h are all greater than 0.

[0124] By selecting the parameters k 3h , k 4h , k 5h and T h , the "platforms" are made non-intersecting but close to each other. Therefore, Φ(t) takes a certain value within one time interval and quickly switches to another value within another time interval, which can meet the required agile formation requirements. Therefore, the geometric shape of the desired formation is expressed as:

[0125] f h (Y k , t) = 0

[0126] where f h (Y k , t) represents the geometric expression of the desired formation shape, h = 1, 2,..., l represents the index of the activation function, and l represents the total number of platform activation functions that activate different formation states or time periods. Therefore, the combined formation constraint function, that is, the overall constraint that the UAV swarm needs to satisfy, can be expressed as:

[0127]

[0128] where Ω(Y k , t) represents the overall constraint that the UAV swarm needs to satisfy, f1(Y k , t), f2(Y k , t),..., f l (Y k , t) determine the formation shape, and the platform activation functions φ1(t), φ2(t),..., φ l (t) take specific values within different time intervals, determining when to switch between shapes and how long each formation shape lasts.

[0129] Therefore, the agile formation constraint is expressed as: For Ω(Y k , t) = 0.

[0130] S24: Establish an air-ground cooperation constraint between the unmanned control equipment group in the airspace and the unmanned control equipment on the ground according to the distance between the unmanned control equipment in the airspace and the unmanned control equipment on the ground.

[0131] Specifically, the distance Δd between the drone ko (t) and the unmanned vehicle o (o ∈ Γ) is expressed as:

[0132] Δd ko (t) = ||Y k (t) - Y o (t)||

[0133] where ||Y k (t) - Y o (t)|| represents the Euclidean distance between the drone k and the unmanned vehicle o. The ideal state, that is, the ideal locomotive distance, is expressed as Specifically as follows:

[0134]

[0135] where, represents the ideal distance between the drone k and the unmanned vehicle o.

[0136] Furthermore, the air-ground cooperation error e ko (t) is expressed as:

[0137]

[0138] Thus, an air-ground cooperation constraint between the drone group and the unmanned vehicle is constructed, expressed as:

[0139]

[0140] where, θ ko > 0 represents an arbitrarily small constant scalar.

[0141] S25: Construct a trajectory tracking constraint for the unmanned vehicle according to the trajectory tracking error of the unmanned control equipment on the ground.

[0142] Specifically, the distance d between the unmanned vehicle o (o ∈ Γ) and its ideal trajectory o (t) is expressed as:

[0143]

[0144] where, Y o (t) represents the position vector of the unmanned vehicle at time t, represents the desired target position of the unmanned vehicle. Thus, its trajectory tracking error is defined, specifically as shown in the following equation:

[0145]

[0146] Among them, represents the ideal trajectory of the driverless vehicle, that is, the expected target position. represents the square of the Euclidean distance between the current position of the driverless vehicle and the expected target position.

[0147] Based on the above formula, the trajectory tracking constraint of the driverless vehicle is constructed, that is, expressed as:

[0148] For ||e o (t)||≤ψ

[0149] Among them, e o (t) represents the trajectory tracking error, ||e o (t)|| represents the norm of the trajectory tracking error, t≥T o , T o represents the maximum time required for the system to enter and remain within the trajectory tracking target range from the initial state, that is, the convergence time; o∈Γ represents the target of performing the above trajectory tracking process, that is, the subscript index of the driverless vehicle; ψ>0 represents an infinitesimal constant scalar.

[0150] S30: Express the constraint following error as the control following object, and transform the control problem into a solvable constraint following problem.

[0151] Specifically, transforming the control problem into a solvable constraint following problem in step S30 includes the following steps:

[0152] S31: Convert the multi-domain behavior control constraints between unmanned control devices into a multi-domain behavior control constraint matrix.

[0153] In the case of the foregoing solution, it is necessary to transform each control constraint in step S20, as shown below:

[0154] 1. Transform the problem of collision avoidance of UAV swarms during cooperative flight. The specific transformation scheme is as follows:

[0155] For any UAV k (k∈N) in the UAV swarm system, define the function:

[0156] e kj : =ln(d kj )=ln(||Y k -Y j || 2 -s 2 )

[0157] Among them, d kjis the spatial measure of the distance between drones k and j, Y k and Y j respectively represent the position vectors of drones k and j in the inertial coordinate system, e kj represents the logarithmic distance error measurement function between drone k and the nearest drone j, which is used to measure the relative distance between drones. s>0 represents the maximum radius of the drone.

[0158] Taking the first and second derivatives of the above functions respectively, we get the following equations:

[0159]

[0160] where and respectively represent the velocity vectors of drones k and j in the inertial coordinate system, and respectively represent the acceleration vectors of drones k and j in the inertial coordinate system; The collision avoidance constraints of the drone swarm on drone k are reconstructed as follows:

[0161]

[0162] Rewriting the above constraints, specifically as follows:

[0163]

[0164] where respectively represent the constraint matrices of the collision avoidance of the drone swarm on drone k after rewriting. and respectively represent the velocity vector and acceleration vector of drone k.

[0165] After rewriting, for drones k and j, there are the following constraints:

[0166]

[0167]

[0168] Specifically applied to each drone k, it is represented in the form of permutations and combinations as follows:

[0169] If k = 1:

[0170]

[0171] If k = 2, 3, …, N - 1:

[0172]

[0173] If k = N:

[0174]

[0175] Based on this, the constraint conditions of the drone k are rewritten in the following form:

[0176]

[0177] Among them, respectively represent the constraint matrix for collision avoidance of the drone swarm. and respectively represent the velocity vector and acceleration vector of the drone k.

[0178] 2. Transform the collision avoidance problem between the drone and the ground. The specific transformation scheme is as follows:

[0179] For the collision avoidance constraint between the drone and the ground, the position of the drone k (k ∈ 1, 2..., N) is Y k =[x k , y k , z k T , and the spatial measure d kh used in the collision avoidance problem between the drone and the ground. Define the function:

[0180] e kh (k): = ln(z k (t) - h k )

[0181] Among them, e kj represents the logarithmic distance error measurement function between the drone and the ground, which is used to measure the relative distance between the drone and the ground. z k (t) represents the coordinate value of the drone k on the vertical axis, and h k represents the minimum safe distance between the drone and the ground.

[0182] Calculate the first - order and second - order differentials of the above function respectively, as shown below:

[0183]

[0184] Among them, z k (t) represents the coordinate value of the drone k on the vertical axis, represents the coordinate change rate of the drone k on the vertical axis, that is, the velocity value, represents the velocity change rate of the drone k on the vertical axis, that is, the acceleration value. h k >0 represents the minimum safe distance between the drone and the ground. Thus, the constraint of the drone k is constructed:

[0185]

[0186] ​Rewrite the UAV and ground collision avoidance control objective constraints into the following form:

[0187]

[0188] Among them, respectively represent the constraint matrix for UAV and ground collision avoidance, represents the velocity vector of UAV k in the inertial coordinate system, represents the acceleration vector of UAV k in the inertial coordinate system.

[0189] 3. Transform the agile formation constraints of the UAV swarm. The specific transformation scheme is as follows:

[0190] To achieve the agile formation constraints, introduce its servo control equation as follows:

[0191]

[0192] Among them, Ω k (Y k , t) represents under the overall constraint Ω(Y k , t), represents the differential of Ω k (Y k , t) with respect to t, that is, the rate of change of the constraint evolving with time. The local constraint that UAV k needs to follow, l k > 0 represents the decay factor for controlling the rate of the formation constraint approaching zero, which is selected by the designer according to the actual system and is a scalar constant.

[0193] Differentiate both sides of the above servo control equation with respect to t to obtain the second-order form of the agile formation constraint:

[0194]

[0195] Among them, represents the second-order differential of Ω k (Y k , t) with respect to t, and the specific representation is as follows:

[0196]

[0197] Among them, represents the velocity vector of UAV k in the inertial coordinate system, represents the acceleration vector of UAV k in the inertial coordinate system.

[0198] Substitute the first-order and second-order forms of the servo control equation into the above formula respectively to get:

[0199]

[0200] Rewrite the above two equations into the following forms:

[0201]

[0202] where, respectively represent the constraint matrix of the agile formation of the UAV swarm, represents the velocity vector of UAV k in the inertial coordinate system, represents the acceleration vector of UAV k in the inertial coordinate system.

[0203]

[0204] where, Ω k (Y k , t) represents that under the overall constraint Ω(Y k , t), represents the differential of Ω k (Y k , t) with respect to t, that is, the rate of change of the constraint evolving with time. The local constraint that UAV k needs to follow, l k >0 represents the decay factor that controls the rate of approximation of the formation constraint to zero, which is selected by the designer according to the actual system and is a scalar constant.

[0205] 4. Transform the air-ground cooperation constraints between the UAV swarm and the unmanned vehicle. The specific transformation scheme is as follows:

[0206] To achieve the air-ground cooperation constraint, introduce its servo control equation as follows:

[0207]

[0208] where, e ko represents the air-ground cooperation constraint of UAV k on unmanned vehicle o, l ko >0 represents the decay factor that controls the rate of approximation of the cooperation constraint to zero, which is selected by the designer according to the actual system and is a constant scalar; represents the differential of e ko with respect to t, that is, the rate of change of the constraint evolving with time.

[0209] Differentiate both sides of the above servo control scheme with respect to t to obtain the second-order form of the air-ground cooperation constraint:

[0210]

[0211] Differentiate the air-ground cooperation error with respect to the first and second orders:

[0212]

[0213] Substitute and Substitute back into the first - order and second - order forms of the servo - control equation:

[0214]

[0215] Rewrite the above two equations in the following form:

[0216]

[0217] where,

[0218]

[0219] where, respectively represent the constraint matrix of the air - ground cooperation between the UAV swarm and the unmanned vehicle, Y k and Y o respectively represent the position vectors of UAVs k and o in the inertial coordinate system, represents the velocity vector of UAV k in the inertial coordinate system, represents the acceleration vector of UAV k in the inertial coordinate system, represents the ideal distance between UAV k and unmanned vehicle o.

[0220] 5. Transform the trajectory - tracking constraint of the unmanned vehicle o (o ∈ N + 1). The specific transformation scheme is as follows:

[0221] To achieve the trajectory - tracking constraint, introduce its servo - control equation as follows:

[0222]

[0223] where, e o represents the trajectory - tracking error of the unmanned vehicle o, l o represents the decay factor of the control trajectory - tracking constraint approaching zero rate, which is selected by the designer according to the actual system and is a constant scalar; represents the differential of e o with respect to t, that is, the trend of the unmanned vehicle deviating from the target trajectory.

[0224] Differentiate both sides of the above servo - control equation with respect to t to obtain the second - order form of the trajectory - tracking constraint:

[0225]

[0226] Differentiate the trajectory - tracking error with respect to t for the first - order and second - order differentials:

[0227]

[0228] where, Y o (t) represents the position vector of the unmanned vehicle at time t represents the ideal trajectory of the driverless vehicle, i.e., the desired target position, represents the rate of change of the ideal trajectory of the driverless vehicle, i.e., the desired speed, represents the desired acceleration of the driverless vehicle. represents the square of the Euclidean distance between the current position of the driverless vehicle and the desired target position.

[0229] Substitute and back into the first-order and second-order forms of the servo control equation:

[0230]

[0231] Rewrite the constraints in the following form:

[0232]

[0233] where:

[0234]

[0235] where, respectively represent the constraint matrix for the trajectory tracking of the driverless vehicle, Y o (t) represents the position vector of the driverless vehicle at time t, represents the ideal trajectory of the driverless vehicle, i.e., the desired target position, represents the rate of change of the ideal trajectory of the driverless vehicle, i.e., the desired speed, represents the desired acceleration of the driverless vehicle, l o represents the decay factor for the rate at which the control trajectory tracking constraint approaches zero, which is selected by the designer according to the actual system and is a constant scalar.

[0236] Based on the above scheme, five control objectives have been transformed into five constraint conditions. After the transformation of the constraint conditions, it is necessary to integrate the control tasks to simplify the scheme, and the specific implementation steps are S40.

[0237] S40: Integrate the multi-domain behavior control tasks among the unmanned control devices into a problem framework, obtain the binding force within this problem framework, and design an adaptive law and an adaptive robust controller according to this binding force.

[0238] Specifically, the step S40 includes the following steps:

[0239] S41: Integrate the multi-domain behavior control constraint matrix and define the constraint following error according to the integration result.

[0240] Specifically, first transform the constraints into a single representation:

[0241]

[0242] Based on this, the constraints for each drone or unmanned vehicle \(i\) can be expressed as:

[0243]

[0244] where \(S\) i , \(c\) i , \(b\) i respectively represent the constraint matrices after integrating five mission objectives, then the constraint following error can be defined as the following formula:

[0245]

[0246] where \(\chi\) i represents the constraint following error, represents the velocity vector of the drone or unmanned vehicle, \(S\) i = [\(S\) iru m×n , \(b\) i = [\(b\) i1 , \(b\) i2 , …, \(b\) im T respectively represent the weight coefficients when each state component satisfies the constraints and the target value of the quadratic constraint required to be satisfied under the given state \(Y\) i , and \(t\).

[0247] S42: Adopt the Udwadia-Kalaba method to represent the control objective by minimizing the norm;

[0248] The convergence of the constraint following error \(\chi\) i is directly guaranteed by the minimum norm control force of the U-K equation. According to U-K theory, the control objective is transformed into minimizing the norm of \(\chi\) i , and its dynamics can be expressed as follows:

[0249]

[0250] where, is the acceleration of the system under unconstrained conditions, is the generalized inverse of the constraint matrix \(S\) i (\(Y\) i , \(t\)), is the inverse matrix of the mass matrix, \(b\) i is the dynamic compensation term of the constraint, represents the time derivative of the constraint following error, that is, its rate of change, represents the time derivative of the constraint target value.

[0251] S43: Decompose the parameters of the dynamic model and define the dynamic model of the nominal system;

[0252] Through the Lyapunov function It can be proven that the control force design of the UK equation enables Thus ensuring that χ i Asymptotically converges to zero. Therefore, the relevant parameters of the system dynamic model are decomposed as follows:

[0253]

[0254] where M i , C i , g i , F i represent the inertia matrix, Coriolis force matrix, gravity vector, and external disturbance force of the system respectively; represent the nominal parts of the inertia matrix, Coriolis force matrix, gravity vector, and external disturbance force respectively, and ΔM i , ΔC i , Δg i , ΔF i represent the uncertain parts of the inertia matrix, Coriolis force matrix, gravity vector, and external disturbance force respectively; and ΔM i (·), ΔC i (·), Δg i (·), ΔF i (·) are all continuous.

[0255] Based on the above decomposition, the dynamic model of the nominal system can be expressed as the following formula:

[0256]

[0257] where, represent the nominal parts of the inertia matrix, Coriolis force matrix, gravity vector, and external disturbance force respectively, represents the nominal control input with respect to the system control input τ i , that is, the theoretical control input of the system under the condition of no uncertainty, represents the velocity vector of the system;

[0258] S44: Based on the dynamic model of the nominal system and the system constraints of the air-ground collaborative system, obtain the binding force that makes the dynamic model of the nominal system satisfy the system constraints and has the minimum norm according to the d'Alembert principle in Lagrangian form.

[0259] Specifically, making the dynamic model of the nominal system satisfy the system constraints and have the binding force with the minimum norm in step S44 means the control input that can make the nominal system satisfy the system constraints, and its acquisition method depends on the following formula:

[0260]

[0261] where τ i represents the control input that makes the nominal system satisfy the system constraints, represents the process of calculating the control input based on the state of UAV i. is the square root of the inertia matrix of the nominal system, is the inverse matrix of the square root of the inertia matrix of the nominal system, is the inverse matrix of the inertia matrix of the nominal system; S i (Y i , t), c i (Y i , t) is the constraint matrix of the system, is the transpose of the constraint matrix; is the Coriolis force matrix, gravitational vector and external disturbance force of the nominal system.

[0262] S45: Introduce the control torque that restricts the uncertainty of the system, dynamically compensates the uncertainty of the system and the disturbance to construct a continuously differentiable and concave uncertainty boundary function, and construct the adaptive law of the online uncertainty vector according to the constraint following error.

[0263] First, define the matrix function of the uncertainty of the system as follows specifically:

[0264]

[0265] where P i represents the weight matrix, which is a symmetric positive definite matrix designed by the designer according to the actual system stability requirements, and is used to adjust the weight and stability of the system, is the inverse matrix of the weight matrix; S i (Y i , t) is the constraint matrix of the system, is the transpose of the system constraint matrix; is the intermediate variable of the relevant transformation of the system dynamics matrix; is the nominal part of the system inertia matrix.

[0266] Then, define the control torque that dynamically compensates the uncertainty and disturbance of the system:

[0267]

[0268] Among them, is the square root of the nominal system inertia matrix, is the inverse matrix of the square root of the nominal system inertia matrix, is the inverse matrix of the nominal system inertia matrix; S i (Y i , t) is the constraint matrix of the system, is the dynamic compensation term of the constraint, c i (Y i , t) represents the target value of the constraint that the system motion needs to satisfy under the given state Y i and t, is the transpose of the constraint matrix; is the Coriolis force matrix, gravitational vector and external disturbance force of the nominal system; is the inverse matrix of the weight matrix; κ i is a positive design constant used to control the adjustment of the input.

[0269] Design a continuously differentiable and concave uncertainty boundary function The specific form is designed by the designer based on the system uncertainty, so that the uncertainty is limited within a controllable range, that is:

[0270]

[0271] Among them, the uncertainty vector α i is a constant vector related to the system uncertainty, which is usually unknown in practical applications and is dynamically adjusted by the designer through estimation or adaptive control algorithms. S i (Y i , t), c i (Y i , t) is the constraint matrix of the system; C i , g i , F i are the Coriolis force matrix, gravitational vector and external disturbance force of the system, ΔM i , ΔC i , Δg i , ΔF i respectively represent the uncertain parts of the inertia matrix, Coriolis force matrix, gravitational vector and external disturbance force; P i is the weight matrix; represents the uncertainty in the unmanned cluster system, is tight but unknown, representing the potential boundary of σ i ; the uncertainty boundary function and is the control torque for dynamically compensating system uncertainties and disturbances; ρ iE represents a constant related to the uncertainty, satisfying:

[0272]

[0273] It is necessary for the designer to estimate and determine it online based on this condition through adaptive control or other estimation methods, where λ m represents the minimum eigenvalue of the uncertainty matrix W i ; represents the transpose of the uncertainty matrix W i .

[0274] And for the uncertainty vector α i the first and second components of

[0275]

[0276] And the uncertainty boundary function is non-decreasing for each component of the uncertainty vector α i ; represents the partial derivative of the uncertainty boundary function with respect to the uncertainty vector α i .

[0277] Based on the constraint following error χ i , design the adaptive law of the online uncertainty vector, specifically including the following formula:

[0278]

[0279] where is the j-th component of the uncertainty vector ; is the constraint following error, represents the derivative of the uncertainty boundary function with respect to the transpose of α i , k i1 , k i2 ∈R, k i1 , k i2 >0 are design parameters adjusted by the designer himself to adjust the update rate and the magnitude of the control correction.

[0280] S46: And construct an adaptive robust controller according to this adaptive law.

[0281] To suppress the uncertainty disturbance, introduce the information of α i into the controller, and further construct the adaptive robust control scheme as follows:

[0282]

[0283] where p i1 and p i2 are control torques for dynamically compensating system uncertainties and disturbances, and p i3 is an adaptive robust control torque based on the adaptive parameter , and its acquisition method depends on the following formula:

[0284]

[0285] where represents an upper bound function of uncertainties previously designed by the designer himself, is the inertia matrix of the nominal system, S i (Y i , t), c i (Y i , t) is a constraint matrix, is the transpose of the constraint matrix, is the inverse matrix of the weight matrix, and ε i represents a tolerance threshold.

[0286] where:

[0287]

[0288] represents an upper bound function of uncertainties previously designed by the designer himself,, is the inertia matrix of the nominal system; S i (Y i , t), c i (Y i , t) is a constraint matrix; is the transpose of the constraint matrix; is the inverse matrix of the weight matrix; ε i represents a tolerance threshold, which is a positive scalar constant used to control the adjustment of the input.

[0289] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Any person familiar with this technology can modify or change the above embodiments without departing from the spirit and scope of the present invention. Therefore, all equivalent modifications or changes made by those with ordinary knowledge in the technical field without departing from the spirit and technical ideas disclosed by the present invention should still be covered by the claims of the present invention.

Claims

1. An air-ground collaborative multi-task constrained following control method based on the Udwadia-Kalaba equation, characterized in that, It includes the following steps: S10: Establish the dynamic model of the air-ground collaborative system composed of unmanned control devices, and establish the system constraints of the air-ground collaborative system according to the Udwadia-Kalaba method; S20: Based on the analysis of the control problem, establish the multi-domain behavior control constraints between the unmanned control devices; S30: Represent the constraint following error as the control following object, and transform the control problem into a solvable constraint following problem; S40: Integrate the multi-domain behavior control tasks between the unmanned control devices into a problem framework, obtain the binding force within the problem framework, and design the adaptive law and adaptive robust controller according to the binding force.

2. A ground-air collaborative multi-task constrained following control method based on the Udwadia-Kalaba equation according to claim 1, characterized in that: The dynamic model in step S10 is as follows: Among them, represents the set of serial numbers assigned to the drones, represents the set of serial numbers assigned to the unmanned vehicles, is the set of serial numbers of all drones and unmanned vehicles. Among them, N is a constant representing the total number of drones; Y i =[x i ,y i ,z i T represents the position of the drone or unmanned vehicle in the inertial coordinate system with the ground as the zero potential energy. Among them, x i ,y i ,z i respectively represent the coordinates of the drone or unmanned vehicle in the three directions x, y, z, represents the rate of change of the position of the drone with time in the inertial coordinate system, represents the rate of change of the velocity of the drone with time in the inertial coordinate system; τ i =[τ xi ,τ yi ,τ zi T represents the control input. Among them, τ xi ,τ yi ,τ zi respectively represent the control inputs in the three directions x, y, z, t∈R represents the time variable, represents the uncertainty in the unmanned cluster system, represents the potential boundary of σ i ; M i (Y i ,σ i ,t)>0 represents the inertia matrix, represents the Coriolis force, g i (Y i (t),σ i (t),t) represents the gravity, represents the external disturbance;​​ The system constraints of the air-ground collaborative system include the following formula: Among them, 1 ≤ m ≤ n, where m represents the number of constraint conditions, n represents the total number of state variables of driverless vehicles or drones in the system, and r represents the index variable of the constraint conditions. represents the rate of change of the velocity of the drone over time in the inertial coordinate system. is the velocity vector of the u-th component, and S iru (·) represents the weight coefficient when each state component satisfies the constraint, and c ir (·) represents the target value of the constraint that the system motion needs to satisfy given the state Y i and t. S i (Y i , t) represents the constraint matrix, and b i is the dynamic compensation term of the constraint. is the acceleration of the system under unconstrained conditions. is for S i (Y i , t) of the generalized inverse, is the inverse matrix of the mass matrix. b i = [b i1 , b i2 , …, b im T respectively represent the weight coefficient when each state component satisfies the constraint and the target value of the quadratic constraint that needs to be satisfied given the state Y i , and t.​ 3. A ground-air collaborative multi-task constrained following control method based on the Udwadia-Kalaba equation according to claim 1, characterized in that: Step S20 includes the following steps: S21: Use the method of spatial measure to construct the collision avoidance constraints between the unmanned control devices in the collaborative flight space-time domain and the collision avoidance constraints between the unmanned control devices in the airspace and the ground during flight; S23: Construct the agile formation constraints of the unmanned control device group in the airspace by introducing the platform activation function method; S24: Establish the air-ground collaboration constraints between the unmanned control device group in the airspace and the unmanned control devices in the region according to the distance between the unmanned control devices in the airspace and the unmanned control devices in the region; S25: Construct the trajectory tracking constraints of the unmanned vehicle according to the trajectory tracking error of the unmanned control device in the region.

4. A ground-air collaborative multi-task constrained following control method based on the Udwadia-Kalaba equation according to claim 3, characterized in that: Step S21 includes the following steps: S211: Obtain the distance representation between any two unmanned control devices in the airspace and the distance representation between any unmanned control device in the airspace and the ground; S212: Define the corresponding spatial measure according to the distance representation, and construct the collision avoidance constraints between the unmanned control devices in the collaborative flight space-time domain and the collision avoidance constraints between the unmanned control devices in the airspace and the ground during flight by the spatial measure.

5. A ground-air collaborative multi-task constrained following control method based on the Udwadia-Kalaba equation according to claim 1, characterized in that: In step S30, transforming the control problem into a solvable constraint following problem includes the following steps: S31: Transform the multi-domain behavior control constraints between the unmanned control devices into a multi-domain behavior control constraint matrix.

6. The method for constrained following control of air-ground coordinated multi-task based on the Udwadia-Kalaba equation according to claim 1 is characterized by: Step S40 includes the following steps: S41: Integrate the multi-domain behavior control constraint matrix, and define the constraint following error according to the integration result; S42: Use the Udwadia-Kalaba method to represent the transformation of the control target as the minimization of the norm; S43: Decompose the parameters of the dynamic model and define the dynamic model of the nominal system; S44: Based on the dynamic model of the nominal system and the system constraints of the air-ground collaborative system, obtain the binding force that makes the dynamic model of the nominal system satisfy the system constraints and has the minimum norm according to the d'Alembert principle in Lagrangian form; S45: Introduce the control torque that limits the uncertainty of the system, dynamically compensates the uncertainty and interference of the system to construct a continuously differentiable and concave uncertainty boundary function, and construct the adaptive law of the online uncertainty vector according to the constraint following error; S46: And construct the adaptive robust controller according to the adaptive law.

7. A ground-air collaborative multi-task constrained following control method based on the Udwadia-Kalaba equation according to claim 6, characterized in that: The constraint following error defined in step S41 is as follows: Among them, χ i represents the constraint following error, represents the velocity vector of the drone or unmanned vehicle, where, S i (Y i , t) represents the constraint matrix, c i represents the constraint target value; The minimized norm in step S42 is expressed as follows: Among them, is the acceleration of the system under unconstrained conditions, is the constraint matrix S i (Y i , t) of the generalized inverse, is the inverse matrix of the mass matrix, b i is the dynamic compensation term of the constraint, represents the time derivative of the constraint following error, that is, its rate of change, represents the time derivative of the constraint target value; The dynamic model of the nominal system in step S43 is shown by the following formula: wherein, respectively represent the inertia matrix, the Coriolis force matrix, the gravity vector, and the nominal part of the external disturbance force, represents the nominal control input with respect to the system control input τ i i.e., the theoretical control input of the system under the condition of no uncertainty, represents the velocity vector of the system; The binding force that makes the dynamic model of the nominal system satisfy the system constraints and has the minimum norm in step S44, that is, the control input that can make the nominal system satisfy the system constraints, and its acquisition method depends on the following formula: where, τ i represents the control input that makes the nominal system satisfy the system constraints, represents the process of calculating the control input based on the state of UAV i; is the square root of the nominal system inertia matrix, is the inverse matrix of the square root of the nominal system inertia matrix, is the inverse matrix of the nominal system inertia matrix; S i (Y i , t) c i (Y i , t) is the constraint matrix of the system, is the transpose of the constraint matrix; is the Coriolis force matrix, gravitational vector, and external disturbance force of the nominal system.

8. A ground-air collaborative multi-task constrained following control method based on the Udwadia-Kalaba equation according to claim 6, characterized in that: The matrix function of the uncertainty of the system is as follows: Among them, P i represents the weight matrix, which is a symmetric positive definite matrix designed by the designer according to the actual system stability requirements to adjust the weight and stability of the system. is the inverse matrix of the weight matrix; S i (Y i , t) is the constraint matrix of the system. is the transpose of the system constraint matrix; is the intermediate variable of the relevant transformation of the system dynamics matrix; is the nominal part of the system inertia matrix. The control torque for dynamically compensating system uncertainties and disturbances: Among them, is the square root of the nominal system inertia matrix, is the inverse matrix of the square root of the nominal system inertia matrix, is the inverse matrix of the nominal system inertia matrix; S i (Y i , t) is the constraint matrix of the system, is the dynamic compensation term of the constraint, c i (Y i , t) represents the target value of the constraint that the system motion needs to satisfy under the given state Y i and t, is the transpose of the constraint matrix; is the Coriolis force matrix, gravity vector and external disturbance force of the nominal system; is the inverse matrix of the weight matrix; κ i is a positive design constant used to control the adjustment of the input; The adaptive law of the online uncertainty vector in step S45 includes the following formula: Among them, is the j-th component of the uncertainty vector ; is the constraint following error, represents the derivative of the uncertainty bound function with respect to the transpose of α i , k i1 , k i2 ∈ R, k i1 , k i2 > 0 is a design parameter adjusted by the designer himself to adjust the update rate of and the magnitude of the control correction.

9. A ground-air collaborative multi-task constrained following control method based on the Udwadia-Kalaba equation according to any one of claims 1-8, characterized in that: The adaptive robust controller includes the following formula: Among them, p i1 and p i2 are control torques for dynamically compensating system uncertainties and disturbances, and p i3 is an adaptive robust control torque based on the adaptive parameter .

Images