CN115857546A - Modular reconfigurable flight array dynamics model and fixed time sliding mode control method - Google Patents
Modular reconfigurable flight array dynamics model and fixed time sliding mode control method Download PDFInfo
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Abstract
The invention discloses a modularized reconfigurable flight array dynamic model and a fixed time sliding mode control method, wherein the dynamic model comprises a coordinate system for describing a modularized reconfigurable flight array; obtaining an inertia tensor of the modularized reconfigurable flight array according to the established coordinate system; and establishing a modularized reconfigurable flight array dynamic model according to the established coordinate system and the inertia tensor. According to the method, a modularized reconfigurable flight array dynamics model is constructed according to the established coordinate system and the inertia tensor, and when the topological configuration of the flight array changes, the mass center does not need to be solved, so that the rapid modeling method is realized; furthermore, the invention adopts a variable exponential coefficient function, and the sliding mode controller designed based on the variable exponential coefficient function has the characteristics of high convergence speed, nonsingularity and strong robustness.
Description
Technical Field
The invention relates to a modularized reconfigurable flight array dynamics model and a fixed time sliding mode control method, and belongs to the field of design and control of unmanned rotor aircrafts.
Background
Due to the advantages of simple structure, portability, capability of vertical take-off and landing, hovering in the air and the like, the multi-rotor aircraft obtains more and more attention and research. The leading edge of current rotor craft research mainly includes multi-motion mode unmanned aerial vehicle, and the robot formation is cooperative and reconfigurable modularization unmanned aerial vehicle, and its physical structure is changed through self-assembling and the self-separation between the module to modularization reconfigurable flight array to dynamic adaptation task and environmental requirement.
In The Research on The modularized reconfigurable flight array, a modularized flight array is proposed in The documents of R.Oung, R D' Andrea, the distributed flight array, design, and analysis of a modulated take-off and driving vehicle [ J ]. The International Journal of Robotics Research,2014,33 (3): 375-400. However, the modeling method needs to know the centroid of the flight array, and when the topological configuration (the number of modules and the configuration of the flight array) of the flight array changes, the centroid of the flight array needs to be recalculated, which increases the complexity of modeling, and cannot realize rapid modeling after the topological configuration changes.
There is less literature in flight array applications for fixed time sliding mode controllers. The existing fixed time sliding mode control method is based on a constant exponential coefficient function y = -k 1 x p (k 1 P is a constant greater than zero), which requires non-singular processing and has a slow convergence speed.
Disclosure of Invention
The invention provides a modularized reconfigurable flight array dynamics model for establishing the modularized reconfigurable flight array dynamics model, and further provides a modularized reconfigurable flight array fixed time sliding mode control method for performing fixed time sliding mode control based on the established modularized reconfigurable flight array dynamics model.
The technical scheme of the invention is as follows: a modular reconfigurable flight array dynamics model, comprising: establishing a coordinate system for describing the modular reconfigurable flight array; obtaining an inertia tensor of the modularized reconfigurable flight array according to the established coordinate system; and establishing a modularized reconfigurable flight array dynamic model according to the established coordinate system and the inertia tensor.
The coordinate system for describing the modular reconfigurable flight array comprises:
module coordinate system O M :{x M ,y M ,z M }; the module coordinate system takes the geometric center of the flight unit module as the origin of coordinates, z M The axis is vertical to the flying unit module and points to the sky, and the coordinate system meets the right-hand rule; the module coordinate system is mainly used for expressing the moment of inertia J = diag (J) of the flight unit module Mx ,J My ,J Mz ) Wherein J Mx Winding x for flying unit module M Moment of inertia of shaft, J My For flying unit module winding y M Moment of inertia of the shaft, J Mz For flying unit module winding z M The rotational inertia of the shaft;
flight array coordinate system
The flight array coordinate system takes any point in the flight array as the origin of coordinates, and the coordinate axis->
The direction of (a) is consistent with the direction of the module coordinate system; the flight array coordinate system is mainly used for describing the position information of the geometric center of the ith flight unit module in the flight array coordinate system>
And respectively wind>
Angular velocity of shaft rotation [ p, q, r] T ；
Inertial coordinate system O E :{x E ,y E ,z E }; the directions of all axes of the inertial coordinate system are consistent with those of all axes of the flight array coordinate system; inertial coordinate system for describing position X of modular reconfigurable flight array in three-dimensional space E ＝[x E ,y E ,z E ] T And attitude angle Θ = [ phi, theta, psi =] T Where phi is winding x E Angle of rotation of the shaft, theta being about y E Angle of rotation of the axis, ψ being about z E The angle of rotation of (c).
The obtaining of the inertia tensor of the modularized reconfigurable flight array according to the established coordinate system comprises the following steps: according to the established coordinate system, the rotational inertia parallel axis theorem is applied to make the rotational inertia J = diag (J) of the flight unit module Mx ,J My ,J Mz ) Calculating the inertia tensor of the modularized reconfigurable flight array based on the number n and the mass m of the single flight unit module
Wherein the content of the first and second substances,
is the position of the geometric center of the ith flight unit module in the flight array coordinate system, J xx 、J yy 、J zz Are respectively wound>
Moment of inertia of the shaft, J xy 、J yx Is relative to>
Shaft and->
The product of inertia of the shaft.
the dynamic model of the translation motion of the modularized reconfigurable flight array is as follows:
in the formula (I), the compound is shown in the specification,
modular reconfigurable flight array along x for inertial frame E Axis, y E Axis, z E Acceleration of the shaft; g is the acceleration of gravity; m is the mass of a single flying unit module; phi, theta, psi] T Respectively winding x for modularized reconfigurable flight array E Axis, y E Axis, z E Attitude angle of shaft rotation; c phi = cos (phi), s phi = sin (phi), c theta = cos (theta), s theta = sin (theta), c psi = cos (psi), s psi = sin (psi); t is z Lift required for flight of the modular reconfigurable flight array;
the dynamic model of the modularized reconfigurable flight array rotary motion is as follows:
in the formula (I), the compound is shown in the specification,
are respectively wound around x E Axial, and axial E Axial, and axial E Angular acceleration of shaft rotation; />
Are respectively wound around x E Axial, and axial E Axial, and axial E Angular velocity of shaft rotation; j. the design is a square xx 、J yy 、J zz Are respectively wound>
Moment of inertia of the shaft, J xy 、J yx Is relative to->
Shaft and->
The product of inertia of the shaft; [ M ] A x ,M y ,M z ] T Are respectively wound around x E Axis, y E Axis, z E The rotational moment of the shaft; />
Lift T in a kinetic model z Moment M x ,M y ,M z The force and moment orthogonal decomposition method is obtained, and the specific implementation form is as follows:
wherein the value of κ is related to the direction of rotation of the propeller: κ × =2 when the propeller is rotating clockwise; κ × =1 when the propeller is rotating counterclockwise; k is a radical of formula F Is the lift coefficient of the propeller; k is a radical of M Is the propeller torque coefficient; u. of i Of the ith flight unit moduleThe actual control input of the actuator, i.e. the pulse width;
respectively representing the position information of the geometric center of the ith flight unit module in a flight array coordinate system.
According to another aspect of the invention, a fixed time sliding mode control method for a modular reconfigurable flight array is provided, which comprises the following steps: extracting a control efficiency matrix B according to the lift force and moment expression e1 And a gravity center compensation matrix B e2 (ii) a According to the control efficiency matrix B e1 And a center of gravity compensation matrix B e2 Establishing a control distribution model; establishing an energy-optimal control distribution strategy according to a control distribution model, and inputting virtual control generated by a fixed-time sliding mode controller with variable exponential coefficients into tau = [ T ] z ,M x ,M y ,M z ] T Are mapped to the individual actuators in an energy-optimal relationship.
Extracting a control efficiency matrix B according to the expression of the lift force and the moment e1 And a gravity center compensation matrix B e2 The method comprises the following steps:
will lift force T z Moment M x ,M y ,M z The expression is rewritten as a matrix:
according to the matrix form of force and moment, the method is used for measuring the virtual control input T z ,M x ,M y ,M z ] T And actuator actual control input u 1 ,u 2 ,…,u n ] T Control efficiency matrix B of the relationship between e1 Writing as follows:
and for describing the virtual control input [ T ] z ,M x ,M y ,M z ] T And a gravity center compensation matrix B of the masses of the individual flight cell modules e2 Write as:
in the formula, the value of κ is related to the rotation direction of the propeller: when the propeller is rotating clockwise, κ × =2; when the propeller is rotating counterclockwise, κ × =1; k is a radical of F Is the lift coefficient of the propeller; k is a radical of formula M Is the propeller torque coefficient; u. u n Actual control input for actuators of the nth flight unit module;
respectively representing the position information of the geometric center of the ith flight unit module in a flight array coordinate system, i =1,2,. The right, n; m is 1 、m 2 、m n Respectively representing the mass of the 1 st, 2 nd and n th flight unit modules; g is the acceleration of gravity.
The dependent control efficiency matrix B e1 And a center of gravity compensation matrix B e2 Establishing a control distribution model, which specifically comprises the following steps: to control the efficiency matrix B e1 And a center of gravity compensation matrix B e2 Is of the form [ T z ,M x ,M y ,M z ] T Mapping to each actuator, and establishing a control distribution model as follows:
τ＝B e1 u+B e2 G
wherein, τ = [ T = z ,M x ,M y ,M z ] T ，G＝[m 1 g,m 2 g,…,m n g] T ；u＝[u 1 ,u 2 ,...u n ]；m n Representing the mass of the nth flying unit module; g is the acceleration of gravity; u. of n Is the actual control input to the actuators of the nth flying unit module.
Establishing an energy optimal control distribution strategy according to the control distribution model, and carrying out virtual control generated by a fixed time sliding mode controller with variable exponential coefficientsSystem input tau = [ T = [ ] z ,M x ,M y ,M z ] T Mapping the energy optimal relationship to each actuator, specifically: according to the established control distribution model, a least square method is adopted to establish an energy optimal control distribution strategy, and virtual control input tau = [ T ] generated by a fixed time sliding mode controller with variable exponential coefficients is input z ,M x ,M y ,M z ] T Mapping into each actuator in an energy-optimal relationship:
s.t.τ＝B e1 u+B e2 G
solving an optimization equation to obtain:
wherein, o (u) represents an objective function, and is called an energy-based optimal allocation strategy;
is a matrix B e1 The right generalized inverse matrix of (d); g = [ m ] 1 g,m 2 g,…,m n g] T ；u＝[u 1 ,u 2 ,...u n ]；m n Representing the mass of the nth flying unit module; g is the acceleration of gravity; u. u n Is the actual control input to the actuators of the nth flying unit module.
The fixed time sliding mode controller with variable exponential coefficients comprises:
and establishing an error dynamic equation of the height and attitude angle according to the dynamic model:
in the formula: in the formula (I), the compound is shown in the specification,
lifting force T z Moment M x ,M y ,M z ；/>
Respectively, a height error, a roll angle error, a pitch angle error and a yaw angle error>
Desired altitude, roll angle, pitch angle, and yaw angle; [ h ] of z (t),h φ (t),h θ (t),h ψ (t)] T Unknown perturbations that are bounded; f. of φ (φ,θ,ψ)、f θ (phi, theta, psi) and f ψ (phi, theta, psi) represents a non-linear function; g 11 、g 21 ，g 22 、g 31 、g 32 And g 41 Representing a control gain function;
designing a sliding mode surface in the following form based on an error equation:
to analyze the dynamic behavior of the sliding mode surface, the derivation can be:
establishing a variable exponential coefficient sliding mode control law:
according to sliding mode control law and
obtaining a virtual control input τ = [ T = [) z ,M x ,M y ,M z ] T ；
In the formula: s 1 、s 2 、s 3 、s 4 Respectively representing sliding mode surfaces for controlling the height, the roll angle, the pitch angle and the yaw angle;
as hyperbolic tangent function, gamma 11 ,γ 12 ,…,γ 41 ,γ 42 >0，κ 1 ,κ 2 ,κ 3 ,κ 4 >1 and 0<ε 1 ,ε 2 ,ε 3 ,ε 4 <1 is a constant; sin (, is a sign function: when>At 0, sgn (= 1); when =0, sgn (= 0); when<At 0, sgn (=) = -1; k is a radical of δ1 ,k δ2 ,k δ3 >0 (δ =1,2,3,4) is the control gain, λ δ >0，μ δ >0，p δ >1 and satisfies lambda δ >p δ μ δ 。
The invention has the beneficial effects that: according to the method, a modularized reconfigurable flight array dynamics model is constructed according to the established coordinate system and the inertia tensor, and when the topological configuration of the flight array changes, the mass center does not need to be solved, so that the rapid modeling method is realized; furthermore, the sliding mode controller designed based on the variable index coefficient function has the characteristics of high convergence speed, nonsingularity and strong robustness.
Drawings
FIG. 1 is a flow diagram of a modular reconfigurable flight array control implementation;
FIG. 2 is a schematic diagram of coordinates of a modular reconfigurable flight array;
FIG. 3 is a schematic view of a flight unit module configuration;
FIG. 4 is an actuator of a modular reconfigurable flight array;
FIG. 5 is a block diagram of altitude and attitude control of a modular reconfigurable flight array;
FIG. 6 is a schematic configuration diagram of an "X" type six-module flight array;
FIG. 7 is a graph of error convergence for different initial conditions;
FIG. 8 is a diagram of an altitude, attitude tracking simulation of the flight array shown in FIG. 6.
Detailed Description
The invention will be further described with reference to the following figures and examples, but the scope of the invention is not limited thereto.
Example 1: as shown in fig. 1-8, a modular reconfigurable flight array dynamics model, comprising: establishing a coordinate system for describing the modular reconfigurable flight array; obtaining an inertia tensor of the modularized reconfigurable flight array according to the established coordinate system; and establishing a modularized reconfigurable flight array dynamic model according to the established coordinate system and the inertia tensor.
Further, the coordinate system for describing the modular reconfigurable flight array comprises:
step 1.1, establishing a coordinate system (as shown in fig. 2) for describing the space position and attitude of the modular reconfigurable flight array, wherein the coordinate system comprises:
1) Module coordinate system O M :{x M ,y M ,z M }; the module coordinate system takes the geometric center of the flight unit module as the origin of coordinates, z M The axis is vertical to the flying unit module and points to the sky, and the coordinate system meets the right-hand rule; the module coordinate system is mainly used for expressing the moment of inertia J = diag (J) of the flight unit module Mx ,J My ,J Mz ) Wherein J Mx Winding x for flying unit module M Moment of inertia of the shaft, J My For flying unit module winding y M Moment of inertia of the shaft, J Mz For flying unit module winding z M The moment of inertia of the shaft; it should be noted that the module coordinate system of each flight unit module is constructed in the above establishing process, and the axis directions of the module coordinate systems of each flight unit module are ensured to be consistent;
2) Flight array coordinate system
The flight array coordinate system takes any point in the flight array as the coordinate origin (usually, any one flight unit model is selected)The geometric center of the block/or the geometric center of the entire flight array selected as the origin of coordinates), the coordinate axis £ greater than or equal to>
The direction of (a) is consistent with the direction of the module coordinate system; the flight array coordinate system is mainly used for describing the position information of the ith flight unit module in the flight array coordinate system>
And respectively wind>
Angular velocity of shaft rotation [ p, q, r] T ；
3) Inertial coordinate system O E :{x E ,y E ,z E }; the directions of all axes of the inertial coordinate system are consistent with those of all axes of the flight array coordinate system; inertial coordinate system for describing position X of modular reconfigurable flight array in three-dimensional space E ＝[x E ,y E ,z E ] T And attitude angle Θ = [ phi, theta, psi =] T Where phi is winding x E Angle of rotation of the shaft (roll angle), theta being around y E The angle of rotation of the shaft (pitch angle), ψ, about z E Angle of rotation (yaw angle);
the modularized reconfigurable flight array is formed by splicing a plurality of flight unit modules. For purposes of illustration and not limitation, the present specification figures show partial configurations, such as fig. 2 showing a modular reconfigurable flight array constructed from 9 regular hexagonal flight cell modules, and fig. 3 giving a corresponding flight cell module structural schematic diagram of fig. 2; FIG. 3 shows a schematic view of a corresponding actuator; fig. 6 is a schematic diagram of another topology.
Further, the obtaining an inertia tensor of the modular reconfigurable flight array according to the established coordinate system includes:
step 1.2 according to the stepThe coordinate system established in step 1.1 applies the theorem of parallel axes of moment of inertia to fly the inertia moment J = diag (J) of the unit module Mx ,J My ,J Mz ) Calculating the inertia tensor of the modularized reconfigurable flight array based on the number n and the mass m of a single flight unit module
Wherein the content of the first and second substances,
is the position of the geometric center of the ith flight unit module in the flight array coordinate system, J xx 、J yy 、J zz Branch and/or device>
Moment of inertia of the shaft, J xy 、J yx Is relative to->
Shaft and->
The product of inertia of the shaft; the mass of each flight unit module in the modularized reconfigurable flight array is the same.
Further, the coordinate system and the inertia tensor are established according to the basis
Establishing a modularized reconfigurable flight array dynamic model, comprising the following steps:
step 1.3, the inertia tensor in step 1.2 is calculated according to the coordinate system established in step 1.1
A Newton-Euler method is used for establishing a dynamic model of the modularized reconfigurable flight array:
the dynamic model of the translation motion of the modularized reconfigurable flight array comprises the following steps:
wherein the content of the first and second substances,
modular reconfigurable flight array along x for inertial frame E Axis, y E Axis, z E Acceleration of the shaft; g is the acceleration of gravity; phi, theta, psi] T Respectively winding x for modularized reconfigurable flight array E Axis, y E Axis, z E Attitude angle of shaft rotation; c phi = cos (phi), s phi = sin (phi), c theta = cos (theta), s theta = sin (theta), c psi = cos (psi), s psi = sin (psi); t is z Lift required for flight of the modular reconfigurable flight array, the lift being generated by high speed rotation of the propellers;
the dynamic model of the rotation motion of the modularized reconfigurable flight array is as follows:
wherein the content of the first and second substances,
are respectively wound around x E Axial, and axial E Axial, and axial E Angular acceleration of shaft rotation; />
Are respectively wound around x E Axial, and axial E Axial, and axial E Angular velocity of shaft rotation; [ M ] x ,M y ,M z ] T Are respectively wound around x E Axis, y E Axis, z E The rotational moment of the shaft;
lift T in a kinetic model z Moment M x ,M y ,M z The force and moment orthogonal decomposition method is obtained, and the specific implementation form is as follows:
wherein the value of κ is related to the direction of rotation of the propeller: when the propeller is rotating clockwise, κ × =2; when the propeller is rotating counterclockwise, κ × =1; k is a radical of F Is the lift coefficient of the propeller; k is a radical of M Is the propeller torque coefficient; u. of i Is the actual control input, i.e. pulse width, of the actuator of the ith flying unit module.
According to another aspect of the invention, a fixed time sliding mode control method for a modular reconfigurable flight array is provided, which comprises the following steps: extracting a control efficiency matrix B according to the lift force and moment expression e1 And a gravity center compensation matrix B e2 (ii) a According to the control efficiency matrix B e1 And a center of gravity compensation matrix B e2 Establishing a control distribution model; establishing an energy-optimal control distribution strategy according to a control distribution model, and inputting virtual control generated by a fixed time sliding mode controller with variable exponential coefficients into tau = [ T ] z ,M x ,M y ,M z ] T Are mapped to the individual actuators in an energy-optimal relationship.
Further, extracting a control efficiency matrix B according to the expressions of the lift force and the moment e1 And a gravity center compensation matrix B e2 The method comprises the following steps:
step 2.1, rewriting the expression of force and moment into a matrix form:
according to the matrix form of force and moment, the method is used for measuring the virtual control input T z ,M x ,M y ,M z ] T And actuator actual control input u 1 ,u 2 ,…,u n ] T Control efficiency matrix B of the relationship between e1 Write as:
and for describing the virtual control input [ T z ,M x ,M y ,M z ] T And the center of gravity compensation matrix B of the mass m of a single flight cell module e2 Write as:
in the formula, the value of κ is related to the rotation direction of the propeller: κ × =2 when the propeller is rotating clockwise; when the propeller is rotating counterclockwise, κ × =1; k is a radical of F Is the lift coefficient of the propeller; k is a radical of M Is the propeller torque coefficient; u. of n Actual control input for actuators of the nth flight unit module;
respectively representing the position information of the geometric center of the ith flight unit module in a flight array coordinate system, i =1,2, ·, n; m is 1 、m 2 、m n Respectively representing the mass of the 1 st, 2 nd and n th flight unit modules; g is the acceleration of gravity.
Further, the control-dependent efficiency matrix B e1 And a center of gravity compensation matrix B e2 Establishing a control distribution model, which specifically comprises the following steps:
step 2.2, by controlling the efficiency matrix B e1 And a center of gravity compensation matrix B e2 Establishing a mathematical model for describing the control distribution problem, i.e. in matrix B e1 Form of (2) will be [ T z ,M x ,M y ,M z ] T Mapping to each actuator, and establishing a control distribution model as follows:
τ＝B e1 u+B e2 G
wherein, τ = [ T = z ,M x ,M y ,M z ] T ，G＝[m 1 g,m 2 g,…,m n g] T ；u＝[u 1 ,u 2 ,...u n ]。
Further, establishing an energy-optimal control distribution strategy according to the control distribution model, and inputting virtual control generated by a fixed-time sliding mode controller with variable exponential coefficients into τ = [ T ] z ,M x ,M y ,M z ] T Mapping the optimal energy relationship to each actuator, specifically:
s.t.τ＝B e1 u+B e2 G
solving an optimization equation to obtain:
wherein O (u) represents an objective function,
is a matrix B e1 The right generalized inverse matrix of (d); referred to as an energy-based optimal allocation strategy.
The steps 1,2 and 3 complete the establishment of a modularized reconfigurable flight array dynamics model and the design of a control distribution strategy, and lay an early-stage foundation for describing the control of the modularized reconfigurable flight array; in the following, we will describe in detail the controller design of a modular reconfigurable flight array:
further, the variable exponential coefficient fixed time sliding mode controller comprises:
step 4.1, establishing an error dynamic equation of the height and attitude angle according to the dynamic model:
wherein the content of the first and second substances,
respectively, a height error, a roll angle error, a pitch angle error and a yaw angle error>
Desired altitude, roll angle, pitch angle, and yaw angle; [ h ] of z (t),h φ (t),h θ (t),h ψ (t)] T Unknown perturbations that are bounded;
non-linear function f φ (φ,θ,ψ)、f θ (phi, theta, psi) and f ψ The expression of (φ, θ, ψ) is as follows:
controlling the gain function g 11 、g 21 ，g 22 、g 31 、g 32 And g 41 The expression of (a) is:
step 4.2, designing the following sliding mode surface based on an error equation:
wherein, the first and the second end of the pipe are connected with each other,
is a hyperbolic tangent function, gamma 11 ,γ 12 ,…,γ 41 ,γ 42 >0，κ 1 ,κ 2 ,κ 3 ,κ 4 >1 and 0<ε 1 ,ε 2 ,ε 3 ,ε 4 <1 is a constant; sin (, is a sign function: when>At 0, sgn (= 1); when =0, sgn (= 0); when<At 0, sgn (=) = -1; s 1 、s 2 、s 3 、s 4 Respectively representing sliding mode surfaces for controlling the height, the roll angle, the pitch angle and the yaw angle;
to analyze the dynamic behavior of the sliding mode surface, the derivation can be:
step 4.3, in order to make the sliding mode surface tend to zero, the invention provides a new variable index coefficient sliding mode control law:
wherein k is δ1 ,k δ2 ,k δ3 >0 (δ =1,2,3,4) is the control gain, λ δ >0，μ δ >0，p δ >1 andsatisfy lambda δ >p δ μ δ ；
According to sliding mode control law and
obtaining a virtual control input τ = [ T = [) z ,M x ,M y ,M z ] T 。
Still further, the following is given:
step 5.1, substituting the sliding mode control law into the sliding mode surface derivative to obtain a sliding mode approach law:
step 5.2, defining a surface s about the sliding form 1 Lyapunov function of (a):
derivation is carried out on the Lyapunov function and the approximation law of the sliding mode is combined to obtain:
according to the derivative of the Lyapunov function, the robustness of the sliding mode controller is mainly determined by
Embodying when we select the parameter k 13 Greater than the perturbation upper bound->
Can ensure that the device is always on>
The situation is always established; therefore, the controller provided by the invention has strong robustness.
Step 5.3, according to the Lyapunov function derivative expression obtained in the step 5.2, the part carries out fixed time stability analysis on the approach law; the derivative of the lyapunov function is rewritten as follows:
two-end integration is carried out:
order to
The definite integral can therefore be written in the form of the following inequality:
as can be seen from the inequality, the,
is a strictly monotonically increasing function with respect to V (t), and the function = is based on a value of V (t) =0>
Therefore, the following limit expression is defined:
wherein, T(s) 1 (0) ) is the settling time (settlingtime), i.e.Slip form surface s 1 Time required for converging on the sliding curved surface S =0;
when in use
In due time, in combination>
Thus T(s) 1 (0) Can be rewritten as:
T(s 1 (0) The upper time bound of) can be defined by a generalized definite integral as follows:
by solving for the generalized definite integral, the upper bound in time is defined as:
i.e. slip form surface s 1 At time T max Internally converging on a sliding curved surface S =0; from the upper bound of the convergence time, the upper bound of the convergence time is independent of the initial condition, i.e., whatever the initial value is, it will be at time T max Inner converges to zero.
In order to verify the effectiveness of the modeling method and the design of the fixed time sliding mode controller, the invention carries out simulation verification on the fixed convergence of the sliding mode approximation rule and the effectiveness of the controller. Fig. 7 shows the convergence characteristics of the sliding mode approach law under different initial values of 5, 10, 15 and 20. Selecting a control parameter of k 11 ＝2，k 12 ＝6，k 13 ＝2，p 1 ＝1.2，λ 1 =0.2, μ =0.1 and d = sin (10 t) (d) z = 1), convergence time T is calculated from the upper time bound max =1.489 seconds. From this watchObviously, the invention still has the fixed time convergence characteristic under the condition that the external interference exists.
For the controller, a six-module reconfigurable flight array as shown in fig. 6 is selected for simulation verification:
fig. 3 shows the structure of one flight unit module in the flight array, and the parameters are selected as shown in table 1.
TABLE 1 flight Unit Module parameter Table
In the flight array coordinate system as shown in fig. 6, the coordinate information of each flight unit module is shown in table 2.
Table 2 coordinate information (unit: meter) of each module of the modularized reconfigurable flight array shown in fig. 6 table 2-1
Tables 2 to 2
Controller parameter selection is shown in table 3:
TABLE 3
The specific implementation block diagram of the flight array control is shown in fig. 5, and the control framework is suitable for the height and attitude control problems of all the flight arrays. Based on the dynamic model and the control method provided by the invention, the six-module flight array shown in FIG. 6 is subjected to altitude and attitude tracking simulation, parameters shown in Table 3 are selected as control parameters, and the simulation result is shown in FIG. 8. Fig. 8 shows trajectory tracking curves of altitude, roll angle, pitch angle, and yaw angle, and it can be seen from the tracking curves that the modeling method and the control method provided by the present invention have good effects.
While the present invention has been described in detail with reference to the embodiments, the present invention is not limited to the embodiments, and various changes can be made without departing from the spirit of the present invention within the knowledge of those skilled in the art.
Claims (9)
1. A modular reconfigurable flying array dynamics model, comprising:
establishing a coordinate system for describing the modular reconfigurable flight array;
obtaining an inertia tensor of the modularized reconfigurable flight array according to the established coordinate system;
and establishing a modularized reconfigurable flight array dynamic model according to the established coordinate system and the inertia tensor.
2. The modular reconfigurable flight array dynamics model according to claim 1, wherein the coordinate system used to describe the modular reconfigurable flight array comprises:
module coordinate system O M :{x M ,y M ,z M }; the module coordinate system takes the geometric center of the flight unit module as the origin of coordinates, z M The axis is vertical to the flying unit module and points to the sky, and the coordinate system meets the right-hand rule; the module coordinate system is mainly used for expressing the moment of inertia J = diag (J) of the flight unit module Mx ,J My ,J Mz ) Wherein J Mx Winding x for flying unit module M Moment of inertia of the shaft, J My For flying unit module winding y M Moment of inertia of the shaft, J Mz For flying unit module winding z M The rotational inertia of the shaft;
flight array coordinate system
The flight array coordinate system takes any point in the flight array as the origin of coordinates and coordinate axes/>
The direction of (a) is consistent with the direction of the module coordinate system; the flight array coordinate system is mainly used for describing the position information of the geometric center of the ith flight unit module in the flight array coordinate system>
And respectively wind>
Angular velocity of shaft rotation [ p, q, r] T ；
Inertial coordinate system O E :{x E ,y E ,z E }; the directions of all axes of the inertial coordinate system are consistent with those of all axes of the flight array coordinate system; inertial coordinate system for describing position X of modular reconfigurable flight array in three-dimensional space E ＝[x E ,y E ,z E ] T And attitude angle Θ = [ phi, theta, psi =] T Where phi is winding x E Angle of rotation of the shaft, theta being about y E Angle of rotation of the axis, psi being about z E The angle of rotation of (c).
3. The model of modular reconfigurable flight array dynamics according to claim 1, wherein the obtaining an inertia tensor for the modular reconfigurable flight array from the established coordinate system comprises: according to the established coordinate system, the rotational inertia parallel axis theorem is applied to make the rotational inertia J = diag (J) of the flight unit module Mx ,J My ,J Mz ) Calculating the inertia tensor of the modularized reconfigurable flight array based on the number n and the mass m of the single flight unit module
Wherein, the first and the second end of the pipe are connected with each other,
is the position of the geometric center of the ith flight unit module in the flight array coordinate system, J xx 、J yy 、J zz Are respectively wound>
Moment of inertia of the shaft, J xy 、J yx Is relative to->
Shaft and->
The product of inertia of the shaft.
4. The model of modular reconfigurable flight array dynamics according to claim 1, wherein the basis is an established coordinate system and an inertia tensor
Establishing a modularized reconfigurable flight array dynamic model, comprising the following steps: establishing a modularized reconfigurable flight array dynamic model by using a Newton-Euler method according to the established coordinate system and the inertia tensor; the modularized reconfigurable flight array dynamic model comprises a dynamic model of the translational motion of the modularized reconfigurable flight array and a dynamic model of the rotational motion of the modularized reconfigurable flight array;
the dynamic model of the translation motion of the modularized reconfigurable flight array is as follows:
in the formula (I), the compound is shown in the specification,
modularized reconfigurable flight array edge x under inertial coordinate system E Axis, y E Axis, z E Acceleration of the shaft; g is the acceleration of gravity; m is the mass of a single flight unit module; phi, theta, psi] T Respectively winding x for modularized reconfigurable flight array E Axis, y E Axis, z E Attitude angle of shaft rotation; c phi = cos (phi), s phi = sin (phi), c theta = cos (theta), s theta = sin (theta), c psi = cos (psi), s psi = sin (psi); t is z Lift required for flight of the modular reconfigurable flight array;
the dynamic model of the modularized reconfigurable flight array rotary motion is as follows:
in the formula (I), the compound is shown in the specification,
are respectively wound around x E Axial, and axial E Axial, around z E Angular acceleration of shaft rotation; />
Are respectively wound around x E Axial, and axial E Axial, and axial E Angular velocity of shaft rotation; j. the design is a square xx 、J yy 、J zz Are respectively wound>
Moment of inertia of shaft, J xy 、J yx Is relative to->
Shaft and->
The product of inertia of the shaft; [ M ] A x ,M y ,M z ] T Are respectively wound around x E Axis, y E Axis, z E The rotational moment of the shaft;
lift T in a kinetic model z Moment M x ,M y ,M z The method is obtained by orthogonal decomposition of force and moment, and the specific implementation form is as follows:
wherein the value of κ is related to the direction of rotation of the propeller: when the propeller is rotating clockwise, κ × =2; when the propeller is rotating counterclockwise, κ × =1; k is a radical of F Is the lift coefficient of the propeller; k is a radical of M Is the propeller torque coefficient; u. of i The actual control input of the actuator of the ith flight unit module, namely the pulse width;
respectively representing the position information of the geometric center of the ith flight unit module in a flight array coordinate system.
5. A fixed time sliding mode control method for a modular reconfigurable flight array is characterized by comprising the following steps:
the expression of lift and moment as claimed in claim 4, extracting the control efficiency matrix B e1 And a gravity center compensation matrix B e2 ；
According to the control efficiency matrix B e1 And a center of gravity compensation matrix B e2 Establishing a control distribution model;
establishing an energy-optimal control distribution strategy according to a control distribution model, and inputting virtual control generated by a fixed time sliding mode controller with variable exponential coefficients into tau = [ T ] z ,M x ,M y ,M z ] T Are mapped to the individual actuators in an energy-optimal relationship.
6. The method of claim 5The fixed time sliding mode control method of the modularized reconfigurable flight array is characterized in that a control efficiency matrix B is extracted according to a lift force and moment expression e1 And a gravity center compensation matrix B e2 The method comprises the following steps:
will lift force T z Moment M x ,M y ,M z The expression is rewritten as a matrix:
according to the matrix form of force and moment, the method is used for measuring the virtual control input T z ,M x ,M y ,M z ] T And actuator actual control input u 1 ,u 2 ,…,u n ] T Control efficiency matrix B of the relationship between e1 Write as:
and for describing the virtual control input [ T z ,M x ,M y ,M z ] T And a gravity center compensation matrix B of the masses of the individual flight cell modules e2 Write as:
in the formula, the value of κ is related to the rotation direction of the propeller: when the propeller is rotating clockwise, κ × =2; when the propeller is rotating counterclockwise, κ × =1; k is a radical of F Is the lift coefficient of the propeller; k is a radical of M Is the propeller torque coefficient; u. u n Actual control input for actuators of the nth flight unit module;
respectively represents the geometric center of the ith flight unit module in the flight arrayPosition information in a coordinate system, i =1,2, ·, n; m is 1 、m 2 、m n Respectively representing the mass of the 1 st, 2 nd and n th flight unit modules; g is the gravitational acceleration.
7. The fixed-time sliding-mode control method for the modular reconfigurable flight array according to claim 5, wherein the control efficiency-based matrix B e1 And a center of gravity compensation matrix B e2 Establishing a control distribution model, which specifically comprises the following steps: to control the efficiency matrix B e1 And a center of gravity compensation matrix B e2 Is of the form [ T z ,M x ,M y ,M z ] T Mapping to each actuator, and establishing a control distribution model as follows:
τ＝B e1 u+B e2 G
wherein, τ = [ T = z ,M x ,M y ,M z ] T ，G＝[m 1 g,m 2 g,…,m n g] T ；u＝[u 1 ,u 2 ,...u n ]；m n Representing the mass of the nth flying unit module; g is the acceleration of gravity; u. of n Is the actual control input to the actuators of the nth flight unit module.
8. The fixed-time sliding-mode control method for the modular reconfigurable flight array according to claim 5, wherein an energy-optimal control distribution strategy is established according to a control distribution model, and virtual control input τ = [ T ] generated by a fixed-time sliding-mode controller with variable exponential coefficient z ,M x ,M y ,M z ] T Mapping the energy optimal relationship to each actuator, specifically: according to the established control distribution model, a least square method is adopted to establish an energy optimal control distribution strategy, and virtual control input tau = [ T ] generated by a fixed time sliding mode controller with variable exponential coefficients is input z ,M x ,M y ,M z ] T Mapping into each actuator in an energy-optimal relationship:
s.t.τ＝B e1 u+B e2 G
solving the optimization equation to obtain:
wherein o (u) represents an objective function, referred to as an energy-optimal-based allocation strategy;
is a matrix B e1 The right generalized inverse matrix of (d); g = [ m ] 1 g,m 2 g,…,m n g] T ；u＝[u 1 ,u 2 ,...u n ]；m n Representing the mass of the nth flying unit module; g is the acceleration of gravity; u. of n Is the actual control input to the actuators of the nth flying unit module.
9. The modular reconfigurable flight array fixed-time sliding-mode control method according to claim 5, wherein the variable-exponent-coefficient fixed-time sliding-mode controller comprises:
establishing an error dynamic equation of the height and attitude angles according to a dynamic model:
in the formula: in the formula (I), the compound is shown in the specification,
lifting force T z Moment M x ,M y ,M z ；/>
Respectively, a height error, a roll angle error, a pitch angle error and a yaw angle error>
Desired altitude, roll angle, pitch angle, and yaw angle; [ h ] of z (t),h φ (t),h θ (t),h ψ (t)] T Unknown perturbations that are bounded; f. of φ (φ,θ,ψ)、f θ (phi, theta, psi) and f ψ (phi, theta, psi) represents a non-linear function; g 11 、g 21 ，g 22 、g 31 、g 32 And g 41 Representing a control gain function;
designing a sliding mode surface in the following form based on an error equation:
to analyze the dynamic behavior of the sliding mode surfaces, the derivation can be:
establishing a variable exponential coefficient sliding mode control law:
according to sliding mode control law and
obtaining a virtual control input τ = [ T = [) z ,M x ,M y ,M z ] T ；
In the formula: s 1 、s 2 、s 3 、s 4 Respectively representing sliding mode surfaces for controlling the height, the roll angle, the pitch angle and the yaw angle;
as hyperbolic tangent function, gamma 11 ,γ 12 ,…,γ 41 ,γ 42 >0，κ 1 ,κ 2 ,κ 3 ,κ 4 >1 and 0<ε 1 ,ε 2 ,ε 3 ,ε 4 <1 is a constant; sin (, is a sign function: when>At 0, sgn (= 1); when =0, sgn (= 0); when<At 0, sgn (=) = -1;
k δ1 ,k δ2 ,k δ3 >0 (δ =1,2,3,4) is the control gain, λ δ >0，μ δ >0，p δ >1 and satisfies lambda δ >p δ μ δ 。
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