CN114879657A - Model-free fully-distributed unmanned ship collaborative time-varying formation control method based on satellite coordinate system - Google Patents

Model-free fully-distributed unmanned ship collaborative time-varying formation control method based on satellite coordinate system Download PDF

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CN114879657A
CN114879657A CN202210324060.3A CN202210324060A CN114879657A CN 114879657 A CN114879657 A CN 114879657A CN 202210324060 A CN202210324060 A CN 202210324060A CN 114879657 A CN114879657 A CN 114879657A
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formation
follows
matrix
unmanned ship
coordinate system
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朱骋
黄兵
张磊
苏玉民
庄佳园
周彬
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Harbin Engineering University
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Harbin Engineering University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course or altitude of land, water, air, or space vehicles, e.g. automatic pilot
    • G05D1/02—Control of position or course in two dimensions
    • G05D1/0206—Control of position or course in two dimensions specially adapted to water vehicles

Abstract

The invention provides a model-free fully-distributed unmanned ship coordinated time-varying formation control method based on a satellite coordinate system, which comprises the following steps: (1) establishing a communication topology based on stress constraints; (2) establishing an expected formation and determining a control target; (3) designing a controller for affine formation; (4) verifying the stability and robustness of the unmanned ship formation system; the invention combines the affine transformation related concept to rapidly carry out formation transformation such as scaling, shearing, rotation and the like. Meanwhile, the boat body in the system can realize formation control under a rejection environment by depending on sensing interaction and inertia information of the boat, the complexity of the controller is effectively reduced, and the system has lower calculation amount.

Description

Model-free fully-distributed unmanned ship collaborative time-varying formation control method based on satellite coordinate system
Technical Field
The invention relates to a model-free fully-distributed unmanned ship coordinated time-varying formation control method based on a body-following coordinate system, which aims at a distributed unmanned ship coordinated control technology and is used for an unmanned ship cluster system.
Background
In recent years, with the deep exploration of oceans by human beings, unmanned boat formation is becoming an object of major attention of scholars. Compared with a single boat, the unmanned boat formation has obvious advantages in tasks such as sea area detection, coastal patrol, target surrounding and the like, so that the research on unmanned boat formation control undoubtedly has extremely high practical significance.
At present, unmanned ship formation is usually designed by a distributed architecture, and in terms of a distributed framework, research on communication topology is a foundation and a core. The distributed controller constructed based on the graph theory can only realize the tracking task of a fixed formation, but the time-varying formation tracking can be completed to avoid obstacles or pass through narrow areas due to the requirement of unmanned boat formation in the actual marine task. The existing control method designed by introducing a distance constraint or an azimuth angle constraint condition into a communication topology is difficult to realize complete formation change of rotation and scaling simultaneously, so that a stress constraint is introduced into the communication topology, an affine transformation related concept is combined to describe a desired formation with time-varying navigational speed and time-varying motion direction, combined transformation including rotation, shearing and scaling can be realized simultaneously, further, additional path planning operation is avoided, and the method has very high practicability. The existing topological structure based on stress constraint only focuses on the aspect of multi-intelligent control, and due to the nonlinear physical characteristics and strong coupling of unmanned boats, the existing related control method has certain complexity and communication burden, and in consideration of rejection phenomena such as magnetic field interference or signal interference existing in the complexity of marine environment, a sliding mode controller is designed by taking azimuth as a tracking error to realize formation control of formation members only through perception interaction (namely, through a laser radar and a visual device, relative positions and relative directions of adjacent boats are obtained) and inertial information of the boat (through a nine-axis sensor, an inertial navigation device and other devices), so that the calculation burden is effectively reduced, and the method is suitable for rejection environment.
Disclosure of Invention
The method aims to realize a control method capable of freely performing time-varying formation such as combination of rotation, shearing, zooming and the like, so that unmanned ship formation can flexibly avoid obstacles or pass through narrow water areas, and additional path planning operation can be effectively avoided. Compared with the traditional method based on distance constraint or azimuth angle constraint, the method introduces stress constraint in the topological structure, combines with the general rigid theory to enable the formation configuration to have affine transformation performance, further carries out controller design based on azimuth information, and can realize formation control only through perception interaction and the inertial information of the ship in a rejection environment.
The purpose of the invention is realized as follows: a model-free fully-distributed unmanned ship coordinated time-varying formation control method based on a satellite coordinate system comprises the following steps:
step 1: establishing a communication topology based on stress constraints;
step 2: establishing an expected formation and determining a control target;
and step 3: designing a controller for affine formation;
and 4, step 4: verifying the stability and robustness of the unmanned ship formation system;
1. a communication topology based on stress constraints is established in step 1, as follows:
firstly, a general Euler-Newton equation is adopted to express a motion model of the unmanned ship as follows:
Figure BDA0003571089910000021
Figure BDA0003571089910000022
in the formula eta i ＝col(p i ,ψ i ) To represent the unmanned boat state vector in the geodetic coordinate system, where p i ＝col(x i ,y i ) Is a position vector, ψ i Is the orientation of the unmanned boat. Theta i ＝col(υ i ,ω i ) Is an unmanned ship velocity vector expressed in a random coordinate system, wherein upsilon i ＝col(u i ,v i ) Is the linear velocity, omega i Is the angular velocity. J (psi) i ) For the rotation matrix, the specific formula is as follows:
Figure BDA0003571089910000023
for brevity of subsequent lines, we use J separately i And R i Alternative J (psi) i ) And R (psi) i )。
Figure BDA0003571089910000024
Expressed as the inertia matrix after the unmanned boat takes into account the additional mass forces.
Figure BDA0003571089910000025
Represented as a coriolis centripetal force matrix.
Figure BDA0003571089910000026
Expressed as a resistance matrix.
Figure BDA0003571089910000027
Expressed as a perturbation interference vector of the model parameters.
Figure BDA0003571089910000028
Represented as an external disturbance vector. Tau is i Represented as control command input.
To achieve affine locatability for unmanned boat formation, the definition of generic overall stiffness is given as follows:
lemma 1. for desired formation (G, p) * ) If the registered stress matrix is a semi-positive definite matrix, and full
Figure BDA0003571089910000029
When the Rank (omega) is N-3, the formation is considered to be generic rigid.
Considering a cluster system comprising N unmanned boats, defining an interaction topology between boats as G ═ (V, E), where V ═ 1, 2.., N } is a set of nodes (one node corresponds to one unmanned boat),
Figure BDA00035710899100000210
is a set of edges (two)The nodes are connected into an edge to represent that the two unmanned boats can interact). Registering a stress for each edge
Figure BDA00035710899100000211
When the stress of the whole formation is in an equilibrium state, the stress of each side satisfies the following formula:
Figure BDA00035710899100000212
the stress matrix obtained in the equilibrium state is:
Figure BDA0003571089910000031
by giving a general configuration (i.e. the basic geometry that the cluster is intended to maintain) of
Figure BDA0003571089910000032
Accordingly, the set of affine changes that can be solved for this configuration is:
Figure BDA0003571089910000033
in the formula (I), the compound is shown in the specification,
Figure BDA0003571089910000034
for performing the rotation, zoom and cut operations,
Figure BDA0003571089910000035
for performing a translation operation.
In order to achieve the requirement of topological affine locatability, the stress matrix is numerically solved according to theorem 1 to obtain a suitable stress constraint:
first, define the incidence matrix
Figure BDA0003571089910000036
Where m is the number of elements in the edge set E. Correlation matrix and Laplace momentsThe relationship of the matrix is H T H is L. The following relationship is obtained:
Figure BDA0003571089910000037
order to
Figure BDA0003571089910000038
Is a group of Null (E) radicals. In practice, we can obtain a set of orthogonal bases for null (E) by computing the singular value decomposition of E. On the other hand, will expand the configuration
Figure BDA0003571089910000039
Is decomposed into singular values
Figure BDA00035710899100000310
Here, we have U ═ col (U ═ col) 1 ,U 2 ) Wherein U is 1 The first 3 columns of U. The following equation is obtained:
Figure BDA00035710899100000311
accordingly, parameter c is obtained by solving the following linear matrix inequality 1 ,...,c o ：
Figure BDA00035710899100000312
To ensure the requirement of affine locatability, we use the following formula to find the stress vector:
Figure BDA00035710899100000313
2. in step 2, a desired formation is established and a control target is determined, as follows:
in order to ensure that the heading of each boat in the formation navigation process is kept consistent, a one-dimensional variable is adopted
Figure BDA00035710899100000314
Describing the expected heading of the formation, and defining the expected formation as (G, p) * ) Wherein
Figure BDA00035710899100000315
Representing the pattern that the unmanned ship group is expected to form during sailing. The time-varying configuration is obtained from the affine variation set as follows:
Figure BDA0003571089910000041
the center of the desired configuration is
Figure BDA0003571089910000042
The direction of motion of the desired formation can thus be:
Figure BDA0003571089910000043
defining a desired direction of the ith boat as
Figure BDA0003571089910000044
From the above definition of the desired trajectory, we can obtain the conventional tracking error for each boat as:
Figure BDA0003571089910000045
the distributed tracking error is defined as:
Figure BDA0003571089910000046
the following relative errors were used as control measures:
Figure BDA0003571089910000047
in the formula, delta ij Represents the coordinate position of the j unmanned boat in the i boat coordinate system (namely the relative position of the j boat), sigma ij Representing the angle of the orientation of the j-th unmanned boat to the longitudinal navigational axis in the i-boat coordinate system (i.e., the relative orientations of the j-boats). From a mathematical point of view, we can know that the relationship between the relative measurement value and the global measurement value is as follows:
Figure BDA0003571089910000048
the control targets are determined as follows:
Figure BDA0003571089910000049
3. the controller design for affine formation in step 3 is as follows:
according to the properties of the Newton-Euler motion model and the rotation matrix, the Euler-Lagrange motion model can be deduced as follows:
Figure BDA00035710899100000410
in the formula (I), the compound is shown in the specification,
Figure BDA00035710899100000411
in the form of an inertia matrix in which,
Figure BDA00035710899100000412
in the form of a centripetal force matrix,
Figure BDA00035710899100000413
in the form of a damping matrix in this manner,
Figure BDA00035710899100000414
to include disturbance lumped.
Based on the relative error given above, the following distributed sliding mode surfaces are designed:
Figure BDA00035710899100000415
in the formula (I), the compound is shown in the specification,
Figure BDA0003571089910000051
is a positive control gain.
The controller is designed as follows:
Figure BDA0003571089910000052
in the formula (I), the compound is shown in the specification,
Figure BDA0003571089910000053
and
Figure BDA0003571089910000054
two adaptive parameters are used for solving the uncertain problem and the completely distributed problem respectively. Gamma-shaped i ＝col(Γ i,1 ,...,Γ i,4 ) Is one and adaptive parameter
Figure BDA0003571089910000055
Related vector by definition
Figure BDA0003571089910000056
Wherein theta is specifically represented as
Figure BDA0003571089910000057
Then we have r i J-th action of (1):
Γ i,j ＝||col(||s i ||Λ i,j ,ε)||
the updating law of the adaptive parameters in the formula is designed as follows:
Figure BDA0003571089910000058
Figure BDA0003571089910000059
Figure BDA00035710899100000510
in the formula I i,j ，q i,j ,j∈1,...,4，ρ i And
Figure BDA00035710899100000511
both are positive control gains.
4. Stability analysis was performed on the unmanned boat formation system at step 4 as follows:
we constructed the following lyapunov function:
Figure BDA00035710899100000512
in the formula (I), the compound is shown in the specification,
Figure BDA00035710899100000513
is a constant that is used only for stability analysis,
Figure BDA00035710899100000514
is an unknown constant, and the specific expression thereof is as follows:
Figure BDA00035710899100000515
all signals that can ultimately result in a closed loop system can ultimately be bounded consistently, and further, we can conclude that:
Figure BDA00035710899100000516
compared with the prior art, the invention has the beneficial effects that: the invention combines the affine transformation related concept to rapidly carry out formation transformation such as scaling, shearing, rotation and the like. Meanwhile, the boat body in the system can realize formation control under a rejection environment by depending on sensing interaction and inertia information of the boat, the complexity of the controller is effectively reduced, and the system has lower calculation amount.
Drawings
FIG. 1 is a flow chart of a radial formation time-varying formation controller method according to the present invention;
FIG. 2 is a conventional configuration and stress distribution of a desired formation;
FIG. 3 is a diagram of the trajectory of the formation in a fleet of unmanned boats;
FIG. 4 is a plot of unmanned boat trajectory tracking error within a cluster system;
fig. 5 control output of a controlled drones within the cluster system.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and specific embodiments.
With reference to fig. 1 to 5, the steps of the present invention are as follows:
in a first step, a communication topology based on stress constraints is established as follows:
firstly, a general Euler-Newton equation is adopted to express a motion model of the unmanned ship as follows:
Figure BDA0003571089910000061
Figure BDA0003571089910000062
in the formula eta i ＝col(p i ,ψ i ) To represent the unmanned boat state vector in the geodetic coordinate system, where p i ＝col(x i ,y i ) Is a position vector, ψ i Is the orientation of the unmanned boat. Theta i ＝col(υ i ,ω i ) Is an unmanned ship velocity vector expressed in a random coordinate system, wherein upsilon i ＝col(u i ,v i ) Is the linear velocity, omega i Is the angular velocity. J (psi) i ) For the rotation matrix, the specific formula is as follows:
Figure BDA0003571089910000063
for brevity of subsequent lines, we use J separately i And R i Alternative J (psi) i ) And R (psi) i )。
Figure BDA0003571089910000064
Expressed as the inertia matrix after the unmanned boat takes into account the additional mass forces.
Figure BDA0003571089910000065
Represented as a coriolis centripetal force matrix.
Figure BDA0003571089910000066
Expressed as a resistance matrix.
Figure BDA0003571089910000067
Expressed as a perturbation interference vector of the model parameters.
Figure BDA0003571089910000068
Represented as an external disturbance vector. Tau is i Represented as control command input.
To achieve affine locatability for unmanned boat formation, the definition of generic overall stiffness is given as follows:
lemma 1. for desired formation (G, p) * ) If the registered stress matrix is a semi-positive definite matrix, and full
Figure BDA0003571089910000069
When the Rank (omega) is N-3, the formation is considered to be generic rigid.
Considering a cluster system comprising N unmanned boats, the interaction topology between boats is defined as G ═ (V, E), where V ═ {1, 2.., N } is the set of nodes (one node corresponds to one unmanned boat)A boat),
Figure BDA00035710899100000610
is an edge set (two nodes are connected to form an edge, which means that two unmanned boats can interact). Registering a stress for each edge
Figure BDA00035710899100000611
When the stress of the whole formation is in an equilibrium state, the stress of each side satisfies the following formula:
Figure BDA0003571089910000071
the stress matrix obtained in the equilibrium state is:
Figure BDA0003571089910000072
by giving a general configuration (i.e. the basic geometry that the cluster is intended to maintain) of
Figure BDA0003571089910000073
Accordingly, the set of affine changes that can be solved to arrive at the configuration may be:
Figure BDA0003571089910000074
in the formula (I), the compound is shown in the specification,
Figure BDA0003571089910000075
for performing the rotation, zoom and cut operations,
Figure BDA0003571089910000076
for performing a translation operation. In order to achieve the requirement of topological affine locatability, the stress matrix is numerically solved according to theorem 1 to obtain a suitable stress constraint:
first, define the incidence matrix
Figure BDA0003571089910000077
Where m is the number of elements in the edge set E. The relation between the incidence matrix and the Laplace matrix is H T H is L. The following relationship is obtained:
Figure BDA0003571089910000078
order to
Figure BDA0003571089910000079
Is a group of Null (E) radicals. In practice, we can obtain a set of orthogonal bases for null (E) by computing the singular value decomposition of E. On the other hand, will expand the configuration
Figure BDA00035710899100000710
Is decomposed into singular values
Figure BDA00035710899100000711
Here, we have U ═ col (U ═ col) 1 ,U 2 ) Wherein U is 1 The first 3 columns of U. The following equation is obtained:
Figure BDA00035710899100000712
accordingly, parameter c is obtained by solving the following linear matrix inequality 1 ,...,c o ：
Figure BDA00035710899100000713
To ensure the requirement of affine locatability, we use the following formula to find the stress vector:
Figure BDA00035710899100000714
second, establish the desired formation and determine the control targets, as follows:
in order to ensure that the heading of each boat in the formation navigation process is kept consistent, a one-dimensional variable is adopted
Figure BDA00035710899100000715
Describing the expected heading of the formation, and defining the expected formation as (G, p) * ) Wherein
Figure BDA00035710899100000716
Representing the pattern that the unmanned ship group is expected to form during sailing. The time-varying configuration is obtained from the affine variation set as follows:
Figure BDA0003571089910000081
the center of the desired configuration is
Figure BDA0003571089910000082
The direction of motion of the desired formation can thus be:
Figure BDA0003571089910000083
defining a desired direction of the ith boat as
Figure BDA0003571089910000084
From the above definition of the desired trajectory, we can obtain the conventional tracking error for each boat as:
Figure BDA0003571089910000085
the distributed tracking error is defined as:
Figure BDA0003571089910000086
the following relative errors were used as control measures:
Figure BDA0003571089910000087
in the formula, delta ij Represents the coordinate position of the j unmanned boat in the i boat coordinate system (namely the relative position of the j boat), sigma ij Representing the angle of the orientation of the j-th unmanned boat to the longitudinal navigational axis in the i-boat coordinate system (i.e., the relative orientations of the j-boats). From a mathematical point of view, we can know that the relationship between the relative measurement value and the global measurement value is as follows:
Figure BDA0003571089910000088
the control targets are determined as follows:
Figure BDA0003571089910000089
thirdly, designing a controller for affine formation, as follows:
according to the properties of the Newton-Euler motion model and the rotation matrix, the Euler-Lagrange motion model can be deduced as follows:
Figure BDA00035710899100000810
in the formula (I), the compound is shown in the specification,
Figure BDA00035710899100000811
in the form of an inertia matrix in which,
Figure BDA00035710899100000812
in the form of a centripetal force matrix,
Figure BDA00035710899100000813
in the form of a damping matrix in this manner,
Figure BDA00035710899100000814
to contain the perturbation lump.
Based on the relative error given above, we design the following distributed sliding mode surfaces:
Figure BDA0003571089910000091
in the formula (I), the compound is shown in the specification,
Figure BDA0003571089910000092
is a forward control gain.
The controller is designed as follows:
Figure BDA0003571089910000093
in the formula (I), the compound is shown in the specification,
Figure BDA0003571089910000094
and
Figure BDA0003571089910000095
two adaptive parameters are used for solving the uncertain problem and the completely distributed problem respectively. Gamma-shaped i ＝col(Γ i,1 ,...,Γ i,4 ) Is one and adaptive parameter
Figure BDA0003571089910000096
Related vector by definition
Figure BDA0003571089910000097
Wherein theta is specifically represented as
Figure BDA0003571089910000098
Then we have r i J-th action of (1):
Γ i,j ＝||col(||s i ||Λ i,j ,ε)||
the updating law of the adaptive parameters in the formula is designed as follows:
Figure BDA0003571089910000099
Figure BDA00035710899100000910
Figure BDA00035710899100000911
in the formula I i,j ，q i,j ,j∈1,...,4，ρ i And
Figure BDA00035710899100000912
both are positive control gains.
Fourthly, verifying the stability of the unmanned ship formation system:
we constructed the following lyapunov function:
Figure BDA00035710899100000913
in the formula (I), the compound is shown in the specification,
Figure BDA00035710899100000914
is a constant that is used only for stability analysis,
Figure BDA00035710899100000915
for unknown constants, the specific expression is as follows:
Figure BDA00035710899100000916
all signals that can ultimately result in a closed loop system can ultimately be bounded consistently, and further, we can conclude that:
Figure BDA00035710899100000917
the performance of the above controller will be demonstrated and verified by the simulation example.
Consider a cluster system comprising 7 drones, of which 3 are pilots and the remaining 4 are followers. The specific ship model parameters are shown in table 1 below, and all parameters not mentioned in the table are default to 0. To fit more closely to the actual scene, we give the actuator saturation constraint, namely
Figure BDA0003571089910000101
TABLE 1
Figure BDA0003571089910000102
The internal disturbance and the external disturbance of the model are respectively set as follows:
Figure BDA0003571089910000103
Figure BDA0003571089910000104
introducing noise generated by second-order Gauss Markov process iteration into external disturbance
Figure BDA0003571089910000105
The specific expression is as follows:
Figure BDA0003571089910000106
in the formula (I), the compound is shown in the specification,
Figure BDA0003571089910000107
is zero mean white gaussian noise.
According to the method for constructing the desired formation, we construct the following desired formation: selection of conventionsConfiguration is r 1 ＝col(32,0)，r 2 ＝col(16,16)，r 3 ＝col(16,-16)，r 4 ＝col(0,16)，r 5 ＝col(0,-16)，r 6 Col (-16,16), and r 7 Col (-16 ). This indicates the expanded conventional configuration
Figure BDA0003571089910000108
Satisfy the requirement of
Figure BDA0003571089910000109
As shown in fig. 2, the registered stress matrix is a semi-positive definite matrix and satisfies Rank (Ω) ═ N-3, and more specifically, we can calculate:
λ(Ω)＝{0,0,0,0.6458,1.1389,2.5230,2.7546}
λ(Ω ff )＝{0.6458,1.1389,2.5230,2.7546}
regarding the selection of control gain, we have a sliding mode surface related parameter of k i 1, and the adaptive law related parameter is l i,1 ＝0.008，l i,2 ＝0.005，l i,3 ＝0.001，l i,4 ＝0.0005，q i,1 ＝20，q i,2 ＝20，q i,3 ＝30，q i,4 ＝200，ρ i 0.001 and
Figure BDA0003571089910000111
adaptive gain
Figure BDA0003571089910000112
And
Figure BDA0003571089910000113
are all set to 0.
The detailed simulation results are shown in fig. 3-5. As can be seen from fig. 3, the drones cluster system implements a combined transformation consisting of rotation, scaling and shearing, where we consider three dark red squares as obstacles to simulate narrow bodies of water and roadblocks. From fig. 4, it can be found intuitively that the tracking error of each boat is stable in a very small area. In fig. 4, there is a case where a step-like increase occurs in the tracking error, because the path information is given in segments (by setting the path information in segments, it is tested whether the control system can quickly respond to a case where a path is expected to suddenly change, so that the capability of the control system to cope with an emergency situation can be embodied). As can be seen from the enlarged partial views of fig. 5 and 4, the controller of the present invention can respond to the tracking error quickly. Furthermore, we can find in conjunction with fig. 1 that the correction of the adaptive parameter is an important reason that the controller can respond to the tracking error quickly, i.e. similar to the high gain controller, when the error becomes large, the adaptive parameter is self-corrected, so as to ensure that the controller can output a more reasonable signal to make the error converge.

Claims (5)

1. A model-free fully-distributed unmanned ship coordinated time-varying formation control method based on a satellite coordinate system is characterized by comprising the following steps:
step 1: establishing a communication topology based on stress constraints;
step 2: establishing an expected formation and determining a control target;
and step 3: designing a controller for affine formation;
and 4, step 4: and verifying the stability and robustness of the unmanned ship formation system.
2. The method for controlling the collaborative time-varying formation of the unmanned ship based on the modeless fully distributed along with the coordinate system of the body as claimed in claim 1, wherein a communication topology based on stress constraint is established in step 1 as follows:
firstly, a general Euler-Newton equation is adopted to express a motion model of the unmanned ship as follows:
Figure FDA0003571089900000011
Figure FDA0003571089900000012
in the formula eta i ＝col(p i ,ψ i ) To represent the unmanned boat state vector in the geodetic coordinate system, where p i ＝col(x i ,y i ) Is a position vector, ψ i Is the orientation of the unmanned boat; theta i ＝col(υ i ,ω i ) Is an unmanned ship velocity vector expressed in a random coordinate system, wherein upsilon i ＝col(u i ,v i ) Is the linear velocity, omega i Is the angular velocity; j (psi) i ) For the rotation matrix, the specific formula is as follows:
Figure FDA0003571089900000013
by J i And R i Alternative J (psi) i ) And R (psi) i )；
Figure FDA0003571089900000014
Representing an inertia matrix after considering the additional mass force for the unmanned boat;
Figure FDA0003571089900000015
expressed as a coriolis centripetal force matrix;
Figure FDA0003571089900000016
expressed as a resistance matrix;
Figure FDA0003571089900000017
expressed as a perturbation interference vector of model parameters;
Figure FDA0003571089900000018
expressed as an external disturbance vector; tau is i Expressed as a control command input;
defining an incidence matrix
Figure FDA0003571089900000019
Wherein m is the number of elements in the edge set E; the relation between the incidence matrix and the Laplace matrix is H T H ═ L, the following relationship is obtained:
Figure FDA00035710899000000110
let z 1 ,...,
Figure FDA00035710899000000111
A group of radicals of null (E); obtaining a set of orthogonal bases for null (E) by computing the singular value decomposition of E, which will expand the configuration
Figure FDA00035710899000000112
Is decomposed into singular values
Figure FDA00035710899000000113
With U ═ col (U) 1 ,U 2 ) Wherein U is 1 For the first 3 columns of U, the following equation is obtained:
Figure FDA0003571089900000021
accordingly, parameter c is obtained by solving the following linear matrix inequality 1 ,...,c o ：
Figure FDA0003571089900000022
To ensure the requirement of affine locatability, the following formula is adopted to obtain the stress vector:
Figure FDA0003571089900000023
3. the method for controlling the time-varying formation in cooperation with the model-free fully-distributed unmanned ship based on the random coordinate system as claimed in claim 1, wherein the desired formation is established and the control target is determined in step 2 as follows:
using one-dimensional variables
Figure FDA0003571089900000024
Describing the expected heading of the formation, and defining the expected formation as (G, p) * ) Wherein
Figure FDA0003571089900000025
Representing a pattern expected to be formed in the sailing process of the unmanned boat group; the time-varying configuration is obtained from the affine variation set as follows:
Figure FDA0003571089900000026
the center of the desired configuration is
Figure FDA0003571089900000027
The direction of motion of the desired formation can thus be:
Figure FDA0003571089900000028
defining a desired direction of the ith boat as
Figure FDA0003571089900000029
From the above definition of the desired trajectory, the conventional tracking error for each boat can be found as:
Figure FDA00035710899000000210
the distributed tracking error is defined as:
Figure FDA00035710899000000211
the following relative errors were used as control measures:
Figure FDA00035710899000000212
in the formula, delta ij Representing the coordinate position of the jth unmanned ship in the i-ship coordinate system ij And representing the included angle between the orientation of the j unmanned ship and the longitudinal navigational speed axis under the i ship coordinate system, the relation between the relative measurement value and the global measurement value is as follows:
Figure FDA00035710899000000213
σ ij ＝ψ i -ψ j ,i∈V f &j∈N i
the control targets are determined as follows:
Figure FDA0003571089900000031
4. the method for controlling the collaborative time-varying formation of the unmanned ship based on the modeless fully distributed along with the coordinate system of the body as claimed in claim 1, wherein the controller design of the affine formation is performed in step 3 as follows:
according to the properties of a Newton-Euler motion model and a rotation matrix, the Euler-Lagrange motion model can be deduced as follows:
Figure FDA0003571089900000032
in the formula (I), the compound is shown in the specification,
Figure FDA0003571089900000033
in the form of an inertia matrix in which,
Figure FDA0003571089900000034
in the form of a centripetal force matrix,
Figure FDA0003571089900000035
in the form of a damping matrix in this manner,
Figure FDA0003571089900000036
to contain perturbation lumped; based on the relative error given above, the following distributed sliding mode surfaces are designed:
Figure FDA0003571089900000037
in the formula (I), the compound is shown in the specification,
Figure FDA0003571089900000038
a control gain in a forward direction;
the controller is designed as follows:
Figure FDA0003571089900000039
in the formula (I), the compound is shown in the specification,
Figure FDA00035710899000000310
and
Figure FDA00035710899000000311
two adaptive parameters are respectively used for solving the uncertain problem and the completely distributed problem; gamma-shaped i ＝col(Γ i,1 ,...,Γ i,4 ) Is one and adaptive parameter
Figure FDA00035710899000000312
Related vector by definition
Figure FDA00035710899000000313
Wherein theta is specifically represented as
Figure FDA00035710899000000314
Then gamma is i J-th action of (1):
Γ i,j ＝||col(||s i ||Λ i,j ,ε)||
the updating law of the adaptive parameters in the formula is designed as follows:
Figure FDA00035710899000000315
Figure FDA00035710899000000316
Figure FDA00035710899000000317
in the formula I i,j ，q i,j ,j∈1,...,4，ρ i And
Figure FDA00035710899000000318
both are positive control gains.
5. The method for controlling the model-free fully-distributed unmanned ship coordinated time-varying formation based on the satellite coordinate system according to claim 1, wherein the stability analysis is performed on the unmanned ship formation system in step 4, as follows:
the following Lyapunov function was constructed:
Figure FDA0003571089900000041
in the formula (I), the compound is shown in the specification,
Figure FDA0003571089900000042
is a constant that is used only for stability analysis,
Figure FDA0003571089900000043
is an unknown constant, and the specific expression thereof is as follows:
Figure FDA0003571089900000044
finally, all signals of the closed-loop system can be obtained to be finally and consistently bounded, and further, the following is obtained:
Figure FDA0003571089900000045
CN202210324060.3A 2022-03-29 2022-03-29 Model-free fully-distributed unmanned ship collaborative time-varying formation control method based on satellite coordinate system Pending CN114879657A (en)

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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115617039A (en) * 2022-09-15 2023-01-17 哈尔滨工程大学 Event trigger based distributed affine unmanned ship formation controller construction method and unmanned ship formation control method
CN115933631A (en) * 2022-09-14 2023-04-07 哈尔滨工程大学 Formation controller construction method and device applied to under-actuated unmanned ship

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115933631A (en) * 2022-09-14 2023-04-07 哈尔滨工程大学 Formation controller construction method and device applied to under-actuated unmanned ship
CN115617039A (en) * 2022-09-15 2023-01-17 哈尔滨工程大学 Event trigger based distributed affine unmanned ship formation controller construction method and unmanned ship formation control method
CN115617039B (en) * 2022-09-15 2023-06-13 哈尔滨工程大学 Event triggering-based distributed affine unmanned aerial vehicle formation controller construction method and unmanned aerial vehicle formation control method

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