CN110879600A - Multi-underwater robot system coordination control method based on distributed predictive control - Google Patents

Multi-underwater robot system coordination control method based on distributed predictive control Download PDF

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CN110879600A
CN110879600A CN201911223257.2A CN201911223257A CN110879600A CN 110879600 A CN110879600 A CN 110879600A CN 201911223257 A CN201911223257 A CN 201911223257A CN 110879600 A CN110879600 A CN 110879600A
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underwater robot
underwater
robot system
module
target object
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陈国军
陈丝雨
陈巍
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Nanjing Institute of Technology
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Nanjing Institute of Technology
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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
    • G05D1/04—Control of altitude or depth
    • G05D1/06—Rate of change of altitude or depth
    • G05D1/0692—Rate of change of altitude or depth specially adapted for under-water vehicles

Abstract

The invention discloses a distributed predictive control-based coordinated control method for a multi-underwater robot system, belongs to the technical field of robots, establishes the multi-underwater robot system, a data acquisition server and a control strategy server, solves the technical problem of a distributed coordinated control strategy of the multi-underwater robot system in a limited working space with static obstacles, and provides a distributed nonlinear model predictive control systemNMPCTo realize giving in a limited work spaceNThe underwater robot system firmly grasps a target object, navigates the underwater robot from an initial position to a final position by using coupling dynamics between the robot and the object and using a certain load distribution coefficient, and controls the underwater robot to move along a path calculated in a working space through a coordinated control strategy.

Description

Multi-underwater robot system coordination control method based on distributed predictive control
Technical Field
The invention belongs to the technical field of robots, and relates to a multi-underwater robot system coordination control method based on distributed predictive control.
Background
Underwater robots have been widely used in various applications such as marine science, ship maintenance, oil and gas facility inspection, and the like. The multiple underwater robot systems are cooperatively controlled, and the requirement on a group control method is high-efficiency and accurate. On the other hand, multi-underwater robots are harsh in working conditions, the most difficult is the strict communication constraint requirement, and the adoption of a communication-based control structure in an underwater environment may cause serious performance problems due to the limited bandwidth and communication rate of underwater acoustic equipment. Furthermore, the number of individuals of the multi-underwater robot system is severely limited due to the narrow bandwidth of the sound communication device.
The coordination control of the underwater robot is divided into a centralized control strategy and a distributed control strategy. The centralized control strategy is efficient, but has poor stability, and as the number of underwater robots increases, the complexity of the system also increases rapidly. While the distributed control strategy usually depends on data communication among the multi-underwater robots, the working space speed is realized by transmitting data among the multi-underwater robots in a coordinated manner, but due to the limitation of communication bandwidth, the number of underwater robots participating in cooperative underwater work is also limited.
Disclosure of Invention
The invention aims to provide a distributed predictive control-based coordinated control method for a multi-underwater robot system, which solves the technical problem of a distributed coordinated control strategy for the multi-underwater robot system in a limited working space with static obstacles.
In order to achieve the purpose, the invention adopts the following technical scheme:
a multi-underwater robot system coordination control method based on distributed predictive control comprises the following steps:
step 1: establishing a multi-underwater robot system, a data acquisition server and a control strategy server, wherein the multi-underwater robot system comprises a plurality of underwater robot systems, each underwater robot system is provided with an underwater robot sensor group, each underwater robot sensor group comprises an inertial measurement unit IMU, an ultra-short baseline positioning system USBL and a Doppler velocimeter DVL, and the underwater robot sensor groups transmit all acquired data to the data acquisition server for data fusion to finally generate position information and speed information of the underwater robot;
step 2: establishing a working area module, a constraint module, a navigation module, a control strategy generation module and a mathematical modeling module in a control strategy server;
and step 3: the method comprises the steps that a data acquisition server presets sampling moments, at each sampling moment, the data acquisition server acquires data of a primary underwater robot sensor group and generates position information and speed information of an underwater robot, a mathematical modeling module reads the position information and the speed information of the underwater robot, and a distributed dynamic mathematical model of each underwater robot system is established according to the position information and the speed information;
and 4, step 4: the working area module carries out sphere modeling on the obstacles, the underwater robot and the working space by adopting a sphere world representation method to generate a working space W:
step A1: setting B (x)O,r0) Is a closed sphere covering the volume of the object and having a radius r0；
Step A2: defining a closed sphere centered on an end effector of each underwater robotic system
Figure BDA0002301438430000021
The closed sphere
Figure BDA0002301438430000022
The underwater robot volume of all possible configurations is covered;
step A3: defining a target object, the distance between the grabbing points on the target object being at least
Figure BDA0002301438430000023
Step A4: is defined at a radius of
Figure BDA0002301438430000031
X of0Ball region B (x)OR) in the spherical region B (x)OR) comprises the volume of the multi-underwater robot system and the volume of the target object;
step A5: will sphere region B (x)OAnd M static obstacles present in R) are defined as being composed of
Figure BDA0002301438430000032
Wherein
Figure BDA0002301438430000033
Is the center of the image, and the center of the image,
Figure BDA0002301438430000034
is a barrier pimM is a positive integer;
and 5: in the working space W, the navigation module designs the desired motion trajectory and velocity of the target object according to the following formulas:
Figure BDA0002301438430000035
wherein
Figure BDA0002301438430000036
Is represented in a workspace
Figure BDA0002301438430000037
The potential of the safe motion vector field is internally derived, k is a design constant and k > 1,
Figure BDA0002301438430000038
indicating the position of the target
Figure BDA0002301438430000039
And γ (0) is 0;
the required motion trail path of the target object is designed as follows:
Figure BDA00023014384300000310
wherein KNFIs a positive gain. Defining a sampling instant tjWherein the sampling time is h, h e (0, T)P) Is constant: t is tj+1＝tj+h；
xO(tj) Is given target object at time tjCurrent position of vO(tj) Is given that the target object is presentTime tjCurrent speed of each underwater robotic system, a time interval S e [ t ] that each underwater robotic system can propagatej,tj+TP]Wherein T isPIs the prediction horizon;
step 6: the constraint module provides a group of system state constraint sets X for each underwater robot systemi；
And 7: the input constraints of each underwater robot system are specified in a constraint module as:
Figure BDA0002301438430000041
and
Figure BDA0002301438430000042
wherein tau isiIs a vector comprising each of the driven joints
Figure BDA0002301438430000043
In the respective limit ranges of (a) and (b),
Figure BDA0002301438430000044
τnis the number of joints that are activated;
defining a control input constraint set of an underwater robot system as Ti：
τi(t)∈Ti；
Wherein:
Figure BDA0002301438430000045
and 8: the control strategy generation module generates an optimal control input track according to the following method:
step B1: setting the current state measurement value to xi(tj) Discrete sampling time tj；
Step B2: for a multi-underwater robotic system, the open-loop input signal applied between sampling instants is given by solving the following equation:
Figure BDA0002301438430000046
wherein:
Figure BDA0002301438430000047
in the formula:
f and E are running and terminal cost functions, respectively, both in quadratic form, i.e.
Figure BDA0002301438430000048
And
Figure BDA0002301438430000049
Px、Qx、Qvand R are respectively positive definite matrices to be properly adjusted,
Figure BDA00023014384300000410
is at time instance tjIn the state of (a) in (b),
Figure BDA00023014384300000411
is the input trajectory of the input object,
Figure BDA00023014384300000412
is the optimal control input trajectory;
step B3: the control strategy generation module sends the optimal control input track to the multi-underwater robot system;
and step 9: at the next sampling instant tj+1，tj+1＝tj+ h, continuing the generation according to the method from step 3 to step 8 at the sampling instant tj+1The input trajectory is optimally controlled.
Preferably, when step 3 is executed, the mathematical modeling module establishes a distributed kinetic mathematical model of each underwater robotic system according to the following formula:
Figure BDA0002301438430000051
wherein the content of the first and second substances,
Figure BDA0002301438430000052
Figure BDA0002301438430000053
Figure BDA0002301438430000054
preferably, when step 6 is executed, the system state constraint set XiComprising a plurality of constraint terms, embodied as:
these constraints are represented by the following equations:
xi(t)∈Xi(ii) a Wherein, XiConsists of the following constraints:
Figure BDA0002301438430000055
wherein the content of the first and second substances,
Figure BDA0002301438430000056
is a singular set of positions for a multi-underwater robotic system,
Figure BDA0002301438430000057
is the set of joint limits of the manipulator:
Figure BDA0002301438430000058
wherein the content of the first and second substances,
Figure BDA0002301438430000061
is corresponding to the joint
Figure BDA0002301438430000062
In the range of the limit (c) of (c),
Figure BDA0002301438430000063
is corresponding to the joint
Figure BDA0002301438430000064
Joint velocity of (1), set XiAll state constraints of the multi-underwater robotic system are centralized.
The invention relates to a coordinated control method of a multi-underwater robot system based on distributed predictive control, which solves the technical problem of a distributed coordinated control strategy of the multi-underwater robot system in a limited working space with a static obstacle, provides a distributed nonlinear model predictive control system NMPC (non-linear predictive control), realizes that N underwater robot systems firmly grab a target object in the limited working space, guides an underwater robot from an initial position to a final position by utilizing the coupling dynamics between the robot and the object and using a certain load distribution coefficient, controls the underwater robot to move along a path calculated in the working space by the coordinated control strategy, and can ensure that the multi-underwater robot system can reduce communication data between the robots during the execution of cooperative underwater work, therefore, the required communication bandwidth is reduced, and the working efficiency of the multi-underwater robot system is improved.
Drawings
Fig. 1 is a schematic diagram of the operation track of the multi-underwater robot system of the present invention.
Detailed Description
Fig. 1 shows a coordinated control method for a multi-underwater robot system based on distributed predictive control, which includes the following steps:
step 1: establishing a multi-underwater robot system, a data acquisition server and a control strategy server, wherein the multi-underwater robot system comprises a plurality of underwater robot systems, each underwater robot system is provided with an underwater robot sensor group, each underwater robot sensor group comprises an inertial measurement unit IMU, an ultra-short baseline positioning system USBL and a Doppler velocimeter DVL, and the underwater robot sensor groups transmit all acquired data to the data acquisition server for data fusion to finally generate position information and speed information of the underwater robot;
step 2: establishing a working area module, a constraint module, a navigation module, a control strategy generation module and a mathematical modeling module in a control strategy server;
and step 3: the method comprises the steps that a data acquisition server presets sampling moments, at each sampling moment, the data acquisition server acquires data of a primary underwater robot sensor group and generates position information and speed information of an underwater robot, a mathematical modeling module reads the position information and the speed information of the underwater robot, and a distributed dynamic mathematical model of each underwater robot system is established according to the position information and the speed information;
for a bounded workspace W ∈ R3N underwater robot systems operating in (1) by first
Figure BDA0002301438430000071
Coordinates representing an end effector of each underwater robot system
Figure BDA0002301438430000072
Neutralization
Figure BDA0002301438430000073
The position and orientation of the w.r.t inertial frame are expressed in terms of euler angles. Is provided with
Figure BDA0002301438430000074
Is a joint state variable of each underwater robot system, wherein
Figure BDA0002301438430000075
Is an inclusion position
Figure BDA0002301438430000076
And underwater robot direction
Figure BDA0002301438430000077
Vector of (A), BiIs a vector of the joint angle positions of the manipulator. In particular, the present invention relates to a method for producing,
Figure BDA0002301438430000078
and
Figure BDA0002301438430000079
the position and orientation of the w.r.t inertial frame are expressed in terms of euler angles. Can also use
Figure BDA00023014384300000710
To define an end effector generalized velocity of an underwater robotic system, wherein
Figure BDA00023014384300000711
And
Figure BDA00023014384300000712
respectively, linear velocity and angular velocity.
Preferably, when step 3 is executed, the mathematical modeling module establishes a distributed kinetic mathematical model of each underwater robotic system according to the following formula:
Figure BDA00023014384300000713
wherein the content of the first and second substances,
Figure BDA00023014384300000714
Figure BDA00023014384300000715
Figure BDA00023014384300000716
Figure BDA0002301438430000081
is a distributed dynamic mathematical model of an underwater robot system,
Figure BDA0002301438430000082
xirepresenting the coordinate position of the underwater robot, wherein
Figure BDA0002301438430000083
And q isiIs a local measurement coordinate value, uiRepresents a control variable;
Figure BDA0002301438430000084
is an underwater robot system coupling kinetic equation, wherein
Figure BDA0002301438430000085
Is a coriolis matrix that is formed by a matrix of coriolis,
Figure BDA0002301438430000086
is the vector of the effects of gravity and buoyancy,
Figure BDA0002301438430000087
is a model of the effects of dissipation and,
Figure BDA0002301438430000088
it is the target object that converts the euler angular rate into velocity that expresses jacobian law.
Figure BDA0002301438430000089
Wherein
Figure BDA00023014384300000810
Is a positive definite inertia matrix.
According to the invention, firstly, the overall dynamics of the multi-underwater robot system is formulated, and then decoupling between a target object and the underwater robot is realized by using a certain load distribution coefficient. Each underwater robot system solves the NMPC at each sampling time according to the corresponding part of the overall dynamics and a plurality of inequality constraints, the underwater robots cooperatively drive the target object and move along the working space, the underwater robots are guided through the calculated feasible paths, and the calculation of the feasible paths is based on the navigation function of the multi-underwater robot system.
And 4, step 4: the working area module carries out sphere modeling on the obstacles, the underwater robot and the working space by adopting a sphere world representation method to generate a working space W:
step A1: setting B (x)O,r0) Is a closed sphere covering the volume of the object and having a radius r0；
Step A2: defining a closed sphere centered on an end effector of each underwater robotic system
Figure BDA00023014384300000811
The closed sphere
Figure BDA00023014384300000812
The underwater robot volume of all possible configurations is covered;
step A3: defining a target object, the distance between the grabbing points on the target object being at least
Figure BDA0002301438430000091
In the present embodiment, the first and second electrodes are,
Figure BDA0002301438430000092
for the working range of each underwater robot end effector, a plurality of grabbing points are arranged on the target object, wherein the distance between any two grabbing points is at least equal to
Figure BDA0002301438430000093
Step A4: is defined at a radius of
Figure BDA0002301438430000094
X of0Ball region B (x)OR) in the spherical region B (x)OR) comprises the volume of the multi-underwater robot system and the volume of the target object;
step A5: will sphere region B (x)OAnd M static obstacles present in R) are defined as being composed of
Figure BDA0002301438430000095
Wherein
Figure BDA0002301438430000096
Is the center of the image, and the center of the image,
Figure BDA0002301438430000097
is a barrier pimM is a positive integer;
and 5: in the working space W, the navigation module designs the desired motion trajectory and velocity of the target object according to the following formulas:
Figure BDA0002301438430000098
wherein
Figure BDA0002301438430000099
Is represented in a workspace
Figure BDA00023014384300000910
The potential of the safe motion vector field is internally derived, k is a design constant and k > 1,
Figure BDA00023014384300000911
indicating the position of the target
Figure BDA00023014384300000912
And γ (0) is 0;
the required motion trail path of the target object is designed as follows:
Figure BDA00023014384300000913
wherein KNFIs a positive gain. Defining a sampling instant tjWherein the sampling time is h, h e (0, T)P) Is constant: t is tj+1＝tj+h；
xO(tj) Is given target object at time tjCurrent position of vO(tj) Is given target object at time tjCurrent speed of each underwater robotic system, a time interval S e [ t ] that each underwater robotic system can propagatej,tj+TP]Wherein T isPIs the prediction horizon;
step 6: the constraint module provides a group of system state constraint sets X for each underwater robot systemiThese constraints are represented by the following equations:
xi(t)∈Xi(ii) a Wherein, XiConsists of the following constraints:
Figure BDA0002301438430000101
wherein the content of the first and second substances,
Figure BDA0002301438430000102
is a singular set of positions for a multi-underwater robotic system,
Figure BDA0002301438430000103
is the set of joint limits of the manipulator:
Figure BDA0002301438430000104
wherein the content of the first and second substances,
Figure BDA0002301438430000105
is corresponding to the joint
Figure BDA0002301438430000106
In the range of the limit (c) of (c),
Figure BDA0002301438430000107
is corresponding to the joint
Figure BDA0002301438430000108
Joint velocity of (1), set XiAll state constraints of the multi-underwater robot system are integrated;
and 7: the input constraints of each underwater robot system are specified in a constraint module as:
Figure BDA0002301438430000109
and
Figure BDA00023014384300001010
wherein tau isiIs a vector comprising each of the driven joints
Figure BDA00023014384300001011
In the respective limit ranges of (a) and (b),
Figure BDA00023014384300001012
τnis the number of joints that are activated;
defining a control input constraint set of an underwater robot system as Ti：
τi(t)∈Ti；
Wherein:
Figure BDA0002301438430000111
in each sampling time, the underwater robot system obtains a distributed dynamics mathematical model through a mathematical modeling module, then solves an NMPC control strategy scheme through a constraint module, and solves a time interval s epsilon [ t ] through a navigation modulej,tj+TP]Track of
Figure BDA0002301438430000112
And velocity
Figure BDA0002301438430000113
Finally, an optimal control input trajectory is formulated by the following step 8.
And 8: the control strategy generation module generates an optimal control input track according to the following method:
step B1: setting the current state measurement value to xi(tj) Discrete sampling time tj；
Step B2: for a multi-underwater robotic system, the open-loop input signal applied between sampling instants is given by solving the following equation:
Figure BDA0002301438430000114
wherein:
Figure BDA0002301438430000115
in the formula:
f and E are running and terminal cost functions, respectively, both in quadratic form, i.e.
Figure BDA0002301438430000116
And
Figure BDA0002301438430000117
Px、Qx、Qvand R are respectively positive definite matrices to be properly adjusted,
Figure BDA0002301438430000118
is at time instance tjIn the state of (a) in (b),
Figure BDA0002301438430000119
is the input trajectory of the input object,
Figure BDA00023014384300001110
is the optimal control input trajectory;
step B3: the control strategy generation module sends the optimal control input track to the multi-underwater robot system;
and step 9: at the next sampling instant tj+1，tj+1＝tj+ h, continuing the generation according to the method from step 3 to step 8 at the sampling instant tj+1The input trajectory is optimally controlled.
Control input τ of the inventioniIn the form of feedback, it can be recalculated based on the current state at each sampling instant.
The invention relates to a coordinated control method of a multi-underwater robot system based on distributed predictive control, which solves the technical problem of a distributed coordinated control strategy of the multi-underwater robot system in a limited working space with a static obstacle, provides a distributed nonlinear model predictive control system NMPC (non-linear predictive control), realizes that N underwater robot systems firmly grab a target object in the limited working space, guides an underwater robot from an initial position to a final position by utilizing the coupling dynamics between the robot and the object and using a certain load distribution coefficient, controls the underwater robot to move along a path calculated in the working space by the coordinated control strategy, and can ensure that the multi-underwater robot system can reduce communication data between the robots during the execution of cooperative underwater work, therefore, the required communication bandwidth is reduced, and the working efficiency of the multi-underwater robot system is improved.

Claims (3)

1. A multi-underwater robot system coordination control method based on distributed predictive control is characterized in that: the method comprises the following steps:
step 1: establishing a multi-underwater robot system, a data acquisition server and a control strategy server, wherein the multi-underwater robot system comprises a plurality of underwater robot systems, each underwater robot system is provided with an underwater robot sensor group, each underwater robot sensor group comprises an inertial measurement unit IMU, an ultra-short baseline positioning system USBL and a Doppler velocimeter DVL, and the underwater robot sensor groups transmit all acquired data to the data acquisition server for data fusion to finally generate position information and speed information of the underwater robot;
step 2: establishing a working area module, a constraint module, a navigation module, a control strategy generation module and a mathematical modeling module in a control strategy server;
and step 3: the method comprises the steps that a data acquisition server presets sampling moments, at each sampling moment, the data acquisition server acquires data of a primary underwater robot sensor group and generates position information and speed information of an underwater robot, a mathematical modeling module reads the position information and the speed information of the underwater robot, and a distributed dynamic mathematical model of each underwater robot system is established according to the position information and the speed information;
and 4, step 4: the working area module carries out sphere modeling on the obstacles, the underwater robot and the working space by adopting a sphere world representation method to generate a working space W:
step A1: setting B (x)O,r0) Is a closed sphere covering the volume of the object and having a radius r0；
Step A2: defining a closed sphere centered on an end effector of each underwater robotic system
Figure FDA0002301438420000011
The closed sphere
Figure FDA0002301438420000012
The underwater robot volume of all possible configurations is covered;
step A3: defining a target object, the distance between the grabbing points on the target object being at least
Figure FDA0002301438420000021
Step A4: is defined at a radius of
Figure FDA0002301438420000022
X of0Ball region B (x)OR) in the spherical region B (x)OR) comprises the volume of the multi-underwater robot system and the volume of the target object;
step A5: will sphere region B (x)OAnd M static obstacles present in R) are defined as being composed of
Figure FDA0002301438420000023
Wherein
Figure FDA0002301438420000024
Is the center of the image, and the center of the image,
Figure FDA0002301438420000025
is a barrier pimM is a positive integer;
and 5: in the working space W, the navigation module designs the desired motion trajectory and velocity of the target object according to the following formulas:
Figure FDA0002301438420000026
wherein
Figure FDA0002301438420000027
Is represented in a workspace
Figure FDA0002301438420000028
The potential of the safe motion vector field is internally derived, k is a design constant and k > 1,
Figure FDA0002301438420000029
indicating the position of the target
Figure FDA00023014384200000210
And γ (0) is 0;
the required motion trail path of the target object is designed as follows:
Figure FDA00023014384200000211
wherein KNFIs a positive gain. Defining a sampling instant tjWherein the sampling time is h, h e (0, T)P) Is constant: t is tj+1＝tj+h；
xO(tj) Is given target object at time tjCurrent position of vO(tj) Is given target object at time tjCurrent speed of each underwater robotic system, a time interval S e [ t ] that each underwater robotic system can propagatej,tj+TP]Wherein T isPIs the prediction horizon;
step 6: the constraint module provides a group of system state constraint sets X for each underwater robot systemi；
And 7: in a constraint moduleInput constraints of each underwater robot system are determined as follows:
Figure FDA0002301438420000031
and
Figure FDA0002301438420000032
wherein tau isiIs a vector comprising each of the driven joints
Figure FDA0002301438420000033
In the respective limit ranges of (a) and (b),
Figure FDA0002301438420000034
τnis the number of joints that are activated;
defining a control input constraint set of an underwater robot system as Ti：
τi(t)∈Ti；
Wherein:
Figure FDA0002301438420000035
and 8: the control strategy generation module generates an optimal control input track according to the following method:
step B1: setting the current state measurement value to xi(tj) Discrete sampling time tj；
Step B2: for a multi-underwater robotic system, the open-loop input signal applied between sampling instants is given by solving the following equation:
Figure FDA0002301438420000036
wherein:
Figure FDA0002301438420000037
in the formula:
f and E are running and terminal costs, respectivelyThe functions being all of quadratic form, i.e.
Figure FDA0002301438420000038
And
Figure FDA0002301438420000039
Px、Qx、Qvand R are respectively positive definite matrices to be properly adjusted,
Figure FDA00023014384200000310
is at time instance tjIn the state of (a) in (b),
Figure FDA00023014384200000311
is the input trajectory of the input object,
Figure FDA0002301438420000041
is the optimal control input trajectory;
step B3: the control strategy generation module sends the optimal control input track to the multi-underwater robot system;
and step 9: at the next sampling instant tj+1，tj+1＝tj+ h, continuing the generation according to the method from step 3 to step 8 at the sampling instant tj+1The input trajectory is optimally controlled.
2. The coordinated control method of the multi-underwater robot system based on the distributed predictive control as claimed in claim 1, wherein: in performing step 3, the mathematical modeling module builds a distributed dynamical mathematical model for each underwater robotic system according to the following formula:
Figure FDA0002301438420000042
wherein the content of the first and second substances,
Figure FDA0002301438420000043
Figure FDA0002301438420000044
Figure FDA0002301438420000045
3. the coordinated control method of the multi-underwater robot system based on the distributed predictive control as claimed in claim 1, wherein: in step 6, the system state constraint set XiComprising a plurality of constraint terms, embodied as:
these constraints are represented by the following equations:
xi(t)∈Xi(ii) a Wherein, XiConsists of the following constraints:
Figure FDA0002301438420000051
wherein the content of the first and second substances,
Figure FDA0002301438420000052
is a singular set of positions for a multi-underwater robotic system,
Figure FDA0002301438420000053
is the set of joint limits of the manipulator:
Figure FDA0002301438420000054
wherein the content of the first and second substances,
Figure FDA0002301438420000055
is corresponding to the joint
Figure FDA0002301438420000056
In the range of the limit (c) of (c),
Figure FDA0002301438420000057
is corresponding to the joint
Figure FDA0002301438420000058
Joint velocity of (1), set XiAll state constraints of the multi-underwater robotic system are centralized.
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