A performance control method for robot formation for heavy workpiece handling

Abstract

The present invention discloses a method for controlling the performance of a robot formation for handling heavy workpieces. First, a control model of the robot formation is established, and state quantities such as target distance, relative position and distance between robots in the robot formation are set. The state quantity error is calculated and constraints are added to the state quantity error of the control system according to the constraints of the robot cooperative handling in the actual scene. The error is converted for the constrained state quantity error to obtain the conversion error variable in the unconstrained space, and the dynamic surface control law of the robot formation is designed and the stability analysis of the closed-loop system is performed. The dynamic surface control law of the robot formation is mathematically simulated in the MATLAB environment to verify the effectiveness of the proposed control method. By designing the dynamic surface control law, the cooperative control of multiple robots can be realized to ensure the accuracy of the robot formation in the cooperative handling process.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-robot collaborative handling control, in particular to the field of mobile robot formation control, and in particular to a robot formation performance control method for heavy workpiece handling. Background Art

[0002] The manufacturing of major equipment involves multiple fields such as aerospace, shipbuilding, energy, and engineering machinery. The workpieces required in the manufacturing process are often heavy and bulky, making it difficult to transport and process them with a single device. The handling of these heavy workpieces is not only a key step in the production process, but also an important factor affecting production efficiency and product quality. Traditional manual handling methods are inefficient and have a high risk factor, and can no longer meet the needs of modern industrial production. Therefore, more and more intelligent means, such as multi-robot collaborative handling technology, are gradually being used in this field. However, due to the complexity of heavy workpiece handling tasks, higher requirements are placed on the stability, flexibility, and accuracy of multi-robot collaborative operations. Among them, how to achieve efficient and stable collaborative handling of heavy workpieces by multiple robots in a complex environment has become a major technical challenge in the current field of major equipment manufacturing, and has also put forward new requirements for multi-robot formation control methods.

[0003] Most of the existing multi-robot formation control algorithms are designed for light object handling or routine tasks. Common control strategies include control based on the leader-follower model, virtual structure method, behavioral control, and distributed control. These methods have their own advantages and disadvantages in different application scenarios. For example, the leader-follower model allows one robot to act as a navigator, and other robots follow according to the navigator's motion trajectory. The advantage is that the system architecture is simple, but it is difficult to cope with sudden environmental changes or complex path planning; the virtual structure method regards the entire robot formation as a whole, and realizes coordinated motion by controlling each point in the virtual structure. It is suitable for tasks that require precise formation shapes, but its computational complexity is high and its real-time performance is poor; behavioral control is based on the local rules of individual robot behavior, and realizes overall coordination by designing local interactive behaviors, but it is easy to lead to poor global coordination when facing multiple environmental variables; distributed control emphasizes the independence of each robot, which is suitable for the collaboration of large-scale, multi-robot systems, but it is easy to reduce control accuracy due to information transmission delays and uneven distribution of computing resources.

[0004] In addition, the existing formation control algorithms still have many technical difficulties when facing the actual scenario of heavy workpiece handling: first, there is the safety issue in the process of heavy workpiece handling. It is usually necessary to ensure that robots do not collide with each other. In addition, common sensors carried by robots are laser trackers or visual cameras, which requires consideration of the limited perception capabilities of robots; second, the existing multi-robot formation control methods require frequent communication between robots and their neighbors, which can easily lead to unnecessary information exchange; finally, the multi-robot formation control has low accuracy, which will produce unreasonable torque and tension, which may cause potential damage to heavy workpieces and robots. Summary of the invention

[0005] In view of the above problems, the present invention proposes a new multi-robot formation control method, which can realize the coordinated movement of multiple robots and ensure the stability and accuracy of the system while ensuring the safety and efficiency of the handling task.

[0006] A multi-robot prescribed performance formation control method for heavy workpiece handling comprises the following steps:

[0007] S1. Preset a robot formation consisting of multiple robots and establish a control model for the robot formation, where the control model includes a kinematic model and a dynamic model for each robot;

[0008] S2. Construct an undirected graph, define the target distance, phase position and distance between the robots, calculate the distance error between the robots based on the target distance and distance, calculate the relative distance square error between the robots based on the target distance and phase position, and define the angle error and speed error of each robot;

[0009] S3, set the safety distance and maximum perception range between robots, set the relative distance square error constraint between robots according to the safety distance, maximum perception range, distance error between robots and relative distance square error, and set the angle error constraint and speed error constraint of each robot according to the angle error and speed error of each robot;

[0010] S4, presetting error variables and designing a specified performance function, designing a transformation error function of the conversion error according to the error variables and the specified performance function, and deriving the conversion error and the derivative of the conversion error based on the monotonicity of the transformation error function;

[0011] S5. Design the dynamic surface control law according to the conversion error and the derivative of the conversion error, perform the stability analysis of the closed-loop system of the dynamic surface control law, preset the initial parameters and expected formation of the robot formation, and use MATLAB software to simulate the dynamic surface control law.

[0012] Preferably, the kinematic model of each robot in S1 is specifically expressed by the formula:

[0013] ;

[0014] In the formula, and Indicates The position coordinates of the robot in the inertial system, Indicates The heading angle of the robot in the inertial frame, Indicates The linear velocity of the robot in the inertial frame is, Indicates The angular velocity of the robot in the inertial frame is , is the total number of robots in the robot formation;

[0015] The dynamic model of each robot in S1 is specifically expressed by the formula:

[0016] ;

[0017] in, ;

[0018] ;

[0019] ;

[0020] , ;

[0021] In the formula, is a symmetric positive definite inertia matrix, , To indicate the The speed of the robot in the inertial system, including the linear speed and angular velocity , , are the centripetal matrix and the Coriolis matrix, , is the gravity vector matrix, , when the i-th robot moves on the plane, , is the input matrix, , is the motor torque of the robot, is the external disturbance, is the total mass of the robot, is the mass of the robot body, is the mass of the robot wheel, is the driving wheel radius of the robot, represents the moment of inertia of the wheel about the axle, represents the moment of inertia of the wheel about the wheel diameter, represents the moment of inertia of the robot body, represents the total moment of inertia of the robot, is half the wheel spacing of the robot, The local coordinate origin of the robot To the centroid distance.

[0022] Preferably, the distance error between the robots in S2 is specifically expressed as:

[0023] ;

[0024] In the formula, Indicates The robot and The distance error between the robots, Indicates and The distance between the robots, Indicates The robot and The target distance between the robots;

[0025] The relative distance square error between robots in S2 is specifically expressed as:

[0026] ;

[0027] In the formula, Indicates The robot and The square error of the relative distance between the robots, Indicates and The relative positions of the robots at time t.

[0028] Preferably, in S3, a safety distance and a maximum sensing range between robots are set, and a relative distance square error constraint between robots is set according to the safety distance, the maximum sensing range, the distance error between robots, and the relative distance square error, which specifically includes the following:

[0029] S31, preset the safe distance and maximum sensing range between robots;

[0030] S32, imposing constraints on the distance between the robots according to the safety distance and the maximum sensing range, and setting a distance error constraint between the robots in combination with the distance error between the robots, and setting a sufficient condition for the robot formation to maintain infinitesimal rigidity according to the distance error constraint between the robots;

[0031] S33. Set a constraint on the squared error of the relative distances between the robots based on the squared error of the relative distances between the robots and a sufficient condition for the robot formation to maintain infinitesimal rigidity.

[0032] Preferably, the relative distance square error constraint between the robots in S33 is specifically expressed by the formula:

[0033] ;

[0034] In the formula, , ,

[0035] , ,

[0036] in, Indicates The robot and The square error of the relative distance between the robots, represents the lower bound of the relative distance square error, represents the upper bound of the relative distance square error, , , All are intermediate variables.

[0037] Preferably, the conversion error and the derivative of the conversion error in S4 are specifically expressed by the formula:

[0038] ;

[0039] ;

[0040] in, , ;

[0041] , ;

[0042] In the formula, represents the conversion error, represents the derivative of the conversion error, represents the error variable, is the actual value of the error variable, is the expected value of the error variable, Relative distance square error , Angle error , speed error The conversion error, represents the specified performance function, represents the lower bound of the error variable, represents the upper bound of the error variable, and The performance functions are specified and error variable The derivative of is the transformation error function of the conversion error.

[0043] Preferably, the dynamic surface control law includes a virtual angular velocity, a reference velocity, a desired angle, a virtual linear velocity and a control law of a dynamic model. In S5, the dynamic surface control law is designed according to the conversion error and the derivative of the conversion error, which specifically includes the following:

[0044] S51, according to the conversion error and its derivatives set up Conversion error and its derivatives , according to the conversion error and its derivatives Design The virtual angular velocity of the robot ;

[0045] S52, according to the conversion error and its derivatives set up Conversion error and its derivatives , according to the conversion error derivative Design The reference speed of the robot and expected angle ;

[0046] S53. According to The reference speed of the robot and expected angle Design Virtual linear speed of the robot ;

[0047] S54, according to the conversion error The first derivative of set up The derivative of the conversion error , according to the conversion error derivative Design control laws for dynamic models.

[0048] Preferably, in S51 The virtual angular velocity of the robot Specifically:

[0049] ;

[0050] in, ;

[0051] ;

[0052] ;

[0053] In the formula, For the The virtual angular velocity of the robot, For the The first adaptive law of a robot, is the specified performance function for the angular error, For the The angular error of the robot, No. The derivative of the desired angle of the robot, is the conversion error of the angle error, , , , , , All are intermediate variables.

[0054] Preferably, in S52 The reference speed of the robot Specifically expressed as:

[0055] ;

[0056] In the formula, For the The reference speed of the robot, and Respectively The reference speed of the robot The velocity components in the x-axis and y-axis, is the conversion error of the relative distance square error, is the prescribed performance function of the relative distance squared error, , , , All are intermediate variables;

[0057] S52 The desired angle of the robot Specifically expressed as:

[0058] ;

[0059] In the formula, Indicates The desired angle of the robot.

[0060] Preferably, the control law of the dynamic model in S54 can be designed as:

[0061] ;

[0062] in, ;

[0063] ;

[0064] ;

[0065] In the formula, For control torque The superposition vector of is the input matrix, is a symmetric positive definite inertia, are the centripetal matrix and the Coriolis matrix, is the specified performance function of the velocity error, for The derivative of It is The speed of the robot The superposition vector of It is The speed error of the robot The superposition vector of is the derivative of the filtering speed, yes The superposition vector of , is the second adaptive law, The second adaptive law The derivative of , , , , , , , , All are intermediate variables.

[0066] The above-mentioned multi-robot prescribed performance formation control method for heavy workpiece handling first establishes a control model of the robot formation, sets the state quantities such as the target distance, relative position and distance between the robots in the robot formation, calculates the state quantity error and adds constraints to the state quantity error of the control system according to the constraints of the robot collaborative handling in the actual scene, performs error conversion on the constrained state quantity error, obtains the conversion error variable in the unconstrained space, designs the prescribed performance function and dynamic surface control law, performs closed-loop system stability analysis on the dynamic surface control law, and mathematically simulates the dynamic surface control law of the robot formation in the MATLAB environment to verify the effectiveness of the proposed control method. By designing the dynamic surface control law, the collaborative control of multiple robots can be realized to ensure the accuracy of the robot formation in the collaborative handling process. BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 It is a flow chart of a multi-robot prescribed performance formation control method for heavy workpiece handling;

[0068] Figure 2 The present invention is a control block diagram of a multi-robot prescribed performance formation control method for heavy workpiece handling;

[0069] Figure 3 It is a schematic diagram of a single robot model and a robot formation;

[0070] Figure 4 It is a topological diagram of the robot formation;

[0071] Figure 5 It is a schematic diagram of the trajectory curve of the robot formation;

[0072] Figure 6 It is a schematic diagram of the curve of the relative distance error between different robots in the robot formation changing over time;

[0073] Figure 7 It is a schematic diagram of the curves of the angle error, linear velocity error and angular velocity error of the robot formation changing with time. DETAILED DESCRIPTION

[0074] In order to enable those skilled in the art to better understand the technical solution of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings.

[0075] See also Figure 1 and Figure 2 A multi-robot specified performance formation control method for heavy workpiece handling includes the following steps:

[0076] S1. Preset a robot formation consisting of multiple robots and establish a control model for the robot formation. The control model includes a kinematic model and a dynamic model of each robot.

[0077] Specifically, see Figure 2 , consider a robot formation consisting of n wheeled robots, where each wheeled robot and the formation structure of five wheeled robots are as follows Figure 2 As shown. A wheeled robot is a typical non-holonomic control system. We generally assume that the wheeled robot's wheels and the ground only undergo pure rolling motion without sliding (including lateral sliding and longitudinal sliding), so that the wheeled robot cannot move sideways (if it wants to move sideways, it needs to control the speed of the left and right wheels). For the sake of simplicity, the wheeled robot will be referred to as a robot in the following.

[0078] Figure 3 middle, is an inertial system, For the The moving coordinate system of a robot, where Axis points to The direction of the robot's movement, Axis and The axis angle is , Indicates The center of mass of the robot, Indicates The local coordinate origin of the robot. The generalized coordinates of a robot in the inertial system can be expressed as:

[0079] (1)

[0080] In the formula, Indicates The generalized coordinates of a robot in the inertial system, Indicates A robot coordinate, Indicates Mobile robots coordinate, Indicates Mobile machines and The angle of the axis, Represents the total number of robots in the robot formation.

[0081] Further, The kinematic model of a mobile robot can be expressed as:

[0082] (2)

[0083] in, ;

[0084] In the formula, and Respectively represent The position and heading angle of each robot in the inertial system, represents the first-order derivative, and Respectively The linear velocity and angular velocity of a robot in the inertial system.

[0085] According to the Euler-Lagrange principle, The dynamic model of a robot can be expressed as:

[0086] (3)

[0087] in, (3-1)

[0088] (3-2)

[0089] (3-3)

[0090] (3-4)

[0091] (3-5)

[0092] In the formula, is a symmetric positive definite inertia matrix, , For the The speed of the robot in the inertial system, including the linear speed and angular velocity , , are the centripetal matrix and the Coriolis matrix, , is the gravity vector matrix, , when When a robot moves on a plane, , is the input matrix, , is the motor torque of the robot, is the external disturbance, is the total mass of the robot, is the mass of the robot body, is the mass of the robot wheel, is the driving wheel radius of the robot, represents the moment of inertia of the wheel about the axle, represents the moment of inertia of the wheel about the wheel diameter, represents the moment of inertia of the robot body, represents the total moment of inertia of the robot, is half the wheel spacing of the robot, The local coordinate origin of the robot To the centroid distance.

[0093] S2. Construct an undirected graph, define the target distance, phase position and distance between robots, calculate the distance error between robots based on the target distance and distance, calculate the relative distance square error between robots based on the target distance and phase position, and define the angle error and speed error of each robot.

[0094] Furthermore, S2 specifically includes the following:

[0095] 1) Construct an undirected graph based on the positions and mutual communication relationships of the robots in the robot formation. The specific process is as follows:

[0096] Use n vertices to represent n robots, The edges represent the positional relationship between adjacent robots. Vertices and An undirected graph with edges is defined as: ,in, is a set of vertices, is a set of undirected edges.

[0097] vertex Neighbor vertices of The set of ,

[0098] definition Vertex The coordinates of is the superposition vector of the coordinates, and the robot formation is defined as , will be Related edge functions Defined as:

[0099] (4)

[0100] in, represents the structure of the robot formation, No. Elements Corresponds to No. An undirected edge, For the The two vertices of an undirected edge.

[0101] Robot formation The rigid matrix for:

[0102] (5)

[0103] in, is the rigidity matrix of the robot formation. From formula (5), we can see that the rigidity matrix of the robot formation is Relative position to the robot only related.

[0104] Known There are , have , then the above degree matrix Each line has the following form:

[0105] (5-1)

[0106] Robot formation The desired formation of can be represented by a minimal and infinitesimal rigid frame:

[0107] (6)

[0108] In the formula, (6-1)

[0109] in, , An undirected graph representing the desired formation and the robot positions.

[0110] 2) Define the target distance, phase position and distance between robots.

[0111] Specifically, no. and The target distance between robots Defined as:

[0112] (7)

[0113] In the formula, Indicates and The target distance between the robots, and Respectively represent and The desired position of the robot.

[0114] No. The robot and The relative positions of the robots Defined as:

[0115] (8)

[0116] In the formula, Indicates The real-time location of each robot, Indicates The real-time location of each robot, Indicates The robot and The real-time relative position of the robots.

[0117] No. The robot and The distance between robots Defined as:

[0118] (9)

[0119] In the formula, Indicates and The distance between the robots, Indicates The coordinates of the robot in the inertial system, Indicates The coordinates of the robot in the inertial system.

[0120] 3) Calculate the distance error between any robots based on the target distance and distance between the robots.

[0121] Specifically, according to The robot and The target distance between robots and distance Calculate the The robot and The distance error between the robots :

[0122] (10)

[0123] In the formula, Indicates The robot and The distance error between the robots.

[0124] 4) Calculate the square error of the relative distance between the robots based on the target distance and phase position between the robots.

[0125] Specifically, according to and The relative positions of the robots at time t And the target distance calculation The robot and The square error of the relative distance between the robots :

[0126] (11)

[0127] In the formula, Indicates The robot and The square error of the relative distance between the robots.

[0128] Substituting formulas (8), (9) and (10) into formula (11), we can obtain:

[0129] (12)

[0130] From formula (12), we can see that when When converged, It is also convergent.

[0131] 5) Define the angle error and angular velocity error of each robot in the robot formation as follows:

[0132] (13)

[0133] In the formula, and Respectively The angular error and velocity error of each robot, and Respectively The angle and speed of the robot, and Respectively The desired angle and virtual velocity of each robot.

[0134] S3. Set the safe distance and maximum perception range between robots, set the relative distance square error constraint between robots according to the safe distance, maximum perception range, distance error between robots and relative distance square error, and set the angle error constraint and speed error constraint of each robot according to the angle error and speed error of each robot.

[0135] Furthermore, in S3, the safety distance and maximum sensing range between the robots are set, and the relative distance square error constraint between the robots is set according to the safety distance, the maximum sensing range, the distance error between the robots, and the relative distance square error, which specifically includes the following:

[0136] S31. Preset safe distance between robots and maximum sensing range .

[0137] S32, according to the safety distance and maximum sensing range A constraint is imposed on the distance between robots, and a distance error constraint between robots is set in combination with the distance error between robots. A sufficient condition for the robot formation to maintain infinitesimal rigidity is set based on the distance error constraint between robots.

[0138] Specifically, the collaborative handling control system requires that there should be no collision between robots in the robot formation. The robot and The relative distance between the robots And apply collision constraints (that is, avoid collision constraints):

[0139] (14)

[0140] in, For the The robot and The safe distance between robots. For the The robot and The distance between the robots.

[0141] In real-world scenarios, the perception capabilities of robots are limited, that is, when the distance between robots is greater than the perception range of the robots, it is assumed that there is no communication between the robots. The robot and The distance between robots To impose connectivity constraints:

[0142] (15)

[0143] in, For the The robot and The maximum sensing range between robots, .

[0144] Set the distance error constraints between robots based on collision constraints, connectivity constraints, and distance errors.

[0145] Specifically, substituting (14) into (10) yields:

[0146] (16)

[0147] Substituting formula (15) into formula (10), we get:

[0148] (17)

[0149] in, .

[0150] From equation (16) and equation (17), we can get The robot and The distance error constraint between robots:

[0151] (18)

[0152] In the formula, ;

[0153] in, represents the lower bound of the distance error, Represents the upper bound of the distance error.

[0154] The necessary condition for the collaborative handling of multiple mobile robots is that the robots form a preset desired formation (corresponding to formulas (6) and (6-1)) and accurately track the desired trajectory. The formation control of multiple mobile robots can be achieved by maintaining infinitesimal rigidity.

[0155] According to the constraint of the distance error between any two robots (Formula (18)), the robot formation The sufficient condition for maintaining infinitesimal rigidity is:

[0156] (19)

[0157] in, is a sufficiently small positive number.

[0158] Assumptions and is also a small enough positive number, then the sufficient condition of formula (19) can be expressed as:

[0159] (20)

[0160] S33. Set a constraint on the squared error of the relative distances between the robots based on the squared error of the relative distances between the robots and a sufficient condition for the robot formation to maintain infinitesimal rigidity.

[0161] Specifically, combining equation (18) and equation (20), The robot and The distance error constraint between the robots can be expressed as:

[0162] (twenty one)

[0163] In the formula, , , , , All are intermediate variables.

[0164] According to the square error of the relative distance between robots (corresponding to formula (12)) and the distance error between any two robots (Corresponding to formula (21)) The relative distance square error constraint between any two robots can be derived:

[0165] (twenty two)

[0166] In the formula, , .

[0167] in, represents the lower bound of the relative distance square error, Represents the upper bound of the squared error of the relative distance.

[0168] The angle error constraint and speed error constraint of each robot are set according to the angle error and speed error of each robot.

[0169] Specifically, in the robot's dynamic control system, a large angle error will seriously affect the robot formation effect. The angular error of each robot is defined as follows:

[0170] (twenty three)

[0171] in, Indicates The angular error of the robot, Indicates The maximum angular error of the robot.

[0172] Line speed and angular velocity Gradually increase or decrease over time, that is and There will be no drastic changes. When the expected trajectory is at time When changing direction suddenly, the linear speed error and angular velocity error There will be great fluctuations. The linear velocity error and angular velocity of a robot define the linear velocity error Angular velocity error The constraints are:

[0173] (twenty four)

[0174] in, and They represent the lower and upper bounds of the linear velocity error respectively.

[0175] (25)

[0176] in, and They represent the lower and upper bounds of the angular velocity error respectively.

[0177] definition For the The speed error of the robot, For the The upper bound of the velocity error of each robot is, No. The lower bound of the velocity error of the robot.

[0178] S4. Preset error variables and design specified performance functions, design a transformation error function of the conversion error according to the error variables and the specified performance function, and derive the conversion error and the derivative of the conversion error based on the monotonicity of the transformation error function.

[0179] Specifically, in order to improve the convergence speed and steady-state accuracy of the robot formation error, a common method is to transform the formation error in the constrained space into the error in the unconstrained space. The specific process is as follows:

[0180] 1) Set the error variable:

[0181] (26)

[0182] in, , which is the error variable Can refer to the relative distance square error Variable, or angular error Variable, or speed error variable, is the actual value of the error variable, is the expected value of the error variable.

[0183] 2) Design specified performance functions :

[0184] The performance function is designed based on the properties of the tuning function. The specific process is as follows: First, the definition of the tuning function is given. When the function meets the following conditions, it is called a tuning function:

[0185] a. exist is monotonically decreasing, and for all , both satisfy ,in, is a constant;

[0186] b. At least ,in is the order of the controlled system. Bounded.

[0187] Using the above definition of the tuning function, the specified performance function is designed To constrain the error variable:

[0188] (27)

[0189] In the formula, (27-1)

[0190] in, To specify the performance function, , , , are process variables, is an arbitrarily chosen constant. It should be noted that , , They are The error variables specifically are angle error, relative distance square error and velocity error which are the specified performance functions.

[0191] 3) According to the error variable and specified performance functions Design the transformation error function of the conversion error:

[0192] (28)

[0193] ;

[0194] In the formula, is the transformation error function of the conversion error, is the conversion error, is the specified performance function, and , the error function of the conversion error is a smooth, continuous, monotonic function with the following boundaries: , and satisfy:

[0195] (29)

[0196] (30)

[0197] In the formula, and They represent the transformation error functions The upper and lower bounds of .

[0198] Considering equations (29) and (30), the conversion error function Can be designed as

[0199] (31)

[0200] From (28) and (31), we can get the conversion error :

[0201] (32)

[0202] Taking the derivative of equation (32), we can get the conversion error: The first derivative of :

[0203] (33)

[0204] Note that, combining equation (32) and equation (27), at the initial moment, when the error variable satisfy When Internal, error variable Converges to the specified performance envelope. Through the above analysis, the effectiveness of the designed new specified performance function (corresponding to formula (27)) is verified.

[0205] S5. Design the dynamic surface control law (DSC Law) based on the conversion error and its derivative, perform a stability analysis of the closed-loop system of the dynamic surface control law, preset the initial parameters and expected formation of the robot formation, and use MATLAB software to simulate the dynamic surface control law.

[0206] Furthermore, the dynamic surface control law includes the control law of the virtual angular velocity, the reference velocity, the desired angle, the virtual linear velocity and the dynamic model. In S5, the dynamic surface control law is designed according to the conversion error and the derivative of the conversion error, which specifically includes the following:

[0207] S51, according to the conversion error and its derivatives set up Conversion error and its derivatives , according to the conversion error and its derivatives Design The virtual angular velocity of the robot .

[0208] Specifically, according to the conversion error (Formula (32)) and its first-order derivative (Formula (33)) set Conversion error and its derivatives :

[0209] (34)

[0210] (35)

[0211] In the formula, is the specified performance function for the angular error.

[0212] According to the conversion error and its derivatives , the virtual angular velocity is designed as:

[0213] (36)

[0214] Among them, the adaptive law is designed as:

[0215] (37)

[0216] In the formula, For the The first adaptive law of a robot, , , , , Is a positive integer.

[0217] Next, according to the conversion error and the first adaptive law The following Lyapunov function is designed to verify the validity of the designed virtual angular velocity (corresponding to formula (36)):

[0218] (38)

[0219] Deriving equation (38) and substituting equations (32), (36) and (37) into equation (38) yields:

[0220] (39)

[0221] In the formula, and are all intermediate variables. , .

[0222] Through analysis, we can know that when hour, At the maximum convergence time Converges within a small area, the maximum convergence time The calculation formula is:

[0223] (40)

[0224] Through formula (38) to formula (40), the designed virtual angular velocity is verified (corresponding to formula (36)) is effective.

[0225] S52, according to the conversion error and its derivatives set up Conversion error and its derivatives , according to the conversion error derivative Design The reference speed of the robot and expected angle .

[0226] Specifically, the square error of the relative distance between any two robots (corresponding to formula (12)) is differentiated, and formula (2), formula (7) and formula (10) are substituted into it to obtain the first-order derivative of the square error of the relative distance between any two robots:

[0227] (41)

[0228] In the formula, For the The robot and The square error of the relative distance between the robots The first derivative of .

[0229] According to the conversion error The first derivative of (corresponding to equations (32) and (33)), when When the conversion error and its derivatives can be rewritten as:

[0230] (42)

[0231] (42-1)

[0232] In the formula, (42-2)

[0233] (42-3)

[0234] In the formula, is the conversion error of the relative distance square error, is the prescribed performance function of the relative distance squared error, and All are intermediate variables.

[0235] according to Design robot formation The reference speed of the robot and expected angle ,in:

[0236] No. The reference speed of the robot The specific expressions are as follows:

[0237] (43)

[0238] No. The desired angle of the robot The specific expressions are as follows:

[0239] (44)

[0240] In the formula, Indicates The desired angle of the robot, , , All are intermediate variables.

[0241] According to (43) and (44), we can get:

[0242] (45)

[0243] In the formula, and They are The speed of the robot Components on the x and y axes.

[0244] S53. According to The reference speed of the robot and expected angle Design Virtual linear speed of the robot .

[0245] Specifically, no. Virtual linear speed of the robot The design process is as follows:

[0246] According to The reference speed of the robot Virtual Line Speed It is expressed as:

[0247] (46)

[0248] In the formula, The function to be designed.

[0249] Deriving the robot's kinematic model (corresponding to formula (2)), we get Combining equation (44) and equation (46), we can get:

[0250] (47)

[0251] In the formula, , is an intermediate variable.

[0252] Notice is a skew-symmetric matrix, for non-zero vectors , whose properties are .

[0253] To simplify the derivation of the formula, let:

[0254] (48)

[0255] In the formula, , obviously for any non-zero vector have and , then formula (41) can be rewritten as:

[0256] (49)

[0257] Since formula (42) is for a single robot The derivative of the conversion error , multiple robots in the robot formation The conversion error derivative at can be equivalent to:

[0258] (50)

[0259] In the formula, and They are and The superposition vector of , is a rigid matrix Abbreviation of , .

[0260] Similarly, from formula (43), we know that the reference speed of multiple robots in the robot formation is It can be expressed as:

[0261] (51)

[0262] In the formula, is the intermediate variable, .

[0263] Substituting equation (51) into equation (50), we can get the relative distance square error The conversion error derivative :

[0264] (52)

[0265] To prove the relative distance square error Convergence of , considering the alternative Lyapunov function as:

[0266] (53)

[0267] When the angle error When converged, there is , , and then we can differentiate equation (53) to get:

[0268] (54)

[0269] Through analysis, we can know that due to is a continuous variable of the closed-loop system, so for all , and yes The sufficient condition for is an intermediate variable greater than 0; when Enough hours, yes . Note that if , then in formula (47) we have This is inconsistent with the fact that the position error of each mobile robot at the initial moment is not 0. So The sufficient condition for can be further described as:

[0270] For all , and That is to say, the function to be designed can be any function greater than 0, for example, .

[0271] Through the above analysis, the effectiveness of the designed virtual linear speed (corresponding to formula (46)) is verified.

[0272] In order to avoid the differential explosion phenomenon in backstepping control, the virtual speed Through the processing of the first-order filter, the first-order filter formula is:

[0273] (55)

[0274] In the formula, is the filtering speed, is the virtual speed, is an intermediate variable.

[0275] According to formula (55), the filtering error is defined as , consider the alternative Lyapunov function as , we can get the derivative:

[0276] ,

[0277] in, is a continuous bounded function. Analysis shows that the filtering error It converges in finite time.

[0278] S54, according to the conversion error The first derivative of set up The derivative of the conversion error , according to the conversion error derivative Design control laws for dynamic models.

[0279] Specifically, according to formula (33), when When the conversion error derivative can be rewritten as:

[0280] (56)

[0281] is equivalent to:

[0282] (57)

[0283] in, , ;

[0284] In the formula, They are The superposition vector of As an auxiliary function, obviously, when When it converges to a small neighborhood around zero, is bounded.

[0285] The control law of the dynamic model can be designed as:

[0286] (58)

[0287] in, ;

[0288] ;

[0289] ;

[0290] In the formula, For control torque The superposition vector of is the input matrix, is a symmetric positive definite inertia, are the centripetal matrix and the Coriolis matrix, is the specified performance function of the velocity error, for The derivative of It is The speed of the robot The superposition vector of It is The speed error of the robot The superposition vector of is the derivative of the filtering speed, yes The superposition vector of , is the second adaptive law, The second adaptive law The derivative of , , , , , , , , All are intermediate variables.

[0291] To verify the speed error Convergence of , considering the alternative Lyapunov function as:

[0292] (59)

[0293] By taking the derivative of formula (59), we can obtain:

[0294] (60)

[0295] In the formula, Is a normal number.

[0296] Through analysis, we can know that the speed error can converge to a bounded set:

[0297] ,

[0298] in, is a finite time constant.

[0299] The convergence time is .

[0300] Therefore, the above analysis verifies the effectiveness of the control law of the designed dynamic model (corresponding to formula (58)).

[0301] Furthermore, in order to verify the effectiveness of the performance control method for robot formation for heavy workpiece handling proposed in this paper, a simulation study was carried out in MATLAB software.

[0302] According to the virtual angular velocity and virtual linear velocity designed previously, the virtual velocity control law is selected as:

[0303] (61)

[0304] Consider a set of five nonholonomic mobile robots modeled on a two-dimensional plane. Figure 4 The square pentagon in the figure is a formation based on the minimum rigid body and infinitesimal rigid body. When the robot is a pentagonal structure, only seven edges are needed to ensure the stability of the robot formation. The set of edges is:

[0305] .

[0306] The expected relative distance between mobile robots is designed as:

[0307] ,

[0308] .

[0309] The expected formation speed is designed as:

[0310] when hour, ;

[0311] when hour, ,when When , the expected trajectory is a straight line,

[0312] when , the expected trajectory is a part of a circle.

[0313] The initial positions of the five robots are:

[0314] , , , , .

[0315] The initial angles of the five robots are:

[0316] .

[0317] The numerical simulation results are as follows Figure 5-7 As shown, where:

[0318] Figure 5 Shows the robot formation in The proposed dynamic surface control law enables the robot formation to consistently generate and maintain the desired formation using the minimum and infinitesimal stiffness principles.

[0319] Figure 6 is the curve of relative distance error changing with time, Figure 6 middle, Indicates The robot and The relative distance error between the robots, ,Depend on Figure 6 It can be seen that the relative distance error between any two robots in the robot formation can converge within the performance envelope, and the accuracy of the relative distance error between the robots is within 10 mm. That is, the proposed dynamic surface control law can ensure that the formation distance error evolves within the range established by the distance error constraint (corresponding to formula (20)), thereby making the formation distance error always within the specified performance envelope, effectively improving the transient and steady-state performance of the formation distance error.

[0320] Figure 7 is the curve of angle error, linear velocity error and angular velocity error changing with time, Figure 7 middle, Indicates The angular error of the robot, , Indicates the linear velocity error, Represents the angular velocity error. It can be seen from the figure that all angle errors, linear velocity errors and angular velocity errors are forced to be limited within the performance envelope, which further illustrates the effectiveness of the dynamic surface control law proposed in this paper.

[0321] The above-mentioned multi-robot prescribed performance formation control method for heavy workpiece handling first establishes a control model of the robot formation, sets the state quantities such as the target distance, relative position and distance between the robots in the robot formation, calculates the state quantity error and adds constraints to the state quantity error of the control system according to the constraints of the robot collaborative handling in the actual scene, performs error conversion on the constrained state quantity error, obtains the conversion error variable in the unconstrained space, designs the prescribed performance function and dynamic surface control law, performs closed-loop system stability analysis on the dynamic surface control law, and mathematically simulates the dynamic surface control law of the robot formation in the MATLAB environment to verify the effectiveness of the proposed control method. By designing the dynamic surface control law, the collaborative control of multiple robots can be realized to ensure the accuracy of the robot formation in the collaborative handling process.

[0322] The above is a detailed introduction to a multi-robot specified performance formation control method for heavy workpiece handling provided by the present invention. This article uses specific examples to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the core idea of ​​the present invention. It should be pointed out that for ordinary technicians in this technical field, without departing from the principles of the present invention, several improvements and modifications can be made to the present invention, and these improvements and modifications also fall within the scope of protection of the claims of the present invention.

Claims

1. A robot formation performance control method for heavy workpiece handling, characterized in that: The method comprises: S1. Preset a robot formation consisting of multiple robots and establish a control model for the robot formation, where the control model includes a kinematic model and a dynamic model for each robot; S2. Construct an undirected graph, define the target distance, phase position and distance between the robots, calculate the distance error between the robots based on the target distance and distance, calculate the relative distance square error between the robots based on the target distance and phase position, and define the angle error and speed error of each robot; S3, set the safety distance and maximum perception range between robots, set the relative distance square error constraint between robots according to the safety distance, maximum perception range, distance error between robots and relative distance square error, and set the angle error constraint and speed error constraint of each robot according to the angle error and speed error of each robot; S4, presetting error variables and designing a specified performance function, designing a transformation error function of the conversion error according to the error variables and the specified performance function, and deriving the conversion error and the derivative of the conversion error based on the monotonicity of the transformation error function; The conversion error and the derivative of the conversion error in S4 are specifically expressed by the formula: ； ； in, , ; ， ； In the formula, represents the conversion error, represents the derivative of the conversion error, represents the error variable, is the actual value of the error variable, is the expected value of the error variable, Relative distance square error , Angle error , speed error The conversion error, represents the specified performance function, represents the lower bound of the error variable, represents the upper bound of the error variable, and The performance functions are specified and error variable The derivative of is the transformation error function of the conversion error; S5. Design the dynamic surface control law based on the conversion error and the derivative of the conversion error, perform a stability analysis of the closed-loop system of the dynamic surface control law, preset the initial parameters and expected formation of the robot formation, and use MATLAB software to simulate the dynamic surface control law.

2. The method for controlling the performance of a robot formation for heavy workpiece handling according to claim 1, characterized in that: The kinematic model of each robot in S1 is specifically expressed by the formula: ； In the formula, and Indicates The position coordinates of the robot in the inertial system, Indicates The heading angle of the robot in the inertial frame, Indicates The linear velocity of the robot in the inertial frame is, Indicates The angular velocity of the robot in the inertial frame is, , is the total number of robots in the robot formation; The dynamic model of each robot in S1 is specifically expressed by the formula: ； in, ; ； ； ， ； In the formula, is a symmetric positive definite inertia matrix, , To indicate the The speed of the robot in the inertial system, including the linear speed and angular velocity , , are the centripetal matrix and the Coriolis matrix, , is the gravity vector matrix, , when the i-th robot moves on the plane, , is the input matrix, , is the motor torque of the robot, is the external disturbance, is the total mass of the robot, is the mass of the robot body, is the mass of the robot wheel, is the driving wheel radius of the robot, represents the moment of inertia of the wheel about the axle, represents the moment of inertia of the wheel about the wheel diameter, represents the moment of inertia of the robot body, represents the total moment of inertia of the robot, is half the wheel spacing of the robot, The local coordinate origin of the robot To the centroid distance.

3. The method for controlling the performance of a robot formation for heavy workpiece handling according to claim 2, characterized in that: The distance error between robots in S2 is specifically expressed as: ； In the formula, Indicates The robot and The distance error between the robots, Indicates and The distance between the robots, Indicates The robot and The target distance between the robots; The relative distance square error between robots in S2 is specifically expressed as: ； In the formula, Indicates The robot and The square error of the relative distance between the robots, Indicates and The relative positions of the robots at time t.

4. The method for controlling the performance of a robot formation for heavy workpiece handling as claimed in claim 3, characterized in that: In S3, the safety distance and maximum sensing range between robots are set. According to the safety distance, maximum sensing range, distance error between robots and relative distance square error, the relative distance square error constraint between robots is set. Specifically, it includes the following: S31, preset the safe distance and maximum sensing range between robots; S32, imposing constraints on the distance between the robots according to the safety distance and the maximum sensing range, and setting a distance error constraint between the robots in combination with the distance error between the robots, and setting a sufficient condition for the robot formation to maintain infinitesimal rigidity according to the distance error constraint between the robots; S33. Set a constraint on the squared error of the relative distances between the robots based on the squared error of the relative distances between the robots and a sufficient condition for the robot formation to maintain infinitesimal rigidity.

5. The method for controlling the performance of a robot formation for heavy workpiece handling according to claim 4, characterized in that: The relative distance square error constraint between robots in S33 is specifically expressed by the formula: ； In the formula, , , ， ， in, Indicates The robot and The square error of the relative distance between the robots, represents the lower bound of the relative distance squared error, represents the upper bound of the relative distance square error, , , are all intermediate variables. For the The robot and The safe distance between robots. For the The robot and The maximum sensing range between robots.

6. The method for controlling the performance of a robot formation for heavy workpiece handling according to claim 5, characterized in that: The dynamic surface control law includes the control law of virtual angular velocity, reference velocity, expected angle, virtual linear velocity and dynamic model. In S5, the dynamic surface control law is designed according to the conversion error and the derivative of the conversion error, which includes the following: S51, according to the conversion error and its derivatives set up Conversion error and its derivatives , according to the conversion error and its derivatives Design The virtual angular velocity of the robot ; S52, according to the conversion error and its derivatives set up Conversion error and its derivatives , according to the conversion error derivative Design The reference speed of the robot and expected angle ; S53. According to The reference speed of the robot and expected angle Design Virtual linear speed of the robot ; S54, according to the conversion error The first derivative of set up The derivative of the conversion error , according to the conversion error derivative Design control laws for dynamic models.

7. The method for controlling the performance of a robot formation for heavy workpiece handling according to claim 6, characterized in that: S51 The virtual angular velocity of the robot Specifically: ； in, ; ； ； In the formula, For the The virtual angular velocity of the robot, For the The first adaptive law of a robot, is the specified performance function for the angular error, For the The angular error of the robot, No. The derivative of the desired angle of the robot, is the conversion error of the angle error, , , , , , All are intermediate variables.

8. The method for controlling the performance of a robot formation for heavy workpiece handling according to claim 6, characterized in that: S52 The reference speed of the robot Specifically expressed as: ； In the formula, For the The reference speed of the robot, and Respectively The reference speed of the robot The velocity components in the x-axis and y-axis, is the conversion error of the relative distance square error, is the prescribed performance function of the relative distance squared error, , , , are all intermediate variables. Indicates The robot and The relative positions of the robots; S52 The desired angle of the robot Specifically expressed as: ； In the formula, Indicates The desired angle of the robot.

9. The method for controlling the performance of a robot formation for heavy workpiece handling according to claim 6, characterized in that: The control law of the dynamic model in S54 can be designed as: ； in, ; ； ； In the formula, For control torque The superposition vector of is the input matrix, is a symmetric positive definite inertia, are the centripetal matrix and the Coriolis matrix, is the specified performance function of the velocity error, for The derivative of It is The speed of the robot The superposition vector of It is The speed error of the robot The superposition vector of is the derivative of the filtering speed, yes The superposition vector of , is the second adaptive law, The second adaptive law The derivative of , , , , , , , , All are intermediate variables.

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