CN112947442B - Finite time convergence vehicle formation controller and design method - Google Patents
Finite time convergence vehicle formation controller and design method Download PDFInfo
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- G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
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Abstract
The invention discloses a finite time convergence vehicle formation controller and a design method, wherein the current pose of a pilot vehicle and the reference pose of a virtual target vehicle to be tracked by a following vehicle are given, an error system is obtained through a control problem conversion module to obtain the error pose, the converted error system is combined with a back-step recurrence technology to obtain an intermediate control law sum with fractional power parameters, the converted error system is combined with the intermediate control law to obtain a finite time actual control law with fractional power, and finally the actual control law is returned to the converted error system, so that the error system is stable in finite time, the following vehicle and the pilot vehicle synchronously move, and effective control of the vehicle formation system is realized.
Description
Technical Field
The invention relates to the technical field of vehicle formation control, in particular to a limited-time convergence vehicle formation controller and a design method.
Background
The vehicle formation control can improve the efficiency of vehicles traveling on roads by improving the flexibility and the flexibility of the vehicle formation. Meanwhile, the vehicle formation can also increase the capacity on the road by increasing the density of vehicles on the road, so that the traffic safety and the road smoothness are improved to a certain extent. In addition, from the energy consumption perspective, the air resistance encountered in the driving process of the formed vehicles can be reduced, so that the fuel consumption of the vehicles is effectively reduced.
Currently, there are several common approaches to vehicle formation control: following a pilot method, a behavior-based method, a virtual structure method, an artificial potential field method, and the like. The formation method based on the behavior is simple to realize and suitable for uncertain environments, but the formation precision is poor and accurate mathematical analysis is difficult to perform; the virtual structure method and the pilot following method respectively need the full state information of the virtual structure and the pilot robot; the artificial potential field method is a method of converting complex environmental information into a repulsive field and a gravitational field model and finding a path from an initial point to a target point through the model. Following the pilot approach has the following problems:
first, in the existing vehicle formation control method, the convergence speed problem of the system is basically not considered. From a practical point of view, it is desirable to achieve the control objective as soon as possible, but due to limitations of technology and the like, the conventional vehicle formation control schemes cannot achieve the desired formation effect within the desired time.
Secondly, although the existing formation control method can realize control of the vehicle, when the situation of complex road conditions such as limiting running time is faced, the safety and reliability of the vehicle system under the running of a large speed range cannot be ensured.
Disclosure of Invention
The invention provides a limited time convergence vehicle formation controller and a design method thereof, which are used for overcoming the technical problems.
The invention discloses a design method of a finite time convergence vehicle formation controller, which comprises the following steps:
establishing a formation vehicle motion model; obtaining vehicle pose information through the formation vehicle motion model; the vehicle pose information includes: piloting the vehicle, following the vehicle and virtual target vehicle pose information;
obtaining the pose of the virtual target vehicle according to the pose information of the virtual target vehicle and the pilot vehicle; calculating the errors of the pose of the virtual target vehicle and the pose of the pilot vehicle, and combining the coordinate system of the following vehicle to obtain a converted error system;
designing a virtual error surface according to the pose of the virtual target vehicle; calculating an intermediate control variable by the error system and a virtual error plane;
obtaining a control law of limited time convergence according to the pose of the virtual target vehicle and the intermediate control quantity;
substituting the control law into the error system so that the error system is stable for a limited time, thereby enabling the following vehicle to move synchronously with the piloted vehicle.
Further, building a movement model of the formation vehicle, including:
the motion model of any vehicle in the formation is expressed as:
setting the gravity center of any vehicle as x, y, and setting the gravity center x of the rear wheels of the vehicle 1 ,y 1 Distance from center of gravity of vehicle and center of gravity x of front wheel of said vehicle 2 ,y 2 The distances from the center of gravity of the vehicle are equal and are half of the distance between the axles of the vehicle, and the distances are expressed as follows:
X＝b 1 x+b 2 y+c 1 cosθ-c 2 sinθ (2)
wherein X= [ X ] 1 ,x 2 ,y 1 ,y 2 ] T ，b 1 ＝[1,1,0,0] T ，b 2 ＝[0,0,1,1] T ，c 1 ＝[-l/2,l/2,0,0] T ，c 2 ＝[0,0,l/2,l/2] T ；
Obtaining a vehicle constraint relation by calculating the formula (2):
in the method, the pose of the vehicle is represented by a vectorX and y represent the coordinates of the position of the vehicle in the coordinate system, θ is the inclination angle of the vehicle body to the x-axis, +.>The steering angle of the front wheel to the vehicle body; v 1 Is the forward speed of the rear wheel of the automobile, v 2 Is the side steering angular velocity of the front wheel, and l represents the distance between the front wheel and the rear wheel.
Further, the pose of the virtual target vehicle is obtained according to the pose information of the virtual target vehicle and the pilot vehicle; calculating the errors of the pose of the virtual target vehicle and the pose of the piloted vehicle, and combining the coordinate system of the following vehicle to obtain a converted error system, wherein the error system comprises the following components:
the vehicle formation pose error equation is established as follows:
in the method, the current pose of the lead vehicle isThe control input is [ u ] 1 ,u 2 ] Τ The reference pose of the virtual target vehicle to be followed by the following vehicle is +.>The control input is [ u ] v1 ,u v2 ] Τ The pose error is +.>
The transformed error system is described as:
wherein u is v1 Is the forward speed of the rear wheels of the virtual target vehicle, u v2 Is the side steering angular velocity of the front wheels of the virtual target vehicle,steering angle of front wheel of leader vehicle to body,/->Steering angle of front wheel pair body of virtual target vehicle, u 1 ，u 2 Is the control rate.
Further, the designing a virtual error plane according to the pose of the virtual target vehicle includes:
according to the position error coordinate x of the rear wheel of the vehicle in a coordinate system e And y e Pose error inclination angle theta of vehicle body to x-axis coordinate e Position and attitude error steering angle of front wheel pair vehicle bodyThe virtual error plane is designed as follows:
wherein α is an introduced intermediate control variable;
said calculating intermediate control variables by said error system and virtual error plane, comprising:
the virtual error plane is derived through a quadratic Lyapunov equation, and the form of the intermediate control variable alpha is as follows:
wherein k is 2 ，k 3 Is a positive design parameter, ρ is a positive fractional power parameter, and 1/2 < ρ < 1;
obtaining a derivative of the virtual error plane by the error system and the virtual error plane, expressed as:
in the method, in the process of the invention,g 1 (·)＝u v1 cosζ 3 +αζ 2 -u 1 ，g 2 (·)＝u v1 sinζ 3 -αζ 1 ，h 1 (·)＝ζ 2 ，h 2 (·)＝-ζ 1 ，h 3 (·)＝-1；
order theFor the intermediate control variable, an intermediate control variable β was designed according to the lyapunov stability theory, expressed as:
wherein k is 4 Is a positive design parameter.
Further, the obtaining a control law of limited time convergence according to the pose of the virtual target vehicle and the intermediate control quantity comprises the following steps:
obtaining a control law u of the error system converged in a limited time according to the formula (5), the formula (7) and the formula (9) 1 And u 2 Expressed as:
wherein k is 1 Is a positive design parameter, sgn (·) is a sign function.
A finite time converging vehicle fleet controller, comprising: the system comprises a control problem conversion module, a virtual control input solving module and an actual control input solving module;
the control problem conversion module is used for obtaining the pose of the virtual target vehicle according to the pose information of the virtual target vehicle and the pilot vehicle; calculating the errors of the pose of the virtual target vehicle and the pose of the pilot vehicle, and combining the coordinate system of the following vehicle to obtain a converted error system;
the virtual control input solving module is used for designing a virtual error face according to the pose of the virtual target vehicle; calculating an intermediate control variable by the error system and a virtual error plane;
the actual control input solving module is used for obtaining a control law with limited time convergence according to the pose of the virtual target vehicle and the intermediate control quantity; substituting the control law into the error system so that the error system is stable for a limited time, thereby enabling the following vehicle to move synchronously with the piloted vehicle.
According to the invention, firstly, the problem of formation control is converted into the problem of track tracking of the following vehicles to the leader vehicles, and the problem of limited time control is combined, so that a dynamic vehicle formation control method under complex road conditions can be realized, and the reliability and anti-interference performance of vehicle formation control are effectively improved. Meanwhile, under the condition of considering the convergence rate, the system can still be controlled by introducing control input based on the fractional power parameter to the control law, and the system has better robustness and faster convergence.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings that are required in the embodiments or the description of the prior art will be briefly described, and it is obvious that the drawings in the following description are some embodiments of the present invention, and other drawings may be obtained according to the drawings without inventive effort to a person skilled in the art.
FIG. 1 is a flow chart diagram of a method of designing a finite time convergence vehicle fleet controller;
FIG. 2 is a schematic diagram of a finite time convergence vehicle fleet controller;
FIG. 3 is a graph showing the effect of controlling the forward speed of the rear wheels of the vehicle in the simulation experiment of the present invention;
FIG. 4 is a graph showing the effect of controlling the steering angle speed of the front wheels of the vehicle in the simulation experiment of the present invention;
fig. 5 is a diagram showing the effect of controlling the steering angle of the front wheel of the vehicle to the body of the vehicle in the simulation experiment of the present invention.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the embodiments of the present invention more apparent, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention, and it is apparent that the described embodiments are some embodiments of the present invention, but not all embodiments of the present invention. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
As shown in fig. 1, the present embodiment provides a method for designing a finite time convergence vehicle formation controller, including:
101. establishing a formation vehicle motion model; obtaining vehicle pose information through a formation vehicle motion model; vehicle pose information, comprising: piloting the vehicle, following the vehicle and virtual target vehicle pose information;
specifically, the motion model of any vehicle in the formation is expressed as:
setting the gravity center of any vehicle as x, y, and setting the gravity center x of the rear wheels of the vehicle 1 ,y 1 Distance to center of gravity of vehicle and front wheel weight of vehicleHeart x 2 ,y 2 The distances from the center of gravity of the vehicle are equal and are half of the distance between the axles of the vehicle, and the distances are expressed as follows:
X＝b 1 x+b 2 y+c 1 cosθ-c 2 sinθ (2)
wherein X= [ X ] 1 ,x 2 ,y 1 ,y 2 ] T ，b 1 ＝[1,1,0,0] T ，b 2 ＝[0,0,1,1] T ，c 1 ＝[-l/2,l/2,0,0] T ，c 2 ＝[0,0,l/2,l/2] T ；
Obtaining a vehicle constraint relation by calculating the formula (2):
in the method, the pose of the vehicle is represented by a vectorX and y represent the coordinates of the position of the vehicle in the coordinate system, θ is the inclination angle of the vehicle body to the x-axis, +.>The steering angle of the front wheel to the vehicle body; v 1 Is the forward speed of the rear wheel of the automobile, v 2 Is the side steering angular velocity of the front wheel, and l represents the distance between the front wheel and the rear wheel.
102. Obtaining the pose of the virtual target vehicle according to the pose information of the virtual target vehicle and the pilot vehicle; calculating the errors of the pose of the virtual target vehicle and the pose of the pilot vehicle, and combining a coordinate system of the following vehicle to obtain a converted error system;
specifically, the vehicle formation pose error equation is established as follows:
in the method, the current pose of the lead vehicle isThe control input is [ u ] 1 ,u 2 ] Τ The reference pose of the virtual target vehicle to be followed by the following vehicle is +.>The control input is [ u ] v1 ,u v2 ] Τ The pose error is +.>
For a pilot following formation system, the main track of the formation is typically determined by the pilot vehicle person, and the reference track of the following vehicle is determined by the pilot vehicle and the virtual vehicle track generated by the structural parameters.
The error system after conversion is described as:
wherein u is v1 Is the forward speed of the rear wheels of the virtual target vehicle, u v2 Is the side steering angular velocity of the front wheels of the virtual target vehicle,steering angle of front wheel of leader vehicle to body,/->Steering angle of front wheel pair body of virtual target vehicle, u 1 ，u 2 Is the control rate.
The method mainly comprises the steps of firstly determining the distance and the angle between a virtual target vehicle and a hope of the virtual target vehicle through the position and the posture of the piloted vehicle, so as to determine the reference position and the posture of the virtual target vehicleSecondly, the position and the posture of the piloting vehicle are differenced with the position and the posture of the virtual target vehicle to obtain position and posture error +.>Obtaining a pose error equation (4) of the pose error under a coordinate system of the following vehicle through coordinate transformation; and finally, deriving the formula (4) and combining a vehicle formation motion model to obtain an error system of the converted formula (5). The transformed error system completes the problem of changing the formation control problem into the track tracking of the following vehicle to the leader vehicle by searching for a proper control law u 1 And u 2 The error system obtained by conversion can be stabilized in a limited time.
103. Designing a virtual error surface according to the pose of the virtual target vehicle; calculating an intermediate control variable through an error system and a virtual error plane;
specifically, the coordinates x are error-corrected according to the position of the rear wheels of the vehicle in the coordinate system e And y e Pose error inclination angle theta of vehicle body to x-axis coordinate e Position and attitude error steering angle of front wheel pair vehicle bodyThe virtual error plane is designed as follows:
wherein α is an introduced intermediate control variable;
the virtual error plane is derived through a quadratic Lyapunov equation, and the form of the intermediate control variable alpha is as follows:
wherein k is 2 ，k 3 Is a positive design parameter, ρ is a positive fractional power parameter, and 1/2 < ρ < 1;
the derivative of the virtual error plane is obtained by the error system and the virtual error plane, expressed as:
in the method, in the process of the invention,g 1 (·)＝u v1 cosζ 3 +αζ 2 -u 1 ，g 2 (·)＝u v1 sinζ 3 -αζ 1 ，h 1 (·)＝ζ 2 ，h 2 (·)＝-ζ 1 ，h 3 (·)＝-1；
order theFor the intermediate control variable, an intermediate control variable β was designed according to the lyapunov stability theory, expressed as:
wherein k is 4 Is a positive design parameter.
104. Obtaining a control law of limited time convergence according to the pose of the virtual target vehicle and the intermediate control quantity;
specifically, according to the formulas (5), (7) and (9), a control law u is obtained in which the error system converges for a finite time 1 And u 2 Expressed as:
wherein k is 1 Is a positive design parameter, sgn (·) is a sign function.
105. Substituting the control law into the error system to stabilize the error system for a limited time, thereby enabling the following vehicle to move synchronously with the piloting vehicle.
Specifically, the control law is substituted into the error system, so that the error system is stable in a limited time, the stability of the error system in the limited time indicates that the converted error system is stable, the stability of the error system indicates that the vehicle follows the pilot vehicle, and the formation of the vehicle formation can be maintained after the vehicle follows the pilot vehicle.
As shown in fig. 2, the present embodiment provides a finite time convergence vehicle formation controller including: the system comprises a control problem conversion module, a virtual control input solving module and an actual control input solving module;
the control problem conversion module is used for obtaining the pose of the virtual target vehicle according to the pose information of the virtual target vehicle and the pilot vehicle; calculating the errors of the pose of the virtual target vehicle and the pose of the pilot vehicle, and combining a coordinate system of the following vehicle to obtain a converted error system; the input end of the control problem conversion module is connected with the pose information of the vehicle (the pose information of the lead vehicle, the following vehicle and the virtual target vehicle) and the output end of the actual control solving module.
The virtual control input solving module is used for designing a virtual error surface according to the pose of the virtual target vehicle; calculating an intermediate control variable through an error system and a virtual error plane; the input end of the virtual control input solving module is connected with the output end of the control problem converting module and the virtual target vehicle pose calculated and output by the vehicle motion model.
The actual control input solving module is used for obtaining a control law with limited time convergence according to the pose of the virtual target vehicle and the intermediate control quantity; substituting the control law into the error system to stabilize the error system for a limited time, thereby enabling the following vehicle to move synchronously with the piloting vehicle. The input end of the actual control input solving module is connected with the output end of the virtual control input solving module and the virtual target vehicle pose calculated and output by the vehicle motion model.
In simulation experiments, it can be seen from fig. 3, 4 and 5 that the following vehicle can track the upper piloted vehicle for a limited time.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solution of the present invention, and not for limiting the same; although the invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical scheme described in the foregoing embodiments can be modified or some or all of the technical features thereof can be replaced by equivalents; such modifications and substitutions do not depart from the spirit of the invention.
Claims (2)
1. A method of designing a finite time convergence vehicle fleet controller, comprising:
establishing a formation vehicle motion model; obtaining vehicle pose information through the formation vehicle motion model; the vehicle pose information includes: piloting the vehicle, following the vehicle and virtual target vehicle pose information;
obtaining the pose of the virtual target vehicle according to the pose information of the virtual target vehicle and the pilot vehicle; calculating the errors of the pose of the virtual target vehicle and the pose of the pilot vehicle, and combining the coordinate system of the following vehicle to obtain a converted error system;
designing a virtual error surface according to the pose of the virtual target vehicle; calculating an intermediate control variable by the error system and a virtual error plane;
obtaining a control law of limited time convergence according to the pose of the virtual target vehicle and the intermediate control quantity;
substituting the control law into the error system so that the error system is stable for a limited time, and thus the following vehicle and the piloting vehicle synchronously move;
establishing a formation vehicle motion model, including:
the motion model of any vehicle in the formation is expressed as:
setting the gravity center of any vehicle as x, y, and setting the gravity center x of the rear wheels of the vehicle 1 ,y 1 Distance from center of gravity of vehicle and center of gravity x of front wheel of said vehicle 2 ,y 2 The distances from the center of gravity of the vehicle are equal and are half of the distance between the axles of the vehicle, and the distances are expressed as follows:
X＝b 1 x+b 2 y+c 1 cosθ-c 2 sinθ (2)
wherein X= [ X ] 1 ,x 2 ,y 1 ,y 2 ] Τ ，b 1 ＝[1,1,0,0] Τ ，b 2 ＝[0,0,1,1] Τ ，c 1 ＝[-l/2,l/2,0,0] Τ ，c 2 ＝[0,0,l/2,l/2] Τ ；
Obtaining a vehicle constraint relation by calculating the formula (2):
in the method, the pose of the vehicle is represented by a vectorX and y represent the coordinates of the position of the vehicle in the coordinate system, θ is the inclination angle of the vehicle body to the x-axis, +.>The steering angle of the front wheel to the vehicle body; v 1 Is the forward speed of the rear wheel of the automobile, v 2 Is the side steering angular velocity of the front wheel, l represents the axial distance between the front wheel and the rear wheel;
the pose of the virtual target vehicle is obtained according to the pose information of the virtual target vehicle and the pilot vehicle; calculating the errors of the pose of the virtual target vehicle and the pose of the piloted vehicle, and combining the coordinate system of the following vehicle to obtain a converted error system, wherein the error system comprises the following components:
the vehicle formation pose error equation is established as follows:
in the method, the current pose of the piloted vehicle isThe control input is [ u ] 1 ,u 2 ] Τ The reference pose of the virtual target vehicle to be followed by the following vehicle is +.>The control input is [ uv ] 1 ,uv 2 ] Τ The pose error is +.>
The transformed error system is described as:
wherein u is v1 Is the forward speed of the rear wheels of the virtual target vehicle, u v2 Is the side steering angular velocity of the front wheels of the virtual target vehicle,steering angle of front wheel of pilot vehicle to vehicle body,/->Steering angle of front wheel pair body of virtual target vehicle, u 1 ，u 2 Is the control rate;
the designing a virtual error plane according to the pose of the virtual target vehicle comprises:
according to the pose error coordinates xe and ye of the rear wheels of the vehicle in the coordinate system and the pose error inclination angle thetae of the vehicle body to the x-axis coordinate, the pose error steering angle of the front wheels to the vehicle bodyThe virtual error plane is designed as follows:
wherein α is an introduced intermediate control variable;
said calculating intermediate control variables by said error system and virtual error plane, comprising:
the virtual error plane is derived through a quadratic Lyapunov equation, and the form of the intermediate control variable alpha is as follows:
wherein k is 2 ，k 3 Is a positive design parameter, ρ is a positive fractional power parameter, and 1/2 < ρ < 1;
obtaining a derivative of the virtual error plane by the error system and the virtual error plane, expressed as:
in the method, in the process of the invention,g 1 (·)＝u v1 cosζ 3 +αζ 2 -u 1 ，g 2 (·)＝u v1 sinζ 3 -αζ 1 ，h 1 (·)＝ζ 2 ，h 2 (·)＝-ζ 1 ，h 3 (·)＝-1；
order theFor the intermediate control variable, an intermediate control variable β was designed according to the lyapunov stability theory, expressed as:
wherein k is 4 Is a positive design parameter;
the control law of finite time convergence is obtained according to the pose of the virtual target vehicle and the intermediate control quantity, and the control law comprises the following steps:
obtaining a control law u of the error system converged in a limited time according to the formula (5), the formula (7) and the formula (9) 1 And u 2 Expressed as:
wherein k is 1 Is a positive design parameter, sgn (·) is a sign function.
2. A finite time convergence vehicle fleet controller, comprising:
the system comprises a control problem conversion module, a virtual control input solving module and an actual control input solving module;
the control problem conversion module is used for obtaining the pose of the virtual target vehicle according to the pose information of the virtual target vehicle and the pilot vehicle; calculating the errors of the pose of the virtual target vehicle and the pose of the pilot vehicle, and combining a coordinate system of the following vehicle to obtain a converted error system; establishing a formation vehicle motion model, including:
the motion model of any vehicle in the formation is expressed as:
setting the gravity center of any vehicle as x, y, and setting the gravity center x of the rear wheels of the vehicle 1 ,y 1 Distance from center of gravity of vehicle and center of gravity x of front wheel of said vehicle 2 ,y 2 The distances from the center of gravity of the vehicle are equal and are half of the distance between the axles of the vehicle, and the distances are expressed as follows:
X＝b 1 x+b 2 y+c 1 cosθ-c 2 sinθ (2)
wherein X= [ X ] 1 ,x 2 ,y 1 ,y 2 ] Τ ，b 1 ＝[1,1,0,0] Τ ，b 2 ＝[0,0,1,1] Τ ，c 1 ＝[-l/2,l/2,0,0] Τ ，c 2 ＝[0,0,l/2,l/2] Τ ；
Obtaining a vehicle constraint relation by calculating the formula (2):
in the method, the pose of the vehicle is represented by a vectorX and y represent the coordinates of the position of the vehicle in the coordinate system, θ is the inclination angle of the vehicle body to the x-axis, +.>The steering angle of the front wheel to the vehicle body; v 1 Is the forward speed of the rear wheel of the automobile, v 2 Is the side steering angular velocity of the front wheel, l represents the axial distance between the front wheel and the rear wheel;
the virtual control input solving module is used for designing a virtual error face according to the pose of the virtual target vehicle; calculating an intermediate control variable by the error system and a virtual error plane; the pose of the virtual target vehicle is obtained according to the pose information of the virtual target vehicle and the pilot vehicle; calculating the errors of the pose of the virtual target vehicle and the pose of the piloted vehicle, and combining the coordinate system of the following vehicle to obtain a converted error system, wherein the error system comprises the following components:
the vehicle formation pose error equation is established as follows:
in the method, the current pose of the piloted vehicle isThe control input is [ u ] 1 ,u 2 ] Τ The reference pose of the virtual target vehicle to be followed by the following vehicle is +.>The control input is [ u ] v1 ,u v2 ] Τ The pose error is +.>
The transformed error system is described as:
wherein u is v1 Is the forward speed of the rear wheels of the virtual target vehicle, u v2 Is the side steering angular velocity of the front wheels of the virtual target vehicle,steering angle of front wheel of pilot vehicle to vehicle body,/->Steering angle of front wheel pair body of virtual target vehicle, u 1 ，u 2 Is the control rate;
the designing a virtual error plane according to the pose of the virtual target vehicle comprises:
according to the position error coordinate x of the rear wheel of the vehicle in a coordinate system e And y e Pose error inclination angle theta e of vehicle body to x-axis coordinate and pose error steering angle of front wheel to vehicle bodyThe virtual error plane is designed as follows:
wherein α is an introduced intermediate control variable;
the actual control input solving module is used for obtaining a control law with limited time convergence according to the pose of the virtual target vehicle and the intermediate control quantity; substituting the control law into the error system so that the error system is stable for a limited time, and thus the following vehicle and the piloting vehicle synchronously move;
said calculating intermediate control variables by said error system and virtual error plane, comprising:
the virtual error plane is derived through a quadratic Lyapunov equation, and the form of the intermediate control variable alpha is as follows:
wherein k is 2 ，k 3 Is a positive design parameter, ρ is a positive fractional power parameter, and 1/2 < ρ < 1;
obtaining a derivative of the virtual error plane by the error system and the virtual error plane, expressed as:
in the method, in the process of the invention,g 1 (·)＝u v1 cosζ 3 +αζ 2 -u 1 ，g 2 (·)＝u v1 sinζ 3 -αζ 1 ，h 1 (·)＝ζ 2 ，h 2 (·)＝-ζ 1 ，h 3 (·)＝-1；
order theFor intermediate control variables, an intermediate is designed according to Lyapunov stability theoryThe control variable β, expressed as:
wherein k is 4 Is a positive design parameter;
the control law of finite time convergence is obtained according to the pose of the virtual target vehicle and the intermediate control quantity, and the control law comprises the following steps:
obtaining a control law u of the error system converged in a limited time according to the formula (5), the formula (7) and the formula (9) 1 And u 2 Expressed as:
wherein k is 1 Is a positive design parameter, sgn (·) is a sign function.
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