Trajectory tracking method of front and rear wheel independent driving robot suitable for pipeline environment

Abstract

The invention belongs to the technical field of robot control, and relates to a trajectory tracking method of a front and rear wheel independent drive robot suitable for a pipeline environment, which comprises the following steps: step 1, analyzing geometric constraints of a pipeline cylindrical surface structure on the robot according to geometric structures of a pipeline and robot wheels, and obtaining a robot motion representation form in a two-dimensional plane; 2, according to the kinematics principle of the incomplete constraint wheeled robot, a nonlinear model of a robot motion system is established; 3, a control target of robot trajectory tracking is determined according to the actual working condition; and 4, designing a kinematics controller by using a Lyapunov direct method in combination with the nonlinear model of the robot motion system and a trajectory tracking control target, and finally raising the dimension of an output result of the controller into a pipeline space to realize trajectory tracking of the mobile robot. According to the invention, rapid and accurate control of the wheeled pipeline robot with disturbance and uncertain items is realized, and the stability of the robot tracking process is ensured.

Description

Technical Field

[0001] The present invention belongs to the technical field of robot control, and relates to a trajectory tracking method for a front and rear wheel independently driven robot applicable to a pipeline environment. Background Art

[0002] In the entire natural gas operation system, natural gas stations are crucial links, mainly playing roles in natural gas processing, distribution, storage, and configuration. With the increase in offshore oil and gas energy development, the construction of coastal oil and gas pipelines has increased. Mobile robots are widely used in the field of pipeline maintenance. Pipeline robots with a Bike configuration can carry various detection devices to any position inside the pipeline, having higher flexibility. However, this type of robot poses higher requirements for trajectory tracking on the inner wall surface of the pipeline. Traditional methods cannot adapt to the three-dimensional space with a relatively large curvature inside the pipeline. Therefore, it is necessary to develop a method applicable to trajectory tracking on the inner wall surface of the pipeline.

[0003] With the development of control technology, from the perspective of trajectory tracking control of surface wheeled robots, designing a reasonable tracking controller has become the focus of attention. Generally speaking, the performance of a trajectory tracking controller designed using a model-based control method depends to a large extent on the accuracy of the model. In the three-dimensional environment of the pipeline, 6 parameters are required to describe the pose of a mobile robot, and at the same time, its speed also has components in 3 directions. This makes the complexity of its model extremely high, which will seriously affect the response time of the controller. And when moving in the pipeline, the mobile robot will face unknown disturbances such as slipping, which further increases the difficulty of trajectory tracking control inside the pipeline. Summary of the Invention

[0004] In order to solve the above-mentioned technical problems existing in the prior art, the present invention proposes a trajectory tracking method for a front and rear wheel independently driven robot applicable to a pipeline environment. According to the special cylindrical structure of the pipeline and the geometric constraints of the robot, the trajectory tracking in a three-dimensional scenario is simplified to a problem in a two-dimensional scenario, and a kinematic controller is designed to ensure the stability and effectiveness of the system, and to achieve the control task of the mobile robot. The specific technical solution is as follows:

[0005] A trajectory tracking method for a front and rear wheel independently driven robot applicable to a pipeline environment, comprising the following steps:

[0006] Step 1: According to the geometric structures of the pipeline and the robot wheels, analyze the geometric constraints of the pipeline cylindrical structure on the robot, and obtain the representation form of the robot motion in a two-dimensional plane;

[0007] Step 2: According to the kinematic principle of a non-holonomic constrained wheeled robot, establish a nonlinear model of the robot motion system;

[0008] Step 3: Determine the control objective for the robot trajectory tracking according to the actual working conditions;

[0009] Step 4: Combine the non - linear model of the robot motion system and the control objective of the trajectory tracking, design a kinematic controller using the Lyapunov direct method, and finally lift the output result of the controller to the pipeline space to achieve the trajectory tracking of the mobile robot.

[0010] Further, the analysis of the geometric constraints of the pipeline cylindrical structure on the robot specifically includes:

[0011] Analyze the relationship between the wheel center speed and the rotational speed;

[0012] Solve the motion speed of a single wheel;

[0013] Analyze the rotational speed relationship between the front and rear wheels.

[0014] Further, the analysis of the relationship between the wheel center speed and the rotational speed specifically includes:

[0015] First, establish a world coordinate system: Take the starting point of the robot as the origin, the pipeline bus direction as the x - axis, the direction vertically upward along the gravity direction as the z - axis, and the direction perpendicular to both x and z as the y - axis;

[0016] Cut and unfold the pipeline cylinder along the bus L at the wheel - pipe contact point A into a two - dimensional plane. According to the definition of the world coordinate system, assume that A is located at the origin of the world coordinate system at the beginning, and the world coordinates of A after t moments are:

[0017]

[0018] Within a very short time, it is considered that v (t) , θ (t) are both constant values. Therefore:

[0019]

[0020] Among them, v refers to the magnitude of the motion speed of the wheel - pipe contact point, θ is the angle between the robot's traveling direction and the pipeline bus at the current position, and R is the pipeline radius;

[0021] Since the running trajectory of the robot in the pipeline is a curve, the speed of the wheel - pipe contact point A is corrected according to the curvature k of the trajectory, and the expression is:

[0022]

[0023] ω represents the rotational speed of the wheel, r represents the radius of the wheel, and η is the defined speed correction coefficient.

[0024] Further, the solution of the motion speed of a single wheel specifically includes:

[0025] An active coordinate system is established with the tangent vector τ, the principal normal n, and the binormal b at the contact point as the coordinate axes. According to the coordinate transformation matrix from the local coordinate system to the global coordinate system, it can be known that:

[0026]

[0027] Among them:

[0028] A ij = e i e jT = [τ n b] T e jT

[0029]

[0030] Among them, v A is the magnitude of the linear velocity of the contact point A:

[0031]

[0032] The coordinates of the wheel center in this local coordinate system are (0, rcosβ, rsinβ), and the coordinates in the world coordinate system are:

[0033]

[0034] Taking the derivative of this coordinate gives the velocity of the wheel center of the wheel, and thus the motion velocity of a single wheel is obtained.

[0035] Furthermore, the analysis of the rotational speed relationship between the front and rear wheels specifically includes: regarding the robot as a rigid body, according to the instantaneous screw axis theorem, the motion of the robot at any moment can be decomposed into a rotational motion around the instantaneous screw axis and a translational motion. Assuming the direction vector of the instantaneous screw axis is:

[0036]

[0037] The position vector of the point on the instantaneous screw axis closest to the origin of the world coordinate system is

[0038] Then there is:

[0039]

[0040]

[0041]

[0042]

[0043] Among them, ω0 is the instantaneous rotational angular velocity of the robot, and its direction is the same as Same, a1 and a2 are the position vectors of the front and rear wheels, and is the velocity vector of the rear and front wheels of the robot;

[0044] After solving equations (1) to (5), we can obtain the front wheel speed v f and rear wheel speed v b The relationship between the front and rear wheel speeds of the robot is:

[0045] v f =αv b

[0046] α=v f / v b .

[0047] Furthermore, the step 2 specifically includes:

[0048] The wheeled robot satisfies the following nonholonomic constraints:

[0049]

[0050] and are the velocity components of the robot in the x-axis and y-axis directions respectively;

[0051] Under nonholonomic constraints, a nonlinear model of the robot motion system is established, including the kinematic model and trajectory tracking error model;

[0052] The kinematic model is:

[0053]

[0054] The desired pose of the mobile robot is q r =[X r Y r θ r ] T ;

[0055] The trajectory tracking error model is:

[0056]

[0057] Where v = v b , φ is the front wheel steering angle, θ is the angle between the robot's travel speed direction and the pipeline busbar at its location, and l is the wheelbase.

[0058] Furthermore, there is a constraint on the speed of the front and rear wheels of the robot: f cosφ=v b .

[0059] Furthermore, the trajectory tracking error model incorporates a slip ratio, which is defined as follows:

[0060]

[0061] are the slip ratios of the front and rear wheels respectively, and the value range is [0, 1);

[0062] When the slip ratio is 0, there is no slip between the wheel and the ground. When the slip ratio is 1, the wheeled robot is in an uncontrollable state;

[0063] According to the definition of the slip ratio, we have:

[0064]

[0065]

[0066] Define Then we have:

[0067]

[0068] Furthermore, in step 4, the expression of the kinematic controller is as follows:

[0069]

[0070] where k x , k y , k θ are all positive real numbers, α = v f / v b is the relationship between the rotational speeds of the front and rear wheels, and represent the derivatives of the horizontal and vertical coordinates in the reference trajectory respectively.

[0071] Furthermore, the Lyapunov function is:

[0072]

[0073] Take the derivative of the above formula with respect to time:

[0074]

[0075] Substitute it into the error model in the kinematic model to obtain:

[0076]

[0077] Beneficial effects: By considering the geometric constraints of the pipeline cylindrical structure on the robot, the present invention describes the movement of the robot in three-dimensional space as movement in a two-dimensional plane, and then designs a kinematic controller for the robot according to the new coordinate representation method of the robot, realizing fast and precise control of the wheeled pipeline magnetic adsorption robot with disturbances and uncertainties, and ensuring the stability of the robot tracking process. Description of the Drawings

[0078] Figure 1 is a schematic structural diagram of the robot in this embodiment;

[0079] Figure 2 is Figure 1 the right view of

[0080] Figure 3 is a schematic diagram of the pipeline cylinder cut and unfolded along the generatrix at the wheel-pipeline contact point in this embodiment;

[0081] Figure 4 is a schematic diagram of the moving coordinate system at the wheel-pipeline contact point in this embodiment;

[0082] Figure 5 is a schematic diagram of the geometric relationship of the circular motion of the robot wheel in this embodiment;

[0083] Figure 6 is a schematic diagram of the kinematic analysis of the robot in a two-dimensional plane in this embodiment;

[0084] Figure 7 is a schematic diagram of the kinematic analysis of the robot in a two-dimensional plane for a short time in this embodiment;

[0085] Figure 8 is a schematic diagram of the mapping relationship between the x-axis and y-axis of the world coordinate system of this embodiment on the circular plane;

[0086] Figure 9 is a schematic diagram of the reference trajectory segment of the robot in this embodiment. Detailed Implementation Modes

[0087] In order to make the objectives, technical solutions, and technical effects of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings of the specification and embodiments.

[0088] This embodiment discloses a trajectory tracking method for a front and rear wheel independently driven robot applicable to a pipeline environment, including the following steps:

[0089] Step 1, according to the geometric structures of the pipeline and the robot wheels, analyze the geometric constraints of the pipeline cylindrical structure on the robot, map the three-dimensional coordinates of the robot moving in the pipeline space to a two-dimensional plane after the pipeline is unfolded, and obtain a two-dimensional representation form of the robot movement.

[0090] Obtain the relevant parameters of the wheeled pipeline magnetic adsorption robot, including the rotational speed of the front wheel, the rotational speed of the rear wheel, the rotational speed of the steering gear, the length of the robot, and the wheelbase of the wheels.

[0091] As Figure 1 and Figure 2 shown, the wheeled pipeline magnetic adsorption robot in this embodiment is a robot with a Bike configuration. It is driven by two magnetic wheels, a front wheel and a rear wheel, and a front wheel steering gear. At the same time, in order to enable the robot to run smoothly in the pipeline, a passive torsion joint is added at the connection between the front wheel and the body.

[0092] When analyzing the motion of the robot, this embodiment focuses on the speed of each wheel of the robot in the world coordinate system. In a two-dimensional plane, the speed of the wheel center is equal to its rotational speed multiplied by the wheel diameter, that is: v = ωr.

[0093] However, in the pipeline, it is necessary to solve the relationship between the rotational speed and the wheel center speed according to the geometric constraints of the pipeline cylinder surface.

[0094] To analyze the relationship between the wheel center speed and the rotational speed, first establish a world coordinate system: Take the starting point of the robot as the origin, the direction of the pipeline bus as the x-axis, the direction vertically upward along the gravity direction as the z-axis, and the direction perpendicular to both x and z as the y-axis to establish the world coordinate system.

[0095] As Figure 3 and Figure 4 shown, cut and unfold the pipeline cylinder surface along the bus L at the wheel-pipe contact point A. According to the definition of the world coordinate system, assume that A is located at the origin of the world coordinate system at the beginning, and the world coordinates of A after t moments are:

[0096]

[0097] In a very short period of time, it is considered that v (t) , θ (t) are both constant values. Therefore:

[0098]

[0099] Here, v refers to the magnitude of the motion speed of the wheel-pipe contact point. Since the running track of the robot in the pipeline is a curve, the speed of the wheel-pipe contact point A should be corrected according to the curvature of the track. The expression is:

[0100]

[0101] ω represents the rotational speed of the wheel, and r represents the radius of the wheel. More specifically, as Figure 5As shown in the figure, consider a certain moment when the robot is moving on a curved trajectory. At this time, the curvature of the trajectory where the wheel is located is k. After a very short time interval of Δt, the wheel rotates from AB to A'B'. Due to the existence of curvature, it can be approximately regarded that the wheel moves on a circle during this time period. From the geometric relationship in the figure, it can be seen that the actual rotation angle of the wheel is Δθ, but the contact point between the wheel and the curve changes from A to B', and this arc length is From the geometric relationship in the figure, it can be known that:

[0102]

[0103] That is:

[0104]

[0105] After arrangement, it is obtained that

[0106]

[0107] Therefore, the actual speed of the contact point between the wheel and the trajectory should be:

[0108]

[0109] The above-mentioned Taking the derivative with respect to time gives the speed of the contact point A between the wheel and the pipeline.

[0110] Next, further solve for the speed of the wheel center. For each wheel plane, an active coordinate system is established with the tangent vector, principal normal, and binormal at the contact point as the coordinate axes, that is, the Frenet active frame. Taking point A as an example, continue to refer to Figure 4 where:

[0111]

[0112]

[0113] n = b × τ

[0114] P is the coordinate of the contact point between the wheel and the pipeline derived above.

[0115] According to the coordinate transformation matrix from the local coordinate system to the global coordinate system, it can be known that:

[0116]

[0117] where:

[0118] A ij = e i e jT = [τ n b] T e jT

[0119]

[0120] where v A is the magnitude of the linear velocity of the contact point A, which has been calculated in the previous derivation:

[0121]

[0122] The coordinates of the wheel center in this local coordinate system are (0, rcosβ, rsinβ), and the coordinates in the world coordinate system are:

[0123]

[0124] Taking the derivative of this coordinate gives the velocity of the wheel center of this wheel.

[0125] Thus, the motion velocity of a single wheel is obtained, denoted here as:

[0126] v = Λωr.

[0127] Since the bicycle configuration robot in this embodiment is independently driven by the front and rear wheels, an incorrect front and rear wheel speed relationship will cause the robot to rotate unnecessarily and increase the torque requirement for the robot to move by a single motor. Therefore, it is necessary to solve for the appropriate front and rear wheel speed relationship.

[0128] Regarding the robot as a rigid body, according to the instantaneous screw axis theorem, the motion of the robot at any moment can be decomposed into a rotational motion about the instantaneous screw axis and a translational motion. Assume that the direction vector of the instantaneous screw axis is:

[0129]

[0130] The position vector of the point on the instantaneous screw axis closest to the origin of the world coordinate system is

[0131] Then there is:

[0132]

[0133]

[0134] where ω0 is the instantaneous angular velocity of the robot, and its direction is the same as , and a1, a2 are the position vectors of the front and rear wheels. and are the velocity vectors of the rear and front wheels of the robot.

[0135] For the above 5 equations, there are e x , e y , e z , ω, v b , vf Six unknown quantities can be combined to obtain v b ,v f The relationship between the front and rear wheel speeds of the robot is α:

[0136] v f =αv b ,

[0137] α=v f / v b .

[0138] Step 2: Based on the kinematic principles of the nonholonomically constrained wheeled robot, a nonlinear model of the wheeled pipeline magnetic adsorption robot motion system is established.

[0139] Specifically, refer to Figure 6 , the wheeled robot satisfies the following nonholonomic constraints:

[0140]

[0141] Under this constraint, the kinematic model is:

[0142]

[0143] The desired pose of the mobile robot is q r =[X r Y r θ r ] T ;

[0144] The trajectory tracking error model of the wheeled mobile robot in the world coordinate system is:

[0145]

[0146] v=v b ,

[0147]

[0148] Where θ is the angle between the robot’s travel speed and the pipeline busbar at its location, v f and v b are the front and rear wheel speeds respectively, l is the wheelbase, and are the velocity components of the robot in the x-axis and y-axis directions respectively.

[0149] Since the rear wheels do not have a steering function, the speed provided by the rear wheels always points along the direction of the robot's body. In addition to driving, the front wheels have a steering angle φ, and the speed component perpendicular to the robot's body direction provided by the front wheels will cause the robot to rotate around its center of mass. The body position sensor is installed directly above the rear wheels. Therefore, taking the rear wheel speed as the reference speed, we have:

[0150] v = v b ,

[0151] Actually, because the robot is a non-stretchable rigid body, there is also a constraint here: v f cosφ = v b .

[0152] In a very short time interval Δt, the mobile robot moves forward with the front wheel speed v f and the front wheel steering angle φ. At the initial state, the angle between the robot and the x-axis of the world coordinate system is θ. The definitions of the various parameters are as Figure 7 shown, and we have:

[0153]

[0154] According to the geometric constraints of the mobile robot, we know that:

[0155] v f cosφ = v,

[0156] y1 - y = lsinθ,

[0157] x1 - x = lcosθ,

[0158] After rearrangement, we get:

[0159]

[0160] Using L'Hopital's rule to calculate and rearrange gives:

[0161]

[0162] The trajectory tracking error model introduces the slip ratio:

[0163] The slip rate is introduced to describe the degree of slipping, and its definition is as follows:

[0164]

[0165] are the slip ratios of the front and rear wheels respectively, and their value ranges are [0, 1). When the slip ratio is 0, there is no slipping between the wheels and the ground. When the slip ratio is 1, the wheeled robot is in an uncontrollable state.

[0166] According to the definition of the slip ratio, we have:

[0167]

[0168]

[0169] Definition

[0170] According to the previous derivation, we have:

[0171]

[0172] In the above coordinate system, the x-axis is the direction of the pipeline busbar in the actual world coordinate system, that is, the x-axis of the world coordinate system, and the y-axis is the value obtained by constraining the y and z parameters in the actual coordinate system through the pipeline cylindrical surface. The mapping relationship between them can be obtained on the circular plane, as Figure 8 shown. Due to the pipeline surface constraint, the movement speed of the robot must be parallel to the tangent plane of the pipeline at the contact point. Therefore, in fact, the movement speed of the robot can be decomposed along two directions: the pipeline busbar and the cross-section at the contact point, that is, decomposed in the two-dimensional plane after the pipeline is unfolded.

[0173] Step 3: Determine the control objective of the trajectory tracking of the wheeled pipeline magnetic adsorption robot according to the actual working conditions.

[0174] In the pipeline detection task of an embodiment, the detection requires the robot to walk to the weld area and then perform a circular motion along the weld to complete the detection task. The general method for wall thickness measurement is to measure at the sampling points on the 4 points of 90° on the circumferential wall of the pipe or at regular intervals of 5-10 mm on regular curves such as helical lines. These tasks all require the robot to be able to perform curved surface motion on the inner wall of the pipeline.

[0175] According to the simplified coordinate expression form, the reference trajectory should be a combined trajectory of a line segment with a fixed angle (depending on the pitch of the equidistant helix) with the x-axis and a line segment along the x-axis, as Figure 9 shown.

[0176] Step 4: Combine the nonlinear model of the motion system of the wheeled pipeline magnetic adsorption robot and the control objective of the trajectory tracking, use the Lyapunov direct method to design a kinematic controller, determine the design parameters by reasonably selecting the Lyapunov function, and prove the stability and effectiveness of the system. Finally, the output result of the controller is dimensionally elevated to the pipeline space to achieve the control task of the mobile robot trajectory tracking.

[0177] The expression of the said controller is as follows:

[0178]

[0179] where k x , k y , kθ are all positive real numbers, α = v f / v b is the rotational speed relationship between the front and rear wheels, and respectively represent the derivatives of the horizontal and vertical coordinates in the reference trajectory.

[0180] The Lyapunov function is:

[0181]

[0182] Taking the derivative of the above formula with respect to time:

[0183]

[0184] Substituting into the error model in the kinematic model, we can obtain:

[0185]

[0186] According to the Lyapunov stability theorem, we know that:

[0187]

[0188] That is, under the aforementioned control law, the error of the kinematic system asymptotically converges to 0.

[0189] In actual control, through the relational expression between the wheel rotational speed and the actual wheel speed, the rotational speeds ω b and ω f of the front and rear wheels that should be had are solved. Adding the value of φ given by the control law can achieve the tracking control of the robot.

[0190] In summary, for the robot trajectory tracking method of the present invention, based on the special cylindrical surface structure of the pipeline, the geometric constraints of the wheeled robot moving in the pipeline are analyzed, and the parameter expression of the trajectory tracking controller is simplified through the wheel-pipeline geometric constraint analysis. A two-dimensional plane after the pipeline cylinder is unfolded is established, and the three-dimensional coordinates of the robot moving in the pipeline are mapped to the two-dimensional plane after the pipeline is unfolded according to the wheel-pipeline geometric constraint relationship. A kinematic model of the robot is established in the virtual two-dimensional plane, and then the control target of the robot trajectory tracking is determined according to the actual working conditions. Combining the nonlinear model of the robot movement and the control target of the trajectory tracking, the kinematic controller is designed using the Lyapunov direct method, and the design parameters are determined by reasonably selecting the Lyapunov function, proving the stability and effectiveness of the system. This method is applicable to various types of traditional wheeled robots, realizing the fast and precise control of the wheeled pipeline magnetic adsorption robot with disturbances and uncertainties, and ensuring the stability of the robot tracking process.

[0191] The above are only the preferred embodiments of the present invention and do not impose any formal restrictions on the present invention. Although the implementation process of the present invention has been described in detail above, those skilled in the art can still modify the technical solutions described in the foregoing examples or make equivalent replacements for some of the technical features. Any modifications, equivalent replacements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A trajectory tracking method for a robot with independent front and rear wheel drive applicable to a pipeline environment, characterized in that, It includes the following steps: Step 1: According to the geometric structures of the pipeline and the robot wheels, analyze the geometric constraints of the pipeline cylindrical surface on the robot, and obtain the motion representation form of the robot in the two-dimensional plane; Step 2: According to the kinematic principle of the nonholonomic constrained wheeled robot, establish the nonlinear model of the robot motion system; Step 3: Determine the control objective of the robot trajectory tracking according to the actual working conditions; Step 4: Combine the nonlinear model of the robot motion system and the control objective of the trajectory tracking, use the Lyapunov direct method to design the kinematic controller, and finally elevate the output result of the controller to the pipeline space to achieve the trajectory tracking of the mobile robot.

2. The trajectory tracking method according to claim 1, wherein The analysis of the geometric constraints of the pipeline cylindrical surface on the robot specifically includes: Analyze the relationship between the wheel center speed and the rotational speed; Solve the motion speed of a single wheel; Analyze the rotational speed relationship between the front and rear wheels.

3. The trajectory tracking method according to claim 2, wherein The analysis of the relationship between the wheel center speed and the rotational speed specifically includes: First, establish the world coordinate system: Take the starting point of the robot as the origin, the pipeline bus direction as the x-axis, the vertical upward direction along the gravity direction as the z-axis, and the direction perpendicular to both x and z as the y-axis; Cut and unfold the pipeline cylindrical surface along the bus L at the wheel-pipeline contact point A into a two-dimensional plane. According to the definition of the world coordinate system, assume that A is located at the origin of the world coordinate system at the beginning, and the world coordinates of A after t moments are: Within an extremely short period of time, it is considered that v (t) , θ (t) are both constant values. Therefore: where v refers to the magnitude of the motion speed of the wheel-pipeline contact point, θ is the angle between the robot's traveling direction and the pipeline bus at its location, and R is the radius of the pipeline. Since the running trajectory of the robot in the pipeline is a curve, the speed of the wheel-pipeline contact point A is corrected according to the curvature k of the trajectory, and the expression is: ω represents the rotational speed of the wheel, r represents the radius of the wheel, and η is the defined speed correction coefficient.

4. The trajectory tracking method according to claim 3, characterized in that The solution of the motion speed of a single wheel specifically includes: Establish a moving coordinate system with the tangent vector τ, the principal normal n, and the binormal b at the contact point as the coordinate axes. According to the coordinate transformation matrix from the local coordinate system to the global coordinate system, it can be known that: where: A ij = e i e jT = [τnb] T e jT where v A is the magnitude of the linear velocity of the contact point A: The coordinates of the wheel center in this local coordinate system are (0, rcosβ, rsinβ), and the coordinates in the world coordinate system are: Taking the derivative of this coordinate is the wheel center speed of this wheel, and thus the motion speed of a single wheel is obtained.

5. The trajectory tracking method according to claim 3, wherein The analysis of the rotational speed relationship between the front and rear wheels specifically includes: Regarding the robot as a rigid body, according to the instantaneous screw axis theorem, the motion of the robot at any moment can be decomposed into a rotational motion around the instantaneous screw axis and a translational motion. Assume that the direction vector of the instantaneous screw axis is: The position vector of the point on the instantaneous screw axis that is closest to the origin of the world coordinate system is Then there is: where ω0 is the instantaneous rotational angular velocity of the robot, and its direction is the same as that of ; a1 and a2 are the position vectors of the front and rear wheels, and are the velocity vectors of the rear and front wheels of the robot; After solving the simultaneous equations (1) to (5), the relationship between the front wheel speed v f and the rear wheel speed v b is obtained, which is the rotational speed relationship α between the front and rear wheels of the robot: v f = αv b α = v f / v b 。 6. The trajectory tracking method according to claim 5, wherein, Step 2 specifically includes: The wheeled robot satisfies the following nonholonomic constraints: and are the velocity components of the robot in the x-axis and y-axis directions, respectively; Under the nonholonomic constraints, the established nonlinear model of the robot motion system includes a kinematic model and a trajectory tracking error model; The kinematic model is: The desired pose of the mobile robot is q r =[[X r Y r θ r T ;​ The trajectory tracking error model is: where v = v b , φ is the front-wheel steering angle, θ is the angle between the robot's traveling speed direction and the axis of the pipeline at its location, and l is the wheelbase of the wheels.

7. The trajectory tracking method according to claim 6, wherein There is a constraint between the speed of the front wheels and the speed of the rear wheels of the robot: v f cosφ = v b .

8. The trajectory tracking method according to claim 6, characterized in that The trajectory tracking error model introduces a slip ratio, which is defined as follows: are the slip ratios of the front and rear wheels respectively, and the value range is [0, 1); When the slip ratio is 0, there is no slip between the wheel and the ground. When the slip ratio is 1, the wheeled robot is in an uncontrollable state; According to the definition of the slip ratio, there is: Definition Then there is:

9. The trajectory tracking method according to claim 6, wherein In Step 4, the expression of the kinematic controller is as follows: where k x , k y , k θ are all positive real numbers, α = v f / v b is the rotational speed relationship between the front and rear wheels, and respectively represent the derivatives of the horizontal and vertical coordinates in the reference trajectory.

10. The trajectory tracking method according to claim 9, wherein The Lyapunov function is: Take the derivative of the above formula with respect to time: Substituting into the error model in the kinematic model gives:

Images