A fixed-time convergence-based intelligent mower constraint control method

Abstract

The application relates to the field of automatic control of intelligent agricultural equipment, and particularly discloses a constraint control method for an intelligent mower based on fixed-time convergence. First, a kinematic model and a heading subsystem of a differential steering intelligent mower are established; a fractional time conversion function is constructed to complete time dimension conversion scheduling of the heading subsystem; a nonlinear mapping function is used to convert the constrained heading system into an unconstrained system; and finally, a nonlinear heading constraint controller based on the backstepping method is designed to realize fixed-time convergence. The application can guarantee that the heading angle converges to the expected value within a fixed time, and the constraint boundary is not exceeded throughout the whole process, thereby improving the response speed, control accuracy and operation safety of the heading control of the intelligent mower.

Description

Technical Field

[0001] This invention belongs to the field of automatic control technology for intelligent agricultural equipment, and specifically relates to a constraint control method for intelligent lawnmowers based on fixed-time convergence. Background Technology

[0002] With the rapid development of smart agriculture and garden automation technologies, intelligent lawnmowers, with their advantages of unmanned operation, high efficiency, and low labor costs, have been widely used in lawn maintenance in various scenarios such as home gardens, municipal parks, and golf courses. Heading control is the core technology for intelligent lawnmowers to achieve autonomous path tracking and precise operation, directly determining the uniformity of mowing coverage, path tracking accuracy, and overall machine safety. Among them, differential steering intelligent lawnmowers have become the mainstream configuration for small intelligent lawnmowers due to their simple structure, flexible steering, and low control costs, and optimizing their heading control performance has become a core research direction in the industry.

[0003] Currently, heading control for differential steering smart lawnmowers mostly employs traditional methods such as PID control, conventional sliding mode control, and backstepping control. These methods generally only achieve asymptotic convergence of heading tracking errors, failing to provide a clear upper bound on the convergence time. The control effect is highly susceptible to factors such as initial heading errors and road disturbances, leading to uncertain convergence times and making it difficult to meet the rapid response and precise control requirements of smart lawnmowers in complex operating scenarios. Furthermore, existing control methods often do not fully consider the physical constraints of the heading angle, easily resulting in heading angle overshooting and excessive steering amplitude during operation. This not only reduces path tracking accuracy but may also cause safety accidents such as lawnmower sideslip and collisions, compromising the stability of the operation.

[0004] Although some studies have introduced fixed-time control theory into the field of motion control for mobile robots, most existing solutions do not incorporate effective state constraint handling mechanisms and cannot simultaneously ensure controllable convergence time and state constraint protection. Conventional constraint control methods such as obstacle Lyapunov functions suffer from complex controller design, difficult parameter tuning, and inability to effectively integrate with fixed-time convergence characteristics. These methods are difficult to directly adapt to the low-cost hardware and low-computing-power control platforms of small intelligent lawnmowers and cannot meet the practical application needs for engineering implementation. Summary of the Invention

[0005] To address the above problems, this invention provides a constraint control method for an intelligent lawnmower based on fixed-time convergence, comprising the following steps: Step 1: Establish a kinematic model of the intelligent lawnmower and construct a heading subsystem of the lawnmower based on the kinematic model; Step 2: Construct a fractional transformation function to limit the steady-state time of error convergence. The fractional transformation function is a piecewise function with the characteristics of completing the transformation within a fixed time and maintaining a stable constant value after the transformation. Step 3: Based on the heading subsystem of Step 1, the heading angle is transformed using the fractional transformation function of Step 2 to construct the transformed lawnmower heading subsystem; Step 4: Apply a nonlinear mapping function to the transformed heading subsystem obtained in Step 3 to construct an unconstrained heading subsystem after mapping, and internalize the state constraints of the original system into mapping relationships; Step 5: Based on the unconstrained heading subsystem in Step 4, a nonlinear heading constraint controller based on fixed-time convergence is constructed using the backstepping control method. The heading of the lawnmower is controlled by the controller output, so as to achieve global fixed-time convergence of heading angle tracking error and full-range constraint without exceeding the limit.

[0006] Preferably, in step 1, before establishing the kinematic model of the lawnmower, a northeast coordinate system and a vehicle body coordinate system are first defined; wherein the northeast coordinate system... The axis points due north. The axis points due east. The x-axis points towards the Earth's center of mass; the vehicle coordinate system has its origin at the center point of the rear axle, with the x-axis pointing towards the front of the vehicle, the y-axis pointing directly to the left of the vehicle, and the z-axis pointing directly above the vehicle.

[0007] Preferably, in step 1, for a differential steering smart lawnmower, the wheelbase between the two drive wheels is set to... The linear velocity of the left drive wheel is The linear velocity of the right drive wheel is Translational speed of the vehicle body , The established kinematic model of the lawnmower in the northeast coordinate system is as follows: ; In the formula, These represent the vehicle's northward position, eastward position, and heading angle in the northeast coordinate system. , , These are the northward linear velocity, the eastward linear velocity, and the heading angular velocity, respectively. For heading controller, The lawnmower's travel speed; the constructed lawnmower heading subsystem is...

[0008] Preferably, in step 2, the constructed fractional transformation function is: ; In the formula, For time variables, For the transformed variables, For function performance parameters and , For the transformation node and .

[0009] Preferably, the fractional transformation function satisfies the following performance: hour, Monotonically increasing; hour, ; hour, first derivative continuous.

[0010] Preferably, in step 3, the converted heading subsystem is constructed as follows: ; In the formula, The transformed heading angle; taking the derivative with respect to time, the transformed heading subsystem is: ; In the formula, for First derivative: , for The first derivative.

[0011] Preferably, the nonlinear mapping function used in step 4 is: ; In the formula, For input variables, For the constraint boundaries of the variables, Let these be the mapped variables; the nonlinear mapping function satisfies: when hour, If and only if hour, .

[0012] Preferably, in step 4, the expression for mapping the transformed heading subsystem using a nonlinear mapping function is as follows: ; In the formula, For the converted heading angle Constraint boundaries, The mapped heading angle; the unconstrained heading subsystem obtained by taking the first derivative with respect to time is: ; In the formula, for The first derivative, the differential gain of the nonlinear mapping .

[0013] Preferably, in step 5, the heading angle tracking error is first defined as: ; In the formula, For heading angle tracking error, This is the reference heading angle after mapping; Lyapunov function as follows: ; Lyapunov function's first time derivative for: ; for The first derivative; Construct a nonlinear heading constraint controller based on fixed-time convergence to obtain the following result. : ; In the above formula, and Represents positive definite gain. This is the derivative of the mapped reference heading angle with respect to time. For high-order convergence parameters, Let be the convergence parameter for the fractional power. , ; Will Substituting, we get: ; In the above formula, and Represents the equivalent gain for fixed-time convergence. , High-order parameters that converge in fixed time Fractional power parameters that converge in fixed time .

[0014] Preferably, the upper limit of the convergence time for the global fixed-time convergence of the heading angle tracking error. ; In the formula, For the maximum convergence time, , For the equivalent gain that converges in a fixed time, For parameters of higher powers, The parameter is a fractional power.

[0015] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. This invention combines a fractional transformation function with a fixed-time convergence control law, which can clearly give the upper bound of the convergence time of the heading tracking error. The convergence performance is not affected by the initial heading error, ensuring that the lawnmower's heading angle accurately tracks the desired heading within a fixed time. This solves the problem of uncertain convergence time in traditional control methods and improves the response speed and control determinism of heading control.

[0016] 2. This invention transforms a constrained heading system into an unconstrained system in the real number domain through a nonlinear mapping function, thereby internalizing the heading angle constraint. This ensures that the heading angle remains within the preset constraint boundary throughout the entire control process, avoiding operational deviations and safety risks caused by heading exceeding the boundary, and improving the stability and safety of lawnmower operations.

[0017] 3. The controller designed in this invention has a regular structure and clear parameter tuning logic. It does not require complex high-computing power calculations and can be adapted to the low-cost control platform of small intelligent lawnmowers. At the same time, it has strong anti-disturbance ability and robustness, which lowers the threshold for engineering applications. Attached Figure Description

[0018] Figure 1 This is a flowchart of the steps of the present invention; Figure 2 This is a heading angle tracking curve diagram for the present invention; Figure 3 This is a diagram of the heading angle tracking curve after mapping according to the present invention; Figure 4 This is a characteristic diagram of the fractional transformation function of the present invention; Figure 5 This is a schematic diagram of the northeast coordinate system of this invention. Detailed Implementation

[0019] The present invention will be further described in conjunction with the accompanying drawings and embodiments, with reference to... Figures 1 to 5 A constraint control method for an intelligent lawnmower based on fixed-time convergence includes the following steps: Step 1: Establish the kinematic model of the lawnmower: To establish the kinematic model of the lawnmower in this embodiment, a chassis kinematic model must first be established. The chassis kinematic model is based on conventional rigid body planar kinematics theory. During modeling, it is assumed that the vehicle body is a rigid body structure, and its motion on a flat road surface can be simplified to rigid body planar motion in a horizontal plane. The following basic assumptions are introduced: Low-speed driving assumption: The vehicle speed is assumed to be low (<5m / s), and tire lateral slippage can be ignored. No sideslip assumption: It is assumed that there is only pure rolling between the wheels and the ground, with no lateral or longitudinal slippage.

[0020] For intelligent lawnmowers with differential steering geometry, the northeast coordinate system and the vehicle body coordinate system are first defined, where the northeast coordinate system is illustrated in the diagram. Figure 5 As shown, Figure 5 middle Pointing due north, Pointing due east, The coordinate system points towards the Earth's center of mass; the vehicle coordinate system takes the center point of the rear axle (i.e., the midpoint of the line connecting the centers of the two drive wheels) as its origin, the x-axis points from the origin towards the front of the vehicle, the y-axis points from the origin towards the left of the vehicle, and the z-axis points from the origin towards the top of the vehicle.

[0021] Based on the geometric relationship of differential drive, the center distance (wheelbase) between the two drive wheels is set as follows: The linear velocity of the left drive wheel is The linear velocity of the right drive wheel is Since the two wheels are rigidly connected to the same vehicle body, the motion of the vehicle body as a whole can be decomposed into the translation of the vehicle body's centerline and the rotation about the origin. The corresponding geometric relationships of the motion parameters are as follows: , ; The kinematic model of the lawnmower in the northeast coordinate system and the vehicle body coordinate system is established as follows: ; In the above formula, These represent the vehicle's northward position, eastward position, and heading angle in the northeast coordinate system, respectively. Represents the northbound linear velocity. Represents the eastward linear velocity. Represents the angular velocity of the heading. For heading controller output, For lawnmower speed; The lawnmower heading subsystem is established as follows: .

[0022] Step 2: Construct the fractional conversion function: To address the steady-state time of convergence of the state tracking error of an intelligent lawnmower, a fractional transformation function is constructed and the conditions that the function satisfies are described. This invention introduces an innovative fractional conversion function as follows: ; In the above formula, For time variables, For the transformed variables, For function performance parameters, , For conversion nodes, ; It is usually selected based on the desired maximum transition time, the control response speed, or the upper bound of the allowed convergence time; The main decision "in "How was the transition completed before?" That is, whether the initial changes were more gradual or more proactive. The larger the value, the slower and closer the function rises in the early stages. It only then did it rapidly approach 1; The smaller the value, the faster the function rises in the early stages.

[0023] The transformation function itself has the property of scheduling functions, and its core function is: The adjustment process, which might otherwise only be completed in an asymptotic sense, is compressed and confined to a fixed time interval. The conversion is completed within the time limit, and in It then remains a stable constant value.

[0024] because exist Reaching 1, and since this time point is predetermined by the design parameters, it provides an explicit upper bound on the time for the control law or error transformation process, allowing the relevant error adjustment terms, gain terms, or constraint terms to... The predetermined evolution was completed ahead of schedule.

[0025] The constructed transformation function has the following performance characteristics: (1) When hour, Monotonically increasing; first derivative ,exist ,because , , Therefore, the numerator and denominator are positive. In the interval It increases strictly with time.

[0026] (2) When hour, ; This conclusion holds directly from the piecewise definition, that is, when time reaches... After that, the function takes a fixed value of 1, indicating that the conversion process has been completed and the system has entered a steady-state maintenance phase.

[0027] (3) When hour, continuous; In the interval Inside: The first derivative is a rational function, and the denominator is... The time is always greater than 0, therefore exist Internal continuity.

[0028] interval Inside: original function (constant), therefore ; The derivative of a constant function is 0. Upper continuous; Verify segmentation points , continuity at the point.

[0029] Left limit ( ): ; Molecules are denominator ,so ; Right limit ( ): interval Inside ,so ; The derivative value at this point: At this point, the right derivative is 0, and the left derivative is also 0. Therefore ; therefore, The derivative function is Continuous.

[0030] In conclusion, exist It is continuous everywhere above.

[0031] This invention employs a fractional transformation function instead of a purely exponential or polynomial function, primarily because: This function can simultaneously satisfy three requirements: smooth change before a preset time T, reaching the target value of 1 at time T, and maintaining a constant value of 1 after time T. Exponential functions typically exhibit asymptotic approximation, and reaching the target value often depends on... It is difficult to obtain the value at the preset T.

[0032] Step 3: Construct the converted lawnmower heading subsystem: The original heading angle is a physical quantity, and its boundedness, periodicity, and smoothness during the conversion process must usually be considered. This is addressed by introducing... The original heading angle tracking process can be embedded into a time-scheduled mapping framework, transforming constraint processing from direct hard constraints into "mapped soft constraints", thereby reducing the difficulty of controller design.

[0033] Based on the lawnmower heading subsystem established in step 1, the fractional transformation function constructed in step 2 is used to transform the lawnmower heading subsystem; ; In the above formula, This is the converted heading angle; The converted heading subsystem is as follows: ; for The first derivative.

[0034] Step 4: Construct the unconstrained system after mapping: Describe the characteristics of the nonlinear mapping function. Based on the transformed heading subsystem constructed in step 3, use the nonlinear mapping function to perform nonlinear mapping on the transformed heading subsystem to construct an unconstrained heading subsystem after mapping. Use the following nonlinear mapping function: ; In the above formula, As variables, For the constraint boundaries of the variables, The mapped variables; This nonlinear mapping function has the following characteristics: when hour, ; If and only if hour, ; The nonlinear mapping function is used to map the transformed heading subsystem from step 3 as follows: ; In the above formula, Represents the converted heading angle Constraint boundaries, This is the mapped heading angle;

[0035] Taking the first derivative with respect to time, the unconstrained heading subsystem is: ; for The first derivative; In the above equation, the differential gain of the nonlinear mapping ; System comparison before and after mapping: (1) Status: Before mapping: Directly constrained Due to limitations, the controller design must explicitly consider the issue of boundaries that cannot be crossed. After mapping: the new variable is defined over the entire real number domain, and the original constraints are implicitly incorporated into the inverse function, thus internalizing the constraints.

[0036] (2) In terms of control difficulty: Before mapping: Controllers often need to balance boundary protection, saturation handling and stability proof, which makes the design more difficult.

[0037] After mapping: mature methods such as backstepping and fixed-time control can be directly used for design, the control law structure is more regular, and the analysis process is clearer.

[0038] (3) Boundary protection capabilities: Before mapping: When the system approaches the boundary, if the control law is not designed properly, there may be a risk of exceeding the boundary.

[0039] After mapping: Since the mapping function diverges at the boundary, the mapped variable is more sensitive to changes in the original variable when it approaches the boundary, thereby enhancing the controller's ability to regulate the boundary neighborhood and improving safety and robustness.

[0040] Step 5: Construct a nonlinear heading constraint controller based on fixed-time convergence: Based on the unconstrained lawnmower heading subsystem in step 4, a nonlinear heading constraint controller based on fixed-time convergence is designed using the backstepping control method. Define the error variable as: ; In the above formula, For heading angle tracking error, This is the reference heading angle after mapping; Consider Lyapunov functions as follows: ; Lyapunov function's first time derivative for: ; for The first derivative; To stabilize the lawnmower's heading angle tracking error, a nonlinear heading constraint controller based on fixed-time convergence is constructed to obtain the following result. : ; In the above formula, and Represents positive definite gain. This is the derivative of the mapped reference heading angle with respect to time. For high-order convergence parameters, Let be the convergence parameter for the fractional power. , ; Will Substituting, we get: ; In the above formula, and Represents the equivalent gain for fixed-time convergence. , High-order parameters that converge in fixed time Fractional power parameters that converge in fixed time ; In summary, we can conclude that the lawnmower's heading angle error converges globally within a fixed time, with an upper limit for the convergence time. ; The maximum time is given. Thus, the stability analysis of the nonlinear heading constraint controller based on fixed-time convergence is complete.

[0041] Step 6: Simulation Verification: The nonlinear heading constraint controller based on fixed-time convergence designed in steps 1 to 5 was simulated on a differential motion model. Simulation platform settings: MATLAB was selected as the simulation platform, RK4 was used as the solver, the step size was 0.01, and the total simulation time was 50s.

[0042] Simulation condition settings: , , , For the desired heading angle, , , , , .

[0043] By using a nonlinear mapping function to constrain the heading angle and taking a differential vehicle as the simulation object, the present invention’s intelligent lawnmower constraint control method based on fixed-time convergence is simulated and verified. Figure 2 The differential vehicle's heading tracking curve under this invention is shown. The curve indicates that the heading angle under this invention can be... The desired angle is tracked forward, and the heading angle response curve remains within the constraint boundaries throughout the simulation process; Figure 3 The heading tracking curve of the differential vehicle after mapping under the present invention is shown. The curve shows that the heading angle after constraint processing can converge to the desired heading angle after mapping. Figure 4The properties of the fractional transformation function are shown in the figure. As can be seen from the figure, the constructed fractional transformation function satisfies the construction conditions, and... The lower the speed regulation coefficient, the faster the response speed. The time function equals 1; Simulation results show that the intelligent lawnmower constraint control method based on fixed-time convergence of the present invention can realize fixed-time constraint control of the differential vehicle's heading angle, and the variables remain within the constraint range and do not exceed the limit throughout the entire control process, thus solving the problem of uncertain convergence time of the initial heading angle of the intelligent lawnmower.

[0044] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0045] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A constraint control method for an intelligent lawnmower based on fixed-time convergence, characterized in that, Includes the following steps: Step 1: Establish a kinematic model of the intelligent lawnmower and construct a heading subsystem of the lawnmower based on the kinematic model; Step 2: Construct a fractional transformation function to limit the steady-state time of error convergence. The fractional transformation function is a piecewise function with the characteristics of completing the transformation within a fixed time and maintaining a stable constant value after the transformation. Step 3: Based on the heading subsystem of Step 1, the heading angle is transformed using the fractional transformation function of Step 2 to construct the transformed lawnmower heading subsystem; Step 4: Apply a nonlinear mapping function to the transformed heading subsystem obtained in Step 3 to construct an unconstrained heading subsystem after mapping, and internalize the state constraints of the original system into mapping relationships; Step 5: Based on the unconstrained heading subsystem in Step 4, a nonlinear heading constraint controller based on fixed-time convergence is constructed using the backstepping control method. The heading of the lawnmower is controlled by the controller output, so as to achieve global fixed-time convergence of heading angle tracking error and full-range constraint without exceeding the limit.

2. The intelligent lawnmower constraint control method based on fixed-time convergence according to claim 1, characterized in that, In step 1, before establishing the kinematic model of the lawnmower, the northeast coordinate system and the vehicle body coordinate system are first defined; where the northeast coordinate system... The axis points due north. The axis points due east. The x-axis points towards the Earth's center of mass; the vehicle coordinate system has its origin at the center point of the rear axle, with the x-axis pointing towards the front of the vehicle, the y-axis pointing directly to the left of the vehicle, and the z-axis pointing directly above the vehicle.

3. The intelligent lawnmower constraint control method based on fixed-time convergence according to claim 1, characterized in that, In step 1, for a differential steering smart lawnmower, the track width between the two drive wheels is set to... The linear velocity of the left drive wheel is The linear velocity of the right drive wheel is Translational speed of the vehicle body , The established kinematic model of the lawnmower in the northeast coordinate system is as follows: ； In the formula, These represent the vehicle's northward position, eastward position, and heading angle in the northeast coordinate system. , , These are the northward linear velocity, the eastward linear velocity, and the heading angular velocity, respectively. For heading controller, The lawnmower's travel speed; the constructed lawnmower heading subsystem is... .

4. The intelligent lawnmower constraint control method based on fixed-time convergence according to claim 1, characterized in that, In step 2, the constructed fractional conversion function is: ； In the formula, For time variables, For the transformed variables, For function performance parameters and , For the transformation node and .

5. The intelligent lawnmower constraint control method based on fixed-time convergence according to claim 4, characterized in that, The fractional transformation function satisfies the following performance: hour, Monotonically increasing; hour, ; hour, first derivative continuous.

6. The intelligent lawnmower constraint control method based on fixed-time convergence according to claim 1, characterized in that, In step 3, the converted heading subsystem is as follows: ； In the formula, The transformed heading angle; taking the derivative with respect to time, the transformed heading subsystem is: ； In the formula, for First derivative: , for The first derivative.

7. The intelligent lawnmower constraint control method based on fixed-time convergence according to claim 1, characterized in that, In step 4, the nonlinear mapping function used is: ； In the formula, For input variables, For the constraint boundaries of the variables, Let these be the mapped variables; the nonlinear mapping function satisfies: when hour, If and only if hour, .

8. The intelligent lawnmower constraint control method based on fixed-time convergence according to claim 7, characterized in that, In step 4, the expression for mapping the transformed heading subsystem using a nonlinear mapping function is as follows: ； In the formula, For the converted heading angle Constraint boundaries, The mapped heading angle; the unconstrained heading subsystem obtained by taking the first derivative with respect to time is: ； In the formula, for The first derivative, the differential gain of the nonlinear mapping .

9. The intelligent lawnmower constraint control method based on fixed-time convergence according to claim 1, characterized in that, In step 5, the heading angle tracking error is first defined as: ； In the formula, For heading angle tracking error, This is the reference heading angle after mapping; Lyapunov function as follows: ； Lyapunov function's first time derivative for: ； In the formula, for The first derivative; Construct a nonlinear heading constraint controller based on fixed-time convergence to obtain the following result. : ； In the above formula, and Represents positive definite gain. This is the derivative of the mapped reference heading angle with respect to time. For high-order convergence parameters, Let be the convergence parameter for the fractional power. , ; Will Substituting, we get: ； In the above formula, and Represents the equivalent gain for fixed-time convergence. , High-order parameters that converge in fixed time Fractional power parameters that converge in fixed time .

10. The intelligent lawnmower constraint control method based on fixed-time convergence according to claim 9, characterized in that, The upper limit of the convergence time for the global fixed-time convergence of the heading angle tracking error. ; In the formula, For the maximum convergence time, , For the equivalent gain that converges in a fixed time, For parameters of higher powers, The parameter is a fractional power.

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