Wheel robot trajectory tracking control and obstacle avoidance method and system

Abstract

The wheel robot trajectory tracking control and obstacle avoidance method and system relate to the mobile robot technical field.The aforementioned method comprises the following steps: 1, establishing a robot error kinematics model and a dynamics model, combining the two and dividing them into a cascade subsystem of second order and third order, respectively constructing a fractional order PID control model, and constructing a dynamic fractional order PID controller to control the linear velocity and angular velocity of the robot;2, establishing a control equation according to the data variables provided by the subsystem, and constructing a finite time trajectory tracking controller to instantly track the linear velocity and angular velocity of the robot;3, establishing an active obstacle avoidance controller with obstacle avoidance compensation and a trajectory controller with disturbance compensation, so that the robot can automatically track the target and avoid obstacles;4, using the fractional order PID controller for joint control to realize the joint control of the robot trajectory control and obstacle avoidance.The application can improve the motion control precision and stability of the robot.

Description

Technical Field

[0001] This invention relates to the field of mobile robot technology, and in particular to a method and system for trajectory tracking control and obstacle avoidance of a wheeled robot. Background Technology

[0002] Compared to stationary robots, mobile robots often have more intelligent control systems to meet the needs of actual work. Advanced mobile robots can often combine environmental perception, path planning, autonomous decision-making, and behavior control, thus playing a crucial role in the field of robotics. With the development of advanced technologies, the functions and applications of mobile robots are constantly improving. However, most existing wheeled mobile robots still suffer from insufficient control precision and stability, making it difficult to accurately follow the desired trajectory and maintain stable movement. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to provide a method for trajectory tracking control and obstacle avoidance of wheeled robots, thereby improving control accuracy and stability.

[0004] To achieve the above objectives, the present invention adopts the following technical solution: a wheeled robot trajectory tracking control and obstacle avoidance method, comprising the following steps:

[0005] 1. Establish the robot error kinematic model and dynamic model, combine the two and divide them into second-order and third-order cascaded subsystems, construct fractional-order PID control models respectively, and build a dynamic fractional-order PID controller to control the robot's linear velocity and angular velocity.

[0006] 2. Based on the data variables provided by the subsystem, establish control equations and construct a finite-time trajectory tracking controller to track the robot's linear and angular velocities in real time.

[0007] Third, establish an active obstacle avoidance controller with obstacle avoidance compensation and a trajectory controller with disturbance compensation, so that the robot can automatically track the target and avoid obstacles;

[0008] Fourth, a fractional-order PID controller is used for joint control to achieve joint control of robot trajectory control and obstacle avoidance.

[0009] Preferably, in step one, the robot error kinematic model is as follows:

[0010]

[0011] Where, [x,y,θ] τ Indicates the current orientation of the robot, [x r y r θr ] τ Indicates the expected orientation of the robot, [x e y e θ e ] τ This indicates the difference in posture and orientation between the current robot and the expected robot.

[0012] Preferably, in step one, the robot's dynamic model is as follows:

[0013]

[0014] Wherein, the control range of u1 is u1=τ r +τ t The control range of u2 is u2 = τ r -τ t The control range of δ1 is The control range of δ2 is

[0015] Preferably, in step one, the fractional-order PID control model of the second-order subsystem is:

[0016]

[0017] Where X = [x1, x2, ..., x n ] τ Represents the operating status, u(t)∈R r To control the input, B∈R n×m Let f(x) ∈ R be a constant matrix, f(x) ∈ R be a continuous nonlinear function, and d(t) ∈ R be a constant matrix. n External interference.

[0018] Preferably, in step one, the fractional-order PID control model of the third-order subsystem is:

[0019]

[0020] in, κ1,γ1,β1, All are normal numbers.

[0021] Preferably, in step two, the control equation of the finite-time trajectory tracking controller is:

[0022]

[0023]

[0024]

[0025]

[0026]

[0027] Preferably, in step three, the control equation of the active obstacle avoidance controller is:

[0028]

[0029] The active obstacle avoidance controller includes an obstacle avoidance compensator for obstacle avoidance compensation, as follows:

[0030]

[0031] in, This represents the difference between the robot's linear velocity direction and the repulsive force direction; l1 and l2 are both positive constants, F r This represents the repulsive force exerted by obstacles on the robot during its operation.

[0032] Preferably, F r In a repulsive potential field, the gradient remains negative throughout, and its geometric representation is as follows:

[0033]

[0034] The repulsive potential field is as follows:

[0035]

[0036] Where, η r For the gain of repulsive force, G i (q) represents the distance between the robot and the i-th obstacle, and Q is the threshold value of the obstacle.

[0037] Preferably, in step three, the control equation of the trajectory controller is:

[0038]

[0039]

[0040] Where c1 and c2 are both positive numbers; ε1, ε2, k1, and k2 are all positive constants; Q = [l, ψ, θ] T This indicates the robot's position in polar coordinates.

[0041] Another object of the present invention is to provide a trajectory tracking control and obstacle avoidance system for a wheeled robot, comprising:

[0042] The trajectory control unit is used to establish the robot's error kinematic model and dynamic model, combine the two and divide them into second-order and third-order cascaded subsystems, construct fractional-order PID controllers respectively, and thus build a dynamic fractional-order PID controller to control the robot's linear velocity and angular velocity.

[0043] The trajectory tracking unit is used to establish control equations based on the data variables provided by the subsystem, construct a finite-time trajectory tracking controller, and perform real-time tracking of the robot's linear and angular velocities.

[0044] The obstacle avoidance unit is used to establish an active obstacle avoidance controller with obstacle avoidance compensation and a trajectory controller with disturbance compensation, enabling the robot to automatically track targets and avoid obstacles.

[0045] The joint control unit is used to perform joint control using a fractional-order PID controller, thereby achieving joint control of robot trajectory control and obstacle avoidance.

[0046] This invention proposes an improved fractional-order PID control method by constructing a robot kinematic and dynamic model, achieving higher asymptotic and finite-time stability. It also features the following characteristics: 1. By constructing a position error model in polar coordinates, setting a state feedback controller, and applying fractional-order PID control to the linear motion of the wheeled mobile robot, this invention ensures reduced lateral slippage during straight-line travel, improving control efficiency and addressing the linear motion control problem of wheeled mobile robots. 2. This invention constructs a new dynamic fractional-order PID controller using a hyperbolic tangent function with cascaded second- and third-order subsystems, introducing the concept of finite-time control, and proposing a finite-time trajectory control method for wheeled mobile robots. A closed-loop error control system is established using the finite-time stability criterion, improving both the accuracy and efficiency of trajectory control and addressing the curved trajectory control problem of wheeled mobile robots. 3. By setting appropriate disturbance observers, a novel dynamic terminal fractional-order PID controller, and an obstacle avoidance compensator, this invention improves the joint control of trajectory control and obstacle avoidance in wheeled mobile robots and reduces the impact of external disturbances. Furthermore, experiments have shown that fractional-order PID control can effectively improve the obstacle avoidance control performance of wheeled mobile robots. Attached Figure Description

[0047] Figure 1 Here is a block diagram of a fractional-order PID controller;

[0048] Figure 2 A schematic diagram of a wheel;

[0049] Figure 3 This is a schematic diagram of the motion model of a two-wheeled differential speed wheeled mobile robot.

[0050] Figure 4 This is the equivalent circuit diagram of a brushless DC motor.

[0051] Figure 5 This is a structural diagram of a DC motor system;

[0052] Figure 6 This is a control block diagram for a two-wheeled differential mobile robot.

[0053] Figure 7 This is a schematic diagram illustrating the pose error between the desired robot and the current robot.

[0054] Figure 8 This is a schematic diagram of obstacle avoidance during trajectory control.

[0055] Figure 9 This is a schematic diagram of a robot control system.

[0056] Figure 10 This is a system architecture diagram for controlling a robot to track a trajectory within a finite time.

[0057] Figure 11 A schematic diagram of the Matlab Simulink simulation model under PID control;

[0058] Figure 12 A schematic diagram of the Matlab Simulink simulation model under fractional-order PID control;

[0059] Figure 13 The diagrams show the response curves for yaw rate and lateral acceleration. (a) shows the response curve for yaw rate, and (b) shows the response curve for lateral acceleration.

[0060] Figure 14 This is a schematic diagram of the response curve for the centroid sideslip angle;

[0061] Figure 15 A schematic diagram of the coordinate system for trajectory tracking of a mobile robot;

[0062] Figure 16 This is a schematic diagram of a simulation model for a finite-time trajectory tracking control system.

[0063] Figure 17 This is a simulation diagram of curve trajectory tracking control;

[0064] Figure 18 Here are schematic diagrams of the angular velocity and slip angle response curves: (a) is a schematic diagram of the angular velocity response curve, and (b) is a schematic diagram of the slip angle response curve.

[0065] Figure 19 The diagrams show the response curves for distance deviation and angle deviation. (a) shows the response curve for distance deviation, and (b) shows the response curve for angle deviation.

[0066] Figure 20 This is a coordinate graph showing the robot's trajectory error in polar coordinates.

[0067] Figure 21 The structure diagram of fractional-order PID control for trajectory control of a mobile robot;

[0068] Figure 22 This is a schematic diagram of a fractional-order PID trajectory control simulation model;

[0069] Figure 23 This is a schematic diagram of autonomous obstacle avoidance in linear trajectory tracking control;

[0070] Figure 24 This is a schematic diagram of the pose error during autonomous obstacle avoidance in linear trajectory control.

[0071] Figure 25 Illustration of linear velocity and angular velocity of a mobile robot Figure 1 (a) shows the linear velocity of the mobile robot. Figure 1 (b) shows the angular velocity of the mobile robot. Figure 1 ;

[0072] Figure 26 This is a trajectory control obstacle avoidance route diagram;

[0073] Figure 27 This is a schematic diagram of the pose error during autonomous obstacle avoidance in curve trajectory control.

[0074] Figure 28 Illustration of linear velocity and angular velocity of a mobile robot Figure 2 , ( a (This is a diagram illustrating the linear velocity of a mobile robot.) Figure 2 (b) shows the angular velocity of the mobile robot. Figure 2 ;

[0075] Figure 29 This is a block diagram of the motion control system for a wheeled mobile robot.

[0076] Figure 30 The figures show experimental diagrams and error diagrams for linear trajectory tracking control based on fractional-order PID control. a (a) is an experimental diagram of linear trajectory tracking control based on fractional-order PID control, and (b) is a schematic diagram of the experimental error of linear trajectory tracking control based on fractional-order PID control.

[0077] Figure 31 The figures show experimental diagrams and error diagrams for curve trajectory tracking control based on fractional-order PID control. a(a) is an experimental diagram of curve trajectory tracking control based on fractional-order PID control, and (b) is a schematic diagram of the experimental error of curve trajectory tracking control based on fractional-order PID control.

[0078] Figure 32 Experimental path diagram for mobile robots trending towards trajectory-free automatic obstacle avoidance;

[0079] Figure 33 The diagram shows the obstacle avoidance experiment path during linear trajectory control and the pose error diagram during linear trajectory tracking control. a (a) is the obstacle avoidance experiment path diagram during the linear trajectory control process, and (b) is the pose error diagram of the obstacle avoidance experiment during the linear trajectory tracking control process.

[0080] Figure 34 This is a diagram showing the obstacle avoidance experiment path and pose error during the curve trajectory tracking control process. a (a) is the obstacle avoidance experiment path diagram in the curve trajectory tracking control process, and (b) is the pose error diagram of the obstacle avoidance experiment in the curve trajectory tracking control process. Detailed Implementation

[0081] To facilitate understanding by those skilled in the art, the present invention will be further described below with reference to embodiments and accompanying drawings. The content mentioned in the embodiments is not intended to limit the present invention.

[0082] This embodiment addresses the problems of low motion control accuracy and low control efficiency in current wheeled mobile robots. Taking a two-wheeled mobile robot as the object, it analyzes and improves the robot's straight-line and curved travel trajectory control and obstacle avoidance methods to improve the accuracy and efficiency of mobile robot control during travel. Details are as follows:

[0083] I. Trajectory Control and Obstacle Avoidance Analysis of Wheeled Mobile Robots

[0084] 1.4 Trajectory control of wheeled mobile robots.

[0085] 1.4.1 Straight-line driving control.

[0086] To address the control problem of autonomous walking, although existing control methods can reduce the error of autonomous walking, wheeled robots are subject to lateral and sideways signal interference under complex working conditions. While conventional PID control systems can simultaneously control the yaw rate and sideslip angle—two different variables—during wheeled robot movement, the control precision remains insufficient. Therefore, this embodiment proposes a fractional-order PID control strategy. A simulation model based on a fractional-order PID yaw angle control system is established to analyze the velocity, angular acceleration, and displacement of the wheeled robot under different working conditions. This enables the wheeled robot to move controllably in linear motion under complex environments, greatly improving the robot's dynamic performance and robustness.

[0087] 1.4.2 Curve driving control.

[0088] When designing a curve-based navigation controller, the relationship between the robot's state and time changes needs to be considered to achieve the desired trajectory control based on the virtual robot's posture and speed. However, in practical applications, in addition to trajectory control, some uncertainties also need to be considered to ensure the accuracy and reliability of the controller. Therefore, the concept of finite-time control is introduced, and a finite-time trajectory control method for wheeled mobile robots is proposed. Dynamic and kinematic models of the mobile robot are constructed separately. Based on this, a new control system is constructed using the hyperbolic tangent function with second- and third-order cascaded subsystems. A fractional-order PID control model for the mobile robot is constructed, a dynamic fractional-order PID controller is designed, and a closed-loop error control system is established using the finite-time stability criterion.

[0089] 1.4.3 Combined control of curved trajectory and obstacle avoidance.

[0090] Obstacle avoidance control is crucial for the safe operation of mobile robots. It helps mobile robots effectively avoid dangers in unknown environments and is more complex than simple trajectory control. Therefore, adopting effective obstacle avoidance control strategies can significantly improve the safety and efficiency of mobile robots. When a mobile robot is performing trajectory tracking, it must constantly check its surrounding environment. Once an obstacle is detected, it needs to adopt appropriate countermeasures to avoid it and will also readjust its posture to continue performing trajectory control tasks. To achieve an effective combination of tracking and obstacle avoidance, these two control strategies can be combined to enable mobile robots to effectively track and avoid obstacles.

[0091] 1.5 Fractional-order PID control.

[0092] like Figure 1As shown, the fractional-order PID controller is a novel control method that extends the order of calculus to the fractional control order. This expands the controller's adjustment range and improves system stability. The advantage of the PID controller lies in its stronger approximation capability. Combining fractional-order PID with a nonlinear controller can significantly improve the controlled performance of wheeled mobile robots.

[0093] In summary, this embodiment aims to solve the motion control problem of a wheeled mobile robot. By constructing its kinematic and dynamic models, an improved fractional-order PID control method is proposed to achieve higher asymptotic stability and finite-time stability. Furthermore, it utilizes M... a tl a The software is validated to ensure its effectiveness. The main steps are as follows:

[0094] (1) To address the control problem of a mobile robot moving in a straight line, this embodiment constructs a pose deviation model using a polar coordinate system and sets up a position feedback controller structure. Furthermore, Lyapunov stability theory is used to verify the reliability and safety of the closed-loop error control system. By adopting the method proposed in this embodiment, it can be ensured that the mobile robot maintains straight-line movement from any initial position, greatly reducing lateral deviation errors and effectively avoiding potential outlier issues.

[0095] (2) To address the curve trajectory control problem of mobile robots, this embodiment employs a hyperbolic tangent function to construct a fractional-order PID control model and a closed-loop error control system with non-singular dynamics, thereby improving the robot's motion performance and stability within a finite time. By adopting the method proposed in this embodiment, not only can the efficiency of mobile robot trajectory tracking be greatly improved, but errors can also be effectively avoided.

[0096] (3) To overcome the impact of external disturbances on the joint control of trajectory control and obstacle avoidance of the mobile robot, this embodiment constructs a new disturbance observer and a fractional-order PID controller for the mobile terminal, and designs an obstacle avoidance compensator using the artificial potential field method to effectively handle obstacle avoidance phenomena. Finally, the safety of the system is verified using Lyapunov stability theory. By adopting the method proposed in this embodiment, it can be ensured that the mobile robot can complete trajectory control and obstacle avoidance tasks.

[0097] II. Wheeled Mobile Robot Model and Control Method.

[0098] 2.1 Incomplete system.

[0099] Nonholonomic systems are a class of physical systems whose states depend on their evolution along paths in a parameter space. These paths are subject to differential constraints, allowing their parameter values ​​to change continuously. Even if, at some point, their parameter values ​​become the same as at the initial moment, they may still return to their initial state. Examples include bicycles, spacecraft, and various wheeled robots, all of which are subject to varying degrees of constraints that can be used to describe the characteristics and behavior of the system.

[0100] F(q,t)=0#(2.1)

[0101] In this system, q = [q1, q2, ..., q n ] τ It is a generalized coordinate vector representing the state of the system. The value of q will change if constraints exist.

[0102]

[0103] If equation (2.1) is converted into a complete constraint, then equation (2.2) can be called a complete constraint; otherwise, it is called a non-complete constraint.

[0104] Figure 2 The single-wheel system shown is a typical nonholonomic system, where v is the velocity of the wheel, θ is the direction of the wheel's velocity, and (x, y) represents the position of the wheel. The wheel can only move in the direction of θ, meaning there is a constraint:

[0105]

[0106] Therefore, a single wheel system is an incomplete system.

[0107] 2.2 Kinematic Model.

[0108] When a wheeled robot moves on the ground, a reference coordinate system needs to be constructed in order to better analyze its motion. Figure 3 The kinematics of a two-wheeled differential wheeled mobile robot are illustrated. XOY represents its motion plane, and P is its center, the midpoint connecting the axes of the two drive wheels, denoted as P(x,y). The distance between the two drive wheels is l, the wheel radius is r, and θ is the angle between the robot's current orientation and the reference X-axis, defined as (-π,π). Therefore, the robot's orientation in the coordinate system should be expressed as q(x,y,θ). In the figure, v L v R Each represents the linear rotational speed of the robot's left and right drive wheels, while v , w Each of these parameters indicates the robot's forward speed and rotational angular velocity.

[0109] The drive system of the wheeled mobile robot consists of two single-closed-loop brushless DC motor modules and a kinematics transformation module, which are respectively powered by G... L G R To describe the open-loop motion of the two drive wheels, where G... L G R The input is voltage U = [u L ,u R ] T The output is the actual linear velocity V = [v L ,v R ] T The combination of these two modules enables more precise movement of wheeled mobile robots, thereby improving the robot's performance and reliability. The input and output modes of the drive system module can be described by the following formula:

[0110]

[0111] By adjusting the rotational speed of the front and rear drive wheels, wheeled mobile robots can achieve various complex movements, thereby improving their posture change patterns. To better describe the kinematic model of a wheeled mobile robot, the input linear velocity can be set as V = [v...]. L ,v R ] T The output pose vector is q = [x, y, θ]. T Thus, based on the relationship between kinematics and coordinate transformation, the kinematic model of the robot's pose change law can be represented as:

[0112]

[0113]

[0114] From equations (2.1) to (2.3), the overall definition of the propulsion control system for a two-wheeled differential wheeled mobile robot can be obtained: V represents the forward speed. w V represents the steering angular rate, l represents the length between the two drive wheels, and V represents the steering angle rate. L V represents the linear velocity of the revolver. R This indicates the linear velocity of the right wheel.

[0115]

[0116]

[0117] In the above formula, x represents the movement of the robot's center of mass P on the X-axis, y refers to the movement on the Y-axis, and θ represents the angle between the robot's orientation and the positive X-axis, which can vary by (-π, π).

[0118] 2.3 Mathematical model of the servo mechanism.

[0119] Brushless DC motors are key drive components for wheeled mobile robots, featuring rapid response, high efficiency, and powerful overload capability. At low speeds, they can generate high torque to assist in performing slow movements. The following section will describe a model of a brushless DC motor.

[0120] 2.3.1 Voltage equation.

[0121] To better analyze the windings of the brushless DC motor in a wheeled mobile robot, we can assume that the windings adopt a star configuration.

[0122] (1) Ignoring the cogging effect, the conductors on the armature surface are evenly arranged together;

[0123] (2) Ignoring factors such as motor coil saturation, wind friction, hysteresis, copper loss and its eddy current loss can yield more accurate results.

[0124] (3) Ignoring the problems of motor armature reflection and leakage flux, the air gap magnetic field distribution and back electromotive force both exhibit good trapezoidal characteristics;

[0125] (4) Electronic power transistors and some integrated circuit components play important roles in the inverter integrated circuit. Their spatial orientation, winding method, and phase distance between windings are all equal. Therefore, the internal equivalent circuit diagram of the brushless DC motor can be obtained through... Figure 4 This means that their inductance, resistance, and mutual inductance are all the same, which enables the brushless DC motor to operate in an ideal switching state.

[0126] exist Figure 4 in,u a ,u b ,u c These represent the stator voltages of each phase, i, and i, respectively. a i b i c Represents current, e a ,e b ,e c Let the back electromotive force, winding resistance R, self-inductance L of the three-phase windings, and mutual inductance M between the windings be the factors. Based on the operating principle of a brushless DC motor, the phase voltage equations of the stator current can be derived:

[0127]

[0128] When the DC motor commutates to the point where phases A and B are conducting, we have:

[0129]

[0130] Then the potential difference U ab We can obtain:

[0131]

[0132] 2.3.2 Back electromotive force equation.

[0133] The winding conductors of a brushless DC motor are affected by a magnetic field, resulting in a back electromotive force (EMF). The back EMF 'e' of one of the conductors can be determined through analysis, and is as follows:

[0134] e = Blv#(2.12)

[0135] There is a close relationship between the magnetic field strength B, the effective shear distance l of the conductor, and the linear velocity v in the shear direction of the conductor, and there is also a certain correlation between the rotational speed n and these parameters.

[0136]

[0137] Assume the winding of the brushless DC motor consists of two wires, and the number of turns in each phase is... The winding rotation radius is R, and the total induced electromotive force is... Its value can be expressed by the following formula:

[0138] E φ =2eW φ # (2.14)

[0139] Substituting equations (2.12) and (2.13) into equation (2.14), we obtain the total electromotive force E. φ The relationship with the rotational speed n is as follows:

[0140]

[0141] 2.3.3 Calculation of electromagnetic torque.

[0142] When voltage flows through the armature windings of a brushless DC motor, these energized conductors experience magnetic interaction, resulting in the total torque of the system. Under normal operating conditions, the electromagnetic torque T of the system is... e From the maximum voltage I p and electromotive force E p It is determined that when only two windings are simultaneously conducting, the electromagnetic power P e =2E p I p Therefore, under ideal conditions, the electromagnetic torque T of a brushless DC motor is... e It should be calculated in the following way:

[0143]

[0144] In the formula, ω represents the angular speed of the generator (rad / s), while This represents the torque coefficient of the generator. They are linearly related to each other, meaning that there is a certain relationship between the current and the torque coefficient of the motor. This relationship can be expressed by the formula in the equation.

[0145] 2.3.4 Equations of motion for the motor.

[0146] By using the equations of motion of the electric motor, we can analyze the specific relationship between torque and acceleration from a physical perspective and solve this problem using differential calculations.

[0147]

[0148] In the formula, T l J represents the load torque, and B represents the rotor's moment of inertia. v This represents the viscosity coefficient of the system. When the motor is running stably, the differential term of the motor speed is zero, which means that the angular acceleration of the motor is zero, so the motor can maintain a stable speed of rotation.

[0149] 2.3.5 Transfer function model.

[0150] By establishing equations for voltage, electromagnetic torque, and motion, the target brushless motor is analyzed, and its driving equation is finally derived:

[0151]

[0152] where U=U ab =U bc =U ca ,R s =2R,L x = 2(LM). Taking the Laplace transform of the above equation, we get:

[0153]

[0154] The structural diagram of the motor system can be obtained from equation (2.19), as follows: Figure 5 .

[0155] Therefore, the motor's speed transfer function can be derived as follows:

[0156]

[0157] 2.3.6 Servo drive model.

[0158] There is a certain proportional relationship between the movement speed of a wheeled robot and the rotational speed of a brushless DC motor. This relationship can be achieved by adjusting the linear speed of the drive wheels.

[0159]

[0160] By changing the radius of the drive wheel, adjusting the reduction ratio of the gearbox, and altering the conversion coefficient between the wheel's linear speed and the motor's angular rate, a pattern for a brushless DC motor voltage driver and a DC motor load torque driver can be constructed, namely:

[0161]

[0162] have:

[0163] v nheel =G w u+G l T l # (2.23)

[0164] Based on the above calculations, the drive model for the robot's left and right wheels can be derived as follows:

[0165]

[0166] In the formula, G wL and G wR Each represents a brushless DC motor voltage-driven model for the front and rear wheels of a wheeled mobile robot, while G IL and G IR Each of these refers to the load torque control mode of the front and rear wheel control systems of a wheeled mobile robot, in order to achieve motion control using artificial intelligence. IL T lR These represent the load torque of the drive motors for the left and right drive wheels, respectively. L u R These are the voltage inputs for the brushless DC motors of the left and right drive wheels, respectively. Equation (2.24) clearly shows that by controlling the voltage signals of the two motors on the brushless DC motor drive board, the speeds of the left and right drive wheels can be precisely adjusted, thereby achieving motion control of the robot.

[0167] 2.4 Robot motion and obstacle avoidance control methods.

[0168] 2.4.1 Robot speed control method.

[0169] By adjusting the pressure information of two brushless DC motors, a two-wheeled differential wheeled mobile robot can move forward and turn, and use static matrix transformation to adjust its position and posture. This type of mobile robot can be adjusted according to actual conditions to achieve optimal motion performance. Figure 6 The open-loop control frame for the speed of a two-wheeled robot is shown.

[0170] Figure 6 in,u L u R For the control input of the motor, its static matrix can be transformed into:

[0171]

[0172] When the motor load torque of the left and right drive wheels is zero, the forward speed and turning speed of the mobile robot can be calculated by applying equation (2.24) and expressed as a mathematical model:

[0173]

[0174]

[0175] According to equation (2.27), it can be observed that the robot's forward and rotation drive modes are coupled to a certain extent, which is due to the transformation of the velocity static matrix. To solve this problem, the control quantities u of the two drive motors can be adjusted. L and u R This enables small robots to move forward and turn.

[0176] 2.4.2 Trajectory control method.

[0177] By designing a control law, a wheeled mobile robot can effectively track the expected movement trajectory of the robot, thereby converging the pose and position errors between the current and expected AI to zero. Figure 7 As shown, where [x,y,θ] τ Indicates the current pose of the robot, [x r y r θ r ] τ This represents the expected pose of the robot, while [x] e y e θ e ] τ This indicates the difference in posture and orientation between the current robot and the expected robot.

[0178] By introducing θ e The sine of / 2 can be used to represent θ. e An angle ∈ (-π, π) can be converted into a monotonic function with a range of (-1, 1), and when the error angle is small, an approximation θ can be obtained. e The convergence effect is observed. To better simulate the kinematic behavior of a real robot, this embodiment establishes an error model to better predict the robot's kinematic behavior.

[0179]

[0180] To study the stability of the model error convergence, it is necessary to solve the differential equation of the error model, where equation (2.28) can be used to represent the differential equation of the error, as shown below:

[0181]

[0182] Through the error differential equation (2.29), it can be found that the model contains three state variables x. e y e θ e However, with only two input control variables, v and w, such a system lacks sufficient power. Therefore, wheeled mobile robots must fully consider the non-holonomic constraints they are subject to during their movement. When the mobile robot moves normally, the relative speed between the wheels and the floor is zero, and the wheels do not slip in the sliding direction or laterally. This is a constraint that must be satisfied to ensure the safe operation of the mobile robot.

[0183]

[0184] By employing different control algorithms, trajectory tracking control can set the input quantity u as u = [v] e w e ] τ This allows for limited control output, thereby achieving precise trajectory tracking control.

[0185]

[0186] By introducing advanced technology, mobile robots can quickly track the desired robot's position even with any initial error, thus achieving precise positioning.

[0187] 2.4.3 Autonomous obstacle avoidance methods in trajectory control process.

[0188] When a wheeled mobile robot is tracking a reference trajectory, if it encounters an obstacle, it must be able to safely and effectively avoid it and continue along the original trajectory. This is the autonomous obstacle avoidance problem, which involves the wheeled mobile robot's self-adjustment capabilities during the tracking process, such as... Figure 8 As shown.

[0189] The key technologies for the autonomous obstacle avoidance capability of wheeled mobile robots when tracking a trajectory lie in the following aspects:

[0190] (1) When encountering obstacles during the tracking process, effective measures should be taken to avoid robot collisions.

[0191] (2) When the robot avoids obstacles, it should carefully plan the best avoidance route based on the position of the reference trajectory in order to minimize energy consumption, rather than blindly avoiding obstacles.

[0192] (3) When the robot is avoiding obstacles, it should be adjusted according to its current position to ensure that the reference track robot can effectively track the target and avoid excessive deviation from the target.

[0193] (4) When the robot can autonomously bypass obstacles and quickly return to the reference trajectory, it can achieve accurate tracking of the target.

[0194] Based on the analysis of the above problems, this embodiment establishes a basic strategy for trajectory tracking monitoring and active obstacle avoidance to ensure that the automated robot can safely and efficiently bypass vehicles and minimize energy consumption.

[0195] In summary, this embodiment aims to construct a kinematic model of a two-wheeled differential-drive wheeled mobile robot and analyze how it changes the robot's position and attitude through kinematic transformations. Furthermore, this embodiment will delve into the mathematical model of brushless DC motor drive, as well as motion control technologies such as robot speed control, trajectory tracking control, and autonomous obstacle avoidance. These aspects provide a foundation for subsequent analyses of the robot's linear and curved trajectory tracking and autonomous obstacle avoidance control.

[0196] III. Fractional-order PID trajectory control of a mobile robot based on Simulink simulation.

[0197] Trajectory tracking control is a crucial component of motion control technology for wheeled mobile robots. It aims to ensure the system's motion trajectory matches a desired path through controller design. While PID control has high requirements in engineering applications, the novel trajectory tracking method using fractional-order PID fuzzy monitoring presented in this embodiment is simple to design, easy to tune parameters, exhibits excellent tracking performance and high stability, and has lower requirements for the main controller. Therefore, it is more readily applicable in engineering practice.

[0198] 3.1 Trajectory tracking control for wheeled mobile robots.

[0199] Mobile robot trajectory tracking control refers to controlling a mobile robot to move from an initial position at a certain point to a predetermined standard trajectory at a predetermined linear velocity and angular velocity, in order to achieve this goal. Effective control of the robot's linear velocity and angular velocity is required.

[0200] This embodiment provides a theoretical analysis of a wheeled mobile robot, neglecting wheel slippage on the plane of motion. Based on the foregoing, the following kinematic model is established:

[0201]

[0202] In the formula, v and w are the linear velocity and angular velocity of the wheeled mobile robot, respectively. Let X be the linear velocity component of the mobile robot along the X-axis. Let be the linear velocity component of the mobile robot along the Y-axis in the coordinate system. The angle between the X-axis and the robot's direction of motion.

[0203] Considering that the trajectory control model of a wheeled mobile robot is subject to parameter disturbances and external disturbances, and that the robot's actuator may also experience constraint oversaturation during movement, the dynamic model and kinematic model of the wheeled mobile robot system are organically combined. By constructing a suitable trajectory tracking controller, the wheeled mobile robot can achieve fast and stable control of its movement trajectory.

[0204] 3.2 Wheeled mobile robot controller.

[0205] 3.2.1 PID controller.

[0206] PID control strategies have advantages such as simple construction, easy operation, and easy implementation. The PID controller determines the direction of operation based on a given parameter y. d The adjustment is achieved by the deviation e(t) between the output value y(t) and the output value y(t), thereby achieving the purpose of the control system.

[0207] e(t) = y d (t)-y(t)#(3.2)

[0208] The control law of the PID controller is given by equation (3.3):

[0209]

[0210] Its transfer function can be written as:

[0211]

[0212] In the formula, K p T represents the proportionality coefficient. d T represents a time constant. i It is an integral constant.

[0213] 3.2.2 Establishment of the mathematical model for fractional-order PID control.

[0214] Transforming equations (2.22) and (2.24) from the previous equations, we obtain:

[0215]

[0216] Differentiating equation (3.5) above, we get:

[0217]

[0218] Here, the state variable X = [βω] r ] τAs an input variable, the input variable is U = [δ f δ r ] τ The output variable is Y = [βω] r α y ] τ ,in,

[0219] The state equation is:

[0220]

[0221] In the formula:

[0222] 3.2.3. Determination of fractional-order PID control parameters.

[0223] By employing fractional-order control principles, the proportional element of a traditional PID controller can be effectively replaced, thereby optimizing the system's transfer function and giving the control model higher dynamic response performance and stronger disturbance suppression capabilities, thus achieving more precise management. In this embodiment, a reasonable yaw rate model is first established. Then, the difference between the actual yaw rate and the reasonable yaw rate is compared, and the comparison result is compared with the reasonable yaw rate. Finally, the comparison result is used as input and fed back to the fractional-order PID controller, thereby achieving control over the robot's sideslip angle, yaw rate, lateral acceleration, and lateral displacement, such as... Figure 9 As shown.

[0224] Because of PI λ D μ The controller is the core of fractional-order PID control, so the PI controller is established first. λ D μ Controller. PI λ D μ The control time-domain expression of the controller is:

[0225] u(t) = K p e(t)+K i D -λ e(t)+K D D μ e(t)#(3.8)

[0226] In the formula: λ is the order of the controller integral term; μ is the order of the controller differential term; K p K is the proportionality coefficient. i K is the integral coefficient; D is the differential coefficient.

[0227] By using the Laplace transform, the feedback function of the fractional-order PID controller can be obtained, enabling more efficient management.

[0228] C(s) = K p +K I s -λ +K D s μ #(3.9)

[0229] In fractional-order PID controllers, the feedback function is a key factor determining the parameters of the controlled object. Therefore, to achieve this goal, it is necessary to derive the relationship between the actual and ideal yaw rates. The two-degree-of-freedom kinematic differential equations of the robot wheeled locomotion mechanism proposed in this embodiment offer an effective method to solve this problem. Through Laplace transform, the relationship between the yaw rate and the rear wheel rotation angle can be derived, thus obtaining the expression of the kinematic differential equations.

[0230]

[0231] In the formula: p =MuI z ;

[0232] The fractional closed-loop transfer function is:

[0233]

[0234] The characteristic equation of the transfer function is:

[0235] Ps 2 + q s+h+(a1s+a0)(K p +K I s -λ +K D s u )=0#(3.12)

[0236] The pole order search method can be used to precisely adjust the fundamental parameters of a fractional-order PID controller, including λ, μ, and K. P K I K D Five basic technical parameters.

[0237] 1) Estimate the proportionality coefficient K P

[0238] proportionality coefficient K p The relationship with the steady-state error E0 is as follows:

[0239] K p =(100 / E0-1)a0 / b0≈(100 / E0)a0 / b0

[0240] In the formula: a0 and b0 are both object parameters.

[0241]

[0242] In the formula: E t The steady-state error reached by the controller can be used to derive the proportional coefficient K of the control system. P .

[0243] 2) Determine the poles.

[0244] From the formula

[0245]

[0246] In the formula: σ is the stability, and ζ is the damping ratio.

[0247] Considering that the search range of ζ has little impact on system performance, ζ can be set to 0.5 to determine the value of σ, where [0.4, 5] represent the range of σ and the system step size, respectively, to minimize the time-squared weighted error square integral of the system's unit step response. If the system can operate stably, then... σ The value of K can then be determined, thus identifying the poles. However, if the system cannot stabilize, it is necessary to continuously adjust K. p The values ​​of the poles and proportional coefficients are adjusted to achieve the optimal state until the system stabilizes. After determining the poles and proportional coefficients, the other control parameters are tuned using the same method.

[0248] 3) Determine the search range of λ and μ for the i-th time, i∈[1,4]. When i=1, μ∈[0.1,1] and λ∈[0.1,1], take the step size Δ. i =0.01; i>1, μ∈[max(Δ i ,μ i-1 -2μ i-1 ),(μ i-1 +2μ i-1 )],λ∈[max(Δ i ,λ i-1 -2λ i-1 ),(λ i-1 +2λ i-1 )],Δ i =Δ i-1 / 10.

[0249] 4) Once the values ​​of the parameters are defined for the i-th time, the characteristic equation can be constructed. If all four indices meet the requirements of the control system, the parameter tuning is complete. However, if any indices still fail to meet the control requirements, λ needs to be selected again. i ,μ iThe values ​​are calculated and summed, and the above step is repeated; if the condition is still not met after i>4, the five parameters need to be redefined and the above step is repeated.

[0250] Outaloup is an Outaloup method for calculating small areas, which utilizes the construction of filters to similarly calculate G(s) = (s / ω). n ) α The transfer function is used to solve for the system's integral and differential terms. PI λ D μ The controller is s α Rationalization can effectively solve problems in practical applications and improve computational efficiency. λ and s μ These represent G1(s) and G2(s), respectively. When constructing the filter solution, only the frequency band is considered. in, In this specific frequency band, a continuous filter can be constructed, with the following transfer function:

[0251]

[0252] Where: K - filter gain; ω k1 - The kth zero point; ω k2 - The kth pole.

[0253]

[0254]

[0255]

[0256] The filter is designed using MATLAB / Simulink software. The N value of the filter determines its accuracy. The larger the N value, the higher the control accuracy. The control signal is processed by the filter to perform α-order calculus calculation, which can achieve the control effect of fractional PID.

[0257] 3.3 Finite-time trajectory tracking controller.

[0258] 3.3.1 Lyapunov stability theory.

[0259] Lyapunov introduced a new concept of system stability:

[0260]

[0261] make The solution x = x0 obtained is the equilibrium point of the system.

[0262] When the equilibrium point x0 of system (2.16) has ε>0, then δ=δ(ε)>0. Then the equilibrium point is stable; and by choosing an appropriate δ, the following condition is met: Then the equilibrium point is asymptotically stable. When the initial conditions are expanded to the entire state space, and the equilibrium state of the system is asymptotically stable, then the equilibrium state is called globally asymptotically stable.

[0263] 3.3.2 Lyapunov stability criterion.

[0264] The stability of a control system decreases over time. Therefore, it is important to construct a Lyapunov function V(x,t) that reflects the changes in system state and time to evaluate the stability of the system.

[0265] This theorem shows that when a nonlinear system And its state x Both time t and time t have a continuous partial derivative, and When the system is negatively definite and bounded, it can be considered stable, as confirmed by Lyapunov's theorem. When the asymptotic stability of the system depends on... x ≠0 o'clock, The value must be less than 0.

[0266] By finding a suitable Lyapunov function, the stability of a control system can be effectively evaluated without having to perform intuitive calculations or linearization operations on the system's differential equations.

[0267] 3.3.3 Finite-time control theory.

[0268] Convergence performance is an indispensable part of control systems, helping them achieve optimal safety and reliability. However, many control system design methods can only achieve asymptotic stability, not optimal time efficiency. Among effective time optimization methods, finite-time control methods can effectively improve the convergence performance of control systems.

[0269] Compared to asymptotic stability, finite-time stability requires that the trajectory of each solution reaches its endpoint within a finite time. The prerequisite for this stability is that the system has sufficient stability.

[0270] The finite-time control theorem states that when a neighborhood U∈R of a non-Lipschitz continuous origin is defined... n When a positive definite function V(x) appears on the expression, and c>0, 0<1, if Then the system possesses local finite-time stability. If U∈R n If there is no boundary between V(x) and V(x), then the system is stable globally.

[0271] To effectively control the finite-time motion of the motion machinery, this embodiment combines the modeling of equations (2.24) and (3.18), dividing them into second-order and third-order cascaded subsystems. Fractional-order PID controllers are designed for each, achieving precise control of the linear and angular velocities of the motion machinery. A motion camera employing a finite-time trajectory tracking control system, such as... Figure 10 As shown, it has good performance.

[0272] Figure 10 The diagram shown is used to compare the poses of a real-world mobile robot with those of a virtual reality mobile robot, thus providing the pose error equation. To better achieve this goal, the system can be divided into second-order and third-order subsystems, and control equations can be designed using the data variables provided by the subsystems. This allows for the design of a finite-time trajectory tracking controller to achieve precise control of the mobile robot. By using the controller output and the dynamic model, the linear and angular velocities of the mobile robot can be calculated, and its actual working position can be determined based on the kinematic model. This allows the construction of a closed-loop system that tracks the mobile robot within a limited time period.

[0273] The mobile robot trajectory tracking management analyzed in this embodiment makes two assumptions: one is based on actual conditions, and the other is based on theoretical analysis.

[0274] Assumption 4.1: The mobile robot can accurately track the velocity and angular velocity, and its first and second derivatives are both continuous and finite.

[0275] Assumption 4.2: The external disturbances experienced by the robot and its first derivative are both continuous and bounded. The angular velocity w of the mobile robot should be expressed by the formula: w = ww r By applying formulas (2.20) and (2.24), the mobile robot system can be modified as follows:

[0276]

[0277]

[0278]

[0279]

[0280]

[0281] The system in equation (3.20) is divided into second-order and third-order systems, and state variables are defined to better understand the behavior and state of the system.

[0282]

[0283]

[0284] By setting up a controller for a second-order subsystem, the angular rate of the motion robot can be effectively adjusted, thereby realizing the derivative of equation (3.26) and achieving the expected goal.

[0285]

[0286] The second-order system of equation (3.27) can be reduced to the following form:

[0287]

[0288] In this system, X = [x1, x2, ..., x n ] τ Represents the operating status of the system, u(t)∈R r It is the control input, B∈R n×m It is a known constant matrix, f(x)∈R is a continuous nonlinear function, and d(t)∈R n The possible external disturbances are that the functions f(x) and d(t) are differentiable with respect to time continuity. Therefore, transforming equation (3.28) into the form (3.29) can better describe the operating status of the system, thereby better controlling the operation of the system.

[0289]

[0290] Differentiate the above expression:

[0291]

[0292] The governing equations are constructed as follows:

[0293]

[0294] Where c1 = [c 11 ,c 12 ,c 13 ] T , Let be a positive constant. Taking the first and second derivatives of the above expression, we get:

[0295]

[0296]

[0297] Furthermore, an improved dynamic fractional-order PID control surface was designed:

[0298]

[0299] A third-order system controller is constructed using equation (3.34), and its equations are as follows:

[0300]

[0301] in, κ1, γ1, β1, All are normal numbers.

[0302] 3.3.4 Stability analysis of finite-time trajectory tracking control system.

[0303] According to the finite-time stability theorem, the trajectory tracking controllers (3.20) and (3.24) can work together to effectively suppress trajectory tracking errors in the mobile robot, thereby achieving global asymptotic stability. Proof: Choose the following Lyapunov function:

[0304]

[0305] Differentiating with respect to σ1 and substituting equations (3.32) and (3.33) into the equations, we get:

[0306]

[0307] Differentiating the above equation and substituting it into (3.36), we get:

[0308]

[0309] Based on assumption 4.2 and controller (3.35), substituting equations (3.36) and (3.37) into equation (3.38) and rearranging, we obtain:

[0310]

[0311] Where: ξ1=2a1η1γ1,

[0312] Choose another Lyapunov function as:

[0313]

[0314] Based on hypothesis 4.2, there exists a positive constant β2 such that Similar to the proof of V1 above, differentiating the above equation yields:

[0315]

[0316] Where: ξ2=2α2η2γ2,

[0317] Then, design a new Lyapunov function:

[0318] V = V1 + V2 (3.42)

[0319] Differentiating the above equation, we get:

[0320]

[0321] Where ξ <min{ξ1,ξ2},ρ<min{ρ1,ρ2}, It is a relatively small positive constant. Therefore, the above equation satisfies the conditions of the finite-time stability theorem. Thus, the finite-time stability of the system is proven.

[0322] 3.4 Simulation and analysis of linear trajectory tracking control based on MATLAB / Simulink.

[0323] The control law of the novel fractional-order PID trajectory tracking controller proposed in this embodiment is as follows:

[0324]

[0325] By analyzing the control law equation, it can be found that when the reference velocity changes, v r The magnitude will affect the error y e The convergence of v has a significant impact; therefore, in the case of a changing reference velocity, v r The size of v can change dynamically, which can lead to decreased tracking efficiency or even make it impossible to track v. r The case where = 0. The advanced trajectory tracking controller proposed in this embodiment has the following control law:

[0326]

[0327] Using a new fractional-order control law can significantly reduce v r The effect on the error convergence speed, when v r =0, w r When the value is not equal to 0, the robot can maintain its rotation in place, thus achieving more accurate tracking.

[0328] In this embodiment, a simulation experiment was conducted on the novel trajectory tracking control method under Matlab conditions, using linear tracking and curve tracking sequentially, and comparing the tracking performance with that of direct PID control. To better verify the effectiveness of the two tracking control methods, a simulation was performed at v... r With a velocity of 0.4 m / s, by adjusting the parameters, the error convergence speed of the two tracking control methods can be kept consistent, and v is changed. r The value of v is used to analyze the value of v. r Tracking performance at 0.8 m / s. The control parameters for the two methods are shown in the table below:

[0329]

[0330] To facilitate simulation, the following simulation experiment will analyze the wheel rotation angle step input to assess the driving stability of the wheeled robot. The robot's speed is set to 0.8 m / s, and the amplitude of the step signal during driving is set to 0.1 rad. A MATLAB / Simulink simulation model is established, as follows: Figure 11 , Figure 12 As shown in the figure. Input the corresponding simulation parameters for calculation, and then obtain the response curves for four physical quantities: yaw rate, lateral acceleration, sideslip angle, and lateral displacement. The simulated response curves obtained under fractional-order PID control will be compared with the simulated driving data curves under PID control. The simulation response comparison curves are shown below. Figure 13 (a) Figure 13 (b) and Figure 14 As shown.

[0331] The established Matlab / Simulink simulation model was used to calculate the response curves of three physical quantities: yaw rate, lateral acceleration, and sideslip angle. The simulated response curves obtained under fractional-order PID control were compared with the simulated data curves under PID control. The simulation response comparison curves are shown below. Figure 13 (a) Figure 13 (b) Figure 14 As shown.

[0332] Depend on Figure 13 (a) It can be seen that during linear motion, the yaw rate of the mobile robot under both PID control and fractional-order PID control abruptly changes one second after the step signal is triggered. The yaw rate under PID control reaches its maximum value of 0.55 rad / s at 3 seconds and then remains stable. In contrast, the mobile robot using fractional-order PID control reaches its maximum value of 0.4 rad / s in approximately 2.5 seconds. This demonstrates that the fractional-order PID control motion system can reduce the robot's yaw rate by 23.6% and effectively improve control efficiency.

[0333] according to Figure 13 (b) When the mobile robot under PID control and fractional-order PID control is moving in a straight line, the lateral acceleration of the system begins to change abruptly after 1 second. The lateral acceleration under PID control reaches 0.45 m / s² after 2.5 seconds. 2 The maximum value is reached, and then it remains stable. The mobile robot using fractional-order PID control reaches 0.33 m / s in approximately 2.2 seconds. 2 The maximum value is shown in the left and right directions. This demonstrates that the fractional-order PID control motion system can reduce the robot's lateral acceleration by 22.1% and effectively improve control efficiency.

[0334] Depend on Figure 14 It can be seen that under fractional-order PID control, the sideslip angle of the mobile robot abruptly changes in about 1 second, fluctuates for about 1.5 seconds, and then returns to near zero and remains stable, with a maximum sideslip angle of 0.01 rad. In contrast, under PID control, the sideslip angle of the mobile robot also abruptly changes in about 1 second, with a maximum deviation angle of approximately 0.035 rad, and then gradually decreases to zero and stabilizes after about two seconds. It is evident that fractional-order PID control outperforms PID control in both response time and sideslip overshoot, reducing the sideslip overshoot by 71.4%, significantly improving the robot's motion stability during movement. This enhances the robot's ability to maintain straight-line motion and improves control accuracy.

[0335] 3.5 Simulation and analysis of curve trajectory tracking control based on MATLAB / Simulink.

[0336] 3.5.1 System Model.

[0337] To demonstrate that the novel fractional-order PID trajectory tracking controller can effectively track curved trajectories, this embodiment sets the reference speed to v. r and w r When dynamic changes occur, its tracking performance will be significantly improved.

[0338]

[0339] Based on the kinematic model of equation (2.28) above, and taking into full account the negative impacts that the technical parameters of the mobile robot itself and external disturbances may have on control, this embodiment constructs an error kinematic equation, based on the reference velocity equation (3.46), and performs dynamic simulation on it to obtain more accurate control results. Its kinematic model is as follows: Figure 15 As shown.

[0340] q = [x, y, θ] τ q represents the position of the mobile robot. r =[x r ,y r ,θ r ] τ This indicates the position of the virtual mobile robot. In fact, the kinematic model of the mobile robot is the same as equation (2.29), and the virtual mobile robot model is as follows:

[0341]

[0342] in:

[0343]

[0344]

[0345] v r w represents the linear velocity of the virtual mobile robot. r This represents the angular velocity of the virtual mobile robot.

[0346] By applying the above model, the positioning error of the mobile robot in the global coordinate system can be expressed as:

[0347]

[0348] To accurately depict the motion state of a mobile robot, its motion in the global coordinate system must be transformed into motion in the local global coordinate system, and this transformation must be represented using orthogonal rotation matrix technology in order to better capture the robot's motion characteristics.

[0349]

[0350] Using a local coordinate system, the trajectory tracking deviation of a mobile robot can be accurately represented as:

[0351]

[0352] By establishing the pose relationship between the actual and virtual mobile robots, the motion analysis and trajectory tracking control of the mobile robots can be effectively improved, thereby enhancing their operational performance and reliability.

[0353] By differentiating equation (3.52), the differential equation for the position deviation of the motion robot can be obtained:

[0354]

[0355] 3.5.2 Dynamics model of mobile robot.

[0356] By using dynamic models, the adverse effects of the robot's physical parameters on manipulation can be effectively reduced, thereby improving the system's robustness. Therefore, when designing a controller, external disturbances should be considered and analyzed to better describe the motion characteristics of the controlled object. Through dynamic principles, the position, attitude, and speed of the mobile robot can be adjusted, thus achieving effective drive of the wheels.

[0357] This embodiment will analyze how to establish the dynamic equations of a mobile robot, where the Lagrange dynamic equations can be expressed as follows:

[0358]

[0359] In this equation, M τ(q) is a noncomplete constraint matrix, L is the Lagrangian function, q is the system state variable, λ is the Lagrange multiplier, E is a non-singular transformation matrix, and τ = [τ r ,τ t ] T It is a generalized force vector generated on matter (that is, translational energy and rotational energy).

[0360] The differential drive robot studied in this embodiment:

[0361]

[0362] Further simplification of (3.55) yields:

[0363]

[0364] Here, Representing centrifugal force and Coriolis force, while g(q) indicates the direction of force. This refers to inertial force.

[0365] To solve the problem of M in the above formula T The constraint on (q)λ can be addressed by using a matrix B(q) of dimension n×(nm) that satisfies the following condition:

[0366] B τ (q)M τ (q)=0#(3.57)

[0367] From equations (3.56) and (3.57), we can derive that there exists a (nm)-dimensional vector v such that:

[0368]

[0369] For differential drive robots:

[0370]

[0371] Combining equations (3.56), (3.57), and (3.58), equation (3.59) can be transformed into:

[0372]

[0373] in:

[0374]

[0375] The wheeled mobile drive robot analyzed in this embodiment has horizontal movement capability, so other parameters in equation (3.60) can be ignored, such as... and By adding a perturbation term d = [d1, d2] to the model in equation (3.60) above τ and eliminate it. We can conclude that:

[0376]

[0377] Simplifying equation (3.61) yields the corresponding dynamic model:

[0378]

[0379] By designing suitable controllers u1 and u2, the linear velocity v and angular velocity w of the mobile robot can be effectively controlled, thereby achieving trajectory tracking. The control range of u1 is u1 = τ. r +τ t The control range of u2 is u2 = τ r -τ t The control range of δ1 is The control range of δ2 is

[0380] 3.5.3 Simulation and analysis of curve trajectory tracking control system within a finite time.

[0381] This embodiment designs a control system capable of tracking the trajectory of a mobile robot within a finite time and constructs a Simulink simulation mode, such as... Figure 16 As shown.

[0382] First, the tracking trajectory is determined as needed, and a kinematic model simulating the mobile robot's motion is established for simulation analysis. By analyzing the desired tracking trajectory and the actual trajectory provided by the kinematic template, an error differential equation template can be constructed. These templates are then used to construct second-order and third-order subsystem templates, as well as a fractional-order PID template, to obtain their first and second derivatives, thereby achieving precise robot control. A finite-time trajectory tracking control system can be constructed based on different variables, and a dynamic template can be constructed through the controller's input and output to achieve real-time tracking of the robot's linear and angular velocities. Finally, the input and output of the dynamic template are combined with the robot's kinematic template to construct a complete simulation model of the trajectory tracking closed-loop control system. This control system consists of two independent loops: one controlled by a fractional-order PID to adjust the rotational speed of the two wheels, and the other to control the robot's position and orientation. The controller parameters are as follows: c1 = [3,3,1], c2 = [1,1], α1 = 2, α2 = 0.1, η1 = η2 = 0.6, λ1 = 3, λ2 = 2. κ1=0.1, κ2=1, β1=5, β2=6, γ1=γ2=0.1. The external disturbance is d1=d2=sint. By adjusting the speed and angular velocity of the mobile robot, it can track the curved track. The initial pose of the desired track is set to (0,0) and the initial pose of the mobile robot is set to (-0.2,0), as follows. Figure 17 As shown.

[0383] Figure 17 A simulation diagram of a mobile robot's path tracking is shown. Solid lines represent the desired path, while dotted lines represent the robot's actual running path. Simulation of this diagram allows for more accurate results and subsequent analysis.

[0384] Figure 18 (a) shows the simulated angular velocity response curves for trajectory control of the mobile robot, where the solid line represents the fractional-order PID control algorithm and the dashed line represents the ordinary PID control algorithm. For the PID control algorithm, the maximum output angular velocity reaches 45° / s during curved trajectory motion, and it is very unstable in the initial stage, dropping sharply and converging to 0 after about 2 seconds. In contrast, the maximum output angular velocity of the fractional-order PID control algorithm can be controlled at 10° / s, and the control quantity converges more smoothly. Clearly, the fractional-order PID algorithm performs better in robot angular velocity control.

[0385] Figure 18 (b) shows the simulated response curves of the mobile robot's trajectory tracking slip angle. The solid line represents the fractional-order PID control algorithm presented in this embodiment, while the dashed lines represent the various responses of the mobile robot under the ordinary PID control algorithm. From the figure, it can be seen that the maximum slip angle of the robot under PID control is approximately 45°, converging to 0 and remaining stable after 2.5 seconds. In contrast, the maximum slip angle of the robot under fractional-order PID control is approximately 33°, decreasing slowly over the first 1.2 seconds and then rapidly converging to 0 within 1 second. Its control effect is significantly better than PID control, and its control efficiency is also higher.

[0386] Figure 19 (a) shows the simulated response curves of the mobile robot's trajectory tracking. The solid line represents the fractional-order PID control algorithm presented in this embodiment, while the dashed lines represent the various responses of the mobile robot under the ordinary PID control algorithm. The fractional-order PID control algorithm can converge the robot's distance deviation from 0.17 to 0 in about 1 second, while the same distance PID control method requires about 2.2 seconds. Clearly, the fractional-order PID control method has higher control efficiency.

[0387] Figure 19(b) shows the simulated response curve of the mobile robot's angle deviation, where the solid line represents the fractional-order PID control algorithm given in this embodiment, while the dashed line represents the response of the mobile robot under the ordinary PID control algorithm.

[0388] from Figure 19 As shown in (b), for the correction of angle deviation, ordinary PID control takes nearly 3 seconds to converge to zero, while fractional-order PID control only takes about 0.5 seconds to converge. Through comparison, the fractional-order PID method proposed in this embodiment is significantly superior to traditional PID control calculations. It can achieve the control objective more smoothly and quickly, and the path required from the starting point to the target trajectory is shorter, thus ensuring both the stability and speed of accuracy.

[0389] In summary, this embodiment, based on the Lyapunov equations and finite-time theory, designs a novel fractional-order PID trajectory tracking control method. The original system is divided into two subsystems, a second-order and a third-order subsystem, and a new fractional-order PID control model is designed. The control method is then compared and verified with the ordinary PID control method in both linear and curvilinear domains.

[0390] (1) In linear trajectory control, the fractional-order PID control method can significantly reduce the yaw rate and lateral acceleration of the robot. The yaw rate is reduced by 23.6%, the lateral acceleration is reduced by 22.1%, and the side slip angle overshoot is reduced by 31.4%. Under the same travel distance, the lateral offset of the robot under the fractional-order PID control automatic driving system is significantly reduced by 29.5% compared with the robot under PID control.

[0391] (2) Under curve trajectory control, the maximum angular velocity of the PID control algorithm is 45° / s, while the maximum angular velocity output by the fractional-order PID control algorithm is only 10° / s. Furthermore, the control error can converge smoothly. The distance deviation correction time under fractional-order PID control is 1.5 seconds faster than that under PID control, and the angle deviation correction time is 2.5 seconds faster. This indicates that the driving stability and tracking ability of the mobile robot system have been significantly improved. This demonstrates that the designed control method has a faster response speed and more precise control effect.

[0392] IV. Simulation and Analysis of Automatic Obstacle Avoidance Control for Mobile Robots Based on Disturbance Observer.

[0393] Changes in the external environment can significantly impact the stability of mobile robot systems. The novel fractional-order PID controller proposed earlier can effectively suppress this instability. However, when faced with complex environmental changes, such as external disturbances, the system may still exhibit instability. Therefore, it is essential to employ targeted strategies to assess external disturbances and further improve the system's anti-interference level. This embodiment uses a disturbance observer to study the external disturbances experienced by the mobile robot's trajectory control system. To achieve this, a fractional-order PID controller can be designed based on the robot's dynamic model and deviation pattern in polar coordinates, replacing the sign function with a saturation function. To enhance system robustness, improve error convergence rate, and increase the system's resistance to small external disturbances, this embodiment uses a novel nonlinear disturbance observer. This observer can monitor constant external disturbances and process the observed values ​​to compensate the controller. Furthermore, this embodiment also employs an improved fractional-order PID control method to better control the trajectory and observe disordered disturbances. Based on mobile robot trajectory tracking technology, this embodiment develops a novel obstacle avoidance compensator, which uses an ultrasonic sensor to effectively avoid simple obstacles, and designs a joint controller structure to meet the trajectory and obstacle avoidance functions of the mobile robot.

[0394] 4.1 Trajectory control of mobile robots based on disturbance observers.

[0395] Based on the pose error differential equation in polar coordinates, this embodiment presents a new system that uses an index-reaching law to make the system's control curve eventually converge, thereby achieving trajectory control. Furthermore, a disturbance observer capable of detecting external constant disturbances is designed to better meet the robot's needs.

[0396] 4.1.1 System Model.

[0397] In the polar coordinate system, the trajectory control error of the mobile robot is as follows: Figure 20 .

[0398] exist Figure 20 In the diagram, ψ represents the angle between the line connecting the center of mass of the real wheeled mobile robot to the origin and the x-axis, while ψ r This represents the angle between the line connecting the center of mass of the virtual wheeled mobile robot to the origin and the x-axis. Furthermore, The lengths of these two parameters are also interdependent. The kinematic model of a real mobile robot can be described by equation (2.28), while the kinematic model of a virtual mobile robot can be described by equation (2.29). Furthermore, the physical parameters of the real mobile robot are Q = [l, ψ, θ]. TThis represents its position in polar coordinates, while the position parameter Q of the virtual mobile robot... r =[l r ,ψ r ,θ r ] T This can be calculated. By differentiating Q and based on equation (2.28) of the real mobile robot model, the pose differential equation of the mobile robot in polar coordinates can be obtained, thus better describing its motion characteristics.

[0399]

[0400] By taking the derivative Q r By combining it with equation (4.1), the pose differential equation of the virtual mobile robot in the polar coordinate system can be obtained.

[0401]

[0402] Based on (4.2), through differential equations, we can express the polar coordinate pose error of the robot as:

[0403]

[0404] By differentiating (4.3), we can derive the differential equation of position deviation in polar coordinates:

[0405]

[0406] 4.1.2 Disturbance observation fractional-order PID controller.

[0407] Due to external influences and uncertainties in system parameters, the controller may become unstable. To effectively assess and mitigate these uncertainties, this embodiment requires the design of a disturbance observer. Fractional-order PID control can be used to reduce this adverse effect. Therefore, a new control structure needs to be established first, such as... Figure 21 As shown.

[0408] By analyzing the kinematic model of a real mobile robot and the running position and speed of a virtual mobile robot, the pose coordinate system and error equation of the mobile robot in polar coordinates can be calculated. Using these variables, a fractional-order PID controller can be designed to achieve precise tracking of the mobile robot. Through the input and output of the controller, the mobile robot can adjust its linear and angular speeds, which are then transmitted to the motion feedback controller to obtain real-time pose coordinates. To resist interference from the external environment, a compensator also needs to be designed to ensure the precise movement of the automated robot. After careful design, a closed-loop control system for mobile robot trajectory tracking using a disturbance observer has been completed.

[0409] By taking into account the position and error of the mobile robot, this embodiment designs a control equation to achieve this goal.

[0410]

[0411] Where c1 and c2 are both positive constants. Differentiating equation (4.5) yields:

[0412]

[0413] in Let equation (4.6) equal the exponential reaching law, that is:

[0414]

[0415] Where ε1, ε2, k1, and k2 are all positive constants. Therefore, combining equations (4.6) and (4.7), the trajectory tracking controller for the mobile robot can be derived:

[0416]

[0417]

[0418] Analysis of equations (4.8) and (4.9) reveals unknown disturbance terms in the designed controller. To overcome this challenge, this embodiment employs a novel disturbance observer that can effectively detect external disturbances and suppress them.

[0419] The dynamic model of the mobile robot (2.28) is transformed into a more concise and efficient expression.

[0420]

[0421] in,

[0422]

[0423] Based on this, the following perturbation observer was then designed:

[0424]

[0425] In equation (4.11), It is an approximation of external disturbances, while z and p(x) are intermediate variables. Furthermore, p(x) = [a1v, a2w]. T , a1<0,a2>0。

[0426] The design of the compensator is as follows:

[0427]

[0428] To make the controller more stable, the sign function in controller (4.12) can be converted into a saturation function, and the controller can be rewritten as follows:

[0429]

[0430]

[0431] From the above formula derivation and analysis, the trajectory controller of the mobile robot after adding disturbance compensation can be expressed as:

[0432]

[0433]

[0434] 4.1.3 Stability analysis of fractional-order PID control under disturbance observation.

[0435] For (4.1), by adopting the controller of equation (4.14), the global asymptotic stability of the mobile robot trajectory tracking error control system can be effectively ensured. To verify this, it is necessary to first study the accuracy of the disturbance observer and determine the detector error as:

[0436]

[0437] For ease of subsequent analysis, we assume that the perturbation remains constant. Therefore, the dynamic equation for the observer error can be expressed as:

[0438]

[0439] Equation (4.18) above shows that the proof is valid, and the error of the perturbation observer can approach zero in the exponential case.

[0440] The Lyapunov function is chosen as follows:

[0441]

[0442] Differentiating equation (4.19) yields:

[0443]

[0444] Substituting (4.13) and (4.14) into equation (4.20) and rearranging, we get:

[0445]

[0446] Design a new Lyapunov function as follows:

[0447]

[0448] Differentiating equation (4.22), we get:

[0449]

[0450] Since a1 < 0 and a2 < 0, then Therefore, the closed-loop system remains asymptotically stable, which proves the hypothesis.

[0451] 4.2 Combined control of mobile robot trajectory control and obstacle avoidance.

[0452] 4.2.1 Trajectory control and obstacle avoidance controller.

[0453] Obstacle avoidance is an indispensable part of motion control analysis for mobile robots. A reasonable obstacle avoidance strategy can help mobile robots efficiently avoid dangers, thereby further improving their practical value. Therefore, this embodiment will analyze how to reasonably implement an obstacle avoidance strategy to further improve the robot's safety and reliability. To better control the mobile robot, this embodiment is based on the new virtual controller proposed in the previous section, and uses a fractional-order PID controller to achieve joint control, enabling it to achieve trajectory tracking and obstacle avoidance. This can more effectively enhance the practical value of the mobile robot.

[0454] In the preceding sections, we analyzed the real speed v and virtual speed v of the mobile robot. c If they are the same, the desired trajectory can be tracked. Similarly, the terminal controller is designed to make the mobile robot's actual velocity vw track the virtual velocity vt. cr w cr It enables combined trajectory control and obstacle avoidance control. First, based on the kinematic modeling of the mobile robot described in the previous section, a virtual reality trajectory tracking obstacle avoidance controller is constructed using the Lyapunov direct method to achieve effective motion control of the mobile robot.

[0455]

[0456] Based on this, the obstacle avoidance compensator is designed as follows:

[0457]

[0458] This represents the difference between the robot's linear velocity direction and the repulsive force direction; l1 and l2 are both positive constants, while F... r This refers to the repulsive force from obstacles encountered by the robot during its operation. It maintains a negative gradient in the repulsive potential field, and its geometric representation is: .

[0459]

[0460] The form of the repulsive potential field is:

[0461]

[0462] η r The gain G represents the repulsive force. i (q) represents the distance between the mobile AI and the i-th obstacle, while Q is the threshold of the obstacle's effect. As the distance increases, the repulsive force on the mobile robot gradually weakens and eventually disappears completely.

[0463] 4.2.2 Stability analysis of trajectory control and obstacle avoidance control system.

[0464] When external interference is present, the designed trajectory tracking controller (4.24) and controller (4.25) can effectively help the mobile robot achieve effective obstacle avoidance, thereby improving its driving efficiency.

[0465] By changing the form of the Lyapunov function and taking its derivative, it can be proven that equation (4.25) is the same as equation (4.27).

[0466]

[0467] k1, k2, and k3 are all positive constants, when the desired linear velocity v r When ≥0, This indicates that V1 is bounded, and qe and All are monotonically bounded, therefore It is also bounded. According to Barbalat's theorem, V 1b →0, which proves qe →0, from which we can conclude that the designed virtual obstacle avoidance controller can bring the system deviation of the mobile robot to converge, thus proving the stability of the mobile robot during operation.

[0468] 4.3 Simulation analysis of fractional-order PID control system for automatic obstacle avoidance trajectory control.

[0469] By analyzing the fractional-order PID trajectory tracking control structure diagram of the robot, a Simulink simulation mode is established to simulate the relationship between the actual pose and the desired trajectory. The pose system of the mobile robot is constructed in polar coordinates to achieve more precise tracking control. Then, using equation (4.14), a trajectory tracking controller is established, providing kinematic functions. The linear and angular velocities of the mobile robot can be measured and transmitted to the controller, thereby constructing the kinematic functions of the robot using equation (2.28), thus achieving precise pose control. By establishing a disturbance observer system, external disturbances are injected into the compensator to effectively supplement the controller, thus establishing a complete simulation model, such as... Figure 22 As shown.

[0470] 4.3.1 Obstacle avoidance in a straight line.

[0471] like Figure 23 As shown, the straight line is the reference trajectory. The control parameters of the control system are as follows: ε1 = 0.5, ε2 = 0.3, k1 = 2, k2 = 10, c1 = 5, c2 = 5. It can be seen from the figure that the artificial intelligence first tracks the reference trajectory, with its starting point at (0,0). The positional deviation of the reference trajectory quickly converges to 0. Figure 24 As shown, this demonstrates that the robot can accurately track the reference trajectory and converge to 0 quickly. When the mobile robot encounters an obstacle at position (0, 2.5) during the tracking process, it automatically takes measures to avoid the obstacle and continues to move along the reference trajectory, as shown. Figure 23 As shown.

[0472] Depend on Figure 23 It can be seen that the mobile robot starts from its initial position and maintains a straight line. When facing a cylindrical obstacle, it automatically turns left to avoid it, with a maximum distance of approximately 0.5 meters from the center of the obstacle. It then turns right and returns to its straight trajectory. The pose errors of the mobile robot's x, y, and θ degrees of freedom during the entire obstacle avoidance process are as follows: Figure 24 As shown: the dotted line represents the pose error in the x-direction, the solid line represents the pose error in the y-direction, and the dashed line represents the pose error at angle θ.

[0473] according to Figure 24 Under normal circumstances, the wheeled mobile robot's pose error along the x-direction initially approaches zero and remains stable. At the 5-second mark, due to the influence of an obstacle ahead, the robot automatically adjusts its pose, with a maximum error of approximately 0.45 meters in the x-direction. Subsequently, the pose error gradually decreases to zero, while the robot's angular error θ changes when it encounters an obstacle at the 5-second mark, with a maximum variation range of approximately -1 rad to 0.5 rad. After 10 seconds, the robot completes the entire obstacle avoidance process and returns to its original trajectory.

[0474] Figure 25 (a) is the linear velocity simulation curve of the wheeled mobile robot during the linear trajectory control and obstacle avoidance process. In the figure, the robot starts from a standstill and performs uniformly accelerated linear motion for the first 2 seconds. Around the 2nd second, it detects an obstacle in front and begins to decelerate. At the 5th second, it automatically turns and accelerates again until it reaches its maximum value at around the 6th second. At this time, the robot just bypasses the obstacle. Then the robot gradually decelerates and returns to the linear trajectory, and the speed also stabilizes. Figure 25 (b) shows the simulated angular velocity curves of the wheeled mobile robot during linear trajectory control and obstacle avoidance. For the first 5 seconds, the robot's angular velocity remains at 0. After 5 seconds, the robot begins automatic obstacle avoidance, with the angular velocity ranging from -0.5 to 1.5 rad / s. After 10 seconds, the robot completes the obstacle avoidance process, and the angular velocity returns to zero.

[0475] 4.3.2 Curve obstacle avoidance.

[0476] according to Figure 25 (a) The curve (solid line) represents the reference trajectory of the wheeled mobile robot, with a expected linear velocity of v. r =0.8m / s, angular velocity is w r =0.8m / s, while the parameters of the trajectory control obstacle avoidance device are η r =0.8, l1=1.5, l2=0.5, Q=0.8m. In the simulation environment, the robot starts from the initial pose (-2,-1,0) and follows a circular trajectory with three radii of 1m each, starting from (-4,0,0). Figure 25 As shown in (a), three circular obstacles are installed on the trajectory to verify the robot's autonomous obstacle avoidance and self-adjustment capabilities.

[0477] Depend on Figure 26 It can be seen that in the simulation environment, after the mobile robot starts moving from the initial point, it can move relatively quickly along the desired trajectory. During the movement, if it encounters obstacles, it can automatically adjust its trajectory to avoid them. After avoiding the obstacles, it can quickly readjust its posture and continue moving along the desired trajectory. The pose errors of the mobile robot's x, y, and θ degrees of freedom during the entire obstacle avoidance process are as follows: Figure 27 As shown: the dotted line represents the pose error in the x-direction, the solid line represents the pose error in the y-direction, and the dashed line represents the pose error at angle θ.

[0478] like Figure 27As shown, when the mobile robot first starts moving, its pose error converges quickly due to the absence of obstacles. At the 15-second mark, the robot encounters its first obstacle, resulting in pose errors of approximately 0.25 meters in both the x and y directions, and an angular error θ of approximately 0.8 rad. The robot then completes its first obstacle avoidance maneuver and returns to its standard trajectory. Around the 28-second mark, the robot performs its second obstacle avoidance maneuver, with pose errors of approximately 0.3 meters in both the x and y directions, and an angular error θ reaching approximately 1 rad. Around the 43-second mark, the robot performs its third obstacle avoidance maneuver. Due to the accumulated errors from the previous two maneuvers, the pose error in the y direction reaches 0.5 meters, and the angular error θ reaches approximately 1.5 rad. The robot then completes the obstacle avoidance maneuver, the pose error curve converges again, and the robot returns to its original curved trajectory. The linear velocity and angular velocity of the robot during curved obstacle avoidance are shown below. Figure 28 (a) and Figure 28 As shown in (b), the solid lines represent the actual linear velocity and angular velocity, and the dashed lines represent the standard reference linear velocity and standard reference angular velocity.

[0479] Figure 28 (a) shows the linear velocity simulation curve of the wheeled mobile robot during the curved trajectory control and obstacle avoidance process. In the figure, the robot starts from a standstill and accelerates to the standard linear velocity in the first 2 seconds. It then maintains a constant speed. Around the 10th second, it detects an obstacle ahead and begins to decelerate. At the 12th second, it automatically turns and accelerates again until it reaches its maximum speed around the 15th second, at which point the robot just bypasses the obstacle. The robot then gradually decelerates and returns to the straight trajectory, and its speed also stabilizes. This obstacle avoidance process is repeated three times until the robot completes the entire obstacle avoidance process. Figure 28 (b) shows the simulated angular velocity curves of the wheeled mobile robot during curved trajectory control and obstacle avoidance. The robot's angular velocity starts from 0 and increases, reaching its first peak of 1.8 rad / s at approximately 4 seconds. The robot then begins to move along a standard trajectory. At the 10th second, it encounters the first obstacle, and the angular velocity begins to change, ranging from approximately 0 to 2 rad / s. The robot completes the first obstacle avoidance process at approximately 15 seconds, and the angular velocity stabilizes. This obstacle avoidance process is repeated three times until the robot completes the entire obstacle avoidance process.

[0480] In summary, this embodiment aims to analyze the challenge of active obstacle avoidance in the path tracking process of wheeled mobile robots. To address this, an active obstacle avoidance controller based on fractional-order PID control theory is proposed. This controller effectively helps the mobile robot automatically track target points and avoid obstacles along the way. By employing a disturbance-resistant dynamic reference trajectory generation method based on a disturbance observer, this embodiment proposes a novel intelligent obstacle avoidance technology for mobile robots. This technology can effectively avoid obstacles and dynamically adjust the time scale and time nodes of reference trajectory generation based on the relative position data between the current mobile robot and a reference mobile robot. This allows the robot to quickly return to its original reference trajectory within 10 seconds after obstacle avoidance.

[0481] During linear obstacle avoidance, the obstacle avoidance pose error along the X and Y axes does not exceed ±0.5m. The linear velocity variation is within 1.5m / s, and the angular velocity variation is within 2.5rad / s. In curved obstacle avoidance, the obstacle avoidance pose error along the X and Y axes is within 0–0.5m. The linear velocity variation is within 2m / s, and the angular velocity variation is within 3rad / s. Autonomous obstacle avoidance of the mobile robot has been successfully achieved.

[0482] V. Wheeled mobile robot system design and trajectory and obstacle avoidance experiments.

[0483] 5.1 Mobile robot system design.

[0484] The motion control of a wheeled mobile robot consists of a chassis structure, servo drive system, control system, sensor data collection network system, and power supply network management system, which can meet various requirements of the host computer. To improve the operating performance of the wheeled mobile robot, this embodiment uses a four-wheel chassis structure, with drive wheels at both ends and independent omnidirectional wheels at the top and bottom. By controlling the left and right drive wheels, the robot can complete forward, backward, and turning movements.

[0485] The motion control system of a wheeled mobile robot uses inertial sensors and gyroscopes to monitor the robot's posture and lidar sensors to detect obstacles in the external environment, enabling precise control of the automated robot. When the robot's underlying MCU controller receives control tasks from a remote controller, a host computer, or an NVIDIA TX2, it uses motion control algorithms based on the information collected by the sensors to measure the control inputs of the left and right motors, thereby driving the robot's left and right wheels to rotate, enabling forward, backward, and turning movements, and ultimately executing various control tasks.

[0486] Figure 29The diagram illustrates the overall control block diagram of a wheeled mobile robot. The inner loop employs a decoupling disturbance rejection mechanism, effectively combining forward speed and turning angular velocity to form a closed-loop control system. The outer loop utilizes a fuzzy control obstacle avoidance system to achieve closed-loop control of the robot's trajectory tracking and motion posture.

[0487] The wheeled mobile robot proposed in this embodiment has multiple motion modes, which can meet a variety of application needs.

[0488] (1) Remote control mode. In this mode, the P2P handle telemetry device can use the upper and lower joysticks to control the robot to move forward, backward, and rotate up and down. Moreover, it can adjust the speed according to the force of pushing and pulling the joysticks, thereby greatly improving the posture adjustment efficiency in the process of debugging the automated robot.

[0489] (2) Trajectory tracking mode. The NVIDIA TX2 can capture road information through a camera and convert it into a reference trajectory curve, which enables the robot's underlying motion control system to track the trajectory curve and thus achieve automatic driving.

[0490] (3) Obstacle avoidance by moving toward the target point. In this mode, the host computer or NVIDIA TX2 sends the coordinates of the target point to the underlying motion controller, enabling the robot to move precisely toward the target point using a fuzzy control algorithm. At the same time, the robot can also use its own lidar sensor to detect external obstacles, thereby effectively avoiding possible obstacles.

[0491] (4) Trajectory Tracking Autonomous Obstacle Avoidance Mode. When the host computer or NVIDIA TX2 can send the coordinates of the target point to the underlying motion controller, the robot can track the target point according to the reference trajectory curve, thereby avoiding the uncertainty of fuzzy control in the absence of obstacles. In this way, the robot can control itself more accurately, thus achieving better results. When the robot faces obstacles, it can autonomously perform obstacle avoidance movements to ensure safety.

[0492] 5.1.1 Control system.

[0493] The Hardrock STM32 development board is a motherboard for basic motion management in artificial intelligence, capable of meeting various complex motion control needs of AI. Through this development board, not only can the WIFIESP2866 and SWD programmer ports be expanded, but a comprehensive interface platform is also created for subsequent function development.

[0494] The underlying motion control motherboard has a wealth of resources, enabling automated robots to easily perform multiple functions such as motor drive, information acquisition, and data communication, thereby achieving efficient operation of automated robots.

[0495] 5.1.2 Sensor acquisition system.

[0496] (1) Attitude sensor. The JY60 six-axis motion angle sensor from Witt Intelligent can be used, which can accurately measure the robot's pitch, roll, and yaw angles. Using the six-axis attitude sensor, the direction angle and relative position data of the mobile robot can be obtained in real time, and the Hall signal of the motor can be used for positioning, thereby instantly locating the attitude information of the wheeled mobile robot, thus improving the control performance and safety of the mobile robot.

[0497] (2) LiDAR sensor. The RPLIAR A3 laser scanning ranging radar can be selected. Through the control module, it can emit a continuous laser beam and receive laser signals from the sensor, thereby obtaining the laser emission and reception time to realize laser detection.

[0498] 5.2 Experiment on motion control of mobile robot.

[0499] In a flat laboratory environment, a program was written to control the trajectory tracking of a wheeled mobile robot and enable autonomous obstacle avoidance during the tracking process. A large amount of data was collected through experiments, and the algorithm was tested and analyzed. To demonstrate the feasibility of a novel trajectory tracking control technology based on fractional-order PID control, a novel track tracking controller was built on the two-wheeled differential wheeled mobile robot platform proposed in this embodiment, and experiments were conducted on straight and curved trajectories to prove its feasibility.

[0500] 5.2.1 Straight trajectory tracking control experiment.

[0501] By setting a reference line with initial coordinates (0,0), the robot can perform trajectory tracking based on this reference line, such as... Figure 30 As shown in (a), the solid line represents the desired trajectory of the mobile robot, while the dashed line represents the actual trajectory of the robot. Its tracking error can be determined by... Figure 30 (b) to reflect.

[0502] like Figure 30 (b) In the figure, the solid line represents the pose error in the x-direction, the dashed line represents the pose error in the y-direction, and θ represents the robot's angular pose error. It can be observed that the robot can effectively suppress pose errors. During the movement of the wheeled robot, the pose error (x... e ,y e ,θ eThe error gradually decreases, with the x-direction error starting at 0.5 meters and approaching zero in about 4 seconds; while the y-direction fluctuates in the first 4 seconds, with a fluctuation range of about ±0.5, and also gradually approaches zero after 4 seconds; in terms of angle θ, the robot quickly converges from an initial large deviation of 1.5 rad to close to zero within 4 seconds, indicating that the robot has successfully completed the straight trajectory tracking control and can run on the reference trajectory.

[0503] 5.2.2 Curve trajectory tracking control experiment.

[0504] By examining v r and w r Experiments were conducted to verify the dynamic changes of the reference trajectory. The new trajectory tracking controller can effectively capture the reference trajectory curve and adjust the curve trajectory according to the actual situation to achieve the best results.

[0505]

[0506] In equation (6.1), v R and w r The changes are respectively v R ∈[-0.25,0.25] sine variation and w r The sine variation is ∈ [-0.5, 0.5]. Assume the initial coordinate system of the reference robot is (0, 0), and the initial coordinate system is (-0.2, 0). According to... Figure 31 ( a )and Figure 31 (b) can then be used to determine the robot's tracking deviation.

[0507] according to Figure 30 (a) The actual motion trajectory starts from position (0.0), moves forward along the S-curve, and finally reaches position (-0.52, 1.6). In the initial stage of motion, due to trajectory adjustment, the error curve fluctuates. The error in the x-direction gradually decreases from 0.2 meters, and after 4 seconds, it approaches a small fluctuation of ±0.05. The fluctuation in the y-direction is larger in the first 4 seconds, with a fluctuation range of about ±0.05, and the fluctuation range decreases after 4 seconds. In terms of angle θ, the robot quickly converges from an initial large deviation of 0.5 rad to a smaller range of 0.05 to 0.05 rad within 4 seconds and can maintain relative stability. However, due to the unevenness of the actual road surface, the trajectory tracking control error will still fluctuate within a small range. This shows that the novel fractional-order PID trajectory tracking control method proposed in this embodiment can effectively track and control the curved motion of the wheeled mobile robot.

[0508] 5.2.3 Autonomous obstacle avoidance experiment in trajectory tracking control.

[0509] (1) Target point tracking and autonomous obstacle avoidance experiment.

[0510] In the experiment, the robot was initially set to (0,0). To enable the robot to autonomously move to the target point, a cylindrical obstacle with coordinates (0,2) and a radius of approximately 0.3m was placed along its path. By applying fractional-order PID control, the robot was able to move towards the target point, and better results could be achieved through autonomous obstacle avoidance experiments. Figure 31 As shown in (a).

[0511] according to Figure 32 The artificial intelligence will first move towards the target point. When the lidar detects an obstacle, it will automatically adjust its trajectory, turn right, and move along the tangent of the obstacle. When it is about 0.5 meters away from the center of the obstacle, it will bypass the obstacle. Then the mobile robot will automatically resume its original straight trajectory to achieve the best obstacle avoidance effect.

[0512] (2) Autonomous obstacle avoidance experiment during straight trajectory tracking.

[0513] First, a vertical reference line is set between the starting point (0,0) and the target point (0,5) so that the robot can follow this line. At the coordinate position (0,2), an obstacle with a radius of approximately 0.3m is placed so that the robot can make tracking and obstacle avoidance weighted decisions based on the obstacle information to achieve autonomous obstacle avoidance.

[0514] During operation, the robot first moves along the vertical reference track. When it encounters an obstacle, it automatically adjusts its direction, turns left, and moves along the tangent of the obstacle. When it is about 0.5 meters away from the center of the obstacle, it bypasses the obstacle and then returns to the vertical reference track, finally reaching the target location accurately.

[0515] exist Figure 33 In (a), the robot first moves about 1 meter along the vertical reference track. Then, the lidar sensor detects an obstacle ahead, and the robot automatically adjusts its direction, turning left and moving along the tangent of the obstacle. When it is about 0.5 meters from the center of the obstacle, it bypasses it, then automatically turns and moves about 4.5 meters from the starting point in the y-direction to return to the vertical reference track, finally accurately reaching the target location. The pose errors of the robot's x, y, and θ degrees of freedom during the entire obstacle avoidance process are as follows: Figure 27 As shown: the dotted line represents the pose error in the x-direction, the straight line represents the pose error in the y-direction, and the dashed line represents the pose error at angle θ.

[0516] according to Figure 33(b) During the initial movement of the wheeled mobile robot, it is only affected by ground friction, and the errors in the three degrees of freedom (x, y, and θ) fluctuate within a small range. At the 5-second mark, due to an obstacle ahead, the robot automatically adjusts its posture and turns left to avoid it, with a maximum offset of approximately 0.45 meters in the x-direction. Subsequently, the posture error gradually decreases to within ±0.2 meters; the maximum offset in the y-direction is approximately 0.5 meters, and the posture error gradually decreases to within ±0.15 meters. The robot's angle error θ changes when it encounters an obstacle at the 5-second mark, with a maximum variation range of approximately -1 rad to 0.5 rad, gradually decreasing to within ±0.25 rad after 10 seconds once the robot has completed obstacle avoidance. The robot moves essentially along the desired trajectory, demonstrating that the novel fractional-order PID control method has excellent control performance for the robot's linear obstacle avoidance.

[0517] (4) Autonomous obstacle avoidance experiment during curve trajectory tracking.

[0518] In autonomous obstacle avoidance experiments, the equation of the curved trajectory can be expressed as follows:

[0519]

[0520] In this experiment, the initial position of the camera was set to (0,1), and three obstacles with a radius of approximately 0.1m were set at coordinate system heights of (0.8,2.5), (-0.35,3.5), and (-1,3). Based on this, a tracking and obstacle avoidance weighted decision-making method based on obstacle data was adopted to track the robot's curved trajectory and conduct autonomous obstacle avoidance experiments in order to obtain more accurate results.

[0521] Plotted from experimental data Figure 34 ( a The image shows the robot's actual trajectory, represented by solid lines (standard trajectory) and dashed lines (actual trajectory). The image illustrates that the robot initially makes errors, but over time it automatically converges to the curved trajectory, automatically avoiding obstacles and returning to the curved trajectory after bypassing them until the entire motion is complete. The pose error during the entire motion is shown in the figure. Figure 34 As shown in (b).

[0522] like Figure 34As shown in (b), when the mobile robot first begins its movement, its pose error converges quickly due to the absence of obstacles. The waveform in the figure shows that during the subsequent movement, at the 15-second mark, the robot encounters its first obstacle, resulting in pose errors of approximately 0.2 meters in both the x and y directions, and an angle error θ of approximately 0.9 rad. The robot then completes its first obstacle avoidance maneuver and returns to its standard trajectory. Around the 28-second mark, the robot performs a second obstacle avoidance maneuver, with pose errors of approximately 0.25 meters in both the x and y directions, and an angle error θ reaching approximately 1 rad. Around the 43-second mark, the robot performs a third obstacle avoidance maneuver. Due to the accumulation of errors from the previous two maneuvers, the pose error in the y direction reaches 0.3 meters, and the angle error θ reaches approximately 1.25 rad. The robot then completes the obstacle avoidance maneuver and returns to its original curved trajectory. Due to ground friction and the loosening of the robot's sensors, the pose error fluctuates within a small range. The experiment demonstrates that the fractional-order PID control method can effectively achieve continuous autonomous obstacle avoidance in the curved motion of a wheeled mobile robot, exhibiting excellent control performance.

[0523] In this embodiment, a test control system for a wheeled robot with dual-wheel differential speed was first established, and practical tests were conducted on this test platform. Experimental results show that by applying the novel fractional-order PID trajectory tracking method provided in this embodiment to the wheeled robot control system, the fractional-order PID control technology using Lyapunov equations can not only effectively control the forward and turning speeds of the mobile robot, but also has good anti-interference capabilities. Furthermore, it can make weighted decisions on tracking and obstacle avoidance based on robot information, thereby improving the control efficiency of the mobile robot. It can effectively control both straight and curved trajectories.

[0524] In straight-line trajectory control, the robot can maintain stable straight-line movement with a pose error within ±0.05 meters and converge quickly within 4.5 seconds. In curved trajectory control, the robot can stably complete curved movement according to the set trajectory with a maximum pose deviation of no more than 0.05 meters and converge quickly in about 4 seconds. In straight-line and curved obstacle avoidance experiments, the mobile robot can also successfully avoid obstacles and return to its original trajectory until the entire process is completed. The distance to the center of the obstacle does not exceed 0.5 meters throughout the process, the obstacle avoidance time does not exceed 10 seconds each time, and the pose error curve can always be effectively converged. This proves the effectiveness and feasibility of the method presented in this embodiment.

[0525] In summary, this embodiment addresses the current issues of low control precision and efficiency in the motion control of wheeled mobile robots. It delves into the motion control and autonomous obstacle avoidance technology of a two-wheeled differential wheeled mobile robot, aiming to achieve better control precision and efficiency during movement. The main focus is on the following aspects:

[0526] (1) By analyzing the mathematical characteristics of the two-wheeled differential wheeled mobile robot, the robot error kinematic model and dynamic model were established. Based on the Lyapunov equation and finite time theory, a novel fractional-order PID trajectory tracking control method was designed. The control method was compared with the ordinary PID control method in the straight line and curve domains. In the straight trajectory control, the fractional-order PID control method can significantly reduce the robot's yaw rate and lateral acceleration. The yaw rate was reduced by 23.6%, the lateral acceleration was reduced by 22.1%, and the side slip angle overshoot was reduced by 31.4%. Under the same travel distance, the lateral offset of the robot under the fractional-order PID control automatic driving system was significantly reduced by 29.5% compared with the robot under PID control.

[0527] (2) Under curve trajectory control, the maximum angular velocity output by the fractional-order PID control algorithm is reduced by 35% compared to the PID control algorithm, and the control error can basically converge smoothly. The distance deviation correction time under fractional-order PID control is improved by 1.5 seconds compared to PID control, and the angle deviation correction time is improved by 2.5 seconds. This shows that the driving stability and tracking ability of the mobile robot system have been significantly improved. This proves that the designed control method has a faster response speed and more accurate control effect.

[0528] (3) An active obstacle avoidance controller based on fractional-order PID control theory is proposed. By employing an anti-disturbance dynamic reference trajectory generation method based on a disturbance observer, it can effectively help the mobile robot automatically track target points and effectively avoid obstacles on the road. The robot can quickly return to its original trajectory within 10 seconds after obstacle avoidance. During straight-line obstacle avoidance, the obstacle avoidance pose error generated along the X and Y axes does not exceed ±0.5 m. The linear velocity change is within 1.5 m / s, and the angular velocity change is within 2.5 rad / s. On curved obstacle avoidance, the obstacle avoidance pose error generated along the X and Y axes is within 0 to 0.5 m. The linear velocity change is within 2 m / s, and the angular velocity change is within 3 rad / s. Autonomous obstacle avoidance of the mobile robot has been successfully achieved.

[0529] (4) A test platform for a wheeled mobile robot with dual-wheel differential speed was established to verify the control effect of the fractional-order PID control method on the wheeled mobile robot. Experimental results show that in straight-line trajectory control, the robot can maintain stable straight-line travel with the pose error within ±0.05 meters and can converge quickly within 4.5 seconds. In curved trajectory control, the robot can stably complete curved travel according to the set trajectory with the maximum pose deviation not exceeding 0.05 meters and can converge quickly in about 4 seconds. In straight-line and curved obstacle avoidance experiments, the mobile robot can also successfully avoid obstacles and return to the original trajectory until the entire process is completed. The distance from the center of the obstacle does not exceed 0.5 meters throughout the process, the obstacle avoidance time does not exceed 10 seconds each time, and the pose error curve can always be effectively converged.

[0530] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the product form and style of the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the technical solution of the present invention. As long as the content does not depart from the technical solution of the present invention, it still falls within the patent scope of the technical solution of the present invention.

Claims

1. A method for trajectory tracking control and obstacle avoidance of a wheeled robot, characterized in that, Includes the following steps:

1. Establish the robot error kinematic model and dynamic model, combine the two and divide them into second-order and third-order cascaded subsystems, construct fractional-order PID control models respectively, and build a dynamic fractional-order PID controller to control the robot's linear velocity and angular velocity. The fractional-order PID control model of the second-order subsystem is as follows: ； in, Indicates the operating status. To control the input, It is a constant matrix. It is a continuous nonlinear function. External interference; The fractional-order PID control model of the third-order subsystem is as follows: ； in, , , All are positive numbers; 2. Based on the data variables provided by the subsystem, establish control equations and construct a finite-time trajectory tracking controller to track the robot's linear and angular velocities in real time. Third, establish an active obstacle avoidance controller with obstacle avoidance compensation and a trajectory controller with disturbance compensation, so that the robot can automatically track the target and avoid obstacles; The control equation for the active obstacle avoidance controller is: ； The active obstacle avoidance controller includes an obstacle avoidance compensator for obstacle avoidance compensation, as follows: ； in, This indicates the difference between the robot's online velocity direction and the repulsive force direction. All are positive numbers. This represents the repulsive force exerted on the robot by obstacles during its operation. The control equation for the trajectory controller is: ； ； in, All are positive numbers; , ; , , , All are positive numbers; This indicates the robot's position in polar coordinates; Fourth, a fractional-order PID controller is used for joint control to achieve joint control of robot trajectory control and obstacle avoidance.

2. The wheeled robot trajectory tracking control and obstacle avoidance method according to claim 1, characterized in that, In step one, the robot error kinematic model is as follows: ； in, This indicates the robot's current orientation and position. This indicates the expected orientation and position of the robot. This indicates the difference in posture and orientation between the current robot and the expected robot.

3. The wheeled robot trajectory tracking control and obstacle avoidance method according to claim 1, characterized in that, In step one, the robot's dynamic model is as follows: ； in, The control range is , The control range is , The control range is , The control range is .

4. The wheeled robot trajectory tracking control and obstacle avoidance method according to claim 1, characterized in that, In step two, the control equations for the finite-time trajectory tracking controller are: ； ； ； ； 。 5. The wheeled robot trajectory tracking control and obstacle avoidance method according to claim 1, characterized in that, In a repulsive potential field, the gradient remains negative throughout, and its geometric representation is as follows: ； The repulsive potential field is as follows: ； in, For the gain of repulsive force, For robots and the first A distance between obstacles.

6. A wheeled robot trajectory tracking control and obstacle avoidance system, characterized in that, include: The trajectory control unit is used to establish the robot's error kinematic model and dynamic model, combine the two and divide them into second-order and third-order cascaded subsystems, construct fractional-order PID controllers respectively, and thus build a dynamic fractional-order PID controller to control the robot's linear velocity and angular velocity. The fractional-order PID control model of the second-order subsystem is as follows: ； in, Indicates the operating status. To control the input, It is a constant matrix. It is a continuous nonlinear function. External interference; The fractional-order PID control model of the third-order subsystem is as follows: ； in, , , All are positive numbers; The trajectory tracking unit is used to establish control equations based on the data variables provided by the subsystem, construct a finite-time trajectory tracking controller, and perform real-time tracking of the robot's linear and angular velocities. The obstacle avoidance unit is used to establish an active obstacle avoidance controller with obstacle avoidance compensation and a trajectory controller with disturbance compensation, enabling the robot to automatically track targets and avoid obstacles. The control equation for the active obstacle avoidance controller is: ； The active obstacle avoidance controller includes an obstacle avoidance compensator for obstacle avoidance compensation, as follows: ； in, This indicates the difference between the robot's online velocity direction and the repulsive force direction. All are positive numbers. This represents the repulsive force exerted on the robot by obstacles during its operation. The control equation for the trajectory controller is: ； ； in, All are positive numbers; , ; , , , All are positive numbers; This indicates the robot's position in polar coordinates; The joint control unit is used to perform joint control using a fractional-order PID controller, thereby achieving joint control of robot trajectory control and obstacle avoidance.

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