Mechanical arm trajectory tracking method combining state observer and improved sliding mode

Abstract

The invention relates to a mechanical arm trajectory tracking method combining a state observer and an improved sliding mode, and belongs to the technical field of mechanical arm trajectory tracking. The mechanical arm trajectory tracking method comprises the steps that S1, a mechanical arm dynamic model is constructed according to related parameters of a mechanical arm; s2, designing an extended state observer; s3, designing an improved superhelix sliding mode surface and an approaching law; s4, trajectory planning is conducted on the mechanical arm through a Cartesian interpolation method; s5, according to the mechanical arm dynamic model, based on the improved super-spiral sliding mode surface and the reaching law thereof, a trajectory tracking control law is determined in combination with an extended state observer; s6, a simulation model of combination of the improved super-spiral sliding mode controller combined with the expansion state observer and the mechanical arm is built; and S7, control parameters are selected, and mechanical arm trajectory tracking control is conducted. The controller provided by the invention can effectively suppress buffeting on the premise of ensuring that a system can reach a stable state, has relatively strong robustness to external disturbance, and ensures trajectory tracking of an end effector of the mechanical arm.

Description

Technical Field

[0001] The present invention relates to the technical field of robot arm trajectory tracking, and in particular to a robot arm trajectory tracking method combining a state observer with an improved sliding mode. Background Art

[0002] The core goal of robot trajectory control is to design a suitable control method that enables the robot to complete a specific task along a predetermined trajectory. Specifically, the robot needs to move accurately from the starting position to the target position and execute motion along the predetermined trajectory, while ensuring accuracy, smoothness, and good dynamic performance during the motion process. In the early stages of the development of robotics, due to the relatively simple tasks of the robot and the single application scenario, PID control was widely used due to its simple structure and easy implementation. However, PID control gradually exposed its limitations of insufficient precision and slow response speed in complex tasks. This prompted researchers to begin exploring more advanced control methods.

[0003] As a nonlinear control technology, sliding mode control has become a highly sought-after control strategy in engineering applications in recent years due to its outstanding anti-interference capabilities, robustness, and reliability, enabling it to effectively address nonlinear system issues. First-order sliding mode controllers, with their low number of parameters and simple structure, offer the advantages of easy parameter adjustment, along with mature controller design schemes and stability analysis methods. Consequently, they have been widely used in robotic arm control. However, the discontinuous switching characteristics of sliding mode control can lead to high-frequency chattering in the system. This chattering phenomenon primarily arises from the fact that when the system's state trajectory approaches the sliding mode, the actual motion process involves continuously crossing both sides of the sliding surface before ultimately reaching an equilibrium point, causing the robotic arm system to oscillate.

[0004] To address the above issues, an improved super-helical sliding mode control method is proposed. This method introduces a high-order sliding surface design and replaces the switching function in traditional super-helical sliding mode control with a continuous reaching law. This method can achieve finite-time convergence while reducing the high-frequency jitter amplitude of the control input. Furthermore, an optimization strategy combining an extended state observer is proposed: the total disturbance of the system (including model uncertainty, external disturbances, and friction nonlinearity) is estimated in real time through the ESO, and the estimated value is fed back into the control law for compensation. This method significantly improves the system's anti-disturbance capability. Summary of the Invention

[0005] (1) Technical problems solved

[0006] In view of the shortcomings of the prior art, the present invention provides a robot arm trajectory tracking method combining a state observer with an improved sliding mode, which solves the problems raised in the above background technology.

[0007] (2) Technical solution

[0008] In order to achieve the above-mentioned purpose, the present invention specifically adopts the following technical solutions:

[0009] A robot arm trajectory tracking method combining a state observer and an improved sliding mode includes the following steps:

[0010] S1. Construct a dynamic model of the manipulator based on the manipulator's related parameters, DH parameter method and Lagrangian method;

[0011] S2. Design an extended state observer based on the dynamic model of the manipulator;

[0012] S3. Based on the dynamic model of the manipulator, an improved super-helical sliding surface and reaching law are constructed;

[0013] S4, using Cartesian interpolation method to plan the trajectory of the robot arm;

[0014] S5. According to the dynamic model of the manipulator, the trajectory tracking control law is determined based on the improved super-helical sliding surface and its reaching law, combined with the extended state observer;

[0015] S6. Build a simulation model of the improved super-helical sliding mode controller combined with the extended state observer and the robotic arm;

[0016] S7. Select control parameters and perform robot arm trajectory tracking control.

[0017] Furthermore, in S1, the following constraints are imposed on the manipulator: first, all mechanical structures of the manipulator are assumed to be rigid bodies, that is, the influence of the deformation of the manipulator structure on the motion accuracy is not considered; on this basis, the system model of the n-DOF robot is determined, the specific DH parameters of each joint of the manipulator are obtained, and an accurate DH parameter table is established; then, the Lagrangian dynamics method is used to model the dynamic characteristics of the manipulator system, and the dynamics model of the system is obtained:

[0018]

[0019] in , represent the positive defined moment of inertia, Coriolis centripetal force and gravitational force respectively. , and Represent the joint angular position, angular velocity and angular acceleration respectively. is the control input torque, and d is the disturbance in the robotic system.

[0020] Furthermore, in S2, an extended state observer is designed, specifically including:

[0021] Defining state variables , , 、 、 、 As an intermediate variable used to simplify the formula, the dynamic equation of the manipulator should be written in the following state space form:

[0022]

[0023] in, .

[0024] Define the state observation variable value as , , ,in, is the joint number of the robotic arm, is the joint angle of the system Estimates, is the angular velocity of the joints of the system Estimates, is an estimate of the total disturbance of the system, for The instantaneous rate of change, that is, the speed at which the estimated output value changes, for The instantaneous rate of change, that is, the acceleration or trend of the estimated "speed" itself, for The instantaneous rate of change of the total disturbance is estimated, and the expression of the extended state observer is designed as:

[0025]

[0026] in , is the robot arm joint position error vector is the linear interval threshold, for The nonlinear exponent of the function.

[0027] Furthermore, in S3, based on the dynamic model of the manipulator, an improved superhelical sliding mode surface and reaching law are constructed, specifically including:

[0028] Determining a joint angle tracking error and an angular velocity tracking error during tracking control of the manipulator based on a dynamic system model of the manipulator to generate a joint angle tracking error set and an angular velocity tracking error set;

[0029] The super-helical sliding surface is constructed based on the joint angle tracking error set and the angular velocity tracking error set. as follows:

[0030]

[0031] in, is the sliding surface parameter, is the error between the desired angle and the actual angle of the robot arm, is the error between the desired angular velocity of the robot and its actual angular velocity.

[0032] In order to alleviate the chattering phenomenon in the system, the traditional super-helical sliding mode control is improved, the switching function is replaced by a hyperbolic function, and a hyperbolic function-based The super spiral sliding mode controller is designed and improved. The reaching law of the super spiral sliding mode control is as follows:

[0033]

[0034]

[0035] Where, is the derivative of the sliding surface, 、 is the parameter to be designed, satisfying .

[0036] Furthermore, in S5, according to the dynamic model of the manipulator, based on the improved super-helical sliding mode surface and its reaching law, combined with the extended state observer, the trajectory tracking control law is determined as follows:

[0037]

[0038] in, is the actual angular acceleration of the robotic arm, is the desired angular acceleration of the manipulator, is the actual angular acceleration of the robotic arm, is the desired angular acceleration of the robot arm.

[0039] Furthermore, in S4, S6, and S7, the desired trajectory of the robotic arm is obtained by the Cartesian space interpolation method, and a superhelical sliding mode controller combined with an extended state observer is constructed to perform trajectory tracking control on the controlled object robotic arm; wherein the desired trajectory position information is input and converted into a desired angle signal through the forward kinematics inverse solution, and by improving the control of the superhelical sliding mode controller and the extended state observer, the robotic arm can better track the input signal and run the desired trajectory.

[0040] (3) Beneficial effects

[0041] Compared with the prior art, the present invention provides a robot arm trajectory tracking method that combines a state observer with an improved sliding mode, which has the following beneficial effects:

[0042] The difference between the present invention and the traditional super-helical sliding mode control is that the approach rate of the super-helical sliding mode control is improved, and an extended state observer is combined to apply it to the trajectory tracking control of the robot arm.

[0043] By introducing an extended state observer, this method can estimate the system state in real time, thereby eliminating state observation errors and effectively suppressing the system's chattering phenomenon. At the same time, the improved superhelical sliding mode control method designs a continuous control law and uses the hyperbolic function stan(s) to replace the sign function sign(s) in the traditional superhelical algorithm. This weakens the chattering problem existing in traditional sliding mode control while achieving more accurate trajectory tracking of the robotic arm. It avoids the discontinuous control input in traditional sliding mode control and further improves the smoothness and control accuracy of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 Flowchart of a robot trajectory tracking method using an improved super-helical sliding mode control method combined with an extended state observer according to an embodiment of the present invention;

[0045] Figure 2 A schematic diagram of the structure of the improved super-helical sliding mode control robot arm trajectory tracking method combined with the extended state observer provided by the present invention;

[0046] Figure 3 A comparison diagram of the improved superhelical sliding mode control surface and the superhelical sliding mode control surface without the extended state observer provided by the present invention;

[0047] Figure 4 A comparison diagram of the angular errors of the robot arm trajectory tracking under the improved super-helical sliding mode control without the extended state observer provided by the present invention and the super-helical sliding mode control;

[0048] Figure 5 The improved super-helical sliding mode trajectory tracking curve diagram provided by the present invention is combined with the extended state observer;

[0049] Figure 6 A comparison diagram of the angular error of the trajectory tracking of the improved super-helical sliding mode controlled manipulator with and without the extended state observer provided by the present invention;

[0050] Figure 7 A comparison diagram of the angular error of the trajectory tracking of the improved super-helical sliding mode controlled manipulator with and without the extended state observer after adding a disturbance signal of 25% of the input value provided by the present invention. DETAILED DESCRIPTION

[0051] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0052] Example

[0053] An embodiment of the present invention provides a robot arm trajectory tracking method combining a state observer and an improved sliding mode, comprising the following steps:

[0054] S1. Establish the dynamics and kinematics model of the robotic arm;

[0055] The following constraints are imposed on the robotic arm. First, all mechanical structures of the robotic arm are assumed to be rigid bodies, that is, the influence of the deformation of the robotic arm structure on the motion accuracy is not considered. On this basis, the system model of the n-DOF robot is determined, the specific DH parameters of each joint of the robotic arm are obtained, and an accurate DH parameter table is established. Subsequently, the Lagrangian dynamics method is used to model the dynamic characteristics of the robotic arm system, and the dynamics model of the system is obtained:

[0056]

[0057] in , represent the positive defined moment of inertia, Coriolis centripetal force and gravitational force respectively. , and Represent the joint angular position, angular velocity and angular acceleration respectively. is the control input torque, and d is the disturbance in the robotic system.

[0058] S2. Based on the dynamic system model of the robot arm, an extended state observer is designed, which includes:

[0059] Defining state variables , , 、 、 、 As an intermediate variable used to simplify the formula, the dynamic equation of the manipulator should be written in the following state space form:

[0060]

[0061] in, .

[0062] Define the state observation variable value as , , ,in, is the joint number of the robotic arm, is the joint angle of the system Estimates, is the angular velocity of the joints of the system Estimates, is an estimate of the total disturbance of the system, for The instantaneous rate of change, that is, the speed at which the estimated output value changes, for The instantaneous rate of change, that is, the acceleration or trend of the estimated "speed" itself, for The instantaneous rate of change of is the estimated rate of change of the total disturbance, and the expression of the extended state observer is designed as:

[0063]

[0064] in , is the robot arm joint position error vector, is the linear interval threshold, for The nonlinear exponent of the function.

[0065] S3. Based on the dynamic model of the manipulator, an improved super-helical sliding surface and reaching law are constructed;

[0066] Get the desired angle of the robot arm With actual angle difference and its derivative, the desired angular velocity of the manipulator The actual angular velocity difference Constructing superhelical sliding surface as follows:

[0067]

[0068] in, is the sliding surface parameter, is the error between the desired angle and the actual angle of the robot arm, is the error between the desired angular velocity of the robot and its actual angular velocity.

[0069] Using hyperbolic function The super spiral sliding mode controller is designed and improved. The reaching law of the super spiral sliding mode control is as follows:

[0070]

[0071]

[0072] Where, is the derivative of the sliding surface, is the parameter to be designed, satisfying .

[0073] S4, using Cartesian interpolation method to plan the trajectory of the robot arm;

[0074] To achieve precise path control during Cartesian trajectory planning for a robotic arm, an interpolation algorithm is required to calculate the coordinates of intermediate points, or interpolation points. The position information for these intermediate points is generated based on the relationship between the trajectory's starting and ending points. With these intermediate points, the position of the robotic arm's end point at each intermediate point is then converted into the corresponding joint angles. By precisely controlling the changes in the joint angles, the robotic arm can accurately move to each target point along the predetermined trajectory.

[0075] Common interpolation methods include linear interpolation and circular interpolation. In Cartesian space linear interpolation, given the known positions (position and orientation) of the starting and ending points of a line, the positions of the interpolated points along the line can be calculated. This process requires not only a precise interpolation algorithm but also the feasibility and accuracy of the robot's motion, ensuring that the robot can move smoothly and accurately along the planned trajectory to meet the requirements of task execution.

[0076] Assume that a straight line starts and ends at two points M ( ), N ( ) is the coordinate pose relative to the base coordinate system. is the required speed along the straight line, For the interpolation time interval, we can find the length of the straight line , Distance within the interval and the interpolation number as follows:

[0077]

[0078]

[0079]

[0080] Then the increments of each axis of adjacent interpolation points and the coordinate values ​​of each axis interpolation point are as follows:

[0081] Increment of adjacent interpolation points:

[0082]

[0083]

[0084]

[0085] in, 、 、 are the increments of horizontal, vertical and vertical coordinates respectively. .

[0086] Coordinates of interpolation points:

[0087]

[0088]

[0089]

[0090] in 、 、 are the horizontal, vertical and vertical coordinates of the interpolation respectively.

[0091] To improve computational efficiency, coordinate system transformation techniques are also required. First, a new rectangular coordinate system is established on the plane where the arc lies. Within this new coordinate system, the coordinates of each interpolated point on the arc are calculated. Once the calculations are complete, the coordinates of these intermediate interpolated points are converted back to the original Cartesian space coordinate system using a transformation matrix, completing the path planning.

[0092] In practice, three points can define the trajectory of a circular arc. Suppose the robot's end effector moves from a starting position P1 (x1, y1, z1), passes through an intermediate point P2 (x2, y2, z2), and finally reaches an end point P3 (x3, y3, z3). If these three points are not collinear, then they must define a circular path from P1 through P2 to P3. During arc trajectory planning, the algorithm typically follows these steps:

[0093] (1) Solve for the center and radius:

[0094] Calculate the arc's center point P0 (X0, Y0, Z0) and radius through geometric deduction or analytical methods. The center point is the center of the arc path, while the radius determines the size of the arc.

[0095] (2) Establish a new spatial coordinate system:

[0096] Create a new rectangular coordinate system on the plane where the arc lies. This step simplifies the calculation of the arc trajectory. Then, calculate the mapping relationship between this new coordinate system and the reference coordinate system for subsequent data conversion.

[0097] (3) Calculate the total angle of the arc:

[0098] Using the known coordinates of the starting point P1, the middle point P2, and the end point P3, we can find the starting and ending angles of the arc. The total angle of the arc is an important parameter of the arc, which determines the starting and ending positions of the arc.

[0099] (4) Calculate the coordinates of the interpolation points:

[0100] Using trigonometric relationships, the coordinates of each interpolated point on the arc are calculated based on the total angle of the arc and the parameters of the center and radius. The coordinates of these interpolated points are calculated based on the new coordinate system and then need to be converted back to the original Cartesian coordinate system using the coordinate transformation matrix.

[0101] Through this method, the robotic arm can not only accurately control the arc trajectory, but also efficiently plan the path to ensure the smoothness and precision of the movement process.

[0102] S5. According to the dynamic model of the manipulator, the trajectory tracking control law is determined based on the improved super-helical sliding surface and its reaching law, combined with the extended state observer;

[0103] It is known that the variables that control the motion trajectory of the robot arm are the angular velocity and angular acceleration of each joint. Combined with the robot arm dynamics model, the trajectory tracking control law is determined as follows:

[0104]

[0105] in, is the actual angular velocity of the robotic arm, is the actual angular acceleration of the robotic arm, is the desired angular acceleration of the manipulator, is the actual angular acceleration of the robotic arm, is the desired angular acceleration of the robot arm.

[0106] S6. Build a simulation model of the improved super-helical sliding mode controller combined with the extended state observer and the robotic arm;

[0107] An extended state observer and a super-helical sliding mode controller with improved control law are constructed, and a simulation model is established to perform trajectory tracking control on the controlled object robotic arm; the desired trajectory position information is input and converted into the desired angle signal through the inverse solution of forward kinematics. The improved super-helical sliding mode controller and the extended state observer are used for interference compensation, so that the robotic arm can better track the input signal and run the desired trajectory.

[0108] S7, select control parameters and perform trajectory tracking control;

[0109] The extended state observer parameters in step S2 are determined through engineering experiments. ,satisfy ,in, The smaller it is, the lower the nonlinear gain is, the smoother the response is, and overshoot is avoided. The larger the value, the higher the nonlinear gain, which speeds up the convergence speed when the error is large, but may cause oscillation. To satisfy , The smaller it is, the smaller the linear region is, the earlier the nonlinear segment intervenes, and the faster the convergence of large errors is. The larger the value, the wider the linear region, which suppresses high-frequency noise, but may reduce dynamic performance.

[0110] Determine the sliding surface parameters in step S3 through engineering tests. , Improved super-helical sliding mode control reaching law parameters 、 satisfy , The selection principle is to ensure that the system state point has a faster approach speed when it is far away from the switching surface, avoiding too small a speed causing too slow approach speed and too large a speed causing violent vibration. By comparing and observing the experimental data, an appropriate selection is made to make the system approach the switching surface at an appropriate speed.

[0111] An embodiment of the present invention provides a robot arm trajectory tracking method that combines a state observer with an improved sliding mode. The desired trajectory input of the robot arm is obtained through the Cartesian interpolation method, and then controlled by an improved superhelical sliding mode controller. The trajectory tracking of the robot arm is achieved by combining the interference compensation of the extended state observer.

[0112] By combining an extended state observer with an improved superhelical sliding mode controller, the present invention achieves trajectory tracking control of the robotic arm in the presence of modeling errors. Compared with traditional superhelical sliding mode control, it reduces system chattering, improves the tracking accuracy of the robotic arm trajectory, and verifies the anti-disturbance performance of the extended state observer when interference is added.

[0113] Finally, it should be noted that the above descriptions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art will be able to modify the technical solutions described in the aforementioned embodiments or substitute equivalents for some of the technical features. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention shall be included within the scope of protection of the present invention.

Claims

1. A robot arm trajectory tracking method combining a state observer and an improved sliding mode, characterized in that: The following steps are involved: S1. Construct a dynamic model of the manipulator based on the manipulator's related parameters, DH parameter method and Lagrangian method; S2. Design an extended state observer based on the dynamic model of the manipulator; S3. Based on the dynamic model of the manipulator, an improved super-helical sliding surface and reaching law are constructed; S4, using Cartesian interpolation method to plan the trajectory of the robot arm; S5. According to the dynamic model of the manipulator, the trajectory tracking control law is determined based on the improved super-helical sliding surface and its reaching law, combined with the extended state observer; S6. Build a simulation model of the improved super-helical sliding mode controller combined with the extended state observer and the robotic arm; S7. Select control parameters and perform robot arm trajectory tracking control.

2. The robot arm trajectory tracking method combining a state observer and an improved sliding mode according to claim 1, characterized in that: In S1, the following constraints are imposed on the manipulator. First, all mechanical structures of the manipulator are assumed to be rigid bodies, that is, the influence of the deformation of the manipulator structure on the motion accuracy is not considered. On this basis, the system model of the n-DOF robot is determined, the specific DH parameters of each joint of the manipulator are obtained, and an accurate DH parameter table is established. Subsequently, the Lagrangian dynamics method is used to model the dynamic characteristics of the manipulator system, and the dynamics model of the system is obtained: ； in , represent the positive definition of inertia moment, Coriolis centripetal force and gravitational force respectively; , and represent joint angular position, angular velocity, and angular acceleration, respectively; is the control input torque, and d is the disturbance in the robotic system.

3. A robot arm trajectory tracking method combining a state observer and an improved sliding mode according to claim 1, characterized in that: In S2, an extended state observer is designed, specifically including: Defining state variables , , 、 、 、 As an intermediate variable used to simplify the formula, the dynamic equation of the manipulator should be written in the following state space form: ； in, ; Define the state observation variable value as , , ,in, is the joint number of the robotic arm, is the joint angle of the system Estimates, is the angular velocity of the joints of the system Estimates, is an estimate of the total disturbance of the system, for The instantaneous rate of change, that is, the speed at which the estimated output value changes, for The instantaneous rate of change, that is, the acceleration or change trend of the estimated "speed" itself, for The instantaneous rate of change of is the estimated rate of change of the total disturbance, and the expression of the extended state observer is designed as: ； in , is the robot arm joint position error vector, is the linear interval threshold, for The nonlinear exponent of the function.

4. A robot arm trajectory tracking method combining a state observer and an improved sliding mode according to claim 1, characterized in that: In S3, based on the dynamics model of the manipulator, constructing an improved super-helical sliding mode surface and reaching law specifically includes: Determining the joint angle tracking error and the angular velocity tracking error during the tracking control process of the manipulator based on the dynamic system model of the manipulator to generate a joint angle tracking error set and an angular velocity tracking error set; The super-helical sliding surface is constructed based on the joint angle tracking error set and the angular velocity tracking error set. as follows: ； in, is the sliding surface parameter, is the error between the desired angle and the actual angle of the robot arm, is the error between the desired angular velocity of the robot and the actual angular velocity; In order to alleviate the chattering phenomenon in the system, the traditional super-helical sliding mode control is improved, the switching function is replaced by a hyperbolic function, and a hyperbolic function-based The super spiral sliding mode controller is designed and improved; the reaching law of the super spiral sliding mode control is as follows: ； ； Where, is the derivative of the sliding surface, 、 is the parameter to be designed, satisfying .

5. The robot arm trajectory tracking method combining a state observer and an improved sliding mode according to claim 1, characterized in that: In S5, based on the improved super-helical sliding surface and its reaching law, the trajectory tracking control law is determined in combination with the extended state observer as follows: ； in, is the actual angular acceleration of the robotic arm, is the desired angular acceleration of the manipulator, is the actual angular acceleration of the robotic arm, is the desired angular acceleration of the robot arm.

6. The robot arm trajectory tracking method combining a state observer and an improved sliding mode according to claim 1, characterized in that: In the above-mentioned S4, in the Cartesian space trajectory planning stage, an interpolation algorithm is used for parametric modeling based on the position and posture changes of the end effector from the starting point to the end point in the three-dimensional space; the Cartesian space trajectory points are converted to the joint space coordinate system through the inverse kinematics solution algorithm, thereby ensuring that the end effector can accurately reproduce the predetermined trajectory.

7. The robot arm trajectory tracking method combining a state observer and an improved sliding mode according to claim 1, characterized in that: In S6 and S7, a super-helical sliding mode controller with improved approach rate is constructed and applied to the robotic arm system to realize the trajectory tracking control function; by receiving the position parameters of the target trajectory, the corresponding expected joint angle signal is calculated using the inverse kinematics algorithm. The controller combines the extended state observer with the improved super-helical sliding mode control algorithm, and estimates the system disturbance in real time and compensates for it, ensuring that the robotic arm can track the target trajectory with high precision and accurately execute the preset motion path.

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