Trajectory tracking control method for rope-driven manipulator based on adaptive time delay estimation

Abstract

The application discloses a rope-driven mechanical arm trajectory tracking control method based on adaptive time delay estimation and belongs to the technical field of flexible mechanical arm control. The method first establishes the dynamics equation of an n-degree-of-freedom mechanical arm; further, an adaptive time delay estimation term is determined; then, an adaptive time delay estimation controller is determined according to the adaptive time delay estimation term and a sliding surface; finally, the adaptive time delay estimation controller is used to perform trajectory tracking control on the mechanical arm. The method sets the adaptive time delay estimation controller, improves the control performance, reduces the control error, has good adaptive capacity and can adapt to different working conditions of the mechanical arm.

Description

Technical Field

[0001] This application relates to the field of flexible robotic arm control technology, and in particular to a trajectory tracking control method for a rope-driven robotic arm based on adaptive time delay estimation. Background Technology

[0002] With the advent of Industry 4.0, the tasks that robotic arms need to perform are becoming increasingly diverse and complex. Traditional robotic arms cannot meet the operational needs of some special environments, while rope-driven robotic arms have the advantages of being lightweight and having a high load-to-weight ratio, which can meet the operational needs of underwater operations and other special environments. The trajectory tracking control of rope-driven robotic arms aims to precisely adjust the tension and length of each rope to ensure that the robotic arm can track the preset trajectory with excellent dynamic response characteristics during movement. However, because rope-driven robotic arms are easily affected by various factors such as external environmental interference and rope elastic deformation during operation, their trajectory tracking control is relatively difficult.

[0003] Currently, controllers designed using integral sliding mode algorithms have low adaptability and cannot adapt to operational conditions with large disturbances. Robotic arms are often subjected to constant or directional forces during operation, such as gravity when the robotic arm moves only on one side and the impact force of water flow when the robotic arm is operating underwater. Sliding surface designs without integral terms cannot compensate for the effects of constant and directional forces, resulting in insufficient control performance.

[0004] Therefore, there is an urgent need to develop a trajectory tracking control method for rope-driven robotic arms based on adaptive time delay estimation to solve the above problems. Summary of the Invention

[0005] In view of this, this application provides a trajectory tracking control method for a rope-driven robotic arm based on adaptive time delay estimation, which improves control performance, reduces control error, has good adaptability, and can adapt to different working conditions of the robotic arm.

[0006] Specifically, the following technical solutions are included:

[0007] This application provides a trajectory tracking control method for a cable-driven robotic arm based on adaptive time delay estimation, the method comprising:

[0008] Establish the dynamic equations of the n-degree-of-freedom robotic arm;

[0009] Determine the adaptive delay estimation term;

[0010] The adaptive delay estimation controller is determined based on the adaptive delay estimation term and the sliding surface;

[0011] The robotic arm is tracked and controlled based on an adaptive time delay estimation controller.

[0012] In some embodiments, the dynamic equation of a robotic arm with n degrees of freedom is established according to the following formula:

[0013]

[0014] where, is the position vector of the robotic arm joint, is the first-order derivative of the position vector of the robotic arm joint with respect to time, is the second-order derivative of the position vector of the robotic arm joint with respect to time, is the inertia matrix, is the centripetal force and Coriolis force vector, is the gravity vector, is the Coulomb friction and viscous friction vector, is the external disturbance force vector, is the generalized output vector of the controllers of each joint of the robotic arm.

[0015] In some embodiments, determining the adaptive time-delay estimation term includes:

[0016] Rewriting the dynamic equation of the robotic arm according to the following formula:

[0017]

[0018] where, τ , m , , , (t-η) ,

[0023] ,

[0022] , is the generalized output vector of the controllers of each joint of the robotic arm, N is the adaptive control parameter, 0 < N < l, θ is the position vector of the robotic arm joint, is the first-order derivative of the position vector of the robotic arm joint with respect to time, is the second-order derivative of the position vector of the robotic arm joint with respect to time, M is the inertia matrix, C is the centripetal force and Coriolis force vector, G is the gravity vector, F is the Coulomb friction and viscous friction vector;

[0019] Performing adaptive time-delay estimation on H to determine the adaptive time-delay estimation term H (t-η) :

[0020]

[0021] where, τ <00​​​​​​​​​​​​​

[0024] In the formula, c1 and c2 are constant control parameters, c1>0, c2>0, and e represents the control error between the desired rotation angle and the actual rotation angle of the robotic arm joint. Let e ​​be the derivative of e with respect to time t, and let e(0) represent the value of the control error at the initial time. express The value at the initial time, where e = θ d -θ, θ d To enable the robotic arm to track the desired trajectory, For θ d The first derivative with respect to time t.

[0025] In some embodiments, determining an adaptive delay estimation controller based on an adaptive delay estimation term and a sliding surface includes:

[0026] Control the robotic arm to track the desired trajectory and record the trajectory tracking error. The recorded trajectory tracking error is denoted as...

[0027] The adaptive time delay estimation controller with fixed gain is obtained according to the following formula:

[0028]

[0029] In the formula, For θ d The second derivative with respect to time t, Let c3 be the derivative of e with respect to time t, and c4 be the derivative of e with respect to time t. All data are sorted in ascending order, and the mean of the smallest 25% of data is taken as the average. Take the mean of the largest 25% of the data as

[0030] In some embodiments, the method further includes: dynamically adjusting the determined adaptive control parameters according to the following formula:

[0031]

[0032] In the formula, R represents the maximum rate of change of N, and R>0;

[0033] The minimum value of N that allows the robotic arm to operate normally under undisturbed conditions is selected, and this value is used as the reference value for the dynamic adjustment of parameter N, denoted as . The maximum value of N that allows the robotic arm to operate normally under undisturbed conditions is selected, and the threshold proportion of this value is taken as the maximum allowable value of N, denoted as . Calculate N using the following formula. d :

[0034]

[0035] In some embodiments, the dynamically adjusted adaptive delay estimation controller is as follows:

[0036]

[0037] In some embodiments, the method further includes: performing a stability analysis on the dynamically adjusted adaptive delay estimation controller, specifically including:

[0038] According to the following formula (1), set the variable P:

[0039]

[0040] Rewriting formula (1) yields formula (2):

[0041]

[0042] The adaptive delay estimation controller with dynamic adjustment is rewritten as follows, resulting in formula (3):

[0043]

[0044] Based on the rewritten robotic arm dynamics equations and the rewritten adaptive time delay estimation controller with dynamic adjustment, formula (4) is derived:

[0045]

[0046] The error in the time delay estimation is determined to be D = HH. (t-η) ;

[0047] Let H, for e = θ d The derivative of -θ is obtained Substituting D = HH (t-η) Thus, we obtain the following formula (5):

[0048]

[0049] Select the following Lyapunov function formula (6) for... Perform stability analysis:

[0050]

[0051] In some embodiments, for Stability analysis was performed, including:

[0052] The Lyapunov function formula (6) is differentiated with respect to time to obtain the following formula (7):

[0053]

[0054] Substitute the fourth equation in Equation (3) into Equation (7) to obtain Equation (8):

[0055]

[0056] Substitute Equation (5) into Equation (8) to obtain Equation (9):

[0057]

[0058] Transform Equation (9) through the defining formula of sgn(s) to obtain Equation (10):

[0059]

[0060] Where the defining formula of sgn(s) is:

[0061]

[0062] According to Obtain Equation (11):

[0063]

[0064] According to |sgn(s)|≤1, |sat(|s|)|≤1, obtain Equation (12):

[0065]

[0066] When the system composed of the robotic arm and the adaptive time-delay estimation controller is asymptotically stable and |s| converges;

[0067] When 0 < N < 1, Where is the i-th eigenvalue of the matrix I m - N -1 N, and I m represents the m-order identity matrix; the error D of the time-delay estimation is bounded, and it is further obtained that |s| is bounded.

[0068] In some embodiments, after determining the adaptive time-delay estimation controller according to the adaptive time-delay estimation term and the sliding mode surface, the method further includes:

[0069] Perform stability verification on the adaptive time-delay estimation controller.

[0070] The beneficial effects of the technical solution provided by the embodiments of the present application at least include:

[0071] This application provides a trajectory tracking control method for a cable-driven robotic arm based on adaptive time delay estimation. First, the dynamic equations of the robotic arm with n degrees of freedom are established. Then, an adaptive time delay estimation term is determined. Next, an adaptive time delay estimation controller is determined based on the adaptive time delay estimation term and the sliding surface. Finally, trajectory tracking control of the robotic arm is performed based on the adaptive time delay estimation controller. This method improves control performance and reduces control error by setting an adaptive time delay estimation controller. It has good adaptability and can adapt to different working conditions of the robotic arm. Attached Figure Description

[0072] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0073] Figure 1 A flowchart illustrating the trajectory tracking control method for a rope-driven robotic arm based on adaptive time delay estimation provided in this application;

[0074] Figure 2 A schematic diagram of the two-degree-of-freedom rope-driven manipulator in the trajectory tracking control method for rope-driven manipulators based on adaptive time delay estimation provided in this application;

[0075] Figure 3 A comparison diagram showing the effect of the controller in the adaptive time delay estimation-based cable-driven robotic arm trajectory tracking control method provided in this application on controlling the motion trajectory of the robotic arm joint 1 with other controllers;

[0076] Figure 4 A comparison diagram showing the effect of the controller in the adaptive time delay estimation-based cable-driven robotic arm trajectory tracking control method provided in this application on controlling the motion trajectory of the robotic arm joint 2 with other controllers;

[0077] Figure 5 A comparison diagram showing the control error of the controller and other controllers controlling the joint 1 of the robotic arm in the rope-driven robotic arm trajectory tracking control method based on adaptive time delay estimation provided in this application;

[0078] Figure 6 A comparison diagram showing the control error of the controller and other controllers controlling the joint 2 of the robotic arm in the rope-driven robotic arm trajectory tracking control method based on adaptive time delay estimation provided in this application;

[0079] Figure 7A comparison diagram showing the local amplification of the control error of the controller and other controllers controlling the joint 1 of the robotic arm in the rope-driven robotic arm trajectory tracking control method based on adaptive time delay estimation provided in this application;

[0080] Figure 8 A comparison diagram showing the local amplification of the control error of the controller and other controllers controlling the joint 2 of the robotic arm in the rope-driven robotic arm trajectory tracking control method based on adaptive time delay estimation provided in this application;

[0081] Figure 9 A comparison diagram showing the control torque of the controller and other controllers controlling the joint 1 of the robotic arm in the rope-driven robotic arm trajectory tracking control method based on adaptive time delay estimation provided in this application;

[0082] Figure 10 A comparison diagram showing the control torque of the controller and other controllers controlling the joint 2 of the robotic arm in the rope-driven robotic arm trajectory tracking control method based on adaptive time delay estimation provided in this application. Detailed Implementation

[0083] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0084] To make the technical solutions and advantages of this application clearer, the embodiments of this application will be described in further detail below with reference to the accompanying drawings.

[0085] This application provides a trajectory tracking control method for a cable-driven robotic arm based on adaptive time delay estimation, such as... Figure 1 As shown, the method includes:

[0086] Step 101: Establish the dynamic equations of the n-degree-of-freedom robotic arm.

[0087] In some embodiments, the dynamic equations of an n-degree-of-freedom robotic arm are established according to the following formula:

[0088]

[0089] In the formula, The position vector of the robotic arm joint. Let be the first derivative of the position vector of the robotic arm joint with respect to time. is the second derivative of the position vector of the robotic arm joint with respect to time. In the trajectory control of the robotic arm, θ is the current angle value of the rotating joint of the robotic arm, which is generally obtained by reading the output of the angle sensor at the robotic arm joint; is obtained by differentiating θ with respect to time; is obtained by taking the second derivative of θ with respect to time, is the inertia matrix, is the centripetal force and Coriolis force vector, is the gravity vector, is the Coulomb friction and viscous friction vector, is the external disturbance force vector, is the generalized output vector of the controllers of each joint of the robotic arm.

[0090] It should be noted that the above robotic arm dynamics equation summarizes the general form of the robotic arm dynamics model for facilitating the subsequent derivation of the controller. However, the subsequent obtained controller does not contain the above dynamics equation. Therefore, when applying the rope-driven robotic arm trajectory tracking control method based on adaptive time-delay estimation provided by this application, there is no need to perform the robotic arm dynamics modeling work.

[0091] Step 102, determine the adaptive time-delay estimation term.

[0092] By introducing the control parameter with adaptive time-delay estimation in Step 102, the robotic arm is enabled to have the ability to adapt to different operating conditions, including ocean current disturbance changes, load changes, etc. The controller with adaptive control parameters can meet the control requirements of various different operating conditions through adaptive adjustment.

[0093] In some embodiments, Step 102 includes:

[0094] Step 1021, rewrite the robotic arm dynamics equation according to the following formula:

[0095]

[0096] In the formula, τ m is the generalized output vector of the controllers of each joint of the robotic arm, N is the adaptive control parameter, 0 < N < 1, θ is the position vector of the robotic arm joint, is the first derivative of the position vector of the robotic arm joint with respect to time, is the second derivative of the position vector of the robotic arm joint with respect to time, M is the inertia matrix, C is the centripetal force and Coriolis force vector, G is the gravity vector, and F is the Coulomb friction and viscous friction vector.

[0097] Since H in step 1021 is relatively complex and changes over time, the rope-driven robotic arm trajectory tracking control method based on adaptive time delay estimation provided in this application uses adaptive time delay estimation to estimate it, thereby introducing step 1022, so that the robotic arm dynamic equation can be rewritten subsequently. The specific parameters of the robotic arm dynamic equation are not required; the stability of the controller can be proved using only the general form of the robotic arm dynamic equation.

[0098] Step 1022: Perform adaptive time delay estimation on H to determine the adaptive time delay estimation term H. (t-η) :

[0099]

[0100] In the formula, τ m,(t-η) τ represents time t-η m Value, N (t-η) This represents the value of N at time t-η. Let t be the first derivative of the position vector of the robotic arm joint with respect to time, where η is the sampling period.

[0101] Existing adaptive delay estimation does not delay N, but uses N with adaptive delay. (t-η) and Multiplication is performed to better estimate the dynamic characteristics of the robotic arm, and the related stability effect is demonstrated in subsequent steps.

[0102] η represents the sampling period. When using a microcontroller to control the robotic arm, the microcontroller should run some code in a loop. The time required for each loop is the sampling period. This allows the designed H to effectively estimate the dynamic characteristics of the robotic arm without needing to know the robotic arm's dynamic model beforehand. This enables the subsequently designed controller to achieve trajectory tracking of the robotic arm with smaller control parameters, which makes the controller more stable.

[0103] In some embodiments, the sampling period can be 1 to 10 ms.

[0104] In some embodiments, the sliding surface is defined according to the following formula:

[0105]

[0106] In the formula, c1 and c2 are constant control parameters, c1>0, c2>0, and e represents the control error between the desired rotation angle and the actual rotation angle of the robotic arm joint. Let e ​​be the derivative of e with respect to time t, and let e(0) represent the value of the control error at the initial time. express The value at the initial time, where e = θ d -θ, θ d To enable the robotic arm to track the desired trajectory, For θ d The first derivative with respect to time t.

[0107] Using a sliding surface with an integral term can eliminate steady-state errors when a robotic arm tracks a trajectory. In practical applications, robotic arms often repeatedly run fixed trajectories, and an appropriate scoring term can effectively improve the tracking performance of the robotic arm when repeatedly running fixed trajectories. Furthermore, e(0), Add it to the sliding surface to prevent the sliding surface s from becoming too large at the initial moment of the robot arm's operation due to excessive initial error or its derivative, which would cause the controller to output excessive torque. e(0), The influence of the sliding surface s will gradually decrease as the robotic arm moves. After the robotic arm has been running for a period of time, the influence of these two factors can be ignored, and the integral term will be used to determine the effect. It plays a major role.

[0108] Step 103: Determine the adaptive time delay estimation controller based on the adaptive time delay estimation term and the sliding surface.

[0109] Adaptive parameters enable robotic arms to adapt to different operating conditions, including variations in ocean currents and load. Controllers equipped with adaptive control parameters can adjust themselves to meet the control requirements of various operating conditions.

[0110] In some embodiments, step 103 includes:

[0111] Step 1031: Control the robotic arm to track the desired trajectory and record the trajectory tracking error. The recorded trajectory tracking error is denoted as...

[0112] Step 1032: Obtain the adaptive time delay estimation controller with fixed gain according to the following formula:

[0113]

[0114] In the formula, For θ d The second derivative with respect to time t, Let c3 be the derivative of e with respect to time t, and c4 be the derivative of e with respect to time t. All data are sorted in ascending order, and the mean of the smallest 25% of data is taken as the average. Take the mean of the largest 25% of the data as

[0115] In some embodiments, the above two parameters can be adjusted according to the actual situation. Adjustments will be made.

[0116] Control experiments were conducted using fixed-gain time delay estimation control to obtain an adaptive time delay estimation controller with fixed gain, which can serve as a reference for designing an adaptive time delay estimation controller.

[0117] Step 104: Based on the adaptive time delay estimation controller, perform trajectory tracking control on the robotic arm.

[0118] In some embodiments, the method further includes: dynamically adjusting the determined adaptive control parameters according to the following formula:

[0119]

[0120] In the formula, R represents the maximum rate of change of N, and R>0;

[0121] The minimum value of N that allows the robotic arm to operate normally under undisturbed conditions is selected, and this value is used as the reference value for the dynamic adjustment of parameter N, denoted as . The maximum value of N that allows the robotic arm to operate normally under undisturbed conditions is selected, and the threshold proportion of this value is taken as the maximum allowable value of N, denoted as . Calculate N using the following formula. d :

[0122]

[0123] Designing N in the above form effectively avoids controller instability caused by excessively rapid changes in N, ensuring the stability of the obtained adaptive time delay estimation controller. By introducing intermediate variables x and k, a method for calculating the adaptive control parameter N through the sliding surface s is presented, thereby enabling adaptive changes in controller parameters and giving the controller better environmental adaptability.

[0124] Compared to adaptive strategies designed using fuzzy logic algorithms, neural networks, etc., this approach can effectively save the controller's computing resources and is easy to apply to embedded control systems with limited computing power, such as microcontrollers.

[0125] In some embodiments, the threshold ratio can be adjusted according to the reliability requirements of the system consisting of the robotic arm and the adaptive delay estimation controller during actual operation, and can be 80%.

[0126] In some embodiments, the dynamically adjusted adaptive delay estimation controller is as follows:

[0127]

[0128] The controller does not contain the robotic arm dynamics equations of step 101. Therefore, the controller is not based on a dynamics model. When using the controller, it is not necessary to establish a dynamics model for the robotic arm.

[0129] In some embodiments, the method further includes: performing a stability analysis on the dynamically adjusted adaptive delay estimation controller, specifically including:

[0130] According to the following formula (1), set the variable P:

[0131]

[0132] Due to N (t-η) -N and Since it is bounded, P is bounded.

[0133] Rewriting formula (1) yields formula (2):

[0134]

[0135] The adaptive delay estimation controller with dynamic adjustment is rewritten as follows, resulting in formula (3):

[0136]

[0137] Based on the rewritten robotic arm dynamics equations and the rewritten adaptive time delay estimation controller with dynamic adjustment, formula (4) is derived:

[0138]

[0139] The error in the time delay estimation is determined to be D = HH. (t-η) ;

[0140] Let H, for e = θ d The derivative of -θ is obtained Substituting D = HH (t-η) Thus, we obtain the following formula (5):

[0141]

[0142] Select the following Lyapunov function formula (6) for... Perform stability analysis:

[0143]

[0144] In some embodiments, for Stability analysis was performed, including:

[0145] The Lyapunov function formula (6) is differentiated with respect to time to obtain the following formula (7):

[0146]

[0147] Substitute the fourth equation in formula (3) into formula (7) to obtain formula (8):

[0148]

[0149] Substitute formula (5) into formula (8) to obtain formula (9):

[0150]

[0151] Transform formula (9) through the definition formula of sgn(s) to obtain formula (10):

[0152]

[0153] Where the definition formula of sgn(s) is:

[0154]

[0155] According to Obtain formula (11):

[0156]

[0157] According to |sgn(s)|≤1, |sat(|s|)|≤1, obtain formula (12):

[0158]

[0159] When The system composed of the robotic arm and the adaptive time-delay estimation controller is asymptotically stable, and |s| converges;

[0160] When 0 < N < 1, Where, Is the i-th eigenvalue of the matrix I m -N -1 N, and I m Represents the m-order identity matrix; According to the design process, it can be known that the error D of the time-delay estimation is bounded. Combining [[ID=​​​​​​​​​

[0163] The stability of the adaptive time delay estimation controller is verified.

[0164] The adaptive time delay estimation controller is compared with an existing sliding mode controller based on time delay estimation technology through simulation. In addition to the adaptive time delay estimation controller, two other controllers are included in the simulation comparison. For ease of description, the adaptive time delay estimation controller proposed in this application will be referred to as controller 1 below, and the controllers used for comparison will be referred to as controller 2 and controller 3. Controller 2 and controller 3 are detailed below.

[0165] Controller 2 is an adaptive time delay estimation controller using a common sliding surface, and its expression is as follows:

[0166]

[0167] Controller 3 is a fixed-gain time-delay estimation controller that uses an integral sliding mode sliding surface, the expression of which is as follows:

[0168]

[0169] The simulation platform is Matlab software under the Win10 x64 operating system, version R2022b, and the simulation object is a two-degree-of-freedom robotic arm, such as... Figure 2 As shown. The simulation uses the ode4 fixed-step solver with a step size of 0.001s and a total simulation duration of 16s.

[0170] To make the expression more concise, the following abbreviation is used: s i =sin(θ) i ), c i =cos(θ) i ), c ij =cos(θ) i +θ j The dynamic equations of the two-degree-of-freedom manipulator used in the simulation are as follows:

[0171]

[0172] In the formula

[0173]

[0174] The specific parameters of the two-degree-of-freedom robotic arm are shown in Table 1.

[0175] Table 1 Specific parameters of the robotic arm

[0176]

[0177]

[0178] The controller parameters are shown in Table 2 below. These are the control parameter values ​​for fixed gain time delay estimation.

[0179] Table 2 Control Parameter Values

[0180]

[0181] Simulation results are as follows Figure 2 As shown. θ1 and θ2 represent the rotation angles of joint 1 and joint 2, respectively. Figure 3 and Figure 4 As can be seen from the diagram, the cable-driven robotic arm trajectory tracking control method based on adaptive time delay estimation provided in this application can achieve accurate tracking of the desired trajectory. e1 and e2 represent the control errors of joint 1 and joint 2, respectively. Figure 5 and Figure 6 As shown, the control error of joints 1 and 2 controlled by controller 1 (corresponding to adaptive time delay estimation) is significantly lower than the control errors of joints 1 and 2 controlled by controller 2 (corresponding to ordinary adaptive time delay estimation) and controller 3 (corresponding to fixed gain time delay estimation). Figure 7 and Figure 8 From the magnified error diagram, it can be seen that the maximum error of controller 1 controlling joint 1 is 0.74°, while the maximum error of controller 2 is 0.77°, and the maximum error of controller 3 is 0.9°. The maximum error of controller 1 controlling joint 2 is 1.37°, the maximum error of controller 2 is 1.43°, and the maximum error of controller 3 is 1.74°. Figure 9 and Figure 10 As shown, the control torque curves of joints 1 and 2 demonstrate that controller 1 can achieve stable torque output. Simulation results show that the cable-driven robotic arm trajectory tracking control method based on adaptive time delay estimation provided in this application has good control performance, and the estimated N value of the time delay has excellent adaptability, effectively adapting to different working conditions of the robotic arm.

[0182] In summary, the cable-driven robotic arm trajectory tracking control method based on adaptive time delay estimation provided in this application improves control performance and reduces control error by setting an adaptive time delay estimation controller. It has good adaptability and can adapt to different working conditions of the robotic arm.

[0183] Other embodiments of this application will readily occur to those skilled in the art upon consideration of the specification and practice of the application disclosed herein. This application is intended to cover any variations, uses, or adaptations of this application that follow the general principles of this application and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only.

[0184] It should be understood that this application is not limited to the precise structure described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of this application is limited only by the appended claims.

Claims

1. A trajectory tracking control method for a cable-driven robotic arm based on adaptive time delay estimation, characterized in that, The method includes: Step 101, establish the dynamic equations of the n-degree-of-freedom robotic arm: Based on the following formula, establish the dynamic equations of the n-degree-of-freedom robotic arm: In the formula, The position vector of the robotic arm joint. Let be the first derivative of the position vector of the robotic arm joint with respect to time. Let be the second derivative of the position vector of the robotic arm joint with respect to time. It is the inertia matrix. The centripetal force and Coriolis force are vectors. It is a gravity vector. It is the vector of Coulomb friction and viscous friction. The external disturbance force vector, For each joint controller of the robotic arm, there is a generalized output vector. Step 102, determine the adaptive delay estimation term, including: Rewrite the dynamic equations of the robotic arm based on the following formula: In the formula, Let N be the generalized output vector of each joint controller of the robotic arm, and let N be the adaptive control parameter. , The position vector of the robotic arm joint. Let be the first derivative of the position vector of the robotic arm joint with respect to time. Let be the second derivative of the position vector of the robotic arm joint with respect to time. It is the inertia matrix. The centripetal force and Coriolis force are vectors. It is a gravity vector. It is the vector of Coulomb friction and viscous friction; right Perform adaptive time delay estimation and determine the adaptive time delay estimation term. : In the formula, express Moment value, express Moment value, for The first derivative of the position vector of the robotic arm joint with respect to time. The sampling period; Step 103: Determine the adaptive delay estimation controller based on the adaptive delay estimation term and the sliding surface. The sliding surface is defined according to the following formula: In the formula and For constant control parameters, , This represents the control error between the desired rotation angle and the actual rotation angle of the robotic arm joint. for Regarding time The derivative of This represents the value of the control error at the initial moment. express The value at the initial time, where , To enable the robotic arm to track the desired trajectory, for Regarding time The first derivative; Based on the adaptive time delay estimation term and the sliding surface, an adaptive time delay estimation controller is determined, including: Control the robotic arm to track the desired trajectory and record the trajectory tracking error. The recorded trajectory tracking error is denoted as... ; The adaptive time delay estimation controller with fixed gain is obtained according to the following formula: In the formula, for Regarding time The second derivative, for Regarding time The derivative of ,Will All data are sorted in ascending order, and the mean of the smallest 25% of data is taken as the average. Take the average of the largest 25% of the data as... ; Step 104: Based on the adaptive time delay estimation controller, perform trajectory tracking control on the robotic arm.

2. The trajectory tracking control method for a cable-driven robotic arm based on adaptive time delay estimation according to claim 1, characterized in that, The method further includes: dynamically adjusting the determined adaptive control parameters according to the following formula: In the formula, R represents the maximum rate of change of N, where R > 0; The minimum value of N that allows the robotic arm to operate normally under undisturbed conditions is selected, and this value is used as the reference value for the dynamic adjustment of parameter N, denoted as . The maximum value of N that allows the robotic arm to function normally under undisturbed conditions is selected, and the threshold proportion of this value is taken as the maximum allowable value of N, denoted as . Calculate according to the following formula : 。 3. The trajectory tracking control method for a cable-driven robotic arm based on adaptive time delay estimation according to claim 2, characterized in that, The adaptive delay estimation controller after introduction of dynamic adjustment is as follows: 。 4. The cable-driven robotic arm trajectory tracking control method based on adaptive time delay estimation according to claim 3, characterized in that, The method further includes: performing stability analysis on the dynamically adjusted adaptive time delay estimation controller, specifically including: According to the following formula (1), set the variable P: Rewriting formula (1) yields formula (2): The adaptive delay estimation controller with dynamic adjustment is rewritten as follows, resulting in formula (3): Based on the rewritten robotic arm dynamics equations and the rewritten adaptive time delay estimation controller with dynamic adjustment, formula (4) is derived: The error in the time delay estimation is determined to be ; Will ,right The derivative is obtained Substitution The following formula (5) is obtained: Select the following Lyapunov function formula (6) for... Perform stability analysis: The Lyapunov function formula (6) is differentiated with respect to time to obtain the following formula (7): Substituting the fourth equation in formula (3) into formula (7), we get formula (8): Substituting formula (5) into formula (8), we get formula (9): pass The definition of formula (9) is used to transform formula (10): in, The defining formula is: according to Thus, we obtain formula (11): according to Thus, we obtain formula (12): when At that time, the system consisting of the robotic arm and the adaptive time delay estimation controller is asymptotically stable. convergence; when hour, ,in, It is a matrix The i-th eigenvalue, Let represent the m-order identity matrix; the error D of the time delay estimation is bounded, and further, we derive... It is bounded; After determining the adaptive delay estimation controller based on the adaptive delay estimation term and the sliding surface, the method further includes: performing stability verification on the adaptive delay estimation controller.

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