Rope-driven mechanical tracking control method based on TDE and recursive nonsingular terminal sliding mode

Abstract

The application discloses a rope driving mechanical tracking control method based on TDE and recursive non-singular terminal sliding mode, and relates to the technical field of mechanical hand control. The application comprises the following steps: through mechanical dynamics analysis, the dynamics standard equation of the mechanical hand is determined and is converted into a TDE simplified equation, and a tracking error is defined; an FNTSM function containing a proportional control parameter is constructed, a recursive non-singular terminal sliding mode variable is formed in combination with a recursive integral term; an interference adaptive law is designed to dynamically adjust the proportional control parameter, the convergence speed and the chattering characteristics of the tracking error are analyzed, and a composite reaching law is generated; based on the composite reaching law and the updated TDE simplified equation, a generalized output vector of a joint controller containing a fluctuation parameter is extracted, a complete control law is formed, the generalized vector is output in real time through the control law, the motion trajectory of the mechanical hand is simulated, and high-precision tracking control is realized.

Description

Technical Field

[0001] This invention relates to the field of robotic arm control technology, specifically to a rope-driven mechanical tracking control method based on TDE and recursive non-singular terminal sliding mode. Background Technology

[0002] Traditional underwater robotic arms are typically based on direct motor drives mounted within sealed joints, resulting in large size, heavy weight, and high energy consumption, making them unsuitable for compact, energy-constrained AUV (Autonomous Underwater Vehicle) platforms. Rope-driven robots, due to their inherent advantages including large workspace, high payload-to-weight ratio, and low structural inertia, have gained widespread attention in robotics research. These characteristics have enabled their successful application in various fields such as large-scale operations, aerial photography, and industrial automation. In recent years, the application of rope-driven mechanisms has expanded to underwater environments, particularly in the development of lightweight manipulators for autonomous underwater vehicle-manipulator systems. To ensure that rope-driven mechanisms can handle complex nonlinear couplings and time-varying loads and to simplify controller design, existing technologies generally employ Time Delay Estimation (TDE) to design the controllers for rope-driven mechanisms.

[0003] However, the practical application of rope-driven systems in underwater environments presents complex challenges. The dynamics of such systems become a complex mixture of rigid body mechanics, flexible rope dynamics, and strong fluid-structure interaction, constituting a rigid-flexible-fluid coupling problem. Furthermore, underwater operations are subject to significant external disturbances, such as time-varying hydrodynamics, ocean currents, and rope tension variations. Therefore, achieving high-precision trajectory tracking control for rope-driven underwater manipulators is a formidable task, requiring control strategies that are robust to inherent model uncertainties arising from complex coupled dynamics and external environmental disturbances. While advanced control techniques such as sliding mode control offer robustness, their direct application is hindered by the need for accurate dynamic models and the persistent problem of chattering. Therefore, a practical, high-performance control framework is urgently needed to ensure the precise and stable operation of rope-driven robots in harsh and uncertain environments such as underwater.

[0004] The information disclosed in the background section is only intended to enhance the understanding of the background of this disclosure, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0005] The purpose of this invention is to provide a rope-driven mechanical tracking control method based on TDE and recursive non-singular terminal sliding mode to solve the problems mentioned in the background art.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A cable-driven mechanical tracking control method based on TDE and recursive nonsingular terminal sliding mode, the specific steps of which include:

[0008] The degrees of freedom of the controlled rope-driven manipulator are analyzed by mechanical dynamics. The standard equation of dynamics of the controlled rope-driven manipulator under the generalized output vector of the joint controller is determined. The standard equation of dynamics is transformed into the simplified equation of TDE by the TDE estimation algorithm. The tracking error is defined based on the trajectory of the manipulator joints.

[0009] Based on the tracking error of the robot joint, an FNTSM function containing proportional control parameters is constructed. Based on the FNTSM function, a recursive integral term is introduced. By integrating the FNTSM function with the recursive integral term, a recursive non-singular terminal sliding mode variable is constructed.

[0010] Based on the recursive non-singular terminal sliding mode variables, an interference adaptation law is designed. The proportional control parameters are dynamically adjusted through the interference adaptation law. The convergence speed and chattering characteristics of the tracking error of the controlled rope-driven manipulator are analyzed. A composite reaching law is generated based on the recursive non-singular terminal sliding mode variables.

[0011] Based on the composite reaching law, combined with the FNTSM function and the recursive non-singular terminal sliding mode variable, the TDE simplified equation is updated, and the generalized output vector of the joint controller containing the turbulence parameter is extracted based on the updated TDE simplified equation, forming a complete control law for the generalized output vector of the joint controller in the controllable rope-driven manipulator.

[0012] Based on the complete control law, the joint controller outputs a real-time generalized output vector to simulate the motion trajectory of the robot driven by the rope under control, so as to achieve actual tracking control.

[0013] Furthermore, the logic for determining the degrees of freedom of the rope-driven manipulator is as follows: determine the number of joints and the number of joint motion constraints of the rope-driven manipulator, and analyze the degrees of freedom of the rope-driven manipulator based on the number of joints and the number of joint motion constraints.

[0014] The logic underlying the construction of the standard dynamic equations for the controlled rope-driven manipulator is as follows: Using the mass inertia matrix, combined with the joint position vectors, and incorporating the centrifugal force vector, gravity vector, and viscous friction vector, the standard dynamic equations for the controlled rope-driven manipulator are comprehensively constructed, specifically expressed as follows:

[0015]

[0016] In the formula, For the controllable rope-driven manipulator joints in position vector The mass inertia matrix at that location, Let be the position vector of the joint of the robot arm driven by the control cable. The second time derivative of the position vector. The first time derivative of the position vector. This represents the centrifugal force vector of the joint of the controlled cable-driven manipulator. Let be the gravity vector of the joint of the robotic arm driven by the control cable. Let be the viscous friction vector of the joint of the cable-driven robotic arm to be controlled. This is the external interference vector. The mass inertia matrix is ​​the generalized output vector of the joint controller, and it is specifically a matrix whose number of rows and columns are the same as the number of degrees of freedom.

[0017] Furthermore, the logic behind transforming the standard dynamic equations into simplified TDE equations using the TDE estimation algorithm is as follows: the standard dynamic equations are simplified using the TDE estimation algorithm, and the specific expression of the simplified TDE equations is:

[0018]

[0019] In the formula, The constant inertial gain matrix is... For the lumped unknown dynamic term in the tracking control of the cable-driven manipulator. For injecting motion interference terms into the joints;

[0020] The tracking error is defined based on the trajectory of the robotic arm joints, and the tracking error is specifically expressed as follows:

[0021]

[0022] In the formula, Let be the tracking error of the i-th joint of the robot arm. Let be the position vector of the desired motion trajectory of the i-th joint of the robot. Let be the actual motion trajectory position vector of the i-th joint of the robot arm. The actual motion trajectory position vector and the desired motion trajectory position are specifically represented by a position function with time as the independent variable, where i is the joint index of the cable-driven robot arm to be controlled. N represents the total number of joints in the cable-driven robotic arm to be controlled.

[0023] Furthermore, based on the tracking error of the robotic arm joints, an FNTSM function containing proportional control parameters is constructed, wherein the FNTSM function is specifically expressed as follows:

[0024]

[0025] In the formula, For FNTSM function value, To track the first derivative vector of the error with respect to the time variable, This is the first proportional control parameter. This is the second proportional control parameter, and and All are greater than 0. It is an exponential vector. To track the error vector, It is a nonlinear control function;

[0026] The first derivative vector of the tracking error with respect to the time variable is specifically represented as follows:

[0027]

[0028] In the formula, Let be the first derivative of the tracking error of the i-th joint of the robot with respect to the time variable. Let be the first derivative of the tracking error of the Nth joint of the robot with respect to the time variable;

[0029] The tracking error vector is similarly represented as follows:

[0030]

[0031] In the formula, Let N be the tracking error of the robot's Nth joint;

[0032] The exponential vector Specifically, it is expressed as follows:

[0033]

[0034] In the formula, Let be the exponential component of the i-th joint of the robotic arm. Let N be the exponential component of the Nth joint of the robotic arm, and let the exponential component value satisfy... ;

[0035] The nonlinear control function is specifically expressed as follows:

[0036]

[0037] In the formula, Let be the absolute value of the tracking error of the i-th joint of the robotic arm. It is a symbolic function.

[0038] Furthermore, the specific formula upon which the recursive non-singular terminal sliding mode variables are constructed is as follows:

[0039]

[0040] In the formula, For recursive non-singular terminal sliding mode variables, For recursive integral terms, This is a recursive matrix, specifically a positive definite diagonal matrix, represented as follows:

[0041]

[0042] In the formula, Describes the constructor for a diagonal matrix. These are the elements on the main diagonal of the recursive matrix;

[0043] The derivative of the recursive integral term is specifically expressed as:

[0044]

[0045] In the formula, The derivative of the recursive integral term, The recursive derivative matrix is ​​a positive definite diagonal matrix, specifically represented as:

[0046]

[0047] In the formula, These are the diagonal elements of the recursive derivative matrix.

[0048] Furthermore, the interference adaptation law is specifically expressed as follows:

[0049]

[0050] In the formula, Let be the disturbance adaptation law at time t. For scaling adjustment factors, and , Let be the adaptive variable at time t, where t is the time variable of the tracking control process;

[0051] The specific formula used to dynamically adjust the proportional control parameters through the interference adaptation law is as follows:

[0052]

[0053] In the formula, and These are the update values ​​of the first and second proportional control parameters at time t, respectively. and These are the nominal values ​​of the first and second proportional control parameters, respectively. and All are greater than 0;

[0054] The specific update rule for the adaptive variable at time t is as follows:

[0055]

[0056] In the formula, To adapt to the scaling factor, and Greater than 0, This represents the magnitude of the sliding mode variable at time t, which is a recursive non-singular terminal variable. For the upper bound of the adaptive variable, Threshold for sliding mode variables of recursive non-singular terminals;

[0057] The magnitude of the sliding mode variable of the recursive non-singular terminal at time t is specifically represented as follows:

[0058]

[0059] In the formula, The margin constant is... .

[0060] Furthermore, the specific formula upon which the composite reaching law is based is:

[0061]

[0062] In the formula, For the compound reaching law, Let be the first composite approaching matrix. This is the second composite approaching matrix. It is a composite proportionality coefficient, and The first and second composite approach matrices are specifically represented as follows:

[0063]

[0064] In the formula, These are the main diagonal elements of the composite approaching matrix.

[0065] Furthermore, the logic behind extracting the generalized output vector of the joint controller containing turbulence parameters based on the updated TDE simplified equation is as follows: update the lumped unknown dynamic terms and the motion disturbance terms injected into the joint in the cable-driven manipulator tracking control in the TDE simplified equation to update the generalized output vector of the joint controller.

[0066] The updated motion disturbance term for the injected joint is specifically represented as follows:

[0067]

[0068] In the formula, Let be the second derivative of the desired motion trajectory position of the robotic arm joint with respect to time;

[0069] The updated lumped unknown dynamic term for the tracking control of the cable-driven manipulator is specifically represented as follows:

[0070]

[0071] In the formula, for The generalized output vector of the moment joint controller. For turbulence parameters, for The second derivative of the desired motion trajectory position of the manipulator joint with respect to time at any given moment;

[0072] Based on the updated motion disturbance terms of the injected joints and the updated lumped unknown dynamic terms of the tracking control of the rope-driven manipulator, a complete control law is formed for the generalized output vector of the joint controller of the rope-driven manipulator to be controlled.

[0073] Compared with the prior art, the beneficial effects of the present invention are:

[0074] This scheme combines the standard equations of dynamics established by mechanical dynamics with the TDE estimation algorithm, which can accurately reflect the dynamic characteristics of the robot and transform them into simplified equations. This helps to improve the modeling accuracy of the control system and significantly reduce the computational complexity, enabling the controller to respond to changes in the environment more quickly.

[0075] Secondly, the recursive non-singular terminal sliding mode variable formed by combining the FNTSM function and the recursive integral term not only enhances the system's adaptive capability but also improves its robustness to external disturbances. By introducing the disturbance adaptation law, the proportional control parameters can be dynamically adjusted, enabling the controller to adapt to environmental changes in real time, thereby effectively reducing tracking errors, maintaining high-precision tracking performance, and overcoming the limitations of traditional control methods in the face of time-varying and uncertain environments.

[0076] This scheme effectively extracts the generalized output vector of the joint controller containing turbulence parameters by generating a composite reaching law and updating the simplified TDE equation. This enables real-time updating and optimization of the control law, improves the system's adaptability to rapidly changing environments, and ensures the stability and accuracy of the robot arm when performing tasks. It also effectively solves the persistent chattering problem, providing a strong guarantee for achieving more complex automated tasks. Attached Figure Description

[0077] Figure 1 This is a schematic diagram of the overall method flow of the present invention. Detailed Implementation

[0078] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to specific embodiments.

[0079] It should be noted that, unless otherwise defined, the technical or scientific terms used in this invention should have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms "first," "second," and similar terms used in this invention do not indicate any order, quantity, or importance, but are merely used to distinguish different components. Terms such as "comprising" or "including" mean that the element or object preceding the word encompasses the elements or objects listed following the word and their equivalents, without excluding other elements or objects. Terms such as "connected" or "linked" are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect. Terms such as "upper," "lower," "left," and "right" are used only to indicate relative positional relationships; when the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0080] Example:

[0081] Please see Figure 1 The present invention provides a technical solution:

[0082] A cable-driven mechanical tracking control method based on TDE and recursive nonsingular terminal sliding mode, the specific steps of which include:

[0083] Step 1: Analyze the degrees of freedom of the controlled rope-driven manipulator through mechanical dynamics analysis, determine the standard equation of dynamics of the controlled rope-driven manipulator under the generalized output vector of the joint controller, transform the standard equation of dynamics into a simplified equation of TDE through the TDE estimation algorithm, and define the tracking error based on the trajectory of the manipulator joints.

[0084] The logic for determining the degrees of freedom of the controlled rope-driven manipulator is as follows: First, determine the number of joints and the number of joint motion constraints of the manipulator. Then, analyze the degrees of freedom based on these factors, specifically by counting the number of motion constraints for each joint, including directional constraints and fixation constraints, which limit the independent movement capabilities of the joints. Finally, verify the rationality of the degrees of freedom by considering the manipulator's workspace and the limitations of the rope-driven system (such as rope tension and length). Alternatively, the degrees of freedom can be determined using the specific design drawings of the controlled rope-driven manipulator.

[0085] The logic underlying the construction of the standard dynamic equations for the controlled cable-driven manipulator is as follows: Using the mass inertia matrix, combined with the joint position vectors, and incorporating the centrifugal force vector, gravity vector, and viscous friction vector, the standard dynamic equations for the controlled cable-driven manipulator are comprehensively constructed, specifically expressed as follows:

[0086]

[0087] In the formula, For the controllable rope-driven manipulator joints in position vector The mass inertia matrix at that location, Let be the position vector of the joint of the robot arm driven by the control cable. The second time derivative of the position vector. The first time derivative of the position vector. This represents the centrifugal force vector of the joint of the controlled cable-driven manipulator. Let be the gravity vector of the joint of the robotic arm driven by the control cable. Let be the viscous friction vector of the joint of the robot arm driven by the controllable rope. This is the external interference vector. The mass inertia matrix is ​​the generalized output vector of the joint controller, and it is specifically a matrix whose number of rows and columns are the same as the number of degrees of freedom.

[0088] It should be noted that, Represents the position vector of the robotic arm The mass distribution and inertial characteristics of the system are shown below. The number of rows and columns is the same as the number of degrees of freedom, reflecting the inertial coupling relationship between the various joints of the system. This matrix is ​​the basis of dynamic analysis and ensures that the system's response to various inputs conforms to physical laws.

[0089] It is the second derivative of the position vector, representing the acceleration of the robotic arm; The first derivative of the position represents the robot's velocity. These two variables are key parameters describing the robot's dynamic behavior and affect the calculation of various forces in the dynamic equations.

[0090] This reflects the centrifugal effect generated by the robot during movement, especially the influence of centrifugal force at higher joint velocities. This term is introduced to ensure that the dynamic equations accurately describe the robot's behavior in dynamic motion.

[0091] This item is essential to represent the effect of gravity on the joints of the robotic arm, especially the force in the vertical direction. Gravity is a key factor that the robotic arm must overcome during movement.

[0092] This describes the frictional force during joint movement, which is usually related to speed. Introducing friction allows for a more realistic reflection of the dynamic characteristics of a robotic arm, especially in terms of control and motion precision.

[0093] This represents external disturbances that may affect the movement of the robotic arm, including wind and load changes. Introducing these disturbances makes the equations more realistic and can handle uncertainties in various practical applications.

[0094] Based on the principles of Newton-Euler dynamics, this equation describes the relationship between forces and motion of an object, ensuring that the motion of the robot follows the laws of classical mechanics. By introducing the mass inertia matrix, the influence of the robot's mass distribution on motion can be effectively considered, which is crucial for improving the stability and control accuracy of the system. Adding centrifugal force, gravity, and friction terms can more comprehensively consider the effects of various forces that the system may encounter during dynamic operation, reflecting the true performance of the robot in actual work.

[0095] Based on the robot's motion trajectory and control input, the velocity and acceleration of the joints are calculated using kinematic equations; for the centrifugal force vector... Based on the kinematic characteristics and velocity state of the robot, the expression for centrifugal force is derived using formulas. This requires calculation by combining the mass inertia matrix and the current velocity information, and then using dynamic simulation software such as MATLAB / Simulink or ADAMS to simulate the robot's motion and obtain the centrifugal force in the dynamic behavior.

[0096] Gravity vector The effect of gravity on each joint is calculated based on the mass of the joints and links. The gravity vector should be related to the joint position, so the effect of gravity needs to be calculated at different positions.

[0097] Viscous friction vector Through experimental setup, the frictional force of joint motion is measured at a known speed. Typically, frictional force can be measured by applying a known force, establishing the relationship between speed and frictional force. Based on experimental data, a friction model, such as a viscous friction model, is fitted to obtain a speed-dependent expression for frictional force.

[0098] External interference vector An empirical model is established by analyzing the external interference encountered by the robotic arm during actual work.

[0099] The logic behind transforming the standard dynamic equations into simplified TDE equations using the TDE estimation algorithm is as follows: The simplified TDE equations are expressed as follows:

[0100]

[0101] In the formula, The constant inertial gain matrix is... For the lumped unknown dynamic term in the tracking control of the cable-driven manipulator. For injecting motion interference terms into the joints;

[0102] It should be noted that in practical applications, complex dynamic equations can make the control system difficult to implement. Therefore, the TDE estimation algorithm uses a pseudo-constant inertial gain matrix. The constant value near a specific operating point makes the control algorithm simpler and more efficient; the dynamic behavior of the robot is decomposed into known and unknown parts. The known part is represented by a constant inertial gain matrix, while the unknown part is determined by... Modeling and decomposing operations enable the controller to prioritize known dynamic characteristics and estimate or compensate for unknown dynamics.

[0103] The TDE algorithm is based on the idea that by filtering and estimating the tracking error of the system output, it can more effectively obtain disturbance information in joint motion. By estimating these disturbances, the control input can be adjusted in real time. This improves the accuracy of tracking and control.

[0104] The tracking error is defined based on the trajectory of the robotic arm joints, and the tracking error is specifically expressed as follows:

[0105]

[0106] In the formula, Let be the tracking error of the i-th joint of the robot arm. Let be the position vector of the desired motion trajectory of the i-th joint of the robot. Let be the actual motion trajectory position vector of the i-th joint of the robot arm. The actual motion trajectory position vector and the desired motion trajectory position are specifically represented by a position function with time as the independent variable, where i is the joint index of the cable-driven robot arm to be controlled. N represents the total number of joints in the cable-driven robotic arm to be controlled.

[0107] It should be noted that the expected and actual position vectors are represented as time functions. and Tracking error is used for real-time tracking and error analysis in dynamic control; tracking error is a key indicator for evaluating the performance of a control system. By defining and analyzing tracking error, the effectiveness of the control algorithm can be quantified, thus providing a basis for subsequent control strategy improvements. In dynamic control, real-time monitoring of the tracking error of each joint can help the controller adjust the control input to optimize the motion trajectory; the tracking error term... The introduction of this allows the control algorithm to make feedback adjustments based on the deviation.

[0108] Step 2: Based on the tracking error of the robot joint, construct the FNTSM function containing proportional control parameters. Based on the FNTSM function, introduce a recursive integral term. By integrating the FNTSM function with the recursive integral term, construct a recursive non-singular terminal sliding mode variable.

[0109] The FNTSM function, specifically FNTSM (Fast Nonsingular Terminal Sliding Mode), provides several key advantages for high-performance tracking in rope-driven robots: First, FNTSM ensures finite-time convergence of tracking errors through its nonlinear terminal term, achieving faster and more accurate stabilization compared to traditional linear sliding surfaces.

[0110] Secondly, this design inherently avoids the singularities that plague traditional terminal sliding modes, thereby improving the reliability of the controller. Furthermore, FNTSM integrates proportional and terminal error terms, jointly improving transient response speed and steady-state accuracy; these characteristics make FNTSM particularly suitable for systems with complex dynamics and significant uncertainties, such as cable-driven underwater manipulators, where robust and rapid error convergence is crucial.

[0111] Based on the tracking error of the robotic arm joints, an FNTSM function containing proportional control parameters is constructed. The FNTSM function is specifically expressed as follows:

[0112]

[0113] In the formula, For FNTSM function value, To track the first derivative vector of the error with respect to the time variable, This is the first proportional control parameter. This is the second proportional control parameter, and and All are greater than 0. It is an exponential vector. To track the error vector, It is a nonlinear control function;

[0114] It should be noted that the formula also includes a linear control term. and nonlinear control terms The linear part ensures the basic ability of the system error to converge, while the nonlinear part introduces an exponential function. It can accelerate error convergence, especially when the error is small, it exhibits strong convergence ability;

[0115] Introducing the time derivative of the error It can reflect dynamic information about error changes, allowing the sliding surface design to not only focus on the current error but also adjust for the error change trend, thus improving the dynamic performance of the control system through nonlinear terms. The sliding mode controller maintains high control accuracy even with small errors, while avoiding chattering issues present in traditional sliding mode control, thus improving the robustness of the control system. Nonlinear sliding mode control, such as terminal sliding mode control, may suffer from singularity problems, which the FNTSM design addresses by using non-singular functions. This avoids the problem, allowing the controller to function properly globally.

[0116] The first derivative vector of the tracking error with respect to the time variable is specifically represented as follows:

[0117]

[0118] In the formula, Let be the first derivative of the tracking error of the i-th joint of the robot with respect to the time variable. Let be the first derivative of the tracking error of the Nth joint of the robot with respect to the time variable; organize the error derivatives of all joints into a vector form so that the control design can handle them uniformly. The state of each joint and the error derivative are important input variables for dynamic control. They can capture the rate of error change, thereby helping the sliding mode controller to react and adjust more quickly.

[0119] The tracking error vector is similarly represented as follows:

[0120]

[0121] In the formula, Let be the tracking error of the Nth joint of the robot arm; the tracking error vector uniformly includes the errors of all joints, which helps in designing a global controller. The error vector is the most basic variable of the controller, and the goal of the controller is to achieve [the desired result] through sliding mode control. Approaching 0, thus achieving trajectory tracking.

[0122] The exponential vector Specifically, it is expressed as follows:

[0123]

[0124] In the formula, Let be the exponential component of the i-th joint of the robotic arm. Let N be the exponential component of the Nth joint of the robotic arm, and let the exponential component value satisfy... ;

[0125] Indicates the first robotic arm The nonlinear exponential components of each joint satisfy the following conditions: Each joint can be individually set with different index values. To meet the specific control requirements of different joints, when Exponential control can accelerate the error convergence speed, when This avoids the problem of excessive nonlinearity making the controller difficult to implement.

[0126] The nonlinear control function is specifically expressed as follows:

[0127]

[0128] In the formula, Let be the absolute value of the tracking error of the i-th joint of the robotic arm. It is a symbolic function.

[0129] It should be noted that, It is the nonlinear power term of the error, reflecting the characteristics of nonlinear control. It is a sign function that ensures the correctness of the direction information and is used to indicate the sign of the error. The nonlinear control term of each joint can be designed separately according to its own dynamic characteristics to enhance the adaptability of the control.

[0130] The sign function The specific expression is:

[0131]

[0132] The core of sliding mode control is to design the dynamic characteristics of the sliding surface so that the tracking error and error derivative can quickly approach zero, ensuring that the system reaches the desired state; nonlinear terms are introduced. This can accelerate error convergence while ensuring the robustness of the controller;

[0133] Nonlinear control function It is the core component of nonlinear sliding mode control, which uses the exponential component of the i-th joint of the robot arm. This avoids the singularity problem in traditional sliding mode control and achieves error convergence within a finite time.

[0134] By setting the tracking error, derivative vector, exponential vector, and nonlinear control function, the core of the FNTSM sliding mode control is formed. This design enables high-precision, fast-convergence, and robust manipulator trajectory tracking control, suitable for dynamic control of multi-joint systems, while also meeting the flexible adjustment requirements for single joints; specifically, it is achieved through sliding mode control theory, nonlinear control theory, and dynamic characteristic analysis in practical applications.

[0135] In this embodiment, the FNTSM function is integrated with a recursive integral term to form a recursive nonsingular terminal sliding mode variable. The recursive integral term is determined by the Recursive Terminal Sliding Mode (RTSM) representation. RTSM ensures a hierarchical convergence process, with the sliding surface being reached sequentially, ultimately driving the tracking error to zero within a finite time. Therefore, the introduction of a recursive integral term not only improves the convergence speed but also enhances the robustness to system uncertainties (including parameter variations, nonlinear friction, and external disturbances). Compared with traditional and nonsingular terminal sliding methods, it provides superior tracking performance, wider control bandwidth, and stronger disturbance suppression capabilities.

[0136] The specific formula upon which the recursive non-singular terminal sliding mode variables are constructed is as follows:

[0137]

[0138] In the formula, For recursive non-singular terminal sliding mode variables, For recursive integral terms, This is a recursive matrix, specifically a positive definite diagonal matrix, represented as follows:

[0139]

[0140] In the formula, Describes the constructor for a diagonal matrix. The elements on the main diagonal of the recursive matrix are designed to be positive to ensure the positive definiteness of the entire matrix, enabling the system to independently adjust each joint during the control process, thereby achieving targeted control.

[0141] The derivative of the recursive integral term is specifically expressed as:

[0142]

[0143] In the formula, The derivative of the recursive integral term, The recursive derivative matrix is ​​a positive definite diagonal matrix, also designed to have positive values, specifically represented as:

[0144]

[0145] In the formula, These are the diagonal elements of the recursive derivative matrix.

[0146] It should be noted that by using the sliding surface function With recursive integral terms By combining these methods, new sliding mode variables are constructed. This is to effectively improve the control performance of the system, especially when facing external disturbances and model uncertainties;

[0147] Recursive matrix and recursive derivative matrix The positive definiteness of the matrix ensures the stability of the control system. When all matrix elements are positive, it guarantees the positive feedback characteristics of the control input, thereby promoting error convergence. The use of nonlinear functions... As a recursive integral term, it enables the controller to exhibit nonlinear characteristics when dealing with errors, enhancing the system's sensitivity to small-range errors, which is crucial for achieving fast convergence. The introduction of the recursive integral term helps suppress the system's tracking error and ensures that the controller can adjust its behavior in real time during long-term operation to cope with possible dynamic changes.

[0148] Step 3: Design an interference adaptation law based on the recursive non-singular terminal sliding mode variables, dynamically adjust the proportional control parameters using the interference adaptation law, analyze the convergence speed and chattering characteristics of the tracking error of the controlled rope-driven manipulator, and generate a composite reaching law based on the recursive non-singular terminal sliding mode variables.

[0149] To enhance the controller's adaptability to time-varying uncertainties (such as model parameter variations and external disturbances), an adaptive law was designed to dynamically adjust the first and second proportional control parameters. and This mechanism is based on sliding mode variables. Real-time size operation: when When the gain is large, increase the gain to strengthen the control effect and accelerate error convergence; when When the gain is low, it is reduced to mitigate control chattering. This strategy ensures an effective balance between fast transient response and smooth steady-state performance, thereby improving the robustness and applicability of the closed-loop system in real-world scenarios.

[0150] The interference adaptation law is specifically expressed as follows:

[0151]

[0152] In the formula, Let be the disturbance adaptation law at time t. For scaling adjustment factors, and , Let be the adaptive variable at time t, where t is the time variable of the tracking control process;

[0153] It should be noted that external disturbances and uncertainties are common challenges in practical control systems. This can be addressed by introducing a disturbance adaptation law. The controller can dynamically adjust its parameters to cope with disturbances. Adjusting proportional control parameters... and This mechanism enables the controller to automatically update its response capabilities based on changes in the environment, such as load changes and external disturbances, thereby improving the system's adaptability.

[0154] The specific formula upon which the proportional control parameter is dynamically adjusted using the interference adaptation law is based:

[0155]

[0156] In the formula, and These are the update values ​​of the first and second proportional control parameters at time t, respectively. and These are the nominal values ​​of the first and second proportional control parameters, respectively. and All are greater than 0; by using the interference adaptation law Multiply by the nominal control parameters to dynamically adjust the controller gain to adapt to the current interference environment;

[0157] The specific update rule for the adaptive variable at time t is as follows:

[0158]

[0159] In the formula, To adapt to the scaling factor, and Greater than 0, This represents the magnitude of the sliding mode variable at time t, which is a recursive non-singular terminal variable. For the upper bound of the adaptive variable, Threshold for sliding mode variables of recursive non-singular terminals;

[0160] It should be noted that, Used to adjust the update magnitude of adaptive variables. It controls the system's response to disturbances, ensuring the flexibility of updates. It represents the system's state at the current moment, reflecting the tracking error and dynamic characteristics of the control system. Its absolute value is used to calculate the increment of the adaptive variable;

[0161] when When the adaptive variable exceeds the allowed range (less than 0 or greater than the upper limit), it indicates that the current sliding mode variable will be readjusted. The value of is determined using a sign function to ensure that the update direction is consistent with the current state, avoiding unnecessary reverse adjustments; this enables the control system to respond promptly to extreme situations, prevents excessive deviation of the adaptive variable, and thus maintains the stability of the system.

[0162] when When, it means that when the adaptive variable is within the allowable range, the sliding mode variable's size and a set threshold are used. To determine the direction and magnitude of the update, if If the threshold is exceeded, the adaptive variable is increased; otherwise, it remains unchanged or is decreased. This ensures that the adjustment of the adaptive variable is based on the actual sliding mode dynamic state, making the system response more flexible and targeted. By comparing with the threshold, the controller can dynamically adapt to the current conditions and improve performance.

[0163] Through adaptive variable dynamic update rules, the controller can adapt to different disturbances and state changes in real time. This flexible adaptability is crucial for ensuring that the control system maintains performance in changing environments; an upper bound is introduced. Limiting the sign function helps prevent system instability caused by excessively large or small adaptive variables. Limiting the range of variation of the adaptive variables ensures the reliability of the controller. This is the maximum allowable value of the adaptive variable, mainly used to limit the range of the adaptive variable and avoid system instability caused by over-adjustment. It should be set according to the design requirements and performance requirements of the control system. The physical limitations and operating range of the system need to be considered to ensure that the output of the controller does not exceed the actual achievable range.

[0164] By adapting the proportional coefficient The introduction of recursive nonsingular terminal sliding mode variables allows the controller to adjust its response intensity according to different disturbance magnitudes and rates of change, thereby achieving higher accuracy and faster response time. The combination of the introduction and update mechanism of these variables improves the nonlinear characteristics of the control system and enhances its adaptability in complex environments; their values ​​typically range from [value range missing]. The specific value needs to be adjusted according to the characteristics and requirements of the system.

[0165] This is the threshold value for the sliding mode controller, used to determine when adjustments to the adaptive variable are needed. It is set based on the system's dynamic characteristics and the desired control accuracy. If the threshold is set too low, it may lead to frequent adjustments, causing control system jitter; if it is set too high, it may cause sluggish system response. It is typically set as a percentage of the system's dynamic response, and its value is generally chosen to adapt to different control environments. Within the range.

[0166] The magnitude of the sliding mode variable of the recursive non-singular terminal at time t is specifically represented as follows:

[0167]

[0168] In the formula, The margin constant is... margin constant This is a small constant introduced to avoid singularities and ensure a smooth transition; it generally does not exceed the constant 1.

[0169] The formula upon which the composite reaching law is based is as follows:

[0170]

[0171] In the formula, For the compound reaching law, Let be the first composite approaching matrix. This is the second composite approaching matrix. It is a composite proportionality coefficient, and The first and second composite approach matrices are specifically represented as follows:

[0172]

[0173] In the formula, These are the main diagonal elements of the composite approaching matrix.

[0174] It should be noted that in sliding mode control, the approach law is designed to ensure that the system state converges quickly and stably to the sliding surface; a composite approach law is used. By combining linear and nonlinear feedback, dynamic global control of the system is achieved, ensuring... and It is a positive definite diagonal matrix (i.e., all diagonal elements are positive), which guarantees the stability of the system response. Positive definiteness can avoid the unstable behavior of the system and ensure that the system can still converge in the presence of disturbances and uncertainties.

[0175] Composite proportionality coefficient The setting can effectively adjust the strength of nonlinear feedback, prevent chattering or instability caused by excessive nonlinear feedback, ensure that the system maintains appropriate nonlinear characteristics during the control process, and promote smooth convergence.

[0176] By using different composite convergence matrices and This allows for the reasonable combination of different control requirements within the same control strategy, such as different control requirements for the main joint and other joints, forming a hierarchical control strategy and improving control flexibility.

[0177] Step 4: Based on the composite reaching law, combined with the FNTSM function and the recursive non-singular terminal sliding mode variable, the TDE simplified equation is updated, and the generalized output vector of the joint controller containing the turbulence parameter is extracted based on the updated TDE simplified equation, forming a complete control law for the generalized output vector of the joint controller in the manipulator driven by the rope to be controlled.

[0178] The logic behind extracting the generalized output vector of the joint controller containing turbulence parameters based on the updated TDE simplified equation is as follows: update the lumped unknown dynamic terms and the motion disturbance terms injected into the joint in the cable-driven manipulator tracking control in the TDE simplified equation to update the generalized output vector of the joint controller.

[0179] The updated motion disturbance term for the injected joint is specifically represented as follows:

[0180]

[0181] In the formula, Let be the second derivative of the desired motion trajectory position of the robotic arm joint with respect to time;

[0182] It should be noted that, Let represent the second derivative of the desired motion trajectory position of the robotic arm joint with respect to time, corresponding to the desired acceleration. and These represent the proportional control terms for speed and position, respectively, with the aim of reducing tracking error and improving the dynamic performance of the system.

[0183] By adjusting the tracking error using nonlinear terms, the controller's adaptability to dynamic changes and nonlinear characteristics is enhanced.

[0184] Introducing a recursive nonsingular terminal sliding mode variable term for sliding mode control This is to ensure the stability of the system on the sliding surface.

[0185] The updated lumped unknown dynamic term for the tracking control of the cable-driven manipulator is specifically represented as follows:

[0186]

[0187] In the formula, for The generalized output vector of the joint controller for the control cable-driven manipulator at all times reflects the actual control requirements of the system. For turbulence parameters, for The second derivative of the desired motion trajectory position of the manipulator joint with respect to time at any given moment;

[0188] Indicates at time The second derivative of the desired motion trajectory position of the robotic arm joint is used to obtain the dynamic error by subtracting the target dynamic term, thus reflecting the unknown dynamic characteristics.

[0189] Based on the updated motion disturbance term of the injected joint and the updated lumped unknown dynamic term of the tracking control of the rope-driven manipulator, a complete control law is formed for the generalized output vector of the joint controller of the rope-driven manipulator to be controlled.

[0190] By updating the motion interference items injected into the joints The system can respond to changes in joint position and velocity in real time, making the controller more flexible and precise in adjusting dynamic errors and ensuring rapid target tracking. By combining the recursive nonsingular terminal sliding mode variables of sliding mode control, the system can maintain high performance stability when facing external disturbances and model uncertainties. The nonlinear and time-varying characteristics of dynamic systems are handled through sliding mode control theory.

[0191] Turbulence parameters Specifically, the set time delay reflects the system's uncertainty and dynamic change capability. By appropriately introducing it into the control law, the controller's adaptability to sudden changes can be further improved, thereby enhancing overall performance and reducing turbulence parameters. Set to a multiple of the system time constant, specifically 0.1 to 0.5 times the system time constant.

[0192] Step 5: Based on the complete control law, control the joint controller to output a real-time generalized output vector to simulate the motion trajectory of the robot driven by the controllable rope, so as to achieve actual tracking control.

[0193] Specifically, the calculated control signals are transmitted to the joint controller, which converts them into specific joint motion commands. These commands, via drive motors or other actuators, drive the joints of the robotic arm to perform corresponding movements. Under the control algorithm, the motion trajectory of the robotic arm is simulated in real time. Simulation software such as MATLAB / Simulink, ROS, and Gazebo can be used for visualization.

[0194] The system monitors the deviation between the motion trajectory and the desired trajectory in real time and dynamically adjusts the control signal based on the deviation to ensure tracking performance. The control effect is evaluated periodically by calculating indicators such as tracking error, response time, overshoot, and steady-state error to determine the performance of the control system.

[0195] The above formulas are all dimensionless calculations. The formulas are derived from software simulations based on a large amount of collected data to obtain the most recent real-world results. The preset parameters in the formulas are set by those skilled in the art according to the actual situation.

[0196] The above embodiments can be implemented, in whole or in part, by software, hardware, firmware, or any other combination thereof. When implemented in software, the above embodiments can be implemented, in whole or in part, as a computer program product. Those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented by electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution.

[0197] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment, depending on actual needs.

[0198] The above description is merely a specific embodiment of this application, but the scope of protection of this application is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application.

Claims

1. A cable-driven mechanical tracking control method based on TDE and recursive nonsingular terminal sliding mode, characterized in that, The specific steps include: The degrees of freedom of the controlled rope-driven manipulator are analyzed by mechanical dynamics. The standard equation of dynamics of the controlled rope-driven manipulator under the generalized output vector of the joint controller is determined. The standard equation of dynamics is transformed into the simplified equation of TDE by the TDE estimation algorithm. The tracking error is defined based on the trajectory of the manipulator joints. Based on the tracking error of the robot joint, an FNTSM function containing proportional control parameters is constructed. Based on the FNTSM function, a recursive integral term is introduced. By integrating the FNTSM function with the recursive integral term, a recursive non-singular terminal sliding mode variable is constructed. Based on the recursive non-singular terminal sliding mode variables, an interference adaptation law is designed. The proportional control parameters are dynamically adjusted through the interference adaptation law. The convergence speed and chattering characteristics of the tracking error of the controlled rope-driven manipulator are analyzed. A composite reaching law is generated based on the recursive non-singular terminal sliding mode variables. Based on the composite reaching law, the TDE simplified equation is updated by combining the FNTSM function and the recursive non-singular terminal sliding mode variable, and the generalized output vector of the joint controller containing the turbulence parameter is extracted based on the updated TDE simplified equation to form a complete control law. Based on the complete control law, the joint controller outputs a real-time generalized output vector to simulate the motion trajectory of the controlled rope-driven manipulator, thereby achieving actual tracking control. The interference adaptation law is specifically expressed as follows: In the formula, Let be the disturbance adaptation law at time t. For scaling adjustment factors, and , Let be the adaptive variable at time t, where t is the time variable of the tracking control process; The specific formula used to dynamically adjust the proportional control parameters through the interference adaptation law is as follows: In the formula, and These are the update values ​​of the first and second proportional control parameters at time t, respectively. and These are the nominal values ​​of the first and second proportional control parameters, respectively. and All are greater than 0; The specific update rule for the adaptive variable at time t is as follows: In the formula, To adapt to the scaling factor, and Greater than 0, This represents the magnitude of the sliding mode variable at time t, which is a recursive non-singular terminal variable. For the upper bound of the adaptive variable, Threshold for sliding mode variables of recursive non-singular terminals; The magnitude of the sliding mode variable of the recursive non-singular terminal at time t is specifically represented as follows: In the formula, The margin constant is... .

2. The rope-driven mechanical tracking control method based on TDE and recursive non-singular terminal sliding mode according to claim 1, characterized in that, The logic for determining the degrees of freedom of the rope-driven manipulator is as follows: determine the number of joints and the number of joint motion constraints of the rope-driven manipulator, and analyze the degrees of freedom of the rope-driven manipulator based on the number of joints and the number of joint motion constraints. The logic underlying the construction of the standard dynamic equations for the controlled rope-driven manipulator is as follows: Using the mass inertia matrix, combined with the joint position vectors, and incorporating the centrifugal force vector, gravity vector, and viscous friction vector, the standard dynamic equations for the controlled rope-driven manipulator are comprehensively constructed, specifically expressed as follows: In the formula, For the controllable rope-driven manipulator joints in position vector The mass inertia matrix at that location, Let be the position vector of the joint of the robot arm driven by the control cable. The second time derivative of the position vector. The first time derivative of the position vector. This represents the centrifugal force vector of the joint of the controlled cable-driven manipulator. Let be the gravity vector of the joint of the robotic arm driven by the control cable. Let be the viscous friction vector of the joint of the robot arm driven by the controllable rope. This is the external interference vector. The mass inertia matrix is ​​the generalized output vector of the joint controller, and it is specifically a matrix whose number of rows and columns are the same as the number of degrees of freedom.

3. The rope-driven mechanical tracking control method based on TDE and recursive non-singular terminal sliding mode according to claim 2, characterized in that: The logic behind transforming the standard dynamic equations into simplified TDE equations using the TDE estimation algorithm is as follows: The simplified TDE equations are expressed as follows: In the formula, The constant inertial gain matrix is... For the lumped unknown dynamic term in the tracking control of the cable-driven manipulator. For injecting motion interference terms into the joints; The tracking error is defined based on the trajectory of the robotic arm joints, and the tracking error is specifically expressed as follows: In the formula, Let be the tracking error of the i-th joint of the robot arm. Let be the position vector of the desired motion trajectory of the i-th joint of the robot. Let be the actual motion trajectory position vector of the i-th joint of the robot arm. The actual motion trajectory position vector and the desired motion trajectory position are specifically represented by a position function with time as the independent variable, where i is the joint index of the cable-driven robot arm to be controlled. N represents the total number of joints in the cable-driven robotic arm to be controlled.

4. The cable-driven mechanical tracking control method based on TDE and recursive non-singular terminal sliding mode according to claim 3, characterized in that: Based on the tracking error of the robotic arm joints, an FNTSM function containing proportional control parameters is constructed. The FNTSM function is specifically expressed as follows: In the formula, For FNTSM function value, To track the first derivative vector of the error with respect to the time variable, This is the first proportional control parameter. This is the second proportional control parameter, and and All are greater than 0. It is an exponential vector. To track the error vector, It is a nonlinear control function; The first derivative vector of the tracking error with respect to the time variable is specifically represented as follows: In the formula, Let be the first derivative of the tracking error of the i-th joint of the robot with respect to the time variable. Let be the first derivative of the tracking error of the Nth joint of the robot with respect to the time variable; The tracking error vector is similarly represented as follows: In the formula, Let N be the tracking error of the robot's Nth joint; The exponential vector Specifically, it is expressed as follows: In the formula, Let be the exponential component of the i-th joint of the robotic arm. Let N be the exponential component of the Nth joint of the robotic arm, and let the exponential component value satisfy... ; The nonlinear control function is specifically expressed as follows: In the formula, Let be the absolute value of the tracking error of the i-th joint of the robotic arm. It is a symbolic function.

5. The rope-driven mechanical tracking control method based on TDE and recursive non-singular terminal sliding mode according to claim 4, characterized in that: The specific formula upon which the recursive non-singular terminal sliding mode variables are constructed is as follows: In the formula, For recursive non-singular terminal sliding mode variables, For recursive integral terms, This is a recursive matrix, specifically a positive definite diagonal matrix, represented as follows: In the formula, Describes the constructor for a diagonal matrix. These are the elements on the main diagonal of the recursive matrix; The derivative of the recursive integral term is specifically expressed as: In the formula, The derivative of the recursive integral term, The recursive derivative matrix is ​​a positive definite diagonal matrix, specifically represented as: In the formula, These are the diagonal elements of the recursive derivative matrix.

6. The cable-driven mechanical tracking control method based on TDE and recursive non-singular terminal sliding mode according to claim 5, characterized in that: The formula upon which the composite reaching law is based is as follows: In the formula, For the compound reaching law, Let be the first composite approaching matrix. This is the second composite approaching matrix. It is a composite proportionality coefficient, and The first and second composite approach matrices are specifically represented as follows: In the formula, These are the main diagonal elements of the composite approaching matrix.

7. The cable-driven mechanical tracking control method based on TDE and recursive non-singular terminal sliding mode according to claim 6, characterized in that: The logic behind extracting the generalized output vector of the joint controller containing turbulence parameters based on the updated TDE simplified equation is as follows: update the lumped unknown dynamic terms and the motion disturbance terms injected into the joint in the cable-driven manipulator tracking control in the TDE simplified equation to update the generalized output vector of the joint controller. The updated motion disturbance term for the injected joint is specifically represented as follows: In the formula, Let be the second derivative of the desired motion trajectory position of the robotic arm joint with respect to time; The updated lumped unknown dynamic term for the tracking control of the cable-driven manipulator is specifically represented as follows: In the formula, for The generalized output vector of the moment joint controller. For turbulence parameters, for The second derivative of the desired motion trajectory position of the manipulator joint with respect to time at any given moment; Based on the updated motion disturbance terms of the injected joints and the updated lumped unknown dynamic terms of the tracking control of the rope-driven manipulator, a complete control law is formed for the generalized output vector of the joint controller of the rope-driven manipulator to be controlled.

Images