Adaptive fractional order nonsingular terminal sliding mode control method based on time delay estimation

Abstract

The application provides an adaptive fractional order non-singular terminal sliding mode control method based on time delay estimation, relates to the technical field of manipulator control, and establishes a dynamics model of joint and motor coupling; a non-singular terminal sliding mode surface containing fractional order differentiation and power transformation is designed, and then a reaching law with variable gain is constructed; then, unknown dynamics and disturbance of the system are compensated in real time by using historical control input and state information through time delay estimation technology; finally, the control law is generated by comprehensively synthesizing the sliding mode surface, the reaching law and the time delay estimation compensation term, high-precision and strong-robust control of the uncertain nonlinear system is realized, and the tracking precision and anti-interference ability of the manipulator in the uncertain environment are effectively improved.

Description

Technical Field

[0001] This invention relates to the field of robotic arm control technology, specifically to an adaptive fractional-order nonsingular terminal sliding mode control method based on time delay estimation. Background Technology

[0002] With the rapid development of industrial automation technology, robotic arms have been widely used in underwater construction, manufacturing, and daily services. However, traditional underwater robotic arms have significant drawbacks due to their heavy weight and bulky structure: the large mass can cause significant disturbances to underwater vehicles, affecting their balance and stability; the bulky structure can easily obstruct ocean currents, causing significant hydrodynamic effects, thus limiting their motion performance and application range. To address these issues, a rope-driven robotic arm has been proposed as a novel solution. By arranging all the drive motors inside the vehicle's cabin, the moving parts of the robotic arm are significantly lighter, and it can be designed with an open structure, significantly reducing interference from ocean currents and hydrodynamics, and improving the system's environmental adaptability and motion flexibility.

[0003] Despite the structural advantages of rope-driven manipulators, they still face significant control challenges. The long, flexible drive cables introduce elastic deformation and clearance, causing the system to exhibit non-isotropic characteristics, making it impossible for the motor-side encoder to accurately reflect joint positions. Friction and clearance at the pulleys vary with tension, temperature, and ambient pressure, introducing nonlinear factors that are difficult to model accurately. Furthermore, external disturbances such as ocean currents or contact forces directly act on the linkages, further increasing the complexity of controller design. These factors collectively constitute a highly nonlinear, uncertain, and time-varying dynamic system. Traditional integer-order sliding mode control or PID strategies struggle to achieve a good balance between tracking accuracy, robustness, and control smoothness, often requiring high gain to suppress uncertainty, which can easily lead to chattering or performance degradation.

[0004] To address the aforementioned issues, researchers have recently introduced fractional calculus into the sliding mode control framework, proposing a fractional sliding mode control method. This method enhances the system's adjustment flexibility through non-integer derivatives and integrals, achieving finite-time convergence while avoiding singularities. Furthermore, time delay estimation techniques have rapidly developed due to their model-free and computationally simple characteristics. They can utilize historical states and control inputs to estimate and compensate for unmodeled dynamics and integrated external disturbances in the system, effectively reducing dependence on precise models. Against this backdrop, this invention combines fractional non-singular terminal sliding mode control with time delay estimation, aiming to design a highly adaptive, robust, and easily implemented control strategy for cable-driven manipulators to improve their trajectory tracking performance and anti-interference capabilities in complex underwater environments. Summary of the Invention

[0005] The purpose of this invention is to provide an adaptive fractional-order nonsingular terminal sliding mode control method based on time delay estimation to solve the problems mentioned in the background art.

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

[0007] An adaptive fractional-order nonsingular terminal sliding mode control method based on time delay estimation includes the following steps:

[0008] Step 1: Establish the joint dynamic equations and motor dynamic equations of the cable-driven manipulator, and combine the two to construct a dynamic model;

[0009] Step 2: Calculate the tracking error and its first derivative of the joint trajectory of the cable-driven manipulator, and then construct a fractional-order non-singular terminal sliding surface. The fractional-order non-singular terminal sliding surface consists of a tracking error velocity term, a first processing term and a second processing term obtained based on two fractional-order differential processing of the tracking error position.

[0010] Step 3: Construct a sliding surface reaching law based on a fractional-order nonsingular terminal sliding surface. The reaching law includes a linear gain term and a nonlinear power term.

[0011] Step 4: Compensate the dynamic model based on historical control inputs to generate time delay estimation compensation terms. Generate the control law for the rope-driven manipulator based on the fractional non-singular terminal sliding surface, the sliding surface reaching law, and the time delay estimation compensation terms.

[0012] Furthermore, the principle for constructing the dynamic model is as follows:

[0013] The joint dynamic equation is:

[0014]

[0015] in, Denotes the positive definite inertia matrix. Represents the joint position vector. Represents the joint velocity vector. Represents the joint acceleration vector. Represents the matrix of Coriolis force and centrifugal force. Represents the gravity vector. Represents the friction force vector. Represents the lumped interference vector. Represents the joint compliance torque vector;

[0016] The dynamic equation of the motor is:

[0017]

[0018] in, Represents the motor inertia matrix. Represents the motor velocity vector. This represents the motor acceleration vector. This represents the motor damping matrix. Represents the motor torque vector;

[0019] By combining the joint dynamic equations and the motor dynamic equations, and introducing a constant matrix, a dynamic model is constructed, as follows:

[0020]

[0021]

[0022] in, Represents a constant matrix. This represents complex dynamic terms in the system.

[0023] Furthermore, the formula used to construct the fractional-order nonsingular terminal sliding surface is as follows:

[0024]

[0025]

[0026]

[0027]

[0028]

[0029]

[0030] in, Denotes a fractional-order non-singular terminal sliding surface. Indicates the location of the tracking error. Represents the desired joint trajectory vector. This represents the tracking error velocity vector, i.e., the tracking error velocity term. for The derivative of represents the desired joint velocity vector. This represents a constant matrix whose diagonal elements are pre-defined positive real numbers. Indicates order is Fractional differential operators, Indicates order is Fractional differential operators, The fractional order of the differential. To improve the convergence parameter vector of control performance, and , , Indicates the first Convergence parameters for each degree of freedom The index representing the degrees of freedom of the cable-driven manipulator, and , This indicates the number of degrees of freedom of a cable-driven manipulator. Indicates the first The convergence parameters for each degree of freedom, and , , Indicates the first The tracking error location for each degree of freedom is... In the The component of each degree of freedom, express The absolute value, Indicates the first processing item. This indicates the second processing item.

[0031] Furthermore, the expression for the sliding surface reaching law is:

[0032]

[0033]

[0034]

[0035] in, This represents the sliding surface approach law. Represents the linear gain coefficient. Represents the linear gain term. Represents a variable gain function. Represents a nonlinear power term. Let represent the power parameter vector, and , Indicates the first A power-order parameter vector with n degrees of freedom, and , Indicates the first The fractional-order non-singular terminal sliding surface corresponding to each degree of freedom is: In the The component of each degree of freedom, express The absolute value, Represents the base gain coefficient. This indicates the adjustment gain coefficient. express The length of the mold, The critical value representing the size of the sliding surface. This represents the sensitivity control coefficient.

[0036] Furthermore, the formula for constructing the delay estimation compensation term is as follows:

[0037]

[0038] in, Indicates the current time The delay estimation compensation term, Indicates the current moment. Indicates the delay time. Indicates the current time interval A historic moment Representing historical moments The motor torque vector, Representing historical moments The joint acceleration vector.

[0039] Furthermore, the principle underlying the control law for generating the rope-driven manipulator is as follows:

[0040]

[0041] in, Indicates the current time The motor torque vector, i.e., the current moment. The corresponding control law for the cable-driven manipulator. The auxiliary control vector is represented by the auxiliary control vector, which is calculated from the fractional-order non-singular terminal sliding surface and the sliding surface reaching law.

[0042] Furthermore, the principle underlying the calculation of the auxiliary control vector is as follows:

[0043] An auxiliary control vector is constructed based on the derivative and reaching law of the fractional-order non-singular terminal sliding surface. The specific formula is as follows:

[0044]

[0045] in, Let the desired joint acceleration vector be denoted as . The derivative of .

[0046] Compared with the prior art, the beneficial effects of the present invention are:

[0047] This invention couples joint dynamics with motor dynamics to construct a dynamic model that more realistically reflects the actual motion characteristics of the cable-driven manipulator. This provides accurate mathematical model support for subsequent sliding surface design and time delay estimation and compensation, enabling the systematic handling of the dynamic coupling between the joint and the motor according to the control law, thereby improving the adaptability and accuracy of the overall control. Furthermore, it constructs an adaptive fractional-order non-singular terminal sliding surface, introducing fractional-order differential operators and power transformations. While ensuring non-singularity, it achieves rapid convergence of tracking errors within a finite time. The introduction of fractional-order operators enhances the system's dynamic adjustment capability, avoids singularities, and improves transient and steady-state performance.

[0048] This invention also constructs an adaptive reaching law comprising linear terms, a variable gain function, and nonlinear power terms. This law adaptively adjusts the convergence speed and gain intensity based on the size of the sliding surface, achieving dynamic adjustment that enables rapid approach when far from the sliding surface and smooth transition when approaching it. This reduces the inherent chattering phenomenon of sliding mode control while maintaining strong robustness, allowing the system to converge stably and quickly even under uncertainties. Furthermore, by utilizing historical control inputs and acceleration information to construct a time delay estimation compensation term, it compensates for unknown dynamics and disturbances in the system in real time. This reduces the dependence on an accurate system model and enhances the controller's robustness to unmodeled dynamics and external disturbances. Combining the time delay estimation compensation term with a fractional-order sliding surface for control improves the system's tracking accuracy in complex environments. Attached Figure Description

[0049] Figure 1 This is a schematic diagram of the method flow of an embodiment of the present invention. Detailed Implementation

[0050] 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.

[0051] 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.

[0052] Please see Figure 1 The present invention provides a technical solution:

[0053] An adaptive fractional-order nonsingular terminal sliding mode control method based on time delay estimation includes the following steps:

[0054] Step 1: Establish the joint dynamic equations and motor dynamic equations of the cable-driven manipulator, and combine the two to construct a dynamic model;

[0055] In this embodiment, the principle for constructing the dynamic model is as follows:

[0056] The joint dynamic equation is:

[0057]

[0058] in, Denotes the positive definite inertia matrix. Represents the joint position vector. Represents the joint velocity vector. Represents the joint acceleration vector. Represents the matrix of Coriolis force and centrifugal force. Represents the gravity vector. Represents the friction force vector. Represents the lumped interference vector. Represents the joint compliance torque vector;

[0059] The joint dynamics equations reflect the multi-physics coupling characteristics of the cable-driven manipulator. Specifically, these include: the existence of inertia, centrifugal force, and Coriolis force coupling between the joints of the manipulator, where the movement of one joint affects the dynamic behavior of other joints; the concentration of uncertainties, including friction and external disturbance forces, which are represented as a lumped disturbance vector; and control coupling, where the joint compliance torque is transmitted by the motor through a flexible cable, resulting in elastic deformation and transmission delay, leading to coupling and time delay between the control input and the joint response.

[0060] This represents the inertial force generated by the system during acceleration, and... Proportional This represents the joint acceleration vector, with dimensions of... ,and for The derivative, This represents the joint acceleration vector, with dimensions of... , for The derivative, This represents the joint position vector, with dimensions of... , This indicates the number of degrees of freedom of a cable-driven manipulator. It is a size of The symmetric positive definite matrix, called the inertia matrix, is used to describe the inertial distribution and coupling relationship of each link in the manipulator during motion. It is calculated using the Lagrange method. Taking a two-degree-of-freedom manipulator as an example, there are links 1 and 2, and the mass of link 1 is... , length is The moment of inertia is The joint rotation angle is ,and The mass of link 2 corresponding to the joint between link 1 and the base is... , length is The moment of inertia is The joint rotation angle is ,and For the joints of link 1 and link 2, the formula used to calculate the position and velocity of the center of mass of each link is:

[0061]

[0062]

[0063]

[0064]

[0065] in, These represent the x and y coordinates of the center of mass of link 1, respectively. These represent the x and y coordinates of the center of mass of link 2, respectively.

[0066] To each Taking the first derivative, we obtain the center of mass of link 1 at... direction and The linear velocities in the directions are respectively , respectively Taking the first derivative, we obtain the center of mass of link 2 at... direction and The linear velocities in the directions are respectively ;

[0067] The kinetic energy of connecting rod 1 and connecting rod 2 is then calculated using the following formula:

[0068]

[0069]

[0070] in, This represents the kinetic energy of link 1. This represents the angular velocity of joint 1. This represents the kinetic energy of link 2. This represents the angular velocity of joint 2;

[0071] The total kinetic energy of the system is: ;

[0072] Summarized as follows: ,in This represents the equivalent moment of inertia of link 1. This indicates inertial coupling between the links. This represents the equivalent moment of inertia of link 2; and , , .

[0073] Coriolis force and centrifugal force matrix This term describes the velocity coupling between joints. When there is relative motion between different joints of the robot, a Coriolis force is generated due to the influence of the rotating reference frame. When the joints rotate at a certain speed, a centrifugal force is generated due to inertia. It is a function of joint position and joint velocity, calculated using Christofel's symbolic formula.

[0074] Gravity vector This represents the force generated by gravity acting on each joint of the manipulator, reflecting the static influence of gravity on each joint under different postures. During movement, gravity affects the acceleration and torque requirements of the joints, which must be compensated for in the control system; otherwise, it will lead to an increase in tracking error. This is obtained through simulation modeling of the manipulator.

[0075] Friction force vector This represents the vectorized representation of the frictional forces generated by the joints of a cable-driven manipulator during its movement due to factors such as contact, lubrication, and material deformation. It reflects the nonlinear frictional effects experienced by the manipulator during actual operation, including Coulomb friction and viscous friction. Friction can lead to increased tracking errors and response lag. If compensation is not performed in the control, it will reduce the tracking performance and robustness of the system. By collecting the manipulator's position and velocity under a known input torque, the frictional force parameters are fitted using the least squares method.

[0076] Lumped interference vector The dimension is the same as the degrees of freedom of the manipulator, and it includes all disturbance factors that are not considered separately in the dynamic model. It is the sum of these disturbance factors, including external disturbances such as ocean current impact, environmental erosion, and wind disturbance; internal unmodeled dynamics such as elastic deformation of the transmission mechanism, clearance changes, and parameter drift caused by temperature; and model errors, including deviations between the dynamic model and the actual system caused by inaccurate parameters. In the early stage of controller design, common disturbance situations are simulated using the multibody dynamics simulation software ADAMS to obtain the data characteristics of the disturbances and construct a lumped disturbance vector.

[0077] Joint compliance torque vector This represents the torque vector transmitted from the motor to the joint via a flexible cable. It reflects the torque transmission characteristics of the cable-driven manipulator during transmission due to nonlinear factors such as the elastic deformation, gaps, and friction of the cable. When the cable is subjected to force, it elastically expands and contracts, causing a difference between the motor's output torque and the actual torque received by the joint. This reflects the impact of this elastic deformation on torque transmission; due to factors such as pulley friction and clearance variations, torque transmission is not perfectly rigid and linear. This includes the effects of these nonlinear transmission characteristics; at the same time, This also reflects the kinematic coupling between joints and the difference in dynamic response between the motor and the joint. In the joint dynamic equation, It is the external input torque vector that drives the joint movement. The controller indirectly controls the joint by adjusting the motor torque. .

[0078] The joint dynamics equations fully describe the dynamic behavior of the cable-driven manipulator, including inertial coupling, velocity coupling, gravity effects, friction effects, external combined disturbances, as well as transmission elasticity and transmission backlash.

[0079] The dynamic equation of the motor is:

[0080]

[0081] in, Represents the motor inertia matrix. Represents the motor velocity vector. This represents the motor acceleration vector. This represents the motor damping matrix. Represents the motor torque vector;

[0082] The motor dynamics equations reflect the dynamic coupling relationship between the motor and the joint in a cable-driven manipulator system, describing the dynamic behavior of the motor. Specifically, they reflect the following physical processes: motor inertia effect, representing the inertial torque generated by the motor rotor during acceleration or deceleration; motor damping effect, representing the viscous damping torque experienced by the rotor during motion; and the balance between the motor output torque and the load torque. By combining the motor-side dynamics and the joint-side dynamics through the motor dynamics equations, a unified control model is formed, facilitating the design of overall control strategies. Furthermore, in cable-driven systems, the cable exhibits nonlinear factors such as elasticity and clearance, meaning the motor output torque is not directly equal to the torque received by the joint. The motor dynamics equations reflect this torque transmission loss.

[0083] The motor inertia matrix is ​​a matrix of size 1. A positive definite matrix, usually a diagonal matrix, where each element on the diagonal represents the moment of inertia of the motor, reflecting the inertial characteristics of the motor rotor and its transmission components in rotational motion, that is, the amount of torque required by the motor when accelerating or decelerating. This represents the inertial torque required for the motor to accelerate, reflecting the influence of the motor rotor inertia on the system's dynamic response. A larger torque indicates a slower motor response, requiring more torque to quickly track the target trajectory. A smaller value indicates that the motor responds faster and is more sensitive to external disturbances. The motor inertia matrix can be obtained by consulting the motor's moment of inertia.

[0084] The motor velocity vector is The vector represents the angular velocity of each motor rotor. In a rope-driven system, the motor drives the joint via a cable. The motor speed and joint speed are correlated through the transmission ratio, reflecting the motion coupling between the motor and the joint. The motor speed and output torque together determine the motor's output power, affecting the system's dynamic response and energy consumption. The motor speed vector is used as a feedback signal to calculate the tracking error and sliding surface. The motor speed is measured by a Hall sensor.

[0085] Motor damping matrix It is a size of The diagonal matrix represents the viscous damping effect experienced by the motor during rotation, that is, the resistance torque generated by the motor rotor due to factors such as friction, air resistance, and electromagnetic damping, which is proportional to the speed. The damping torque is proportional to the speed and opposite in direction to the speed. It reflects the energy consumed by the motor during its movement due to internal friction and the resistance of the external medium, and embodies the energy dissipation characteristics of the manipulator system. Larger damping means that the system will consume kinetic energy faster during movement, which helps to suppress oscillations, but will also reduce the response speed. Smaller damping may cause the system to respond too quickly, easily resulting in overshoot or oscillation. The method to obtain the motor damping matrix is ​​as follows: under no-load or known load conditions, apply a sinusoidal torque input to the motor, measure the motor speed and acceleration, and fit the motor damping using the least squares method.

[0086] Motor torque vector This represents the total driving torque applied by the motor to the joints of the cable-driven robotic arm through the drive circuit and transmission device; its dimension is the same as the robotic arm's degrees of freedom, with each component corresponding to the driving torque of one joint; it reflects the actual torque output by the motor, which is used to overcome the following three parts, including the motor's own inertial torque, i.e. Motor damping torque, i.e. and the load torque transmitted to the joint. In a rope-driven system, and They are not equal; the difference between the two reflects nonlinear characteristics such as elastic deformation, friction loss, and clearance during the transmission process.

[0087] By combining the joint dynamic equations and the motor dynamic equations, and introducing a constant matrix, a dynamic model is constructed, as follows:

[0088]

[0089]

[0090] in, Represents a constant matrix. This represents complex dynamic terms in the system.

[0091] The principle of constructing a dynamic model is as follows:

[0092] Based on the solution of the motor dynamics equations ,have to The solution will be obtained Substituting into the joint dynamics equation, we get: Rearranging the terms, we get: In the typical working environment of a robotic arm, calculation The values ​​at different joint positions are averaged and used as elements in the constant matrix. If the complete inertia matrix were used directly, the subsequent control law would involve calculating its inverse matrix, increasing computational complexity. Introducing the constant matrix simplifies the calculation. Split into ,in For the known part, The modeling error component is incorporated into the system's complex dynamics. In the middle, the equation is simplified to: Complex dynamic terms of the system Includes except All dynamics other than those modeled are the sum of all model errors, unmodeled dynamics, and external disturbances, and are a single-dimensional dynamic. In real systems, these terms do not act independently but are coupled together to affect system dynamics, so they are estimated as a whole.

[0093] The dynamic model reflects the comprehensive dynamic characteristics of the cable-driven manipulator system in terms of multi-physics coupling, uncertainty concentration, and nonlinear transmission. Physical coupling characteristics include joint-motor coupling, specifically combining joint dynamics with motor dynamics to reflect the dynamic coupling relationship between the motor's output torque and the actual torque received by the joint. Nonlinear factors such as cable elastic deformation and gaps cause transmission delays and torque losses between the two. Inertial coupling between joints reflects the inertial distribution and mutual influence of each joint during motion; that is, the movement of one joint affects the acceleration requirements of other joints. Velocity coupling effects, with the Coriolis force and centrifugal force matrices demonstrating the influence of joint velocity on inter-joint interactions, are particularly significant during high-speed or multi-degree-of-freedom motion. Uncertainties are concentrated in the model, including nonlinear friction and gravity, as well as lumped disturbances. The model includes gravity and friction vectors, reflecting the nonlinear static and dynamic forces experienced by the manipulator under different postures and velocities. All unmodeled or difficult-to-model external disturbances and internal dynamics are represented in a concentrated manner, enhancing the model's robustness. Nonlinear transmission includes the effects of elastic transmission characteristics and motor dynamics; the joint compliance torque vector reflects the influence of cable elastic deformation on torque transmission, indicating a nonlinear mapping between the motor output torque and the actual joint torque. The motor inertia and damping matrices reflect the motor's dynamic response characteristics, including rotor inertial effects and energy dissipation. The known and unknown complexities are discussed separately to facilitate real-time compensation using time delay estimation techniques. It provides the foundation.

[0094] Step 2: Calculate the tracking error and its first derivative of the joint trajectory of the cable-driven manipulator, and then construct a fractional-order non-singular terminal sliding surface. The fractional-order non-singular terminal sliding surface consists of a tracking error velocity term, a first processing term and a second processing term obtained based on two fractional-order differential processing of the tracking error position.

[0095] In this embodiment, the formula used to construct the fractional-order non-singular terminal sliding surface is as follows:

[0096]

[0097]

[0098]

[0099]

[0100]

[0101]

[0102] in, Denotes a fractional-order non-singular terminal sliding surface. Indicates the location of the tracking error. Represents the desired joint trajectory vector. This represents the tracking error velocity vector, i.e., the tracking error velocity term. for The derivative of represents the desired joint velocity vector. This represents a constant matrix whose diagonal elements are pre-defined positive real numbers. Indicates order is Fractional differential operators, Indicates order is Fractional differential operators, The fractional order of the differential. To improve the convergence parameter vector of control performance, and , , Indicates the first Convergence parameters for each degree of freedom The index representing the degrees of freedom of the cable-driven manipulator, and , This indicates the number of degrees of freedom of a cable-driven manipulator. Indicates the first The convergence parameters for each degree of freedom, and , , Indicates the first The tracking error location for each degree of freedom is... In the The component of each degree of freedom, express The absolute value, Indicates the first processing item. This indicates the second processing item.

[0103] The purpose of constructing a fractional-order nonsingular terminal sliding surface is to achieve high-precision tracking within a finite time, while avoiding singularity problems and enhancing the system's adaptability to nonlinearity, uncertainty, and external disturbances. The fractional-order nonsingular terminal sliding surface includes a tracking error velocity term, a first processing term, and a second processing term. Traditional sliding mode control often only guarantees asymptotic convergence, meaning the error tends to zero over time but not within a finite time. Terminal sliding mode control, by introducing a power term, enables the system state to converge to the sliding surface within a finite time, thus achieving finite-time tracking. Once the sliding surface converges to zero, it means the system state has entered a preset ideal trajectory, at which point the tracking error will dynamically converge according to a predetermined path. Therefore, the core objective of the controller design is to drive... Converging to zero, The expression is an error dynamic equation that defines how the error evolves over time and determines the convergence path of the system from the current state to the target state. This is achieved by setting... , This ensures that the control law does not diverge when the error is close to zero, and also achieves convergence to zero within a finite time.

[0104] Tracking error speed term This represents the deviation between the desired joint velocity and the actual joint velocity, reflecting the current trend of the system's error. Introducing the velocity error into the sliding surface makes the sliding surface not only dependent on the position error but also directly related to the rate of change of the error, which helps to achieve a faster convergence speed near the sliding surface. , representing the error vector between the expected joint velocity vector and the joint velocity vector. This represents the desired velocity vector, with dimension . Taking its derivative yields a dimension of The expected joint acceleration vector , For the dimension of The expected joint trajectory vector The derivative is obtained.

[0105] First processing item This is used to introduce fractional-order dynamic adjustment capability, through fractional-order differential operators. For power functions that exhibit errors Differentiation introduces the dynamic properties of fractional order. The power parameter in Combined with fractional derivatives, this term provides a strong regulating effect when the error is large, driving the system to converge to the sliding surface in a finite time. Its design principle encompasses both mathematical form and physical meaning. The mathematical form includes: This is to perform a nonlinear transformation on the tracking error, enhancing the ability to adjust the weight of the error in the sliding surface, and employing... The fractional derivative of the transformed error is given by the order of... Stay To achieve smooth dynamic adjustment and avoid high-frequency jitter that may be caused by integer-order derivatives, its initial value is set to 0.01. This is a diagonal positive definite matrix used to adjust the weights of this term across different degrees of freedom, achieving decoupled control of the multi-degree-of-freedom system. Its physical meaning includes: the first processing term is essentially a dynamic compensation term for position errors, introducing historical references for error changes through fractional derivatives, enabling the controller to not only respond to the current error but also consider the trend of error changes. Singularity problems often occur in traditional terminal sliding mode, especially when the error approaches zero. Here, a combination of fractional derivatives and power functions is used to construct a function that remains well-defined and continuous even when the error is zero, thus avoiding singularities. When the error is large, it ensures strong convergence; when the error approaches zero, the fractional derivative maintains smooth adjustment, avoiding chattering.

[0106] Second processing item Its main functions are to achieve smooth convergence and suppress chattering, enhance steady-state accuracy, work with the first processing term to achieve two-stage convergence, and provide fractional-order dynamic compensation; the second processing term plays a dominant role when the error is small, because When the error When approaching zero, The derivative is infinite, which is achieved here by using the fractional differential operator. Smoothing is achieved to avoid singularities, while ensuring smooth changes in control quantities as the system approaches equilibrium, effectively suppressing chattering in sliding mode control. As the system approaches the desired trajectory, the second processing term, through the memory effect of fractional derivatives and nonlinear characteristics with powers less than 1, provides finer error adjustment capabilities, thereby improving the steady-state tracking accuracy of the system. When the error is large in the first stage, the first processing term dominates, driving the system to quickly approach the sliding surface; when the error is small in the second stage, the second processing term dominates, achieving smooth, singular-free terminal convergence. (Setting...) This is to construct a terminal attractor that still has a preferential convergence time when the error approaches zero, while combining fractional derivatives to avoid singular problems in traditional terminal sliding mode. The order of its fractional derivative is... , usually take , The first term is a diagonal positive definite matrix used to adjust the gain of this term at different degrees of freedom, so as to achieve independent adjustment of the multi-degree-of-freedom system; the second term, when the error is small, adjusts the weight of the error to achieve fine compensation of the system dynamics.

[0107] A fractional-order non-singular terminal sliding surface is constructed based on the tracking error velocity term, the first processing term, and the second processing term. The tracking error velocity term reflects the current dynamic deviation of the system from the desired trajectory, ensuring that the sliding surface contains first-order dynamic information, enabling the control law to directly respond to the velocity error. The first processing term introduces fractional-order derivatives and power transformations to enhance the dynamic adjustment capability of the position error, providing strong convergence force when the error is large and accelerating the approach to the sliding surface. The second processing term is also based on fractional-order derivatives and plays a dominant role when the error is small, achieving smooth, non-singular terminal convergence.

[0108] Step 3: Construct a sliding surface reaching law based on a fractional-order nonsingular terminal sliding surface. The reaching law includes a linear gain term and a nonlinear power term.

[0109] In this embodiment, the expression for the sliding surface reaching law is:

[0110]

[0111]

[0112]

[0113] in, This represents the sliding surface approach law. Represents the linear gain coefficient. Represents the linear gain term. Represents a variable gain function. Represents a nonlinear power term. Let represent the power parameter vector, and , Indicates the first A power-order parameter vector with n degrees of freedom, and , Indicates the first The fractional-order non-singular terminal sliding surface corresponding to each degree of freedom is: In the The component of each degree of freedom, express The absolute value, Represents the base gain coefficient. This indicates the adjustment gain coefficient. express The length of the mold, The critical value representing the size of the sliding surface. This represents the sensitivity control coefficient.

[0114] The sliding surface reaching law defines fractional-order nonsingular terminal sliding surfaces. How to converge to zero over time, so that the system can approach from any initial state in a finite amount of time? and remain The upward motion achieves convergence of the tracking error; the sliding surface approach law reflects the balance between rapid convergence and smooth adjustment of the controller. When far from the sliding surface, rapid convergence is dominant; when approaching, smooth adjustment is dominant; near the sliding surface, the nonlinear term dominates, achieving convergence within a finite time. This is achieved by setting... and variable gain function The system constructs a smooth transition mechanism, which suppresses chattering caused by high-frequency switching in traditional sliding mode control; it also reflects the decoupling capability of multi-degree-of-freedom systems, with each degree of freedom corresponding to an independent gain and power function, which can be independently adjusted for different joint dynamic characteristics.

[0115] The sliding surface reaching law consists of a linear gain term and a nonlinear power term. The linear gain term reflects the exponential convergence trend of the sliding surface; that is, when the system state deviates far from the sliding surface, the controller will quickly pull it back to the sliding surface in a linear manner, ensuring system stability and fast initial convergence. The linear gain term ensures global stability, guaranteeing that the system converges to the sliding surface under any initial state, avoiding divergence. It also ensures a smooth transition; near the sliding surface, the linear gain term dominates the convergence process, making the control variable change smoothly and reducing chattering. The design of the linear gain term is based on classical sliding mode control theory. It converges to zero at an exponential rate, with the convergence speed increasing from... control, The larger the value, the faster the convergence, but... An excessively large value will cause drastic changes in the control variable, easily leading to chattering. Its value range is... The initial value is set to 1; the nonlinear power term reflects the idea of ​​sliding mode control converging within a finite time and suppressing chattering. It has nonlinear characteristics, when the sliding surface When it is large, It provides strong convergence capability, accelerating the system's approach to the sliding surface, when When the value is small, the function region is smooth, avoiding abrupt changes in the control signal and suppressing chattering; variable gain function The gain coefficient is dynamically adjusted based on the size of the sliding surface; when constructing the variable gain function, the base gain coefficient is... To ensure that the system maintains minimum gain even with a small sliding surface, and to avoid slow or stalled convergence due to excessively low gain, its value is determined based on the system's minimum convergence requirements. Simultaneously, the magnitude of external disturbances is considered to ensure that the base gain is sufficient to offset low-frequency disturbances. Typically, a value of [value missing] is used. ,Will The initial value is set to 1, and gradually increased until the convergence requirement is met; For adaptive adjustment, Normalize the size of the sliding surface. express The length of the mold, Used to determine whether the size of the sliding surface exceeds a threshold. If the index is positive, it indicates that the surface is moving away from the sliding surface. Follow Increasing the gain accelerates convergence; if... A negative exponent indicates that the device is approaching the sliding surface. Follow Decrease the gain, thus suppressing chattering; To adjust the gain coefficient, which controls the range of gain variation with the size of the sliding surface and determines the degree of gain increase as the system moves away from the equilibrium point, in Larger values ​​enhance control effectiveness, and stronger nonlinearity results in greater control action. The larger the value, the faster the required convergence speed. The larger, The range is Sensitivity control coefficient Used to control the sensitivity of gain to changes in the size of the sliding surface. The larger the value, the more sensitive it is to changes in gain. The smaller the value, the smoother the gain change; typically, a value of [value missing] is used. Start with a small value and gradually increase it until the gain change meets the convergence requirement.

[0116] Step 4: Compensate the dynamic model based on historical control inputs to generate time delay estimation compensation terms. Generate the control law for the rope-driven manipulator based on the fractional non-singular terminal sliding surface, the sliding surface reaching law, and the time delay estimation compensation terms.

[0117] In this embodiment, the formula for constructing the delay estimation compensation term is:

[0118]

[0119] in, Indicates the current time The delay estimation compensation term, Indicates the current moment. Indicates the delay time. Indicates the current time interval A historic moment Representing historical moments The motor torque vector, Representing historical moments The joint acceleration vector.

[0120] The purpose of constructing the time delay estimation compensation term is to use the system's historical state and control input information to approximate the unknown dynamics and disturbances at the current moment, thereby achieving real-time compensation for the complex dynamic parts of the system. This is based on the assumption that the unmodeled dynamics and external disturbances of the system change slowly over a very short time interval; therefore, the system response at the previous moment can be used to approximate the dynamics at the current moment. The system's dynamic model is as follows: ,in It includes all unmodeled dynamics, nonlinear terms, friction, and external disturbances, within a very short time delay. Therefore, in the time delay is hour, The time delay estimation compensation term reflects the system's real-time estimation capability for unmodeled dynamic disturbances. The value is set to 1ms. When the robotic arm moves slowly, it can be increased appropriately, but it should be below 10ms.

[0121] The principle underlying the control law for generating the rope-driven robotic arm is:

[0122]

[0123] in, Indicates the current time The motor torque vector, i.e., the current moment. The corresponding control law for the cable-driven manipulator. The auxiliary control vector is represented by the auxiliary control vector, which is calculated from the fractional-order non-singular terminal sliding surface and the sliding surface reaching law.

[0124] The control law reflects the combination of model compensation, time delay estimation and sliding mode control. The control law consists of two parts: a term that compensates for the known dynamics of the system and a term that provides robustness to uncertainties and disturbances. It is a term of the known dynamics of the compensated system. It is a compensation term based on historical information, and the system will calculate the desired acceleration command. The inertia matrix is ​​converted into torque, and then compensation for unknown dynamics is added to obtain the final control law. A constant matrix is ​​used as an estimate of the inertia term, combined with the desired acceleration and fractional-order error correction term, to form the feedforward compensation part, which is used to realize the basic dynamic response of the desired tracking trajectory. This term ensures that the derivative of the sliding surface conforms to a preset reaching law, guaranteeing that the system can reach and maintain the sliding surface within a finite time. It introduces... This method utilizes historical control inputs and acceleration information to estimate and compensate for unknown dynamics and disturbances in the system, ultimately enabling the control law to achieve precise trajectory tracking and disturbance suppression, and ensuring convergence to the sliding surface within a finite time, thereby improving response speed.

[0125] The principle underlying the calculation of the auxiliary control vector is as follows:

[0126] An auxiliary control vector is constructed based on the derivative and reaching law of the fractional-order non-singular terminal sliding surface. The specific formula is as follows:

[0127]

[0128] in, Let the desired joint acceleration vector be denoted as . The derivative of .

[0129] The auxiliary control vector is derived based on the mathematical relationship between the derivative of the fractional nonsingular terminal sliding surface and the sliding surface reaching law. This represents the ideal calculated value derived from the definition of the sliding surface, through the... We use differentiation to make inferences.

[0130] Based on the system dynamics model and control law structure, the model simplifies to the following in an ideal case: ,Right now ,right Differentiate, the result is ,in, for The derivative of represents the error vector between the desired joint acceleration vector and the joint acceleration vector, i.e. Therefore, The result of differentiation is Make it equal to The auxiliary control vector is obtained. This is used to compensate for the deviation between the joint acceleration and the desired value.

[0131] 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.

[0132] 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.

[0133] 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 according to actual needs.

[0134] 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 time delay estimation based adaptive fractional order non-singular terminal sliding mode control method, characterized in that, The specific steps include: Step 1: Establish the joint dynamic equations and motor dynamic equations of the cable-driven manipulator, and combine the two to construct a dynamic model; Step 2: Calculate the tracking error and its first derivative of the joint trajectory of the cable-driven manipulator, and then construct a fractional non-singular terminal sliding surface. The fractional non-singular terminal sliding surface consists of a tracking error velocity term, a first processing term and a second processing term obtained based on two fractional differential processing of the tracking error position. Step 3: Construct a sliding surface reaching law based on a fractional-order nonsingular terminal sliding surface. The reaching law includes a linear gain term and a nonlinear power term. Step 4: Compensate the dynamic model based on historical control inputs to generate time delay estimation compensation terms. Generate the control law for the rope-driven manipulator based on the fractional non-singular terminal sliding surface, the sliding surface reaching law, and the time delay estimation compensation terms. The principle of constructing a dynamic model is as follows: The joint dynamic equation is: in, Denotes the positive definite inertia matrix. Represents the joint position vector. Represents the joint velocity vector. Represents the joint acceleration vector. Represents the matrix of Coriolis force and centrifugal force. Represents the gravity vector. Represents the friction force vector. Represents the lumped interference vector. Represents the joint compliance torque vector; The dynamic equation of the motor is: in, Represents the motor inertia matrix. Represents the motor velocity vector. This represents the motor acceleration vector. This represents the motor damping matrix. Represents the motor torque vector; By combining the joint dynamic equations and the motor dynamic equations, and introducing a constant matrix, a dynamic model is constructed, as follows: wherein, represents a constant matrix, represents a system complex dynamic term; The formula used to construct fractional-order nonsingular terminal sliding surfaces is: in, Denotes a fractional-order non-singular terminal sliding surface. Indicates the location of the tracking error. Represents the desired joint trajectory vector. This represents the tracking error velocity vector, i.e., the tracking error velocity term. for The derivative of represents the desired joint velocity vector. This represents a constant matrix whose diagonal elements are pre-defined positive real numbers. Indicates order is Fractional differential operators, Indicates order is Fractional differential operators, The fractional order of the differential. To improve the convergence parameter vector of control performance, and , , Indicates the first Convergence parameters for each degree of freedom The index representing the degrees of freedom of the cable-driven manipulator, and , This indicates the number of degrees of freedom of a cable-driven manipulator. Indicates the first The convergence parameters of each degree of freedom, and , , Indicates the first The tracking error location for each degree of freedom is... In the The component of one degree of freedom, express The absolute value, Indicates the first processing item. Indicates the second processing item; The expression for the sliding surface reaching law is: in, This represents the sliding surface approach law. Represents the linear gain coefficient. Represents the linear gain term. Represents a variable gain function. Represents a nonlinear power term. Let represent the power parameter vector, and , Indicates the first A power-order parameter vector with n degrees of freedom, and , Indicates the first The fractional-order non-singular terminal sliding surface corresponding to each degree of freedom is: In the The component of each degree of freedom, express The absolute value, Represents the base gain coefficient. This indicates the adjustment gain coefficient. express The length of the mold, The critical value representing the size of the sliding surface. This represents the sensitivity control coefficient.

2. The adaptive fractional order non-singular terminal sliding mode control method based on time delay estimation according to claim 1, characterized in that: The formula for constructing the delay estimation compensation term in step 4 is as follows: in, Indicates the current time The delay estimation compensation term, Indicates the current moment. Indicates the delay time. Indicates the current time interval A historic moment Representing historical moments The motor torque vector, Representing historical moments The joint acceleration vector.

3. The adaptive fractional order non-singular terminal sliding mode control method based on time delay estimation according to claim 2, characterized in that: The principle underlying the generation of the control law for the rope-driven manipulator in step 4 is as follows: in, Indicates the current time The motor torque vector, i.e., the current moment. The corresponding control law for the cable-driven manipulator. The auxiliary control vector is represented by the auxiliary control vector, which is calculated from the fractional-order non-singular terminal sliding surface and the sliding surface reaching law.

4. The adaptive fractional order non-singular terminal sliding mode control method based on time delay estimation according to claim 3, characterized in that: The principle underlying the calculation of the auxiliary control vector is as follows: An auxiliary control vector is constructed based on the derivative and reaching law of the fractional-order non-singular terminal sliding surface. The specific formula is as follows: wherein, represents the desired joint acceleration vector, and is the derivative of

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