Mechanical arm predefined time trajectory tracking control method and system based on time delay estimation

Abstract

The invention provides a mechanical arm predefined time trajectory tracking control method and system based on time delay estimation, and relates to the technical field of multi-degree-of-freedom mechanical arm trajectory tracking control. Constructing a predefined time error conversion function to define a system error and conversion error system, designing a barrier function and a feedback control rate based on the conversion error system, and designing a Lyapunov function to prove the stability of the system so as to track an expected trajectory; according to the method, time delay information is deeply fused, dependence on a complex model approximate structure is not needed, stable convergence of the mechanical arm system state within predefined time can be achieved when a mathematical model is not accurate, the system control performance and robustness are effectively improved, and the requirement of modern industry for mechanical arm high-performance control is met.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of multi-degree-of-freedom robot arm trajectory tracking control, in particular to a robot arm predefined time trajectory tracking control method and system based on time delay estimation. BACKGROUND

[0002] In modern industrial automation, intelligent manufacturing and complex task execution scenarios, robot arms are the core equipment for realizing automatic operation due to their high precision, high flexibility and repeatability. With the continuous iteration and upgrading of robot technology, higher requirements are put forward for the trajectory tracking accuracy, dynamic response speed, stability and reliability of the robot arm control system in complex environments. At present, the control methods of robot arms mainly include traditional PID control, adaptive control, sliding mode control and intelligent control, etc. Although the traditional PID control has a simple structure and is easy to implement in engineering, its control performance is highly dependent on accurate system mathematical models, and its robustness is obviously insufficient when the model parameters fluctuate or face external disturbances; adaptive control can dynamically adjust the controller parameters according to system parameter changes, but when the parameter estimation is inaccurate or the system exhibits strong nonlinear characteristics, the control effect is significantly reduced; sliding mode control has good robustness and fast response capability, but its inherent chattering problem affects the control accuracy and service life of the system; intelligent control methods such as neural network control and fuzzy control do not require accurate mathematical models, but have the defects of long training period, high computational complexity and difficult to guarantee the control performance. The above methods generally depend on accurate mathematical models, and when the model deviates from the actual system, the control effect will be greatly reduced. In recent years, fixed-time control technology has attracted attention due to its unique advantages. This technology breaks through the limitations of traditional control methods, ensuring that the system converges within a pre-set fixed time, and the convergence time is independent of the initial conditions, and has strong robustness to model uncertainty, providing a new idea for robot arm control.

[0003] To solve the above problems, the present application proposes a robot arm predefined time control method with global performance index based on time delay estimation. This method deeply integrates time delay information, without relying on complex model approximation structures such as neural networks and fuzzy logic. Even in the case of inaccurate mathematical models, the method can achieve stable convergence of the robot arm system state within a predefined time, effectively improving the control performance and robustness of the system, and meeting the demand for high-performance control of robot arms in modern industry.

[0004] The above information disclosed in the background section is only used to enhance the understanding of the background of the present disclosure, and therefore it can include information that does not constitute prior art known to those of ordinary skill in the art. SUMMARY

[0005] The purpose of this invention is to provide a method and system for tracking and controlling a robotic arm's predefined time trajectory based on time delay estimation, so as to solve the problems mentioned in the background art.

[0006] To achieve the above objectives, the present invention provides the following technical solution: The robotic arm predefined time trajectory tracking control method based on time delay estimation includes the following steps: S1: Based on the time delay estimation method, a two-dimensional dynamic model of the robot arm is established, and the mathematical assumptions and lemmas required to realize the trajectory tracking algorithm are constructed to support error transformation, barrier function design and stability analysis. The assumptions include that the expected trajectory and its derivative are known, and the lemmas include providing time convergence criteria, error term scaling, logarithmic term processing and cross term decomposition. S2: Based on the two-dimensional dynamic model, a function for error transformation with a predefined time is constructed by processing several terms and using time convergence criteria. This transforms the original trajectory error into a transformed error system that satisfies the predefined time convergence characteristics. S3: Based on the transformation error system, the barrier function and feedback control law calculation formula are designed by scaling the error term and decomposing the cross term. Based on the calculation results, the position, velocity and acceleration errors are suppressed in layers, and the error convergence boundary is dynamically adjusted through the barrier function. S4: Based on the calculation formula of the barrier function and the feedback control law, a Lyapunov function containing the error energy of each layer is constructed by using the error term scaling and time convergence criteria to complete the construction of the trajectory tracking algorithm and achieve high-precision tracking of the desired trajectory.

[0007] Furthermore, the method for establishing a two-dimensional dynamic model of the robot arm based on the time delay estimation method is as follows: The two-dimensional dynamics of a robotic arm are defined as follows: ; In the formula, Represents the inertia matrix. Represents the centrifugal force-Coriolis force matrix. Represents the gravity vector. A vector representing the control input. A function representing the vector of external disturbances. A vector representing the joint position. A vector representing joint velocity. The vector representing joint acceleration, where, ; Define the vectors for joint position, joint velocity, control input, and external disturbance as follows: ; In the formula, Indicates the position of joint 1. Indicates the position of joint 2. Indicates the velocity of joint 1. Indicates the velocity of joint 2. This represents the acceleration of joint 1. This represents the acceleration of joint 2. This indicates the control torque of joint 1. This indicates the control torque of joint 2. This represents the external disturbance torque acting on joint 1. A vector representing external disturbance. This represents the external disturbance torque experienced by joint 2; The two-dimensional dynamics of the robotic arm described above are converted into a first-order state-space form, and the state variables are defined as follows: ; In the formula, Indicates the position status. Representing the velocity state, the joint accelerations are obtained by solving the two-dimensional dynamics of the linkage manipulator: ; Define system synthesis items: ; In the formula, Indicates the system synthesis item. Indicates the control allocation coefficient. Indicates the controller's control rate; The final first-order state space form is: ; Define system state ; Based on the two-dimensional dynamics of linkage machinery, the equations for joint acceleration obtained by solving them yield the transformed system synthesis terms as follows: ; Define the estimated delay time as , , As a system integration item The simplified representation, at this time, The delay value is defined as ,Right now The system synthesis term at a given time is used, therefore past time is adopted. The system synthesis term is used as the estimate of the system synthesis term at the current moment. Based on the transformed system synthesis term, the formula for the estimate of the system synthesis term at the current moment is constructed as follows: ; In the formula, This represents the estimated time delay of the system's comprehensive terms at the current moment. express Joint acceleration at any moment express The controller control rate at any given time; At this point, the time delay error is: ; In the formula, This represents the time delay error, the existence of which is an unknown upper bound. ,Right now ; At this point, the state-space model including time delay is: .

[0008] Furthermore, in constructing the mathematical assumptions and lemmas required to implement the trajectory tracking algorithm, the assumption is the desired trajectory. and its derivative It is known, continuous, and bounded; In Lemma 1, for nonlinear systems There exists a continuous scalar function, the Lyapunov function. and control parameters , so that: ; In the formula, This represents the derivative of the Lyapunov function. Indicates the error decay exponent. Indicates the predefined convergence time. Indicates the upper bound of the perturbation; The convergence region of the system state is defined as: ; In the formula, This represents the interference suppression coefficient, which adjusts the scaling of the interference term in the Lyapunov function. This indicates the preset time, i.e., the upper limit of the convergence time. ; In Lemma 2, for and The following inequalities hold: ; In Lemma 3, for any function The following inequalities hold: ; In Lemma 4, for the variable and , and the given positive constants and The following inequalities hold: .

[0009] Furthermore, by constructing a predefined time-based error transformation function based on several processing terms and time convergence criteria, the method for transforming the original trajectory error into a transformed error system that satisfies the predefined time convergence characteristics is as follows: Constructing the original trajectory error: ; In the formula, This represents the error of the original trajectory, and ; ; In the formula, This represents the original trajectory error of joint 1. This represents the original trajectory error of joint 2. This represents the desired trajectory of joint 1. This represents the desired trajectory of joint 2; For the original trajectory error of each joint, define the error constraint boundary: ; In the formula, Indicates the first The boundary of the original trajectory error of each joint. These represent the design parameters that determine the rate of error convergence. This represents the initial constraints. This indicates the maximum permissible error range during the steady-state tracking phase. The index represents the joint, where, ; Based on the predefined time error transformation function in Lemma 1: ; In the formula, A function representing a predefined time error transformation; Differentiate the function with respect to the predefined time error transformation: ; Define the function of the conversion error system: ; In the formula, The function representing the conversion error system, A function representing the error of the original trajectory; Constructing an error system: ; In the formula, The function representing the first layer of error, i.e., the function for transforming position errors. The function representing the second-level error, that is, the function of the deviation between the actual acceleration of the system and the target acceleration. A function representing the filtering error. A function representing the virtual control law. The function representing the virtual control objective is the output signal of the first-order filter, where... , This represents the virtual control rate of joint 1. This represents the virtual control rate of joint 2, and , Describes a symmetric positive definite matrix. , This represents the filtering error of joint 1. This represents the filtering error of joint 2.

[0010] Furthermore, the method for designing the barrier function and feedback control law calculation formula by scaling the error term and decomposing the cross term is as follows: Construct the barrier function: ; In the formula, Represents the barrier function. Indicates the first Constraint boundaries of each joint. Indicates the first The first layer error of each joint, where... , Indicates the first The lower limit of each joint Indicates the first The upper limit of a joint, ,and , ; According to the chain rule, the derivative of the barrier function is obtained as follows: ; In the formula, Indicates the first Original trajectory error of each joint Indicates the first The second layer error of each joint Indicates the first Virtual control target for each joint. Indicates the first The derivative of the expected trajectory of each joint. This represents a predefined time error conversion factor; Then, by transforming Young's inequality, we get: ; In the formula, This represents the design parameters, which are constant terms. ; Based on the transformation of Young's inequality and the formula for differentiating the barrier function, we obtain: ; make ; In the formula, Indicates the first Virtual control rate of each joint Indicates the first layer controller gain for each joint The first layer error represents the... Robust compensation terms for each joint, among which... , Therefore, the virtual control law formula 1 can be obtained as follows: ; Combining Formula 1 with Lemma 2, we get Formula 2: ; Define the second Lyapunov function: ; In the formula, This represents the second Lyapunov function. Indicates the second layer error; Taking the derivative of the second-level error, we get: ; Differentiating the second Lyapunov function and combining it with the derivative of the second-level error and the result of Equation 2, we obtain Equation 3: ; In the formula, Indicates the first The filtering error of each joint; The controller control rate is designed to be: ; In the formula, This indicates the controller's control rate.

[0011] Furthermore, the trajectory tracking algorithm is constructed by using error term scaling and time convergence criteria to build a Lyapunov function containing the error energy of each layer. Define candidate Lyapunov functions: ; In the formula, Represents a candidate Lyapunov function; Depend on have to ; Differentiate the filtering error and then... Substitute: ; Construct the disturbance term to represent the derivative of the virtual control law: ; In the formula, Let represent the interference term, which is a continuous and bounded function. After simplification, the formula for the derivative with respect to the filtering error is: ; at this time Then, according to Young's inequality, the following transformations are made: ; Differentiate the candidate Lyapunov function and then... Substituting the result of Formula 3 into Formula 4, we get: In the formula, Indicates the first The first layer of error constraint strength for each joint Indicates the second layer The controller gain of each joint, and According to Young's inequality, we get: ; Combining Lemma 3, we obtain Formula 5: ; Therefore, Formula 6 is: ; In the formula, The attenuation coefficient represents the stability of the system. This represents the upper bound of the sum of system disturbances and errors. , ; Multiply formula 6 by Integrating again, we get: ; Therefore, the Lyapunov function is bounded, defined as follows: Error signal , and In scope The interior is bounded, within which, ; Formula 4, Formula 5 and Range Combining these, we get Formula 7: ; Then, according to Lemma 4, make ,get: ; In the formula, This represents the constant term introduced during scaling. Combining formula 7, we get formula 8: ; Substituting formula 8 into formula 7, we get: ; In the formula, This represents the upper bound of the total perturbation. ; Since the controlled system is time-stable, according to Lemma 1, an error signal is constructed. Combined with preset times to form a set: ; Therefore, it is determined that the candidate Lyapunov function, the first-level error, the second-level error, the filtering error, the original trajectory error, and the system state are all bounded.

[0012] Additionally, a robotic arm predefined time trajectory tracking control system based on time delay estimation is provided, characterized in that: the system is used to execute the aforementioned robotic arm predefined time trajectory tracking control method based on time delay estimation, including: The dynamics model module is used to establish a two-dimensional dynamics model of the robot arm based on the time delay estimation method, and to construct the mathematical assumptions and lemmas required to implement the trajectory tracking algorithm, so as to support error transformation, barrier function design and stability analysis. The assumptions include that the expected trajectory and its derivative are known, and the lemmas include providing time convergence criteria, error term scaling, logarithmic term processing and cross term decomposition. The error system module is used to construct a predefined time error transformation function based on the two-dimensional dynamic model by processing several terms and using time convergence criteria, so as to transform the original trajectory error into a transformed error system that satisfies the predefined time convergence characteristics. The control law design module is used to design the barrier function and feedback control law calculation formula based on the transformation error system by scaling the error term and decomposing the cross term. Based on the calculation results, it performs hierarchical suppression of position, velocity and acceleration errors, and dynamically adjusts the error convergence boundary through the barrier function. The stability proof module is used to construct a Lyapunov function containing the error energy of each layer based on the barrier function and the calculation formula of the feedback control law, through error term scaling and time convergence criteria, to complete the construction of the trajectory tracking algorithm and achieve high-precision tracking of the desired trajectory.

[0013] Compared with the prior art, the beneficial effects of the present invention are: The two-dimensional dynamic model constructed by time delay estimation and its accompanying mathematical lemma, combined with a predefined time error transformation function, enables the original trajectory error of the robotic arm to converge stably within a pre-set time. By using the barrier function and the feedback control law calculation formula, even under conditions of model inaccuracy and time delay, the position, velocity, and acceleration errors can still be limited to a safe range. This significantly improves the accuracy, stability, and robustness of the robotic arm trajectory tracking, avoids dependence on complex models, and enhances the applicability of the control method in various industrial environments. Attached Figure Description

[0014] Figure 1 This is a schematic diagram of the overall method flow of the present invention; Figure 2 This is a schematic diagram of the system tracking error of joints 1 and 2 of the two-bar linkage robotic arm of the present invention; Figure 3 This is a schematic diagram of the system response curves of joints 1 and 2 of the two-bar linkage robotic arm of the present invention; Figure 4 This is a schematic diagram of the output of the control force of the controller of the two-bar linkage robotic arm of the present invention; Figure 5 This is a schematic diagram of the overall system structure of the present invention. Detailed Implementation

[0015] 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.

[0016] 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.

[0017] Example: Please see Figures 1 to 4 The present invention provides a technical solution: The robotic arm predefined time trajectory tracking control method based on time delay estimation includes the following steps: S1: Based on the time delay estimation method, a two-dimensional dynamic model of the robot arm is established, and the mathematical assumptions and lemmas required to realize the trajectory tracking algorithm are constructed to support error transformation, barrier function design and stability analysis. The assumptions include that the expected trajectory and its derivative are known, and the lemmas include providing time convergence criteria, error term scaling, logarithmic term processing and cross term decomposition. First, a two-dimensional dynamic model of the robotic arm is established. The two-dimensional dynamics of the robotic arm are defined as follows: ; In the formula, Represents the inertia matrix. Represents the centrifugal force-Coriolis force matrix. Represents the gravity vector. A vector representing the control input. A vector representing external disturbance. A vector representing the joint position. A vector representing joint velocity. The vector representing joint acceleration, where, The dot symbol directly above the scalar symbol represents the derivative; Dual-joint robotic arms belong to low-degree-of-freedom systems. Their dynamic model has low dimensionality, which can significantly simplify the derivation process of control algorithms while retaining the core dynamic characteristics of the robotic arm system. This effectively verifies the universality of the control method and can be directly extended to robotic arm systems with higher degrees of freedom, providing a foundation for complex scenarios. The joint position vector, joint velocity vector, control input vector, and external disturbance vector are defined as follows: ; In the formula, in the formula, Indicates the position of joint 1. Indicates the position of joint 2. Indicates the velocity of joint 1. Indicates the velocity of joint 2. This represents the acceleration of joint 1. This represents the acceleration of joint 2. This indicates the control torque of joint 1. This indicates the control torque of joint 2. This represents the external disturbance torque acting on joint 1. A vector representing external disturbance. This represents the external disturbance torque experienced by joint 2; State-space models are standard tools in control theory for describing the dynamic behavior of systems. To convert the two-dimensional dynamics of the robotic arm described above into a first-order state-space form, we first define the state variables: ; In the formula, Indicates the position status. Representing the velocity state, it fits the standard framework of the first-order state-space model in control theory, thereby completing the subsequent controller design and stability analysis. The joint acceleration is obtained by solving the two-dimensional dynamics of the linkage manipulator: ; Define system synthesis items: ; In the formula, Indicates the system synthesis item. Indicates the control allocation coefficient. Indicates the controller's control rate; The final first-order state space form is: ; Define system state superscript Indicates transpose; Based on the two-dimensional dynamics of linkage machinery, the equations for joint acceleration obtained by solving them yield the transformed system synthesis terms as follows: ; To estimate the impact of time delay, time delay estimation is used to handle the uncertainties of system dynamics, such as model errors and deviations caused by time delay. The estimated delay time is defined as... , , As a system integration item The simplified representation, at this time, The delay value is defined as ,Right now The system synthesis term at a given time is used, therefore past time is adopted. The system synthesis term is used as the estimate of the system synthesis term at the current moment. Based on the transformed system synthesis term, the formula for the estimate of the system synthesis term at the current moment is constructed as follows: ; In the formula, This represents the estimated time delay of the system's comprehensive terms at the current moment. express Joint acceleration at any moment express The controller control rate at any given time, and the error of the system comprehensive term are treated as bounded disturbances. The complex multi-factor influences are transformed into a single comprehensive term and its error, avoiding the tediousness of handling each factor separately, and enabling the control strategy to cope with multiple disturbances at the same time. At this point, the time delay error is: ; In the formula, Indicates the time delay error, where, It has an unknown upper bound. ,Right now ; At this point, the state-space model including time delay is: ; In the assumptions and lemmas required for controller design, the assumption is the desired trajectory. and its derivative The assumptions are known, continuous, and bounded. Regarding the support error transformation, the assumption provides a computable and bounded baseline condition for constructing the original trajectory error and ensures that the constructed error transformation function can satisfy the predefined time convergence criterion. Regarding the support barrier function design, the assumption sets a theoretical bound for the barrier function, that is, by constraining the boundedness of the desired trajectory, it indirectly sets a theoretical bound for the original trajectory error, ensuring that the error constraint boundary in the barrier function can always be greater than the error transformation value, ultimately ensuring that the barrier function can effectively constrain the error from exceeding the bounds in engineering. Regarding the support stability analysis, the assumption ensures that the system state can converge to a bounded set within a predefined time by guaranteeing the boundedness of the derivative of the desired trajectory, thus forming a theoretical closed loop for the trajectory tracking algorithm together with the lemma. The lemma, as a commonly used mathematical tool or proven mathematical conclusion in control theory, belongs to the preliminary knowledge at the pure mathematical level, rather than a definition specific to the robotic arm data. In Lemma 1, for nonlinear systems There exists a continuous scalar function, the Lyapunov function. and control parameters , Indicates the error decay exponent. Indicates the predefined convergence time. Indicates the upper bound of the perturbation, such that: ; In the formula, Represents the derivative of a Lyapunov function; The convergence region of the system state is defined as: ; In the formula, This represents the interference suppression coefficient, which adjusts the scaling of the interference term in the Lyapunov function. This indicates the preset time, i.e., the upper limit of the convergence time. This is the core mathematical basis for determining whether a system can converge stably within a preset time. Subsequent stability analysis needs to verify that the system satisfies the conditions of this lemma. In Lemma 2, for and The following inequalities hold: ; Represents a sequence of real variables. This represents the scaling factor. Representing the absolute values ​​of each term Summing of powers is a key tool in the document for deriving the derivative of Lyapunov functions when dealing with the summation of multiple error terms. It transforms the summation of higher powers into the summation of each higher power, which facilitates the scaling and estimation of inequalities. In Lemma 3, for any function The following inequalities hold: ; This lemma can be used to transform the derivative of the logarithmic term into a fractional term, which facilitates subsequent inequality analysis and control law design. In Lemma 4, for the variable and , and the given positive constants and The following inequalities hold: ; The inequality relationship between the power of the product of two variables and the higher powers of each variable is given, which can be used to scale multivariate product terms and avoid interference from complex cross terms in stability analysis.

[0018] S2: Based on the two-dimensional dynamic model, a function for error transformation with a predefined time is constructed by processing several terms and using time convergence criteria. This transforms the original trajectory error into a transformed error system that satisfies the predefined time convergence characteristics. The original trajectory error is constructed by defining the error based on the difference between the actual position of the robotic arm joints and the desired trajectory: ; ; In the formula, This represents the error of the original trajectory, and ; In the formula, This represents the original trajectory error of joint 1. This represents the original trajectory error of joint 2. This represents the desired trajectory of joint 1. This represents the desired trajectory for joint 2; For the original trajectory error of each joint, set time-varying constraint boundaries, i.e., define the error constraint boundaries: ; In the formula, Indicates the first The original trajectory error of each joint, These represent the design parameters that determine the rate of error convergence. This represents the initial constraints. This indicates the maximum permissible error range during the steady-state tracking phase. The index represents the joint, where, This ensures that the original trajectory error always satisfies the condition... to The interval, i.e., the error, needs to be confined within a dynamic boundary that shrinks over time, eventually converging to a point in steady state. nearby; Based on the predefined time error transformation function in Lemma 1: ; In the formula, This represents a predefined time error conversion function, where the predefined time control requires the error to be within a preset time. During convergence, the converted error is kept within a preset time. Rapid contraction ahead, The time error should have converged to the steady-state range. Therefore, the predefined time error transformation function remains constant. ; Differentiate the predefined time error transformation function: ; Define the conversion error system: ; This allows the original trajectory error to be weighted using a predefined time error transformation function, making its dynamic characteristics adaptable to a predefined time-converged analysis framework, thus constructing an error system: ; In the formula, This represents the first layer of error, i.e., the transformation of position error. This represents the second-level error, which is the deviation between the actual acceleration of the system and the target acceleration. It is a common layered processing method in backstepping control. This represents the filtering error function, used to reduce high-frequency jitter in the control signal. Indicates virtual control rate. This represents the virtual control target, i.e., the output signal of the first-order filter, where... , This represents the virtual control rate of joint 1. This represents the virtual control rate of joint 2, and , Representing a symmetric positive definite matrix, its core function is to... adjust The smoothness of the signal depends on the value; too small a value leads to jitter, while too large a value results in lag. , This represents the filtering error of joint 1. This represents the filtering error of joint 2. As a common method in the control of complex systems, the hierarchical error system is used to process each joint separately, decomposing the complex multivariable system into low-order subsystems, which facilitates the design of controllers layer by layer.

[0019] S3: Based on the transformation error system, the barrier function and feedback control law calculation formula are designed by scaling the error term and decomposing the cross term. Based on the calculation results, the position, velocity and acceleration errors are suppressed in layers, and the error convergence boundary is dynamically adjusted through the barrier function. In robotic arm trajectory tracking, the error needs to be limited to a specific range, therefore a barrier function is constructed: ; In the formula, Represents the barrier function. Indicates the first Constraint boundaries of each joint. Indicates the first The first layer error of each joint, where... , Indicates the first The lower limit of each joint Indicates the first The upper limit of a joint, ,and , It can handle asymmetric constraints, keeping the error within the constraint range; according to and We can obtain: ; Differentiate the error of the original trajectory: ; Replace actual states with virtual control variables: ; state It is the system output that replaces the actual state, transforming the complex and uncontrollable actual state into a designable intermediate goal; At this point, we get: ; pass Transformed to: ; According to the chain rule, the derivative of the barrier function is obtained as follows: ; In the formula, Indicates the first Original trajectory error of each joint Indicates the first The second layer error of each joint Indicates the first Virtual control target for each joint. Indicates the first The derivative of the expected trajectory of each joint; Then, by transforming Young's inequality, we get: ; In the formula, This represents the design parameters, which are constant terms. In order to break it down into square terms related to the design parameter h and containing The constant term, using Young's inequality, for any real number... and The constant term has The solution is obtained.

[0020] Transforming the formula for differentiating the barrier function, and relating it to... The terms are separated to obtain: ; Based on the transformation of Young's inequality and the formula for differentiating the barrier function, we obtain: ; Will Substitute: ; Combination right Transformed to: ; at this time: ; exist In this item, when When it is positive, it will make Increasing this value disrupts stability; therefore, in order to achieve this... Become a negative term, let Meanwhile, to enable the filter to converge dynamically and quickly, let middle , Therefore, the virtual control law formula 1 can be obtained as follows: ; In the formula, Indicates the first layer controller gain for each joint The first layer error represents the... Robust compensation terms for each joint, making proportional term Can be When the value increases, strong negative feedback is output, quickly suppressing error growth. The robust compensation term can offset model uncertainties and enhance system robustness. , ; Combining Formula 1 with Lemma 2, we get Formula 2: ; Define the second Lyapunov function: ; In the formula, This represents the second Lyapunov function. The barrier function can only guarantee that the position error does not exceed the limit, but it cannot constrain the acceleration deviation. The superposition of the second Lyapunov function... After obtaining the energy, position error convergence and acceleration deviation suppression can be analyzed simultaneously; Differentiate the second Lyapunov function: ; Taking the derivative of the second-level error, we get: ; Thus, the control input section is associated. This allows the control law design to be adjusted. inhibition ; Differentiating the second Lyapunov function and combining it with the derivative of the second-level error and the result of Equation 2, we obtain Equation 3: ; According to the proportion The derivative is defined as follows: ; In the formula, The first term of the time delay estimation function One portion, Representing vectors The One portion, The first element representing the time delay error One component; Will Substitution have to: ; All Group the terms that are factors together and combine them into a single sum: ; therefore ; To use the controller to control the rate Distribution and elimination, introduction The feedback control rate involved is: ; In the formula, Indicates the controller gain of the second layer. This represents the robust compensation term for the second-level error. , at this time, and substitute And ignore have to: ; Substitute the formula The transformation yields: ; Barrier function handles position constraints, second layer error Negative feedback processing of acceleration disturbances, robust compensation term for second-layer error. To suppress time delay errors, all three are unified into the Lyapunov framework, achieving synchronous convergence of position, velocity, and acceleration; like Figures 2-3 As shown in the figure, the tracking errors of joints 1 and 2 fluctuate in the initial stage, but gradually converge towards the upper and lower bounds of the error as time goes by, eventually stabilizing in a range close to zero. The actual trajectories of joints 1 and 2 are generally consistent with their respective expected trajectories. Although there is a certain deviation between the actual trajectory and the expected trajectory in the initial stage, the actual trajectory can follow the changes of the expected trajectory well as time progresses. In subsequent periodic motions, the matching degree continuously improves, demonstrating the effectiveness of the control strategy in trajectory tracking. This indicates that the designed control strategy can effectively reduce the tracking error and make the actual output of the system gradually approach the expected output.

[0021] S4: Based on the calculation formula of the barrier function and the feedback control law, a Lyapunov function containing the error energy of each layer is constructed by using the error term scaling and time convergence criteria to complete the construction of the trajectory tracking algorithm and achieve high-precision tracking of the desired trajectory. The second Lyapunov function covers the first and second layer errors, but does not include the filtering error; therefore, a candidate Lyapunov function is defined: ; In the formula, Let represent the candidate Lyapunov function. The filtering error originates from the first-order filter. If this is not included in the analysis, the convergence of the filtering error cannot be guaranteed. This is a typical representation of its energy; Depend on have to ; This establishes a dynamic relationship between the filter output and input, providing a basis for subsequent filter error analysis. Differentiate the filtering error and then... Substitute: ; The derivative of the virtual control law involves the second derivative of the desired trajectory, the derivative of the original trajectory error, and the second derivative of the transformation function. It has a complex form and may contain nonlinear terms. An interference term is constructed to represent the derivative of the virtual control law. ; In the formula, The complex structure representing the disturbance term, which is a continuous and bounded function, does not require precise modeling of the derivative of the virtual control law. This is a common practice in robust control. After simplification, the formula for differentiating the filtering error is: ; at this time Then, according to Young's inequality, the following transformations are made: ; Differentiate the candidate Lyapunov function and then... Substituting the result of Formula 3 into Formula 4, we get: ; In the formula, Indicates the first The first layer of error constraint strength for each joint, and According to Young's inequality, we get: ; Combining Lemma 3, we obtain Formula 5: ; Therefore, Formula 6 is: ; In the formula, The attenuation coefficient represents the stability of the system. This represents the upper bound of the sum of system disturbances and errors. , This ensures that all error terms, including filtering errors, converge within a predefined time. Multiply formula 6 by Integrating again, we get: ; This is a standard technique for integrating differential inequalities, therefore it can be proven that the Lyapunov function is bounded, defined as follows: Error signal , and In scope The interior is bounded, within which, Lyapunov functions are derived from The boundedness of these variables means that their energy is also limited. Formula 4, Formula 5 and Range Combining these, we get Formula 7: ; Then, according to Lemma 4, make ,get: ; In the formula, This represents the constant term introduced during scaling. Combining formula 7, we get formula 8: ; Substituting formula 8 into formula 7, we get: ; In the formula, This represents the upper bound of the total perturbation. Since the controlled system is time-stable, according to Lemma 1, an error signal is constructed. Combined with preset times to form a set: ; According to Lemma 1, the system at a preset time Converging to set After rigorous proof of stability over a predetermined time, it is determined that the candidate Lyapunov function, the first-level error, the second-level error, the filtering error, the original trajectory error, and the system state are all bounded. Figure 4As shown, at the initial moment, the actual trajectory deviates significantly from the desired trajectory. The controller needs to output a large control quantity to quickly adjust the joint motion to reduce the tracking error. After 0.5s, the fluctuation amplitude of the control signals of joint 1 and joint 2 decreases significantly, gradually becomes smooth and changes periodically. The controller output also becomes stable and matches the dynamics of the desired trajectory. This ensures that the system can respond quickly to the initial error and can stably track the desired trajectory in steady state.

[0022] Please see Figure 5 The present invention also provides a robotic arm predefined time trajectory tracking control system based on time delay estimation, for executing the above-mentioned robotic arm predefined time trajectory tracking control method based on time delay estimation, including: The dynamics model module is used to establish a two-dimensional dynamics model of the robot arm based on the time delay estimation method, and to construct the mathematical assumptions and lemmas required to implement the trajectory tracking algorithm, so as to support error transformation, barrier function design and stability analysis. The assumptions include that the expected trajectory and its derivative are known, and the lemmas include providing time convergence criteria, error term scaling, logarithmic term processing and cross term decomposition. The error system module is used to construct a predefined time error transformation function based on the two-dimensional dynamic model by processing several terms and using time convergence criteria, so as to transform the original trajectory error into a transformed error system that satisfies the predefined time convergence characteristics. The control law design module is used to design the barrier function and feedback control law calculation formula based on the transformation error system by scaling the error term and decomposing the cross term. Based on the calculation results, it performs hierarchical suppression of position, velocity and acceleration errors, and dynamically adjusts the error convergence boundary through the barrier function. The stability proof module is used to construct a Lyapunov function containing the error energy of each layer based on the barrier function and the calculation formula of the feedback control law, through error term scaling and time convergence criteria, to complete the construction of the trajectory tracking algorithm and achieve high-precision tracking of the desired trajectory.

[0023] 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.

[0024] 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.

[0025] 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.

[0026] 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 robotic arm predefined time trajectory tracking control method based on time delay estimation, characterized in that, The specific steps include: S1: Based on the time delay estimation method, a two-dimensional dynamic model of the robot arm is established, and the mathematical assumptions and lemmas required to realize the trajectory tracking algorithm are constructed to support error transformation, barrier function design and stability analysis. The assumptions include that the expected trajectory and its derivative are known, and the lemmas include providing time convergence criteria, error term scaling, logarithmic term processing and cross term decomposition. S2: Based on the two-dimensional dynamic model, a function for error transformation with a predefined time is constructed by processing several terms and using time convergence criteria. This transforms the original trajectory error into a transformed error system that satisfies the predefined time convergence characteristics. S3: Based on the transformation error system, the barrier function and feedback control law calculation formula are designed by scaling the error term and decomposing the cross term. Based on the calculation results, the position, velocity and acceleration errors are suppressed in layers, and the error convergence boundary is dynamically adjusted through the barrier function. S4: Based on the calculation formula of the barrier function and the feedback control law, a Lyapunov function containing the error energy of each layer is constructed by using the error term scaling and time convergence criteria to complete the construction of the trajectory tracking algorithm and achieve high-precision tracking of the desired trajectory.

2. The robotic arm predefined time trajectory tracking control method based on time delay estimation according to claim 1, characterized in that: The method for establishing a two-dimensional dynamic model of a robot arm based on time delay estimation is as follows: The two-dimensional dynamics of a robotic arm are defined as follows: ； In the formula, Represents the inertia matrix. Represents the centrifugal force-Coriolis force matrix. Represents the gravity vector. A vector representing the control input. A function representing the vector of external disturbances. A vector representing the joint position. A vector representing joint velocity. The vector representing joint acceleration, where, ; Define the vectors for joint position, joint velocity, control input, and external disturbance as follows: ； In the formula, Indicates the position of joint 1. Indicates the position of joint 2. Indicates the velocity of joint 1. Indicates the velocity of joint 2. This represents the acceleration of joint 1. This represents the acceleration of joint 2. This indicates the control torque of joint 1. This indicates the control torque of joint 2. This represents the external disturbance torque acting on joint 1. A vector representing external disturbance. This represents the external disturbance torque experienced by joint 2; The two-dimensional dynamics of the robotic arm described above are converted into a first-order state-space form, and the state variables are defined as follows: ； In the formula, Indicates the position status. Representing the velocity state, the joint accelerations are obtained by solving the two-dimensional dynamics of the linkage manipulator: ； Define system synthesis items: ； In the formula, Indicates the system synthesis item. Indicates the control allocation coefficient. Indicates the controller's control rate; The final first-order state space form is: ； Define system state ; Based on the two-dimensional dynamics of linkage machinery, the equations for joint acceleration are solved, and the resulting system synthesis terms are: ； Define the estimated delay time as , , As a system integration item The simplified representation, at this time, The delay value is defined as ,Right now The system synthesis term at a given time is used, therefore past time is adopted. The system synthesis term is used as the estimate of the system synthesis term at the current moment. Based on the transformed system synthesis term, the formula for the estimate of the system synthesis term at the current moment is constructed as follows: ； In the formula, This represents the estimated time delay of the system's comprehensive terms at the current moment. express Joint acceleration at any moment express The controller control rate at any given time; At this point, the time delay error is: ； In the formula, This represents the time delay error, the existence of which is an unknown upper bound. ,Right now ; At this point, the state-space model including time delay is: 。 3. The robotic arm predefined time trajectory tracking control method based on time delay estimation according to claim 2, characterized in that: In the mathematical assumptions and lemmas required to construct the trajectory tracking algorithm, the assumption is the desired trajectory. and its derivative It is known, continuous, and bounded; In Lemma 1, for nonlinear systems There exists a continuous scalar function, the Lyapunov function. and control parameters , so that: ； In the formula, This represents the derivative of the Lyapunov function. Indicates the error decay exponent. Indicates the predefined convergence time. Indicates the upper bound of the perturbation; The convergence region of the system state is defined as: ； In the formula, This represents the interference suppression coefficient, which adjusts the scaling of the interference term in the Lyapunov function. This indicates the preset time, i.e., the upper limit of the convergence time. ; In Lemma 2, for and The following inequalities hold: ； In Lemma 3, for any function The following inequalities hold: ； In Lemma 4, for the variable and , and the given positive constants and The following inequalities hold: 。 4. The robotic arm predefined time trajectory tracking control method based on time delay estimation according to claim 3, characterized in that: The method for transforming the original trajectory error into a transformed error system that satisfies the predefined time convergence characteristics is as follows: This is achieved by constructing a predefined time-based error transformation function based on several processing terms and time convergence criteria. Constructing the original trajectory error: ； In the formula, This represents the error of the original trajectory, and ; ； In the formula, This represents the original trajectory error of joint 1. This represents the original trajectory error of joint 2. This represents the desired trajectory of joint 1. This represents the desired trajectory of joint 2; For the original trajectory error of each joint, define the error constraint boundary: ； In the formula, Indicates the first The boundary of the original trajectory error of each joint. These represent the design parameters that determine the rate of error convergence. This represents the initial constraints. This indicates the maximum permissible error range during the steady-state tracking phase. The index represents the joint, where, ; Based on the predefined time error transformation function in Lemma 1: ； In the formula, A function representing a predefined time error conversion; Differentiate the function with respect to the predefined time error transformation: ； Define the function of the conversion error system: ； In the formula, The function representing the conversion error system, A function representing the error of the original trajectory; Constructing an error system: ； In the formula, The function representing the first layer of error, i.e., the function for transforming position errors. The function representing the second-level error, that is, the function of the deviation between the actual acceleration of the system and the target acceleration. A function representing the filtering error. A function representing the virtual control law. The function representing the virtual control objective is the output signal of the first-order filter, where... , This represents the virtual control rate of joint 1. This represents the virtual control rate of joint 2, and , Describes a symmetric positive definite matrix. , This represents the filtering error of joint 1. This represents the filtering error of joint 2.

5. The robotic arm predefined time trajectory tracking control method based on time delay estimation according to claim 4, characterized in that: The method for designing the barrier function and feedback control law calculation formula by using error term scaling and cross term decomposition is as follows: Construct the barrier function: ； In the formula, Represents the barrier function. Indicates the first Constraint boundaries of each joint. Indicates the first The first layer error of each joint, where... , Indicates the first The lower limit of each joint Indicates the first The upper limit of a joint, ,and , ; According to the chain rule, the derivative of the barrier function is obtained as follows: ； In the formula, Indicates the first Original trajectory error of each joint Indicates the first The second layer error of each joint Indicates the first Virtual control target for each joint. Indicates the first The derivative of the expected trajectory of each joint, This represents a predefined time error conversion factor; Then, by transforming Young's inequality, we get: ； In the formula, This represents the design parameters, which are constant terms. ; Based on the transformation of Young's inequality and the formula for differentiating the barrier function, we obtain: ； make ； In the formula, Indicates the first Virtual control rate of each joint Indicates the first layer controller gain for each joint The first layer error represents the... Robust compensation terms for each joint, among which... , Therefore, the virtual control law formula 1 can be obtained as follows: ； Combining Formula 1 with Lemma 2, we get Formula 2 as follows: ； Define the second Lyapunov function: ； In the formula, This represents the second Lyapunov function. Indicates the second layer error; Taking the derivative of the second-level error, we get: ； Differentiating the second Lyapunov function and combining it with the derivative of the second-level error and the result of Equation 2, we obtain Equation 3: ； In the formula, Indicates the first Filtering error of each joint; The controller control rate is designed to be: ； In the formula, This indicates the controller's control rate.

6. The robotic arm predefined time trajectory tracking control method based on time delay estimation according to claim 5, characterized in that: The method for constructing the trajectory tracking algorithm is as follows: Lyapunov functions containing the error energies of each layer are constructed using error term scaling and time convergence criteria. Define candidate Lyapunov functions: ； In the formula, Represents a candidate Lyapunov function; Depend on have to ； Differentiate the filtering error and then... Substitute: ； Construct the disturbance term to represent the derivative of the virtual control law: ； In the formula, Let represent the interference term, which is a continuous and bounded function. After simplification, the formula for the derivative with respect to the filtering error is: ； at this time Then, according to Young's inequality, the following transformations are made: ； Differentiate the candidate Lyapunov function and then... Substituting the result of Formula 3 into Formula 4, we get: In the formula, Indicates the first The first layer of error constraint strength for each joint Indicates the second layer The controller gain of each joint, and According to Young's inequality, we get: ； Combining Lemma 3, we obtain Formula 5: ； Therefore, Formula 6 is: ； In the formula, The attenuation coefficient represents the stability of the system. This represents the upper bound of the sum of system disturbances and errors. , ; Multiply formula 6 by Integrating again, we get: ； Therefore, the Lyapunov function is bounded, defined as follows: Error signal , and In scope The interior is bounded, within which, ; Formula 4, Formula 5 and Range Combining these, we get Formula 7: ； Then, according to Lemma 4, make ,get: ； In the formula, This represents the constant term introduced during scaling. Combining formula 7, we get formula 8: ； Substituting formula 8 into formula 7, we get: ； In the formula, This represents the upper bound of the total perturbation. ; Since the controlled system is time-stable, according to Lemma 1, an error signal is constructed. Combined with preset times to form a set: ； Therefore, the candidate Lyapunov function, the first-level error, the second-level error, the filtering error, the original trajectory error, and the system state are all determined to be bounded.

7. A robotic arm predefined time trajectory tracking control system based on time delay estimation, characterized in that: The system is used to execute the robotic arm predefined time trajectory tracking control method based on time delay estimation as described in any one of claims 1-6, including: The dynamics model module is used to establish a two-dimensional dynamics model of the robot arm based on the time delay estimation method, and to construct the mathematical assumptions and lemmas required to implement the trajectory tracking algorithm, so as to support error transformation, barrier function design and stability analysis. The assumptions include that the expected trajectory and its derivative are known, and the lemmas include providing time convergence criteria, error term scaling, logarithmic term processing and cross term decomposition. The error system module is used to construct a predefined time error transformation function based on the two-dimensional dynamic model by processing several terms and using time convergence criteria, so as to transform the original trajectory error into a transformed error system that satisfies the predefined time convergence characteristics. The control law design module is used to design the barrier function and feedback control law calculation formula based on the transformation error system by scaling the error term and decomposing the cross term. Based on the calculation results, it performs hierarchical suppression of position, velocity and acceleration errors, and dynamically adjusts the error convergence boundary through the barrier function. The stability proof module is used to construct a Lyapunov function containing the error energy of each layer based on the barrier function and the calculation formula of the feedback control law, through error term scaling and time convergence criteria, to complete the construction of the trajectory tracking algorithm and achieve high-precision tracking of the desired trajectory.

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