A multi-constraint-oriented manipulator tracking control method
By integrating a general obstacle function, a radial basis function neural network, and Nussbaum gain technology, a virtual control law and an adaptive neural network controller were designed to solve the problem of high-precision trajectory tracking of a robotic arm under strong nonlinearity and physical constraints. This achieved strict satisfaction of all-state constraints and fixed-time convergence, improving the reliability and engineering applicability of the system.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SICHUAN UNIV
- Filing Date
- 2026-05-21
- Publication Date
- 2026-07-10
AI Technical Summary
Existing robotic arm control methods struggle to achieve high-precision trajectory tracking when faced with strong nonlinearity, parameter uncertainty, and physical constraints. Furthermore, they are insufficient in handling full-state constraints, especially when the control direction is uncertain due to unknown inertial matrices.
By employing a general obstacle function, radial basis function neural network, and Nussbaum gain technique, combined with fixed-time control theory, a virtual control law and an adaptive neural network controller are designed to construct a tracking control method for a robotic arm oriented towards multiple constraints, thereby achieving high-performance control of a robotic arm with unknown dynamics.
It achieves strict satisfaction of all-state constraints, has fixed-time convergence characteristics, adapts to completely unknown dynamics, solves the problem of control direction uncertainty caused by unknown inertia matrix, and improves the reliability and engineering applicability of the system.
Smart Images
Figure CN122353602A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of high-precision control technology for robotic arms, and in particular to a tracking control method for robotic arms oriented towards multiple constraints. Background Technology
[0002] With the deepening development of industrial automation and human-machine collaboration, robotic arms are increasingly widely used in precision manufacturing, medical assistance, and space manipulation. High-precision and high-reliability trajectory tracking control is crucial for mission success. However, practical robotic arm systems generally face two major challenges: first, the system dynamics model contains strong nonlinearity, parameter uncertainties, and external disturbances; second, they are inevitably subject to physical constraints during operation, such as the requirement that state variables like joint angles and angular velocities must be kept within safe ranges. Therefore, designing a control strategy that can effectively suppress uncertainties, strictly satisfy all-state constraints, and possess fast convergence characteristics is of significant engineering importance.
[0003] Existing research on high-precision control of robotic arms mainly focuses on methods such as sliding mode control, adaptive control, and neural network control. However, these methods still have shortcomings in dealing with full-state constraints and uncertainties: (1) Although sliding mode control is robust, it has inherent chattering problems and it is difficult to ensure that the system's maneuverability is met while ensuring that the constraints are met; (2) Adaptive control can cope with some parameter uncertainties, but its ability to suppress unmodeled dynamics and sudden disturbances is limited, and it is difficult to handle complex full-state constraints; (3) Although neural network control can approximate complex nonlinearities, it has problems of high computational complexity and dependence on training samples. In particular, under constraints, how to ensure strict satisfaction of constraints and the unknown control direction (the unknown inertia matrix leads to uncertainty in the control direction) remains an unsolved problem. Existing research on state constraints mainly focuses on output constraints and pays insufficient attention to full-state (position and velocity) constraints. Moreover, it often relies on precisely known dynamic models or inertia matrix information, which limits its application in real complex scenarios. Summary of the Invention
[0004] The purpose of this invention is to address the problems in existing technologies by proposing a multi-constraint robotic arm tracking control method. This method integrates a universal obstacle function, radial basis function neural network, Nussbaum gain technique, and fixed-time control theory to construct a multi-constraint robotic arm tracking control approach, achieving high-performance control of robotic arms with unknown dynamics.
[0005] This invention is achieved through the following technical solution: This invention proposes a robotic arm tracking control method oriented towards multiple constraints, the method comprising: Step 1: Establish the dynamic model and constraint description of the robotic arm; Step 2: Constraint transformation based on the general obstacle function; Step 3: Design the virtual control law and adaptive neural network controller; Step 4: Stability analysis and proof of fixed-time convergence; Step 5: Constraint satisfaction verification.
[0006] Further, step 1 includes: Consider a rigid manipulator with n degrees of freedom, whose dynamic model is as follows:
[0007] in These represent the joint angle vector, angular velocity vector, and angular acceleration vector, respectively. Indicates an unknown external disturbance; Represents control torque; matrix , and These represent the inertia matrix, Coriolis force matrix, and gravity vector of the robotic arm, respectively.
[0008] Furthermore, the tracking error is defined as: , ,in For the desired trajectory, Let i be the virtual control law to be designed; the full-state constraints are expressed as follows: for each joint i, the following conditions are met:
[0009] in, These are the time-varying upper and lower bounds for joint angle and angular velocity, respectively.
[0010] Furthermore, step 2 specifically includes: To handle full-state constraints, a general obstacle function is introduced, and the transformed error variable is defined as follows:
[0011] in, Through the above transformation, it is ensured that when the variable is transformed... When bounded, the original state error strictly satisfies the constraints.
[0012] Furthermore, in step 3, the virtual control law design is specifically as follows: Choosing Lyapunov functions The virtual control law is designed as follows:
[0013] in, It is an invertible diagonal matrix related to the constraints; For design parameters;
[0014] For composite functions used to avoid singularities and achieve fixed-time convergence, here , , It is a positive number.
[0015] Furthermore, in step 3, the design of the neural network and adaptive law is specifically as follows: A radial basis function neural network is used to approximate the total uncertainty of the system, and virtual parameters are introduced. Design an adaptive update law:
[0016] in, for The estimated value, For design parameters, , is the Gaussian function vector.
[0017] Furthermore, in step 3, the controller design is specifically as follows: To address the problem of unknown control direction, the Nussbaum function is introduced. The actual control law is designed as follows:
[0018]
[0019] in, To control the gain, For fixed-time convergence parameters, It is an invertible diagonal matrix related to the constraints.
[0020] Furthermore, step 4 specifically includes: Constructing Lyapunov functions
[0021] By analyzing its derivative, and combining the properties of the Nussbaum function, the boundedness of neural network approximation error, and Young's inequality, it is proved that all signals in the closed-loop system are uniformly bounded, and the conversion error... At a fixed time It converges inward to a small neighborhood near the origin; where, , , Upper limit of convergence time It is independent of the initial state of the system and is determined solely by the controller parameters.
[0022] The present invention also proposes an electronic device, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the steps of the multi-constraint-oriented robotic arm tracking control method.
[0023] The present invention also proposes a computer-readable storage medium for storing computer instructions, which, when executed by a processor, implement the steps of the multi-constraint-oriented robotic arm tracking control method.
[0024] The beneficial effects of this invention are: 1. Strict satisfaction of full-state constraints: Unlike methods that only consider output constraints, this invention directly integrates the full-state constraints of joint angles and angular velocities into the controller design through a general obstacle function, ensuring that all state variables do not violate the preset physical boundaries from the initial moment, which significantly improves the reliability of the system in safety-critical tasks.
[0025] 2. Adaptability to completely unknown dynamics: A radial basis function neural network is used to approximate the total dynamic uncertainty, including the inertia matrix, Coriolis force, gravity, and external disturbances, online without requiring any prior dynamic knowledge. Addressing the fundamental challenge of control direction uncertainty caused by an unknown inertia matrix, a novel Nussbaum gain mechanism is introduced for effective compensation, overcoming the limitation of existing methods that require knowledge of the inertia matrix or the sign of the control gain.
[0026] 3. Fixed-time convergence characteristic: Compared with finite-time control, the fixed-time convergence characteristic achieved by this invention makes the upper limit of the convergence time of the tracking error independent of the initial state and can be preset according to the controller parameters, providing a deterministic guarantee for task time planning.
[0027] 4. Simple structure and engineering applicability: Compared with deep learning methods that rely on a large number of training samples and have complex structures, this invention is based on the classic backstepping framework and combined with a simplified adaptive law (which transforms the weight matrix of a high-dimensional neural network into a scalar parameter estimate). It has high computational efficiency, is easy to implement in practical embedded controllers, and has good potential for engineering application. Attached Figure Description
[0028] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.
[0029] Figure 1 This is a curve showing the joint angle tracking.
[0030] Figure 2 This is a graph showing the joint angular velocity tracking.
[0031] Figure 3 This is a graph showing the joint angle tracking error.
[0032] Figure 4 This is a graph showing the tracking error of joint angular velocity.
[0033] Figure 5 This is an adaptive parameter estimation curve, where the parameter estimates remain bounded during the control process.
[0034] Figure 6 To control the torque curve. Detailed Implementation
[0035] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0036] To address the problems existing in the prior art, this invention aims to provide a multi-constraint robotic arm tracking control method to solve the following technical problems: (1) How to achieve high-precision trajectory tracking when the dynamic model of the robotic arm is completely unknown (including the unknown inertia matrix); (2) How to strictly ensure that the constraints of the entire state, such as joint angle and angular velocity, are not violated during the control process; (3) How to achieve fixed-time convergence of tracking error, that is, the upper limit of convergence time does not depend on the initial state of the system; (4) How to solve the problem of uncertain control direction caused by unknown inertia matrix and ensure system stability.
[0037] Specifically, in combination Figures 1-6 This invention proposes a tracking control method for a robotic arm oriented towards multiple constraints, the method comprising: Step 1: Establish the dynamic model and constraint description of the robotic arm; Further, step 1 includes: Consider a rigid manipulator with n degrees of freedom, whose dynamic model is as follows:
[0038] in These represent the joint angle vector, angular velocity vector, and angular acceleration vector, respectively. Indicates an unknown external disturbance; Represents control torque; matrix , and These represent the inertia matrix, Coriolis force matrix, and gravity vector of the robotic arm, respectively.
[0039] Furthermore, the tracking error is defined as: , ,in For the desired trajectory, Let i be the virtual control law to be designed; the full-state constraints are expressed as follows: for each joint i, the following conditions are met:
[0040] in, These are the time-varying upper and lower bounds for joint angle and angular velocity, respectively.
[0041] Step 2: Constraint transformation based on the general obstacle function; Furthermore, step 2 specifically includes: To handle full-state constraints, a general obstacle function is introduced, and the transformed error variable is defined as follows:
[0042] in, Through the above transformation, it is ensured that when the variable is transformed... When bounded, the original state error strictly satisfies the constraints.
[0043] Step 3: Design the virtual control law and adaptive neural network controller; Furthermore, in step 3, the virtual control law design is specifically as follows: Choosing Lyapunov functions The virtual control law is designed as follows:
[0044] in, It is an invertible diagonal matrix related to the constraints; For design parameters;
[0045] For composite functions used to avoid singularities and achieve fixed-time convergence, here , , It is a small positive number.
[0046] Furthermore, in step 3, the design of the neural network and adaptive law is specifically as follows: A radial basis function neural network is used to approximate the total uncertainty of the system, and virtual parameters are introduced. Design an adaptive update law:
[0047] in, for The estimated value, For design parameters, , is the Gaussian function vector.
[0048] Furthermore, in step 3, the controller design is specifically as follows: To address the problem of unknown control direction, the Nussbaum function is introduced. The actual control law is designed as follows:
[0049]
[0050] in, To control the gain, For fixed-time convergence parameters, It is an invertible diagonal matrix related to the constraints.
[0051] Step 4: Stability analysis and proof of fixed-time convergence; Furthermore, step 4 specifically includes: Constructing Lyapunov functions
[0052] By analyzing its derivative, and combining the properties of the Nussbaum function, the boundedness of neural network approximation error, and Young's inequality, it is proved that all signals in the closed-loop system are uniformly bounded, and the conversion error... At a fixed time It converges inward to a small neighborhood near the origin; where, , , Upper limit of convergence time It is independent of the initial state of the system and is determined solely by the controller parameters.
[0053] Step 5: Constraint satisfaction verification.
[0054] As can be seen from the properties of the universal barrier function, as long as the conversion error... If the state is bounded, the original state error strictly satisfies the preset constraint boundary, thereby ensuring that the full-state constraint is not violated throughout the entire control process.
[0055] Example In the experimental case, specific values for the controller, etc., are shown in Table 1. The Franka Emika Panda robot platform was used, with other joints locked, activating only joints 2 and 3 to form a 2-DOF robotic arm system. The desired trajectory was a sinusoidal signal, with preset full-state constraint boundaries. Experimental data was recorded for 20 seconds.
[0056] Table 1. Parameters for tracking the desired trajectory, fixed-time controller, etc.
[0057] Experimental results show that both the joint angle and angular velocity accurately track the desired trajectory within 3 seconds, and are strictly confined within the preset constraint boundaries throughout the entire process. Even with a large initial error, the tracking error can still converge quickly and remain within the constraint range throughout the transient process, fully verifying the effectiveness of the proposed constraint handling mechanism. Simultaneously, the parameter estimates remain bounded throughout the process, the control torque changes smoothly, and its peak amplitude is far below the actuator's physical limit, indicating that the system has good engineering feasibility. In summary, the method of this invention can strictly satisfy state constraints, avoiding the risk of system downtime or mechanical damage due to state exceeding limits, and also achieves higher tracking accuracy.
[0058] This invention proposes a composite control architecture consisting of a "general obstacle function + RBF neural network + Nussbaum gain". This architecture is the first to simultaneously solve three major challenges within a unified framework: satisfying full-state constraints, approximating completely unknown dynamics, and compensating for unknown control directions, thus forming a complete fixed-time tracking control scheme.
[0059] This invention utilizes a general obstacle function to transform the original state constraint into a bounded problem of transformation variables, and combines it with fixed-time Lyapunov theory to prove that while ensuring the fixed-time convergence of transformation variables, the original state strictly satisfies the preset constraints, thus achieving synergy between constraint handling and control performance.
[0060] This invention addresses the problem of control direction uncertainty caused by a completely unknown inertia matrix by incorporating Nussbaum gain technology into the backstepping framework. Under the premise of ensuring system stability, it achieves dynamic adaptation of the control direction without requiring any prior information.
[0061] This invention simplifies the online learning problem of the weight matrix of a high-dimensional neural network into the estimation of a single scalar parameter by using Young's inequality, which significantly reduces the online computational burden of the control algorithm and enhances the real-time performance and engineering applicability of the method.
[0062] The present invention also proposes an electronic device, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the steps of the multi-constraint-oriented robotic arm tracking control method.
[0063] The present invention also proposes a computer-readable storage medium for storing computer instructions, which, when executed by a processor, implement the steps of the multi-constraint-oriented robotic arm tracking control method.
[0064] The memory in this application embodiment can be volatile memory or non-volatile memory, or it can include both volatile and non-volatile memory. The non-volatile memory can be read-only memory (ROM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), or flash memory. The volatile memory can be random access memory (RAM), which is used as an external cache. By way of example, but not limitation, many forms of RAM are available, such as static random access memory (SRAM), dynamic random access memory (DRAM), synchronous dynamic random access memory (SDRAM), double data rate synchronous dynamic random access memory (DDRSDRAM), enhanced synchronous dynamic random access memory (ESDRAM), synchronous linked dynamic random access memory (SLDRAM), and direct rambus RAM (DR RAM). It should be noted that the memory used in the methods described in this invention is intended to include, but is not limited to, these and any other suitable types of memory.
[0065] In the above embodiments, implementation can be achieved, in whole or in part, through software, hardware, firmware, or any combination thereof. When implemented in software, it can be implemented, in whole or in part, as a computer program product. The computer program product includes one or more computer instructions. When the computer instructions are loaded and executed on a computer, all or part of the processes or functions described in the embodiments of this application are generated. The computer can be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions can be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer instructions can be transmitted from one website, computer, server, or data center to another via wired (e.g., coaxial cable, fiber optic, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium can be any available medium accessible to a computer or a data storage device such as a server or data center that integrates one or more available media. The available media may be magnetic media (e.g., floppy disks, hard disks, magnetic tapes), optical media (e.g., high-density digital video discs (DVDs)), or semiconductor media (e.g., solid-state disks (SSDs)).
[0066] In implementation, each step of the above method can be completed by integrated logic circuits in the processor's hardware or by instructions in software. The steps of the method disclosed in the embodiments of this application can be directly implemented by a hardware processor, or by a combination of hardware and software modules in the processor. The software modules can reside in random access memory, flash memory, read-only memory, programmable read-only memory, electrically erasable programmable memory, registers, or other mature storage media in the art. This storage medium is located in memory, and the processor reads information from the memory and, in conjunction with its hardware, completes the steps of the above method. To avoid repetition, detailed descriptions are omitted here.
[0067] It should be noted that the processor in the embodiments of this application can be an integrated circuit chip with signal processing capabilities. During implementation, each step of the above method embodiments can be completed by the integrated logic circuitry in the processor's hardware or by instructions in software form. The processor can be a general-purpose processor, a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), or other programmable logic devices, discrete gate or transistor logic devices, or discrete hardware components. It can implement or execute the methods, steps, and logic block diagrams disclosed in the embodiments of this application. The general-purpose processor can be a microprocessor or any conventional processor. The steps of the methods disclosed in the embodiments of this application can be directly embodied as execution by a hardware decoding processor, or as a combination of hardware and software modules in the decoding processor. The software modules can be located in random access memory, flash memory, read-only memory, programmable read-only memory, electrically erasable programmable memory, registers, or other mature storage media in the art. This storage medium is located in memory, and the processor reads the information in the memory and, in conjunction with its hardware, completes the steps of the above methods.
[0068] The above provides a detailed description of a multi-constraint robotic arm tracking control method proposed in this invention. Specific examples have been used to illustrate the principles and implementation methods of this invention. The descriptions of the above embodiments are only for the purpose of helping to understand the method and core ideas of this invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this invention. Therefore, the content of this specification should not be construed as a limitation of this invention.
Claims
1. A tracking control method for a robotic arm oriented towards multiple constraints, characterized in that, The method includes: Step 1: Establish the dynamic model and constraint description of the robotic arm; Step 2: Constraint transformation based on the general obstacle function; Step 3: Design the virtual control law and adaptive neural network controller; Step 4: Stability analysis and proof of fixed-time convergence; Step 5: Constraint satisfaction verification.
2. The method according to claim 1, characterized in that, Step 1 includes: Consider a rigid manipulator with n degrees of freedom, whose dynamic model is as follows: in These represent the joint angle vector, angular velocity vector, and angular acceleration vector, respectively. Indicates an unknown external disturbance; Represents control torque; matrix , and These represent the inertia matrix, Coriolis force matrix, and gravity vector of the robotic arm, respectively.
3. The method according to claim 2, characterized in that, The tracking error is defined as: , ,in For the desired trajectory, For the virtual control law to be designed; The full-state constraint is expressed as follows: For each joint i, the following condition is satisfied: in, These are the time-varying upper and lower bounds for joint angle and angular velocity, respectively.
4. The method according to claim 1, characterized in that, Step 2 specifically includes: To handle full-state constraints, a general obstacle function is introduced, and the transformed error variable is defined as follows: in, Through the above transformation, it is ensured that when the variable is transformed... When bounded, the original state error strictly satisfies the constraints.
5. The method according to claim 1, characterized in that, In step 3, the virtual control law design is specifically as follows: Choosing Lyapunov functions The virtual control law is designed as follows: in, It is an invertible diagonal matrix related to the constraints; For design parameters; For composite functions used to avoid singularities and achieve fixed-time convergence, here , , It is a positive number.
6. The method according to claim 5, characterized in that, In step 3, the design of the neural network and adaptive law is as follows: A radial basis function neural network is used to approximate the total uncertainty of the system, and virtual parameters are introduced. Design an adaptive update law: in, for The estimated value, For design parameters, , is the Gaussian function vector.
7. The method according to claim 6, characterized in that, In step 3, the controller design is as follows: To address the problem of unknown control direction, the Nussbaum function is introduced. The actual control law is designed as follows: in, To control the gain, For fixed-time convergence parameters, It is an invertible diagonal matrix related to the constraints.
8. The method according to claim 1, characterized in that, Step 4 specifically includes: Constructing Lyapunov functions By analyzing its derivative, and combining the properties of the Nussbaum function, the boundedness of neural network approximation error, and Young's inequality, it is proved that all signals in the closed-loop system are uniformly bounded, and the conversion error... At a fixed time It converges inward to a small neighborhood near the origin; where, , , Upper limit of convergence time It is independent of the initial state of the system and is determined solely by the controller parameters.
9. An electronic device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the steps of the method according to any one of claims 1-8.
10. A computer-readable storage medium for storing computer instructions, characterized in that, When the computer instructions are executed by the processor, they implement the steps of the method according to any one of claims 1-8.