Usr fixed time path tracking control method with preset performance

Abstract

The application discloses a USR fixed time path tracking control method with preset performance, relates to the technical field of underwater bionic robot motion control, and comprises the following steps: a simplified control guide model is established; based on a fixed time convergence system, an extended state observer combining an integral sliding mode term and adaptive gain adjustment is constructed, a composite controller is constructed, a global expected path is generated, a sensor collects the coordinate position, a heading angle and a joint state of a USR center of mass, the extended state observer is started to capture and quantitatively estimate total disturbance of an underwater environment in real time; the composite controller is based on a dynamics model, a preset performance mapping relationship and an estimated value of the extended state observer on the total disturbance, the system dynamics satisfies a fixed time sliding mode control law, and a final control torque instruction is obtained. The application can provide the USR with the autonomous path following control capability with shorter fixed time convergence characteristics, high anti-interference robustness and actuator protection capability under typical complex underwater environmental conditions.

Description

Technical Field

[0001] This invention relates to the field of underwater biomimetic robot motion control technology, and in particular to a USR fixed-time path tracking control method with preset performance, specifically an anti-saturation robust control method applied to underwater snake robots in the process of performing path following tasks. Background Technology

[0002] In recent years, with the increasing global efforts in marine resource development, underwater robots have played an increasingly crucial role in seabed resource exploration, marine structure inspection and maintenance, and other fields. In confined operating spaces and complex aquatic environments, compared to traditional working-class remotely operated vehicles (ROVs), underwater snake robots (USRs) demonstrate immense application potential due to their slender, bio-inspired structure, high mobility of multiple joints, and superior motion efficiency. To ensure the successful execution of various complex tasks, high-precision centroid path following control technology is a core prerequisite for USRs.

[0003] Regarding the modeling and control of USR, existing technical solutions still have the following common shortcomings when dealing with complex constraints: One representative challenge is the coupling of external environmental disturbances with internal model uncertainties. During motion, underactuated robots (USRs) are not only affected by model parameter perturbations (such as uneven mass distribution and changes in joint friction), but also by strong nonlinear, time-varying external ocean current disturbances. Existing techniques attempt to improve the depth tracking capability of underactuated robots by using second-order sliding mode perturbation observers for estimation or by introducing nonlinear perturbation observers (NDOs). Although these methods compensate for the disturbances to some extent, when faced with high-frequency fluctuating composite disturbances, the convergence speed of existing observers often fails to match the joint oscillation frequency of the USR, leading to decreased estimation accuracy and consequently path-following deviations.

[0004] Another core issue is balancing the convergence efficiency and robustness of control systems. Traditional asymptotically stable control can only guarantee that the error approaches zero over infinite time. To improve response speed, finite-time control has been introduced, allowing users to adjust the convergence time. However, the performance of finite-time control is highly dependent on the initial state; if the initial position deviation is too large, the convergence time will increase significantly. Although fixed-time control has emerged in recent years, ensuring that the upper bound of the convergence time is independent of the initial state, in practical applications, the upper bound of the convergence of traditional fixed-time systems is still not tight enough, causing the USR to lag when dynamically switching tasks.

[0005] Furthermore, actuator saturation is a physical bottleneck restricting system stability. In practical engineering applications, USR joint actuators are limited by physical characteristics and can only provide a limited control torque. When the control command exceeds the torque limit, the actuator enters the saturation region, which leads to a sharp decrease in control gain, deterioration of system performance, and in severe cases, even instability of the closed-loop system. Although there is mature research on anti-saturation compensation for land robotic arms, for USRs with complex hydrodynamic constraints, how to simultaneously handle the dynamic coupling between actuator saturation and preset error performance while ensuring convergence within a fixed time is a question that has not yet been addressed in the published literature with a mature and complete technical solution. Summary of the Invention

[0006] To address the technical problems of slow convergence speed, low steady-state accuracy, and insufficient robustness in existing USR path following control processes due to multiple factors such as environmental time-varying disturbances, model uncertainties, actuator physical saturation, and preset performance constraints, this invention proposes a USR fixed-time path tracking control method with preset performance.

[0007] The technical solution adopted by this invention to solve its technical problem is: a USR fixed-time path tracking control method with preset performance, comprising the following steps: Step 1, USR system modeling and kinematic description: Establish a simplified control-oriented model based on the dynamic equations; Step 2, Design the extended state observer: Construct a fixed-time convergent system containing a variable exponential term, and based on the fixed-time convergent system, construct an extended state observer that combines an integral sliding mode term and adaptive gain adjustment; Step 3, constructing a composite controller: The navigation guidance module converts path deviation into the desired heading angle using an adaptive integral line-of-sight method; the preset performance constraint module uses a time-varying performance function. The transient and steady-state performance of the USR joint angle tracking error is constrained. The constrained tracking error is mapped to an unconstrained conversion error through a transformation function. The adaptive auxiliary module compensates for the actuator saturation effect. The preset performance constraint module and the state of the adaptive auxiliary module are combined to obtain the preset performance mapping relationship. Step 4: Generate the global desired path through the preset path algorithm. The sensor collects the coordinate position, heading angle and joint status of the USR centroid. Simultaneously, the extended state observer obtained in Step 2 is started to capture and quantify the total underwater environmental disturbance in real time. The navigation guidance module outputs data as the input of the preset performance constraint module. Step 5: The composite controller is based on the dynamic model from Step 1, the preset performance mapping relationship from Step 3, and the estimate of the lumped disturbance from the extended state observer from Step 4. By ensuring that the system dynamics satisfy the fixed-time sliding mode control law, the final control torque command can be calculated. .

[0008] In the aforementioned USR fixed-time path tracking control method with preset performance, the dynamic equation of the fixed-time convergent system in step 2 is: ; ; ; ; Where k1 represents the variable exponential coefficient, k2 represents the constant exponential coefficient, m1 represents the system convergence exponent, m2 represents the system convergence exponent, n1 represents the system positive gain coefficient, n2 represents the system positive gain coefficient, e represents the natural constant, and z represents the system state variable. This represents the rate of change of state.

[0009] In the aforementioned USR fixed-time path tracking control method with preset performance, the extended state observer extended system in step 2 is as follows: ; ; ; in, For the improved fixed-time convergent function, p i q i The gain coefficient of the nonlinear function. It is a nonlinear exponential term, which is the power that ultimately acts on the observer error correction function; It is a low power exponent. This is the state estimate. x1 represents the joint angle estimation error, and x1 represents the system state variable. These are estimates of the system state variables. This is the actual joint angle. To estimate joint angles; The observer gain coefficient, For adaptive gain term, To achieve dynamic adjustment of the gain, g is the gain coefficient; m is the link mass; AD is the structure matrix; DD is the structure matrix; u is the torque of the i-th actuator; c n c is the normal friction coefficient. p The correlation coefficient represents the fluid interaction. For the final control torque command, Let k be the tangential relative velocity. i1 k i2These are intermediate mapping parameters used to transform m1 and m2 into transitional variables for the final exponent.

[0010] The aforementioned USR fixed-time path tracking control method with preset performance, wherein the time-varying performance function in step 3... ; in, Let be the initial value of the performance function, b be a constant factor, and t be time. For steady-state error boundary, The attenuation coefficient; The gain coefficient is time-varying and designed in time segments to balance convergence speed and accuracy.

[0011] In the aforementioned USR fixed-time path tracking control method with preset performance, the calculation formula for the adaptive auxiliary module in step 3 is as follows: ; in, , As auxiliary system state variables, , The first derivative of the state variable with respect to time represents the rate of change of the compensation intensity; , The gain coefficient is positive. By selecting appropriate parameters, the system matrix can satisfy the Hurwitz condition, thus ensuring the exponential convergence of the system state. To control the saturation difference of the input, For the final control torque command, This is the torque that actually acts on the robot.

[0012] In the aforementioned USR fixed-time path tracking control method with preset performance, the preset performance mapping relationship in step 3 is specifically as follows: ; ; ; in, , , Used to describe the dynamic mapping relationship from the physical error space to the preset performance constraint space; and Responsible for compensating for performance boundaries The time-varying nonlinear term generated as time converges; By linearizing the joint acceleration information into a preset performance framework, the controller can directly calculate the physical torque command within the transformation space. This represents the time-varying error performance index function, where e represents the joint angle tracking error. Represents performance function The first derivative with respect to time, This represents the first derivative of the tracking error e with respect to time. Represents performance function The second derivative with respect to time.

[0013] The aforementioned USR fixed-time path tracking control method with preset performance, wherein the heading guidance module processing flow in step 4 is as follows: based on the lateral error of the centroid offset from the desired path, the optimal desired heading angle for the regression path is calculated; based on the current heading error, an offset signal is output; combined with the serpentine gait equation, the desired turning angle command that each joint actuator should achieve at the current moment is calculated; and the desired turning angle command is used as the reference input of the preset performance constraint module.

[0014] The aforementioned USR fixed-time path tracking control method with preset performance, wherein the final control torque command in step 5... for: ; ; ; ; ; ; in, It is responsible for resisting residual disturbances and forcing the system state to contract towards the sliding surface. The USR dynamic model is used for feedforward compensation to counteract the effects of nonlinear drag, Coriolis force and acceleration of the reference trajectory. It is a positive gain, and the parameters satisfy... and m5 and m6 represent the reaching law exponent constants, and s represents the sliding surface function. This represents the estimated value of the lumped disturbance. This represents the normal velocity between adjacent links. Indicates the robot's tangential velocity. This represents the desired joint angular acceleration. Represents the state variables of the adaptive auxiliary system. Represents the state variables of the adaptive auxiliary system. Represents nonlinear terms The first derivative, The switching function itself is designed to avoid sliding mode singularity, and its derivative... It is incorporated into the controller solution to ensure the fixed-time convergence of the system; , The intermediate exponent parameter of the reaching law, This is a composite variable that combines the preset performance conversion error with the auxiliary system state. yes The first derivative.

[0015] The beneficial effects of this invention are: (1) It proposes an improved fixed-time convergence system with a shorter convergence time. Although traditional fixed-time control systems can ensure that the convergence time is independent of the initial state, there is still room for improvement in their convergence speed. This paper improves upon the classic fixed-time system and designs a novel fixed-time convergence system, whose upper bound on convergence time is significantly smaller than that of existing systems. This system introduces a variable exponential term to adaptively adjust the convergence behavior at different stages, thereby achieving faster convergence characteristics globally, laying a theoretical foundation for the subsequent design of observers and controllers.

[0016] (2) An extended state observer based on an improved fixed-time system was constructed. To address the external time-varying disturbances and internal model uncertainties encountered by the USR during execution, this paper designs an extended state observer based on a novel fixed-time system. This observer, combining integral sliding mode terms and adaptive gain adjustment, can quickly and accurately estimate lumped disturbances, joint angles, and angular velocities, achieving high-precision real-time observation of the system's critical states. Compared with traditional observers, this method maintains high estimation accuracy even under rapid time-varying disturbances, providing effective disturbance feedforward compensation for the controller and improving the system's robustness in complex underwater environments.

[0017] (3) An adaptive auxiliary system was designed. To address the input saturation problem of the USR actuator, this invention designs a stable adaptive auxiliary system. By dynamically compensating for the saturation error, the impact of saturation on system stability and tracking performance is mitigated. This auxiliary system has a simple structure and adjustable parameters, ensuring that the system remains stable when saturation occurs and that the tracking error converges within a preset performance range, significantly improving the applicability of the controller under actual physical constraints.

[0018] (4) A composite control strategy integrating preset performance function (PPC), fixed-time convergence, extended state observation, and fast non-singular terminal sliding mode control is proposed. This invention is the first to combine the preset performance function (PPC) with the improved fixed-time non-singular terminal sliding mode control (FFNTSMC) and apply it to the centroid path tracking control of the USR. By designing a time-varying error performance function, the system can constrain the tracking error within a preset range while ensuring transient and steady-state performance. Combined with the designed fixed-time extended state observer and adaptive auxiliary system, this controller can still achieve fixed-time convergence under the conditions of actuator saturation, time-varying disturbances, and model uncertainties, and has a faster response speed and higher tracking accuracy. Attached Figure Description

[0019] Figure 1 This is a schematic diagram illustrating the simplification process of the USR model in an embodiment of the present invention; Figure 2 This is a schematic diagram of the control method of the present invention; Figure 3 This is a schematic diagram of the USR centroid trajectory tracking of the present invention; Figure 4 These are schematic diagrams of path tracking and tracking error under different initial states of the present invention, wherein (a) is a schematic diagram of path tracking and (b) is a schematic diagram of tracking error; Figure 5 This is a comparison of path following and tracking errors of different joint controllers under the same conditions, where (a) is a schematic diagram of path following and (b) is a schematic diagram of tracking error. Figure 6 This is a comparison diagram of path following and tracking errors of different joint controllers of the present invention under the same conditions, wherein (a) is a schematic diagram of path following and (b) is a schematic diagram of tracking error; Figure 7 Under the controller designed in this invention, the control torque of the u1-u3 joint actuators of the underwater snake robot is as follows: (a) is the control torque of the u1 joint actuator, (b) is the control torque of the u2 joint actuator, and (c) is the control torque of the u3 joint actuator. Figure 8 Under the controller designed in this invention, the control torque of the u4-u6 joint actuators of the underwater snake robot is shown, where (a) is the control torque of the u4 joint actuator, (b) is the control torque of the u5 joint actuator, and (c) is the control torque of the u6 joint actuator. Figure 9 Under the controller designed in this invention, the control torque of the u7-u9 joint actuators of the underwater snake robot is shown, where (a) is the control torque of the u7 joint actuator, (b) is the control torque of the u8 joint actuator, and (c) is the control torque of the u9 joint actuator. Figure 10 This is the real-time estimation effect of the fixed-time extended state observer on the sinusoidal disturbance in this embodiment, where (a) is the estimation result of the underwater snake robot u1 joint actuator, (b) is the estimation result of the underwater snake robot u5 joint actuator, and (c) is the estimation result of the underwater snake robot u9 joint actuator; Figure 11 This is the real-time estimation effect of the fixed-time extended state observer on the state disturbance in this embodiment, where (a) is the estimation result of the underwater snake robot u1 joint actuator, (b) is the estimation result of the underwater snake robot u5 joint actuator, and (c) is the estimation result of the underwater snake robot u9 joint actuator; Figure 12 This is the real-time estimation effect of the fixed-time extended state observer on the composite disturbance in this embodiment, where (a) is the estimation result of the underwater snake robot u1 joint actuator, (b) is the estimation result of the underwater snake robot u5 joint actuator, and (c) is the estimation result of the underwater snake robot u9 joint actuator. Detailed Implementation

[0020] To enable those skilled in the art to better understand the technical solution of the present invention, the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0021] This embodiment uses the precise path tracking control of an underwater snake robot (USR) with parallel joints under conditions of actuator saturation, time-varying disturbances, and model uncertainties as an example to explain in detail the specific implementation methods of each technical module. For example... Figure 2 As shown, it includes the following steps: Step 1, USR System Modeling and Kinematic Description: Establish a simplified control-guided model based on the dynamic equations, such as... Figure 1 As shown in the diagram, the centroid trajectory tracking diagram is as follows: Figure 3 As shown.

[0022] USR by A link and It consists of joints, each link has a length of and a uniformly distributed mass of . The global coordinate system is... The local coordinate system is These correspond to the tangential and normal directions of the robot, respectively. This embodiment considers a robot composed of... The USR consists of several links, each link having a length of... ,quality Its system dynamics equations are: ; in These are the tangential and normal friction coefficients, respectively; External flow field disturbance; This refers to the lumped disturbance, which includes external time-varying disturbances and model uncertainties; Input torque to the joint actuator; in this embodiment, . The structure matrix is ​​determined by the robot's geometry. Each joint is driven by an independent servo motor, and its dynamics can be simplified as follows: ; in For rotational inertia, The damping coefficient is... This represents the joint disturbance torque.

[0023] Set initial joint angles Joint angular velocity centroid position heading angle The desired path is generated using cubic spline interpolation, and the desired joint swing trajectory is as follows: ; in These are amplitude, frequency, phase difference, and offset angle, respectively. In this embodiment, Offset angle The heading controller is generated online.

[0024] Actuator saturation limit set to The saturation function is defined as: ; The lumped disturbance d takes three forms: Sine perturbation: ; State perturbation: ; Composite disturbance: ; Step 2, Design the extended state observer: Construct a fixed-time convergent system containing a variable exponent term, and based on the fixed-time convergent system, construct an extended state observer that combines an integral sliding mode term and adaptive gain adjustment.

[0025] Design a novel fixed-time convergence system as the basis for the observer and controller: ; The sign power function is defined as follows: The key parameters are designed as follows: exponential parameter , Gain parameters , The nonlinear exponential term is calculated as follows: ; In the specific implementation, simulation parameters are selected. , , , The system can be based on Automatic size switching working mode: When At that time, the system behaves as ;when At that time, the system behaves as This piecewise design allows for faster convergence when the error is large, using terms with powers greater than 1, while avoiding singularity issues when the error is small. Lyapunov analysis proves that the system has fixed-time stability, with an upper bound on the convergence time: ; Calculations show that Second.

[0026] To estimate lumped disturbances in real time and joint angular velocity A third-order fixed-time extended state observer is designed. First, the dynamics of the USR joint subsystem are rewritten in state-space form: ; in In an expanding state, Let be the derivative of the perturbation, and assume it is bounded. .

[0027] The designed Fixed-Time Extended State Observer (FTESO) has the following structure: ; in This represents the joint angle estimation error. (Nonlinear function) A design based on an improved fixed-time system is adopted: ; The exponent parameter is calculated as follows: ; The observer gain parameters are set as follows: number of links n=10, link length l=0.14m, link mass m=1kg, and link moment of inertia J=0.0016. tangential friction coefficient normal friction coefficient Rotational damping coefficient 1 Rotational damping coefficient 2 Actuator output torque limit The integral sliding mode term gain employs an adaptive design. ,in This observer guarantees that the estimation error converges within a fixed time, and the upper bound of the convergence time can be calculated using the following formula.

[0028] ; Step 3, constructing a composite controller: The navigation guidance module converts path deviation into the desired heading angle using an adaptive integral line-of-sight method; the preset performance constraint module uses a time-varying performance function. The transient and steady-state performance of the USR joint angle tracking error is constrained. The constrained tracking error is mapped to an unconstrained conversion error through a transformation function. The adaptive auxiliary module compensates for the actuator saturation effect. The preset performance constraint module and the state of the adaptive auxiliary module are combined to obtain the preset performance mapping relationship.

[0029] In this embodiment, the composite controller includes the following modules: Heading guidance module: Employs an improved adaptive integral line-of-sight method (ILOS) to convert path deviation into the desired heading angle. Gait generation module: Combines heading commands with serpentine gait equations to calculate the desired swing angle signal for each joint. Adaptive auxiliary compensation module: Specifically addresses the physical limitations of the joint motors, compensating for deviations caused by actuator input saturation (insufficient power). Preset performance constraint module (PPC): Utilizes a time-varying performance function to constrain the transient and steady-state behavior of tracking errors. Fixed-time sliding mode control law (FNTSMC): As the core algorithm, it uses an improved fixed-time stabilization system to ensure that all errors converge to zero within a fixed time.

[0030] To achieve precise control over the transient and steady-state performance of joint tracking errors, a time-varying performance function is introduced. .

[0031] The parameters are set as follows: (satisfy ), (for steady-state error boundaries) (Attenuation coefficient). Time-varying gain Designed as a piecewise function to optimize the convergence process: ; Switching time The performance function satisfies:

[0032] Through strictly increasing transformation functions The constrained tracking error Mapping to unconstrained transformation error : ; Therefore, the inverse transformation can be obtained: ; In actual operation of underwater snake robots (USRs), the actuators are limited by physical structure and can only provide a limited control torque. When the control command exceeds the torque limit, a control input constraint problem arises, affecting system stability. To address this, this invention designs the following second-order adaptive auxiliary system to compensate for saturation deviation: ; in: , These are the state variables of the auxiliary system. , The gain coefficient is positive. By selecting appropriate parameters, the system matrix can satisfy the Hurwitz condition, ensuring the exponential convergence of the system state. To control the saturation difference of the input.

[0033] To achieve decoupled control between the saturation compensation state and the preset performance conversion error, this invention defines a composite variable. The preset performance constraint module is combined with the state of the adaptive auxiliary module: ; calculate First and second derivatives: ; ; The intermediate variable is defined as follows: ; ; ; Among them, intermediate variables , , Used to describe the dynamic mapping relationship from the physical error space to the preset performance constraint space. and Responsible for compensating for performance boundaries The time-varying nonlinear term generated by convergence over time ensures that the control process does not become unstable as the boundary shrinks. As a key mapping gain, joint acceleration information is linearly introduced into the preset performance framework, enabling the controller to directly calculate the physical torque command within the transformation space.

[0034] Step 4: Generate the global desired path through the preset path algorithm. The sensor collects the coordinate position, heading angle and joint status of the USR centroid. Simultaneously, the extended state observer obtained in Step 2 is started to capture and quantify the total underwater environmental disturbance in real time. The navigation guidance module outputs data as the input of the preset performance constraint module.

[0035] The specific processing flow of the heading guidance module is as follows: First, define the lateral path tracking error. This error represents the vertical distance from the USR centroid to the virtual guide point on the desired path: ; in The coordinates of the virtual guide point on the desired path. This represents the tangential angle of the path at the guide point. The motion of the virtual guide point is correlated with the forward speed of the USR to ensure that the guide point is always positioned appropriately in front of the USR.

[0036] An improved adaptive integral line-of-sight method is used to calculate the desired heading angle. The traditional LOS method uses forward look distance... The fixed value results in slow convergence with large initial errors and is prone to overshoot with small errors. Therefore, an adaptive forward look distance design is proposed: ; in , The design increases the forward visibility distance. With lateral error Adaptive change: When the error is large, Approaching the minimum value This generates a large steering command, allowing the USR to quickly return to the path; when the error is small... Close to the maximum value This generates smooth steering commands and reduces overshoot.

[0037] Desired heading angle The calculation formula is: ; in The integral coefficient is... For the integral state, its update law is: ; The integral term is introduced to compensate for the steady-state tracking error caused by constant lateral environmental forces. When a constant lateral force is present, the pure proportional LOS method will produce a steady-state error, which the integral term can automatically adjust to eliminate.

[0038] The heading error is defined as the difference between the actual heading angle and the desired heading angle. ; Traditional USR heading control uses simple proportional control. However, this simple proportional control may produce an excessively large bias angle when there is a large error, affecting the smoothness of control.

[0039] Based on this, the calculation method of the present invention is as follows: ; When the error is large, the control output tends to saturate to avoid excessive control commands; at the same time, heading error is also taken into account. and position error This improves tracking accuracy; it provides a large gain near zero error and a limited gain when the error is large.

[0040] The calculated offset angle Substituting into the gait equation of the USR, the desired tracking signals for n-1 joints are generated. : ; This signal serves as a reference input for the preset performance constraint module, forming a complete cascaded control system of "position-heading-joint angle".

[0041] Step 5: The composite controller is based on the dynamic model from Step 1, the preset performance mapping relationship from Step 3, and the estimate of the lumped disturbance from the extended state observer from Step 4. By ensuring that the system dynamics satisfy the fixed-time sliding mode control law, the final control torque command can be calculated. .

[0042] Based on the above variables, a non-singular terminal sliding surface s is designed to avoid the singularity problem of traditional terminal sliding: ; For a piecewise nonlinear function designed based on fixed-time stability theory: ; in: ; ; To enable the system state to converge to the sliding surface with a fixed-time characteristic, and to decouple the convergence time from the initial error state, this invention employs an improved reaching law. : ; in , , , .in, It is a positive gain, and the parameters satisfy... and This design ensures that, regardless of the initial deviation, there is a definite upper bound on the time it takes for the system state to enter the sliding surface.

[0043] Differentiate the sliding mode face over time and substitute it into the USR dynamic equations and the observer estimate: ; The estimation of lumped disturbances is achieved by combining the USR's dynamic model, pre-defined performance mapping relationships, and the Extended State Observer (FTESO). By satisfying the aforementioned reaching law in the system dynamics, the final control torque command of the actuator can be obtained through inverse solving. : ; in: ; It is responsible for resisting residual disturbances and forcing the system state to contract towards the sliding surface. Through nonlinear gain adjustment, the inherent "chattering" phenomenon of sliding mode control is effectively suppressed while ensuring response speed.

[0044] ; Feedforward compensation is performed using the USR dynamics model to counteract the effects of nonlinear drag, Coriolis force, and acceleration of the reference trajectory. This involves introducing... This technology enables precise counterbalancing of external time-varying disturbances such as ocean currents; simultaneously, through... , Corrected the default performance function False dynamics caused by contraction.

[0045] The final actuator input is .

[0046] After deriving the analytical expression for the controller, to verify the stability of the closed-loop system under actuator saturation and external disturbances, a Lyapunov candidate function was selected. By differentiating V and substituting it into the control law, we can obtain that its rate of change satisfies: ; ; According to the fixed-time stability criterion, the above inequality ensures that the system state will converge to the equilibrium point within a fixed time. The upper bound of the system's total convergence time is also provided. Defined as: ; in These represent the upper bounds of the convergence time for the sliding mode motion segment, the approaching motion segment, and the observer estimation segment, respectively, and their specific expressions are as follows: Upper bound of sliding surface convergence time : ; Upper bound of convergence time for the convergence law: ; Upper bound of observer convergence time: ; Finally, considering the physical constraints of the underwater snake robot's motor, the actual actuator's input torque u is handled using the following saturation function: ; The saturation function is: .

[0047] Figure 4 The path tracking and tracking error under different initial states are shown. Figure 4 These three operating conditions represent different initial positions, velocities, and heading angles of the underwater snake robot (USR) when it begins to track the desired sinusoidal trajectory. Specific parameters are shown in Table 1.

[0048] Table 1

[0049] Figure 4 The above three operating conditions were simulated in a simulation environment such as Matlab / Simulink using the fixed-time non-singular terminal sliding mode controller (PPC-FFNTSMC) based on an extended state observer proposed in this embodiment. Figure 4 (a) demonstrates that the robots, starting from different points, can eventually converge accurately on the pre-set sinusoidal path (PlannedPath). Figure 4 (b) shows the change of path tracking error over time, proving that the error can converge rapidly to near zero within a fixed time, regardless of the initial state.

[0050] Figure 5 The comparison of path following and tracking errors of different joint controllers under the same conditions is shown. Figure 5 The results were obtained through simulation comparison, aiming to demonstrate that the heading controller proposed in this embodiment outperforms traditional proportional controllers in path tracking performance. Under the same simulation conditions, three different heading controllers (the method in this embodiment, reference controller 1, and reference controller 2) were run, and the position and error data of the underwater snake robot tracking a sinusoidal path were recorded to demonstrate the superiority of the heading controller proposed in this paper. Compared with the reference controller, the method proposed in this paper has smaller position tracking errors and more significant heading control accuracy. The robot can approach and stabilize on the desired path more quickly, and the more accurate heading tracking reduces the risk of collisions in narrow waters, improving navigation safety.

[0051] Reference controller 1: , based solely on position error The proportional controller. Reference controller 2: To simultaneously base on heading angle error and position error Proportional controller.

[0052] Figure 6 The graph shows a comparison of path following and tracking errors of different joint controllers under the same conditions. Figure 6This was achieved by comparing the control scheme proposed in this embodiment with two existing advanced sliding mode control algorithms in the literature, under the constraints of actuator hardware. Under the same initial state and external disturbances, the path tracking paths (e.g., ...) of the underwater snake robot (USR) joints driven by three different controllers were recorded. Figure 6 (a) shown) and path tracking error (e.g. Figure 6 (b) shows the ability of each controller to maintain tracking accuracy when the actuator output is limited.

[0053] Figure 6 The aim is to demonstrate the superior performance of the PPC-FFNTSMC controller (preset performance fixed-time non-singular terminal sliding mode control) proposed in this embodiment at the joint control level: Analysis Figure 6 (a) As can be seen, the method of this embodiment improves the time required to bring the tracking error to zero by 71.19% compared to reference controller 1 and by 76.22% compared to reference controller 2. Furthermore, the scheme of this embodiment exhibits smaller overshoot during tracking, and the system is more stable. This demonstrates that regardless of the initial error, the system can converge within a predetermined fixed time that is independent of the initial state.

[0054] Reference controller 1 is a Fast Nonsingular Terminal Sliding Mode Control, from the reference: Tian, ​​Y., Cai, Y., & Deng, Y. (2020). A fast nonsingular terminalsliding mode control method for nonlinear systems with Fixed-Time stability guarantees. IEEE Access, 8, 60444–60454.

[0055] Reference controller 2 is an Adaptive Terminal Sliding Mode Observer (NDO) Based Control. (From references) Chen, W., Wei, Y., Zeng, JM, Han, H., & Jia, X. (2016). Adaptiveterminal sliding mode NDO-Based control of underactuated AUV in verticalplane. Discrete Dynamics in Nature and Society, 2016, 1–9. https: / / doi.org / 10.1155 / 2016 / 6590517.

[0056] Figures 7-9 This data was obtained through dynamic simulation of an underwater snake-like robot (USR), primarily demonstrating the performance of the u1-u9 joint actuators (motors) during the control process. The figure shows the results obtained using an attitude angle tracking controller, simulated in a Matlab / Simulink environment. The core objective is to verify the robustness of the control system under actuator saturation conditions, specifically including the following three points: (1) Demonstrating saturation: The curve clearly shows that the actuator frequently reaches saturation during the motion process. The limit. This shows that in actual engineering, the torque required by the algorithm often exceeds the physical limit of the motor.

[0057] (2) Verification of the effectiveness of the auxiliary system: Despite the limited output of the actuator, the robot was still able to successfully complete the control objective. This proves that the adaptive auxiliary system proposed in this embodiment effectively compensates for the negative impact of saturation and maintains the stability of the system.

[0058] (3) Simulate the actual engineering environment: By demonstrating the control performance under saturation conditions, it is shown that this embodiment is not only valid in mathematical theory, but also takes into account the physical constraints of the hardware, and has strong practical application value.

[0059] Figures 10-12 This demonstrates the real-time estimation performance of the fixed-time extended state observer in this embodiment for three different types of disturbances. Figure 10 The (sinusoidal perturbation) simulates periodic environmental disturbances (such as waves and ocean currents) that change over time. Figure 11 State disturbance: Simulate internal state disturbances related to robot joint friction. Figure 12 Composite disturbance: The most complex operating condition, which includes state disturbances, external environmental disturbances, and uncertainties in the model itself.

[0060] Figures 10-12This is based on the operation of an improved fixed-time extended state observer. The figure contains two curves: the actual perturbation value d (black dashed line) and the observer's estimated value. (Red solid line). The observer uses the novel fixed-time convergence system proposed in this embodiment as its theoretical basis. That is... Three perturbation signals were generated based on the mathematical formulas defined in Table 2 and input into the dynamic model of the underwater snake robot (USR).

[0061] The observer reads the system state in real time during the robot's movement and calculates the estimated value of unknown disturbances. Finally, it compares the estimated curve with the preset actual disturbance curve and plots it. As can be seen from the figure, when the red solid line (estimated value) can quickly and accurately coincide with the black dashed line (actual value), it proves that the observer in this embodiment has an extremely short convergence time and extremely high tracking accuracy. It can capture and quantify complex environmental disturbances in real time, providing accurate compensation for the subsequent controller.

[0062] Table 2

[0063] Figures 10-12 The core objective is to verify the estimation performance of the observer. The results show that the estimated curve (red dashed line) can quickly and accurately coincide with the actual disturbance curve (black solid line), proving that the observer has an extremely short convergence time and extremely high tracking accuracy. This observer can "see through" complex environmental disturbances, providing accurate compensation for the subsequent controller, thereby offsetting the impact of disturbances on robot path tracking. Furthermore, it remains stable under complex disturbances, demonstrating the strong robustness (tolerance) of this scheme in complex and variable underwater environments. Figures 10-12 The verification demonstrated that its control system can effectively solve the problem of "not being able to see or sense" interference caused by sensor limitations and environmental complexity in underwater snake robots.

[0064] The above embodiments are merely exemplary embodiments of the present invention and are not intended to limit the present invention. Those skilled in the art can make various modifications or equivalent substitutions to the present invention within its scope and spirit, and such modifications or equivalent substitutions should also be considered to fall within the scope of protection of the present invention.

Claims

1. A USR fixed-time path tracking control method with preset performance, characterized in that, Includes the following steps: Step 1, USR system modeling and kinematic description: Establish a simplified control-oriented model based on the dynamic equations; Step 2, Design the extended state observer: Construct a fixed-time convergent system containing a variable exponential term, and based on the fixed-time convergent system, construct an extended state observer that combines an integral sliding mode term and adaptive gain adjustment; Step 3, Constructing a composite controller: The navigation guidance module converts path deviation into desired heading angle using an adaptive integral line-of-sight method; The preset performance constraint module uses a time-varying performance function. The transient and steady-state performance of the USR joint angle tracking error is constrained. The constrained tracking error is mapped to an unconstrained conversion error through a transformation function. The adaptive auxiliary module compensates for the actuator saturation effect. The preset performance constraint module and the state of the adaptive auxiliary module are combined to obtain the preset performance mapping relationship. Step 4: Generate the global desired path through the preset path algorithm. The sensor collects the coordinate position, heading angle and joint status of the USR centroid. Simultaneously, the extended state observer obtained in Step 2 is started to capture and quantify the total underwater environmental disturbance in real time. The navigation guidance module outputs data as the input of the preset performance constraint module. Step 5: The composite controller is based on the dynamic model from Step 1, the preset performance mapping relationship from Step 3, and the estimate of the lumped disturbance from the extended state observer from Step 4. By ensuring that the system dynamics satisfy the fixed-time sliding mode control law, the final control torque command can be calculated. .

2. The USR fixed-time path tracking control method with preset performance according to claim 1, characterized in that, The dynamic equation of the fixed-time convergent system in step 2 is: ； ; ; ; Where k1 represents the variable exponential coefficient, k2 represents the constant exponential coefficient, m1 represents the system convergence exponent, m2 represents the system convergence exponent, n1 represents the system positive gain coefficient, n2 represents the system positive gain coefficient, e represents the natural constant, and z represents the system state variable. This represents the rate of change of state.

3. The USR fixed-time path tracking control method with preset performance according to claim 2, characterized in that, The extended state observer system in step 2 is as follows: ; ; ; in, For the improved fixed-time convergent function, p i q i The gain coefficient of the nonlinear function. It is a nonlinear exponential term, which is the power that ultimately acts on the observer error correction function; It is a low power exponent. This is the state estimate. x1 represents the joint angle estimation error, and x1 represents the system state variable. These are estimated values ​​of the system state variables. This is the actual joint angle. To estimate joint angles; The observer gain coefficient, For adaptive gain term, To achieve dynamic adjustment of the gain, g is the gain coefficient; m is the link mass; AD is the structure matrix; DD is the structure matrix; u is the torque of the i-th actuator; c n c is the normal friction coefficient. p The correlation coefficient is the fluid interaction coefficient. For the final control torque command, Let k be the tangential relative velocity. i1 k i2 These are intermediate mapping parameters used to transform m1 and m2 into transitional variables for the final exponent.

4. The USR fixed-time path tracking control method with preset performance according to claim 1, characterized in that, The time-varying performance function in step 3 ; in, Let b be the initial value of the performance function, b be a constant factor, and t be time. For steady-state error boundary, The attenuation coefficient; The gain coefficient is time-varying and designed in time segments to balance convergence speed and accuracy.

5. The USR fixed-time path tracking control method with preset performance according to claim 1, characterized in that, The calculation formula for the adaptive auxiliary module in step 3 is as follows: ; in, , As auxiliary system state variables, , The first derivative of the state variable with respect to time represents the rate of change of the compensation intensity; , The gain coefficient is positive. By selecting appropriate parameters, the system matrix can satisfy the Hurwitz condition, ensuring the exponential convergence of the system state. To control the saturation difference of the input, For the final control torque command, This is the torque that actually acts on the robot.

6. A USR fixed-time path tracking control method with preset performance according to claim 3, characterized in that, The preset performance mapping relationship in step 3 is specifically as follows: ; ; ; in, , , Used to describe the dynamic mapping relationship from the physical error space to the preset performance constraint space; and Responsible for compensating for performance boundaries Time-varying nonlinear terms that arise as the time converges; By linearizing the joint acceleration information into a preset performance framework, the controller can directly calculate the physical torque command within the transformation space. This represents the time-varying error performance index function, where e represents the joint angle tracking error. Represents performance function The first derivative with respect to time, This represents the first derivative of the tracking error e with respect to time. Represents performance function The second derivative with respect to time.

7. A USR fixed-time path tracking control method with preset performance according to claim 1, characterized in that, The specific processing flow of the heading guidance module in step 4 is as follows: based on the lateral error of the centroid offset from the desired path, the optimal desired heading angle for the regression path is calculated; based on the current heading error, an offset signal is output; combined with the serpentine gait equation, the desired turning angle command that each joint actuator should achieve at the current moment is calculated; and the desired turning angle command is used as the reference input of the preset performance constraint module.

8. A USR fixed-time path tracking control method with preset performance according to claim 6, characterized in that, The final control torque command in step 5 for: ; ; ; ; ; ; in, It is responsible for resisting residual disturbances and forcing the system state to contract towards the sliding surface. The USR dynamic model is used for feedforward compensation to counteract the effects of nonlinear drag, Coriolis force and acceleration of the reference trajectory. It is a positive gain, and the parameters satisfy... and m5 and m6 represent the reaching law exponent constants, and s represents the sliding surface function. This represents the estimated value of the lumped disturbance. This represents the normal velocity between adjacent links. Indicates the robot's tangential velocity. This represents the desired joint angular acceleration. Represents the state variables of the adaptive auxiliary system. Represents the state variables of the adaptive auxiliary system. Represents nonlinear terms The first derivative, The switching function itself is designed to avoid sliding mode singularity, and its derivative... It is incorporated into the controller solution to ensure the fixed-time convergence of the system; , The intermediate exponent parameter of the reaching law, This is a composite variable that combines the preset performance conversion error with the auxiliary system state. yes The first derivative.

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