A control method and system for a multi-cell based flexible joint robotic arm

Abstract

The application discloses a kind of based on the control method and system of flexible joint mechanical arm of polytope, it is related to mechanical arm control technical field, including the following steps: constructing the polytope model for wrapping system state boundary and external disturbance boundary, with the center of polytope model converging to equilibrium point, radius converges to zero and the performance index for constraining system state amplitude is constraint generation objective function, based on the parameter matrix of discrete state space model to solve target function, obtain the controller gain of state feedback controller;According to the closed-loop control system of discrete state space model flexible joint mechanical arm system with the state feedback controller with controller gain;The control of flexible joint mechanical arm system is realized by closed-loop control system.The application realizes the significant improvement of anti-interference ability and reliable operation level of system under replication working condition.

Description

Technical Field

[0001] This invention relates to the field of robotic arm control technology, and in particular to a control method and system for a flexible joint robotic arm based on multicellular structures. Background Technology

[0002] In recent years, robotics technology has made significant progress in various application fields, particularly in hazardous environments, complex tasks, advanced control methods, and human-robot interaction safety, where numerous innovative achievements have emerged. To meet the complex task requirements of industrial scenarios and emerging fields, human-robot collaboration is often necessary to complete operational goals. Traditional rigid robotic arms, using high-strength alloy materials and rigid joint designs, feature high load capacity, small deformation, high control precision, and fast response speed, and are widely used in scenarios such as automotive welding and machine tool loading and unloading. However, rigid robots also have significant shortcomings: First, in unstructured environments or scenarios involving human collaboration, their lack of flexibility limits their movement and makes them prone to large impacts from collisions, potentially damaging their structure and posing safety hazards. Second, the joints of rigid robots typically use rigid connections and drives, which are prone to significant vibrations and impacts during high-speed, high-load movements. This affects their motion stability and control precision, and also shortens the equipment's lifespan.

[0003] Flexible-joint manipulators (FJMs) have attracted widespread attention due to their compact actuators, high precision, and low energy consumption. FJMs integrate flexible components such as harmonic reducers and torque sensors, exhibiting good compliance, high precision, fast response, and a high load-to-weight ratio. These advantages allow FJMs to be safely adapted to flexible industrial production, minimally invasive medical surgery, home services, and daily assistance scenarios. Therefore, research on FJM systems has significant academic and practical application value. Despite its significant advantages, the FJM (Flexible Joint Machine) is a complex system with multiple inputs and outputs, strong coupling, and nonlinearity, facing numerous uncertainties and challenges. These include: model uncertainty, where friction and coupling effects between the joints of the robotic arm are difficult to accurately describe using dynamic equations and often require simplification; parameter uncertainty, where measurement errors exist in physical quantities such as joint angles, link lengths, mass, and friction coefficients due to sensor accuracy limitations, and these parameters may fluctuate with changing working conditions; output constraints, where mechanical structure and task requirements impose limitations on the robotic arm's range of motion and speed, which, if not handled properly, can hinder task completion; actuator failure, where joint jamming and insufficient power may occur during actual operation, reducing production efficiency and potentially posing safety risks; and input nonlinearity, where input dead zones and saturation phenomena can significantly affect the system's control accuracy and even lead to system instability.

[0004] The prior art provides a fuzzy PID control method for a variable stiffness flexible joint manipulator, including: establishing a dynamic model of the variable stiffness flexible joint manipulator; designing fuzzy rules for the fuzzy PID control method; applying the fuzzy rules to the real-time adjustment of the system's PID parameters; fully considering the system's variable stiffness and flexibility characteristics; applying the fuzzy PID control method to the variable stiffness flexible joint manipulator; using MatlabSimulink software to perform dynamic simulation of the system; and outputting the system's rotation frequency and torque curves. This demonstrates that the method has a fast response speed, small overshoot, and significant and stable vibration suppression effect.

[0005] However, due to their inherent structure and operational characteristics, existing control methods for variable stiffness flexible joint robotic arms present significant challenges in terms of parameter uncertainty and adaptation to external disturbances. The dynamic model parameters of the variable stiffness flexible joint are prone to change during operation, leading to continuous fluctuations in the system's dynamic characteristics. This parameter uncertainty makes it difficult for control strategies to establish stable adaptation relationships, posing a direct challenge to control accuracy and stability. Simultaneously, the elastic characteristics of the robotic arm's flexible joints make them extremely sensitive to external disturbances, which can easily induce elastic vibrations in the joints, resulting in unnecessary impacts on the system. Summary of the Invention

[0006] In view of the defects of the prior art, the present invention provides a control method and system for a flexible joint robotic arm based on multicellular structures, which solves the existing problems.

[0007] The present invention adopts the following technical solution: In a first aspect, the present invention provides a control method for a flexible joint robotic arm based on multicellular structures, comprising the following steps: The dynamic model of the flexible joint robotic arm system is linearized and discretized to obtain a discrete state-space model that includes external disturbances; A multi-cell model is constructed to enclose the system state boundary and external disturbance boundary. The objective function is generated by using the convergence of the center of the multi-cell model to the equilibrium point, the convergence of the radius to zero, and the performance index used to constrain the system state amplitude as constraints. The objective function is solved based on the parameter matrix of the discrete state space model to obtain the controller gain of the state feedback controller. Based on a state feedback controller with controller gain and a discrete state-space model, a closed-loop control system for a flexible joint robotic arm system is implemented; the control of the flexible joint robotic arm system is achieved through the closed-loop control system.

[0008] Preferably, the discrete state-space model is as follows: ; In the formula, and They are respectively Time and The system state at any given moment. For the system matrix, For the input matrix, It is a real matrix. To control the input, External interference; The system status includes the deflection angle of the servo motor, the deflection angle of the connecting rod, the angular velocity of the motor, and the relative angular velocity of the connecting rod. The control input is the motor armature voltage.

[0009] Preferably, the state feedback controller is specifically as follows: ; In the formula, This is the controller gain.

[0010] Preferably, the multicellular model is as follows: ; In the formula, and They are respectively in Time and System status at all times Multicellular center, and They are respectively in Time and System status at all times The multicellular generation matrix, and External interference The multicellular center and generating matrix, For the system matrix, For the input matrix, It is a real matrix.

[0011] Preferably, the performance indicators are: The performance metrics, which are defined by the convergence of the center of the multicellular model to the equilibrium point, the convergence of the radius to zero, and the performance metrics used to constrain the amplitude of the system state, are used to generate the objective function. Specifically, this includes the following steps: The weighted center function and weighted radius function are defined based on the center and generating matrix of the multicellular model; Based on the weighted center function and the weighted radius function, construct constraint expressions that satisfy the central convergence to the equilibrium point, the radius convergence to zero, and the performance index used to constrain the state amplitude of the multicell model; Transform the constraint expression into an objective function.

[0012] Preferably, the constraint expression is as follows: ; in, ; ; In the formula, and They are respectively Weighted radius and weighted center of multicellular bodies at any given time. and They are respectively Weighted radius and weighted center of multicellular bodies at any given time. The robust exponential decay rate for biconvergence of multicellular structures. For disturbance suppression performance indicators, for The convergence term of the multicell radius at time t. For the multicellular center convergence term, for The radius of the multicellular body at any given time, The distance between the centers of the multicellular organisms. for External disturbance energy at any given moment.

[0013] Preferably, the objective function is as follows: ; in, ; In the formula, P To set the matrix, T It is a transpose operator. For the system matrix, For the input matrix, It is a real matrix. It is a discrete matrix. It is the identity matrix. This is the controller gain.

[0014] Preferably, the controller gain is as follows: ; In the formula, For the controller gain matrix, For matrix variables, This is the controller gain.

[0015] Preferably, the flexible joint robotic arm system is controlled through a closed-loop control system, specifically including the following steps: Collect the system status of the flexible joint robotic arm system at the current moment; The current system state is input to the state feedback controller, which generates a control quantity based on the current system state and the controller gain. The flexible joint robotic arm is controlled by outputting control quantities to the flexible joint robotic arm system.

[0016] Secondly, the present invention provides a control system for a flexible joint robotic arm based on multicellular structures, comprising: The module is used to linearize and discretize the dynamic model of the flexible joint robotic arm system, resulting in a discrete state-space model that includes external disturbances. The solver module is used to construct a multi-cell model to wrap the system state boundary and external disturbance boundary. The objective function is generated with the multi-cell model's center converging to the equilibrium point, radius converging to zero, and performance indicators used to constrain the system state amplitude as constraints. The objective function is solved based on the parameter matrix of the discrete state space model to obtain the controller gain of the state feedback controller. The control module is used to implement a closed-loop control system for the flexible joint manipulator system based on a state feedback controller with controller gain and a discrete state-space model; the closed-loop control system enables the control of the flexible joint manipulator system.

[0017] Compared with the prior art, the above-mentioned at least one technical solution adopted by the present invention can achieve the following beneficial effects: This invention constructs a multicellular model, using a center and generating matrix to encapsulate the feasible domain of system states and external disturbances, transforming parameter fluctuations into quantifiable convex boundary sets. Unlike traditional PID control, which heavily relies on precise dynamic models, this scheme eliminates the need for a fixed, precise mathematical model. The multicellular boundary effectively encapsulates all possible operating states, ensuring control accuracy and system stability even under inaccurate model conditions. Furthermore, this invention combines system convergence conditions with performance indicators. The objective function is generated by constraining the multicellular model's center to the equilibrium point, the radius to zero, and performance indicators used to constrain system state amplitudes. The objective function is solved based on the parameter matrix of the flexible joint robotic arm system to generate the controller gain. This design significantly improves the system's tolerance to parameter fluctuations and anti-interference performance when facing bounded external disturbances, achieving "dual convergence" of the state set's center and radius. This theoretically guarantees closed-loop system stability while reducing reliance on precise modeling, effectively decreasing system debugging costs, and improving engineering applicability. Attached Figure Description

[0018] 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 some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0019] Figure 1 This is a flowchart of a control method for a flexible joint robotic arm based on multicellular structures according to the present invention. Figure 2 This is a structural diagram of the closed-loop system of the flexible joint robotic arm of the present invention; Figure 3 The diagram shows the state trajectory of four state variables in the time response of the flexible joint robotic arm system of the present invention. Figure 4 This is a state boundary envelope diagram showing the actual values ​​of different system states and their upper and lower boundaries of the flexible joint robotic arm of the present invention. in, Figure 4 (a): System state , Figure 4 (b): System state , Figure 4 (c): System state , Figure 4 (d): System state ; Figure 5 This is a convergence diagram of the multicell radius and the trajectory convergence of the center of the flexible joint robotic arm system of the present invention. in, Figure 5 (a): radius of the multicellular body, Figure 5 (b): Multicellular center. Detailed Implementation

[0020] 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.

[0021] Existing control methods face significant challenges in terms of parameter uncertainty, handling external disturbances, and robustness adaptation. Specifically, these shortcomings are as follows: First, significant parameter uncertainty. The dynamic model parameters of variable stiffness flexible joints are prone to change during operation, leading to continuous fluctuations in the system's dynamic characteristics. This parameter uncertainty makes it difficult for control strategies to establish stable adaptation relationships, directly challenging control accuracy and stability. Second, sensitivity to external disturbances, easily inducing vibration and errors. The elastic characteristics of the robotic arm's flexible joints make them extremely sensitive to external disturbances, which can easily induce elastic vibrations in the joints, causing unnecessary impacts on the system. Third, high robustness requirements and difficulty in adapting to complex working conditions place extremely high demands on the robustness of the control strategy. PID control heavily relies on accurate system models, but flexible joints contain many unmodeled dynamics, resulting in insufficient model accuracy. This makes it impossible to accurately reflect the actual operating state of the system, making it difficult for control strategies based on this model to achieve the precision standards required for precision operations. These core limitations collectively restrict the application effectiveness of PID control in flexible joint robots.

[0022] To address the aforementioned problems, this invention proposes a control method for a flexible joint robotic arm based on ensemble control, which constructs and integrates a fusion state multicellular object with... Robust control addresses parameter uncertainties and external disturbances that traditional control methods struggle to handle. The polytopic model, based on precise dynamic equations, fully encapsulates the system state, parameter perturbations, and external disturbance boundaries in a convex polyhedral geometry, transforming complex linear control into a convex optimization problem. Building upon this foundation, [further details are needed]. A robust control strategy is designed for multicellular systems to effectively suppress worst-case scenarios of parameter uncertainties. This method significantly improves the accuracy and robustness of flexible joint robotic arm control, and enhances the system's anti-interference capability and reliable operation under complex conditions.

[0023] For the typical controlled object of flexible joint robotic arms, this invention, based on the theory of flexible joint robotic arm systems, first clarifies its state-space model and designs a state feedback controller, deriving the closed-loop system expression; then, combining the structural and kinematic characteristics of the flexible joint robotic arm, it constructs a multi-cell model, defining key indicators such as the multi-cell radius and center distance; further, it conducts dual convergence analysis and... Performance analysis, through the derivation of the performance function, proves that the controller designed for the flexible joint robotic arm system enables the closed-loop system of the flexible joint robotic arm to achieve [performance / performance]. The stable performance indicators provide theoretical support and analytical methods for the high-precision control of flexible joint robotic arms.

[0024] The core mechanical structure of the flexible joint robotic arm includes an SRV02 rotary servo base, a flexible joint module, a linkage, an encoder, and a potentiometer. The SRV02 rotary servo base is the system's power source, containing a DC motor and a high-precision gearbox to provide rotational motion. The flexible joint module is a core feature of the system; instead of a traditional rigid bearing, it consists of a linear torsion spring. This spring provides controllable and measurable flexibility between the motor base and the connecting rod above it. The connecting rod typically comprises two arms, a main arm and a top arm. The main arm is directly mounted on the servo base. The top arm is connected to the main arm via a flexible joint. The mounting position of the top arm can be adjusted, which changes the moment of inertia of the entire linkage system relative to the rotation axis, allowing for the study of the impact of different load dynamics on the system. The encoder is mounted on the servo motor for high-precision measurement of the servo base's deflection angle. Potentiometer: Directly mounted at the flexible joint, used to measure the deflection angle of the connecting rod relative to the base. This angle directly reflects the deformation of the flexible joint.

[0025] To better achieve the collective control of flexible joint robotic arms, refer to Figure 1 The present invention specifically includes the following steps: S1: Establish a dynamic model of the flexible joint robotic arm and transform it into a discrete state-space model that includes external disturbances.

[0026] S11: Establish the dynamic model of the flexible joint robotic arm.

[0027] Based on the Lagrange equations. Consider a second-order system:

[0028] (1); In the formula, The inertia coefficient, The damping coefficient is... This is the stiffness coefficient. Angular acceleration, Angular velocity, For angle.

[0029] Based on equation (1), the dynamic equation of the flexible joint robotic arm is established: (2); From (2), we get: (3); Summarized as follows: (4); in, It is the deflection angle of the servo motor. It is the deflection angle of the connecting rod. and These represent the angular velocity of the motor and the relative angular velocity of the connecting rod, respectively. and These represent the angular acceleration of the motor and the relative angular acceleration of the connecting rod, respectively. It is the equivalent viscous damping coefficient on the servo motor side. It is the equivalent moment of inertia on the servo motor side. It is the moment of inertia of the connecting rod. It refers to the torsional stiffness of the flexible joint robotic arm. It is the torque output by the servo motor.

[0030] S12: Linearized model of flexible joint robotic arm.

[0031] Define the state of the flexible joint robotic arm system Taking into account the uncertainties introduced by flexibility, the output torque ,but (5); (6); In the formula, It has a high overall gear ratio. It is the back electromotive force constant of the motor. It is the motor current torque constant. It's the motor efficiency. It's gearbox efficiency. It is the armature resistance of the motor. It controls the voltage.

[0032] Linearizing the dynamic equations of the flexible joint manipulator yields a linear state-space model of the flexible joint manipulator: (7); in, for t The derivative of the system state at time t. To linearize the system matrix, To linearize the input matrix, the control quantity , It is the armature voltage of the motor.

[0033] Linearized system matrix for: (8); Linearized input matrix for: (9); Discretize using a zero-order hold, and then add an external disturbance to obtain: (10); in, It is the system's state variable. For real numbers, for n Vioclimatic space For the system matrix, For the input matrix, It is a real matrix with appropriate dimensions. To control the quantity; To mitigate external disturbances, linearization transforms the nonlinear system into a standard state-space form, facilitating analysis and controller design using linear system theory.

[0034] S2: Mechanism modeling of the flexible joint robotic arm. The detailed steps can be broken down as follows: S21~S22.

[0035] S21: First, the model expression of the flexible joint robotic arm system: (11); The following state feedback controller is designed for the flexible joint robotic arm system: (12); in, It is the controller gain.

[0036] The resulting closed-loop system expression is: (13); Considering process noise State feedback control is used to achieve system stabilization. A closed-loop system, such as... Figure 2 As shown.

[0037] S22: Constructing a multicellular model based on a closed-loop system: (14); in, Indicates in System status at all times Multicellular center, Indicates in System status at all times The multicellular generation matrix, and These represent external interference. The multicellular centers and generating matrix.

[0038] Proof: From multicellular organisms ,according to We can obtain: (15); in, It is a perturbation of multicellular organisms. It is a state multicellular organism.

[0039] Therefore, a state-bounded multicellular structure was established. its center and generating matrix Equation (14) provides the basis for further derivation of equation (15), which lays the foundation for subsequent analysis of double convergence and... Performance analysis provides the mathematical foundation.

[0040] Based on the dynamic equations of flexible joints, the multi-cell model combines finite element analysis and modal order reduction techniques. It precisely encapsulates the state space, parameter uncertainty range, and external disturbance boundaries through the geometric form of convex polyhedra, transforming complex nonlinear characteristics into convex set constraints. This model extracts the variation ranges of key parameters such as joint stiffness and damping coefficients, constructing a multi-cell set covering all possible operating conditions. This transforms the control problem of the nonlinear system into a solvable convex optimization problem, preserving the core dynamic characteristics of the system while simplifying uncertainty handling through geometric constraints, thus laying a solid foundation for ensemble control design.

[0041] S3: Double convergence and Performance analysis.

[0042] This section mathematically analyzes the convergence and robustness of the system and provides sufficient conditions (16) for controller design to ensure that the system can still meet performance requirements under disturbances.

[0043] definition and Define the radius and weighted radius of a multicellular body:

[0044] (16); in, and These are the state components and the disturbance components. yes 1D unit hypercube , and These represent the maximum radius of the multicellular body under the Euclidean norm and the weighted norm, respectively, quantifying the range of uncertainty.

[0045] The distances from the center to the equilibrium point and the weighted distances are respectively: (17); in, and Characterizes the degree to which the center of a multicellular organism deviates from its equilibrium point. and Let represent the Euclidean vector norm and the weighted Euclidean vector norm, respectively.

[0046] The radius of the multicellular body and the distance from its center to the equilibrium point are calculated as follows: (18); in, and It reflects the square of the radius of the state multicell and the square of the distance from the center to the equilibrium point; Construction of weighted radius and central function: (19); in, The weighted radius squared of the system state or disturbance response is constrained. This represents the squared weighted distance from the system center to the equilibrium point. In set-based control schemes, the convergence of the system state is guaranteed by ensuring that the radius converges to the center, and the center converges to 0, thus ensuring the system's stability.

[0047] S31: Double convergence and Performance analysis.

[0048] By defining the radius and central function of the multicellular body, the convergence behavior of the system under both with and without disturbance is analyzed. Performance provides a theoretical guarantee for the stable operation of the system.

[0049] (20); in, and , This represents the energy of external disturbances. and These represent the center and radius of the multicellular body, respectively. It is the robust exponential decay rate of multicellular biconvergence. for The convergence term of the multicell radius at time t. For the multicellular center convergence term, for The radius of the multicellular body at any given time, The distance between the centers of the multicellular organisms. It is a disturbance suppression performance indicator.

[0050] prove: The first step is to determine the convergence of the multicell radius and... Performance. Based on (20), it can be known that:

[0051] (twenty one); From the above formula (21), we can obtain: ; In the formula, k For the time of the discrete system, s The index for the summation count.

[0052] From the above, we can conclude that: (twenty two); The second step will consider the convergence of the multicellular center and Performance. Similar to the proof in the first step, based on equation (20), it can be derived that:

[0053] (twenty three); when hour, (twenty four); If there is no interference, the system performance function and central function The fact that the value will not exceed the initial value indicates that the system state will not diverge and is stable. For a flexible joint robotic arm, this means that the system's vibration amplitude and tracking error will not increase in the absence of external torque interference or load disturbance.

[0054] when and hour, (25); (26); Through rigorous mathematical derivation, the closed-loop control system of the flexible joint robotic arm was analyzed. Performance: Under undisturbed conditions, the system is Lyapunov stable, and its state does not diverge. Under disturbed conditions, the system meets the performance indicators, and the upper bounds of the state variables are clearly defined by the initial conditions and the magnitude of the disturbance. This provides a solid theoretical guarantee for the reliable and stable operation of the flexible joint robotic arm in uncertain environments. Subsequent work will be based on this performance analysis framework to design and optimize the controller parameters.

[0055] In control systems, multi-celled systems exhibit unique comprehensive advantages due to their powerful ability to handle nonlinearity and uncertainty. Their core value lies in providing a unified and rigorous mathematical framework, transforming complex global problems into manageable linear models. This allows controller design to be grounded in solid linear theory while effectively covering the global behavior of the system, ultimately optimizing control performance while ensuring robustness. This makes them a key tool for addressing the core challenges of modern control systems. Unlike traditional control, the control objective of this design is not the convergence of a single state trajectory, but rather the "double convergence" of the entire state set (Zonotope): first, central convergence, meaning the center of the Zonotope converges to the origin; and second, radial convergence, meaning the radius of the Zonotope converges to zero. This invention proves the dual convergence of the centrosymmetric polyhedron by constructing suitable radius and center functions, thereby ensuring the stability of the closed-loop system. Meanwhile, multi-cell control aims for "double convergence" and theoretically ensures the stability of the closed-loop system by constructing a dedicated radius and center function. Compared with the limitations of PID, which relies on empirical parameter tuning and cannot fundamentally solve the problem of nonlinear system stability, it achieves a comprehensive improvement in robustness and accuracy, making it an advanced control solution more suitable for flexible joint robotic arms.

[0056] Performance is one of the key guarantees for the reliable and high-precision operation of flexible joint robots in complex scenarios. Its core advantage lies in the fact that, when faced with bounded disturbances, the system state amplitude can be controlled within a defined limit related to the disturbance amplitude, thus endowing the system with strong robustness. This performance focuses on the worst-case gain of the system output and external disturbances. Even when encountering sudden impacts in unstructured environments, model uncertainties, or external disturbances such as sudden load changes or collisions, it ensures that the motion accuracy and vibration suppression effect of the flexible joint do not exceed the preset performance boundaries. Simultaneously, The performance analysis and design approach can be directly adapted to the nonlinear and time-varying characteristics brought about by the multi-cell structure of flexible joint robots. In scenarios such as event-triggered control, it can strictly constrain the maximum amplitude of triggering errors such as command deviation, and take into account the real-time performance and accuracy of control signals, thereby fundamentally ensuring the stable performance of the robot in terms of safety, dynamic performance and other dimensions.

[0057] S32: Next, we provide a method to ensure that the multicellular body has dual convergence and Sufficient condition for performance, assuming the existence of a matrix. It satisfies the linear matrix inequality (LMI) constraint as shown in formula (27): (27); In the formula, For an identity matrix of appropriate dimension, .

[0058] Prove the convergence of the radius and The performance-based weighted radius function for the flexible joint robotic arm system is constructed as follows: (28); Therefore, by definition For each time step, inequality (20) can be rewritten as: (29); In the formula, Given an n-dimensional vector, .

[0059] Therefore, we can conclude that: (30); Will Replacing it with equation (14), and then combining it with equations (13) and (28), we get: (31); make , ,have to (32); Clearly, according to Schur, equation (32) implies equation (27). Therefore, condition (20) is proved. The convergence of the center and radius of the multicellular body and Performance is guaranteed.

[0060] S33: Controller Design.

[0061] Based on the foregoing analysis, a method for calculating the gain of the state feedback controller is presented to ensure stable control of the system while satisfying performance constraints. It is assumed that a matrix exists. The controller design is satisfied as shown in formula (33): (33); In the formula, This is the controller gain matrix.

[0062] For any abstract data type that satisfies equation (33), the polytope of the flexible joint system is biconvergent, and Performance is satisfied; furthermore, the state feedback controller gain is calculated using the following formula: (34); Proof: Let From equation (32), we will multiply the left side of equation (27) by... And right multiplication Then, it can be immediately deduced that: (35); Using Schur, we can obtain equation (31). Therefore, condition (31) implies (25), and we can conclude that if system (4) satisfies equation (35), then the flexible joint robotic arm system has performance stability, and its state feedback controller gain design is shown in equation (33).

[0063] This section focuses on a rigorous mathematical analysis of the convergence and robustness performance of the closed-loop system, culminating in the design and solution of the controller. By defining metrics such as the radius of the polytope and the center distance, the dual convergence characteristics of the system under both disturbance-free and disturbance-free conditions are quantitatively analyzed. The performance was demonstrated, proving the system's stability and performance upper bound under disturbances. Furthermore, by constructing a linear matrix inequality (LMI) and applying Schur's complement lemma, sufficient conditions for the system to possess the required performance were given, and finally, the formula for solving the state feedback controller gain was derived.

[0064] Addressing the control challenges of variable stiffness flexible joint robotic arms, a control scheme based on ensembles of multiple cells exhibits significant advantages. Firstly, it accurately accommodates parameter uncertainties, eliminating reliance on precise models. During operation, the dynamic model parameters of variable stiffness flexible joints are easily affected, leading to fluctuations in the system's dynamic characteristics. The ensemble control and multiple cell-based scheme eliminates the need for a fixed, precise dynamic model. Instead, through the convex set description of the multiple cells, the parameter fluctuation range is transformed into a quantifiable ensemble boundary—regardless of parameter drift, it is effectively contained within the multiple cell boundary. The control strategy can dynamically adapt to parameter changes within the ensemble domain, without relying on precise models or frequent parameter adjustments as with traditional PID control, ensuring control accuracy and stability. Secondly, it rapidly suppresses external disturbances, breaking the "disturbance-error-fluctuation" chain reaction. The elastic characteristics of flexible joints make them sensitive to external disturbances such as sudden load changes and environmental shocks, easily triggering vibration propagation and trajectory deviation. Ensemble control, combining the convexity and dynamic adjustment capabilities of multiple cells, can incorporate the impact of disturbances into the ensemble constraint range in real time when they occur. Through rapid updates of the multiple cell boundary, the vibration amplitude caused by the disturbance is limited. Third, it exhibits strong robustness to adapt to the complex characteristics of a single system, ensuring continuous and stable operation. During the operation of a variable stiffness flexible joint robotic arm system, subtle parameter drifts inherently occur, demanding robustness from the control strategy. Based on a combination of ensemble control and multi-cell design, the subtle parameter drifts within a single system can be uniformly incorporated into the ensemble constraint framework through the convex set characteristics of the multi-cell and the dynamic optimization of the ensemble domain. The multi-cell can accurately encompass all possible operating states of a linear system in a convex set form, without requiring additional simplification of linear characteristics, thus avoiding control deviations caused by insufficient model accuracy. Ensemble control can optimize the control output in real time within the domain defined by the multi-cell. Even if the system experiences minor state fluctuations due to its own characteristics, it can quickly adapt through dynamic adjustments of the ensemble domain, always maintaining stable control performance.

[0065] Example S1: Parameter settings for the flexible joint robotic arm.

[0066] A flexible joint robotic arm system, defined as subsystem 1, was adopted. Based on actual physical parameters, a state-space model of the flexible joint robotic arm was systematically constructed, and the spring stiffness was defined. High gear ratio motor back electromotive force constant Motor current torque constant Equivalent moment of inertia Equivalent viscous damping coefficient motor armature resistance Gearbox efficiency motor efficiency Moment of inertia of the connecting rod Key parameters are obtained, and fixed elements in the system matrix are calculated to form a continuous state-space model.

[0067] S2: Design of a flexible joint robotic arm model.

[0068] The state-space model constructed using the actual physical parameters of the flexible joint robotic arm is discretized using the zero-order preservation method, resulting in the following matrix: ; ; By setting the convergence rate parameter and Performance indicators ,in Construct a matrix containing the inverse of the Lyapunov matrix. and auxiliary variables The LMI condition unifies system stability, convergence, and interference suppression performance into a convex optimization problem to be solved. Restore the state feedback controller gain to ensure that the closed-loop system simultaneously satisfies the dual convergence characteristics and the specified robust performance boundary under load changes and external disturbances.

[0069] S3: External Disturbances and Simulation Settings.

[0070] The simulation is set to the initial state. This means that the servo motor deflection angle, flexible joint deflection angle, and their corresponding angular velocities are all initially deviated from the equilibrium position, simulating the non-zero initial conditions caused by actual startup or external disturbances, effectively verifying the controller's ability to recover from an unbalanced state to the equilibrium point.

[0071] Figure 3The state trajectory diagrams of the four state variables are shown. All state components fluctuate greatly at the initial moment, but gradually converge to near 0 and remain stable as time progresses, indicating that the system state can quickly suppress the initial disturbance and eventually achieve stability. Figure 4 The system state and upper and lower bounds of subsystem 1 are shown. It is evident that the lower and upper bounds derived based on the polytope method always encompass the system state, and the state boundaries converge rapidly, ensuring the convergence of the system state. Furthermore, this invention compares the convergence of the polytope radius and center (…). Figure 5 The results show that the multicellular organism exhibits dual convergence properties, confirming the effectiveness and correctness of the proposed method.

[0072] In the field of flexible joint robot control, the ensemble-based control method proposed in this invention shows significant improvements in robustness, control accuracy, and model adaptability compared to traditional PID control. Specific advantages are as follows: Superior Robustness and Higher Control Accuracy: Addressing the complex issues commonly found in flexible joint robot systems, such as strong nonlinearity, dynamic coupling, load fluctuations, and parameter perturbations, this invention employs a set-based control architecture. It abandons reliance on fixed-parameter models and describes the system's uncertainty boundaries through a set of multiple cells, achieving dynamic adaptation to various operating conditions. Even near singularity regions, the system effectively suppresses overshoot and oscillations, avoiding response hysteresis. The control strategy can be adjusted online without relying on empirical parameter tuning, ensuring stable control performance under a wide range of operating conditions. This overcomes the inherent defects of traditional PID control, such as fixed parameters and insufficient adaptive capability. Regarding control accuracy, this invention breaks through the limitations of traditional PID control based on the rigid system assumption. It actively integrates the elastic deformation characteristics of flexible joints, effectively suppressing structural vibrations caused by flexibility by applying precise constraints within the set domain, and quickly eliminating the impact of vibrations on the end-effector trajectory accuracy.

[0073] System Modeling and Uncertainty Handling Capabilities Based on Polytopes: This invention, based on flexible joint dynamics and combining finite element analysis and modal order reduction techniques, constructs a polytope model capable of encompassing the system's state space, parameter uncertainties, and external disturbance boundaries. This model characterizes the perturbation range of key parameters such as joint stiffness and damping coefficients as a set of convex polyhedra, thereby transforming the original nonlinear control problem into a solvable convex optimization problem. While preserving the core dynamic characteristics of the system, it provides a rigorous mathematical framework for the design of ensemble controllers.

[0074] "Dual Convergence" Control Objective and Closed-Loop System Stability Guarantee: Unlike traditional single-state trajectory tracking, the control objective of this invention is set as "dual convergence" of the zonotope: that is, the center of the zonotope converges to the equilibrium point, and the radius of the zonotope converges to zero, thereby shrinking the entire zonotope to the origin. By constructing dedicated radius and center functions, the stability of the closed-loop system is rigorously proven theoretically, effectively solving the problem of PID control's difficulty in guaranteeing stability in nonlinear systems.

[0075] have Strong disturbance suppression capability of performance: This invention integrates Performance metrics ensure that, under bounded disturbances, the amplitude of the system state is always confined within a defined boundary related to the disturbance amplitude. This performance allows the system to strictly guarantee motion accuracy and vibration suppression within preset performance ranges even when facing unstructured disturbances such as sudden impacts, load changes, and external collisions. In applications such as event-triggered control, this performance can effectively constrain the maximum amplitude of triggering errors, balancing control real-time performance and accuracy, thus providing reliable assurance in terms of safety and dynamic performance.

[0076] This invention, based on a multicellular model and ensemble control framework, does not rely on precise system parameters and can accommodate uncertainties and unmodeled dynamics inherent in flexible joint robotic arms. This significantly reduces the dependence on high-precision modeling and complex parameter identification found in traditional control methods, shortens the system debugging cycle, and reduces manpower and time costs during the research and development and implementation phases.

[0077] Through dual convergence design and Performance is guaranteed; the system maintains high-precision trajectory tracking and vibration suppression capabilities even under complex operating conditions such as external interference and load variations. This feature is particularly suitable for industrial scenarios such as high-precision assembly and precision machining, helping to improve product quality and operational consistency.

[0078] This method exhibits good robustness and adaptability, making it suitable for various loads and operating conditions without requiring repeated adjustments to controller parameters for different tasks. This enables the flexible joint robotic arm to be rapidly deployed in diverse industrial environments, improving equipment reuse rates.

[0079] Based on the same concept, the present invention also provides a control system for a flexible joint robotic arm based on multicellular structures, including a construction module, a solution module and a control module.

[0080] The building blocks are used to linearize and discretize the dynamic model of the flexible joint robotic arm system, resulting in a discrete state-space model that includes external disturbances.

[0081] The solver module is used to construct a multi-cell model to encapsulate the system state boundary and external disturbance boundary. The objective function is generated by using the convergence of the center of the multi-cell model to the equilibrium point, the convergence of the radius to zero, and the performance index used to constrain the system state amplitude as constraints. The objective function is solved based on the parameter matrix of the discrete state space model to obtain the controller gain of the state feedback controller.

[0082] The control module is used to implement a closed-loop control system for the flexible joint manipulator system based on a state feedback controller with controller gain and a discrete state-space model; the control of the flexible joint manipulator system is realized through the closed-loop control system.

[0083] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0084] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A control method for a flexible joint robotic arm based on multicellular structures, characterized in that, Includes the following steps: The dynamic model of the flexible joint robotic arm system is linearized and discretized to obtain a discrete state-space model that includes external disturbances; A multi-cell model is constructed to enclose the system state boundary and external disturbance boundary. The objective function is generated by using the convergence of the center of the multi-cell model to the equilibrium point, the convergence of the radius to zero, and the performance index used to constrain the system state amplitude as constraints. The objective function is solved based on the parameter matrix of the discrete state space model to obtain the controller gain of the state feedback controller. Based on a state feedback controller with controller gain and a discrete state-space model, a closed-loop control system for a flexible joint robotic arm system is implemented; the control of the flexible joint robotic arm system is achieved through the closed-loop control system.

2. The control method for a flexible joint robotic arm based on multicellular structures as described in claim 1, characterized in that, The discrete state-space model is shown below: ； In the formula, and They are respectively Time and The system state at any given moment. For the system matrix, For the input matrix, It is a real matrix. To control the input, External interference; The system status includes the deflection angle of the servo motor, the deflection angle of the connecting rod, the angular velocity of the motor, and the relative angular velocity of the connecting rod. The control input is the motor armature voltage.

3. The control method for a flexible joint robotic arm based on multicellular structures as described in claim 2, characterized in that, The specific state feedback controller is as follows: ； In the formula, This is the controller gain.

4. The control method for a flexible joint robotic arm based on multicellular structures as described in claim 1, characterized in that, The specific multicellular model is shown below: ； In the formula, and They are respectively in Time and System status at all times Multicellular center, and They are respectively in Time and System status at all times The multicellular generation matrix, and External interference The multicellular center and generating matrix, For the system matrix, For the input matrix, It is a real matrix.

5. The control method for a flexible joint robotic arm based on multicellular structures as described in claim 4, characterized in that, The performance indicators are: The performance metrics, which are defined by the convergence of the center of the multicellular model to the equilibrium point, the convergence of the radius to zero, and the performance metrics used to constrain the amplitude of the system state, are used to generate the objective function. Specifically, this includes the following steps: The weighted center function and weighted radius function are defined based on the center and generating matrix of the multicellular model; Based on the weighted center function and the weighted radius function, construct constraint expressions that satisfy the central convergence to the equilibrium point, the radius convergence to zero, and the performance index used to constrain the state amplitude of the multicell model; Transform the constraint expression into an objective function.

6. The control method for a flexible joint robotic arm based on multicellular structures as described in claim 5, characterized in that, The constraint expression is as follows: ； in, ； ； In the formula, and They are respectively Weighted radius and weighted center of multicellular bodies at any given time. and They are respectively Weighted radius and weighted center of multicellular bodies at any given time. The robust exponential decay rate for biconvergence of multicellular structures. For disturbance suppression performance indicators, for The convergence term of the multicell radius at time t. For the multicellular center convergence term, for The radius of the multicellular body at any given time, The distance between the centers of the multicellular organisms. for External disturbance energy at any given moment.

7. The control method for a flexible joint robotic arm based on multicellular structures as described in claim 6, characterized in that, The specific objective function is as follows: ； in, ； In the formula, P To set the matrix, T It is a transpose operator. For the system matrix, For the input matrix, It is a real matrix. It is a discrete matrix. It is the identity matrix. This is the controller gain.

8. The control method for a flexible joint robotic arm based on multicellular structures as described in claim 7, characterized in that, The controller gain is specifically shown below: ； In the formula, For the controller gain matrix, For matrix variables, This is the controller gain.

9. The control method for a flexible joint robotic arm based on multicellular structures as described in claim 1, characterized in that, The control of the flexible joint robotic arm system through a closed-loop control system includes the following steps: Collect the system status of the flexible joint robotic arm system at the current moment; The current system state is input to the state feedback controller, which generates a control quantity based on the current system state and the controller gain. The flexible joint robotic arm is controlled by outputting control quantities to the flexible joint robotic arm system.

10. A control system for a flexible joint robotic arm based on multicellular structures, characterized in that, include: The module is used to linearize and discretize the dynamic model of the flexible joint robotic arm system, resulting in a discrete state-space model that includes external disturbances. The solver module is used to construct a multi-cell model to wrap the system state boundary and external disturbance boundary. The objective function is generated with the multi-cell model's center converging to the equilibrium point, radius converging to zero, and performance indicators used to constrain the system state amplitude as constraints. The objective function is solved based on the parameter matrix of the discrete state space model to obtain the controller gain of the state feedback controller. The control module is used to implement a closed-loop control system for the flexible joint manipulator system based on a state feedback controller with controller gain and a discrete state-space model; the closed-loop control system enables the control of the flexible joint manipulator system.

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