Three-stage bolt tightening method based on Kalman filtering and predictive control

Abstract

The invention belongs to the technical field of mechanical assembly automation and intelligent control, and particularly relates to a three-stage bolt tightening method based on Kalman filtering and predictive control. The method comprises the steps that a man-machine coupling system state space model is constructed, and bias slowly changing along with time in sensor measurement is added to be a new state variable; collecting a motor rotation angle, a motor angular speed and an output torque containing fixed delay; performing state prediction based on the optimal state estimation value of the previous moment and the actual control input of the current moment by using the man-machine coupling system state space model after dimension expansion, and correcting the predicted value by using the current observation vector to obtain the optimal state estimation value of the current moment; the tightening process is divided into three stages, and smooth switching of the three stages is executed through a state machine provided with a hysteretic interval and a tolerance neighborhood. Accurate, stable and intelligent control over the tightening process is achieved.

Description

Technical Field

[0001] This invention belongs to the field of mechanical assembly automation and intelligent control technology, specifically relating to a three-stage bolt tightening method based on Kalman filtering and predictive control. Background Technology

[0002] In industries such as mechanical assembly, automotive manufacturing, and aerospace, bolted connections are one of the most widely used connection methods, and their quality directly affects the reliability and safety of the entire product structure. As the core equipment for automatic bolt tightening, the control precision of bolt tightening machines, especially the precision of the final tightening torque, has always been a focus of industry attention. Traditional tightening equipment typically uses pneumatic wrenches or simple power tools, applying torque through compressed air drive or basic motor control, which suffers from problems such as inaccurate torque control, high noise, and easy damage to workpieces. In recent years, with the development of servo control technology, tightening equipment using servo motors has gradually become the mainstream choice for high-precision tightening operations due to its advantages such as controllable torque and rapid response.

[0003] In existing high-precision tightening technologies, various solutions have been proposed to improve torque control accuracy. For example, patent CN222104936U discloses a torque testing and calibration device for tightening equipment. This device monitors the actual output torque of the tightening equipment through the cooperation of a torque sensor, a simulated bolt, and a fixing fixture, and compares and calibrates it with the theoretical output torque to ensure the accuracy and reliability of the output torque. This technology focuses on post-process calibration and verification, but does not address real-time prediction and dynamic control during the tightening process.

[0004] Another patent, CN106371315A, proposes a tightening control method for threaded connections based on the compression ratio of the sealing ring. This method establishes a torque-angle model through mechanistic analysis and uses an expert prediction method to identify the zero point of the sealing ring online, performing segmented torque-angle control. While this technology introduces the concepts of online identification and segmented control, its control model relies on the identification of specific characteristic points (such as the zero point of the sealing ring) and does not involve forward-looking prediction of the overall dynamic behavior of the system. Furthermore, it is somewhat lacking in handling the inertial impact of the system under high-speed tightening conditions.

[0005] Furthermore, patent US7467669B2 relates to a control method for a pneumatic impact wrench. This patent addresses the problem that, when tightening so-called "hard connections," the steep torque increase characteristic can lead to overtightening during the initial torque impact. This prior art reveals that equipment inertia is a key factor affecting the final torque accuracy towards the end of the tightening process; however, its solution is designed for the characteristics of pneumatic tools and its dynamic model and control method are not directly applicable to precision electric tightening systems composed of servo motors, reducers, couplings, etc.

[0006] Based on the above analysis, existing technical solutions in this field may encounter the following technical problems in practical applications: First, most control methods lack accurate modeling of the overall dynamics of the tightening system (including the servo motor, reducer, transmission chain, and load), making it impossible to accurately predict future torque trends, resulting in lag in control response. Second, especially near the target torque, after a sudden stop of the servo motor, the rotational kinetic energy stored in the components of the transmission chain (such as the motor rotor, reducer, coupling, and bit) will continue to be released due to inertia, thus converting into additional tightening torque, causing the final torque to exceed the preset target value, resulting in overshoot. This overshoot phenomenon can cause the bolt preload to exceed the design range, which may cause the bolt itself to yield and fracture, or the connecting parts to be crushed, seriously affecting product safety and reliability.

[0007] Therefore, there is an urgent need for an advanced control method for servo tightening machines that can predict torque change trends based on a precise dynamic model of the system and control the motor to stop when the torque is determined to be close to the target value, thereby ensuring that the final torque value is precisely controlled within the target range. Summary of the Invention

[0008] The purpose of this invention is to solve the problems of torque overshoot and backshoot during the tightening process, and to propose a three-stage bolt tightening method based on Kalman filtering and predictive control, which is suitable for industrial assembly scenarios with high requirements for bolt tightening torque accuracy.

[0009] To achieve the above objectives, the technical solution provided by this invention is as follows:

[0010] A three-stage bolt tightening method based on Kalman filtering and predictive control includes:

[0011] A state-space model of the human-machine coupling system is constructed by adding the time-varying bias in the sensor measurement as a new state variable, thus obtaining the expanded-dimensional state-space model of the human-machine coupling system.

[0012] Collect motor rotation angle, motor angular velocity, and output torque with a fixed delay;

[0013] Using the expanded state-space model of the human-machine coupled system, state prediction is performed based on the optimal state estimate of the previous time step and the actual control input at the current time step. The predicted value is then corrected using the current observation vector to obtain the optimal state estimate at the current time step.

[0014] The tightening process is divided into three stages, and a state machine with hysteresis intervals and tolerance neighborhoods is used to smoothly switch between the three stages: In the first stage, a proportional-integral controller is used to perform speed tracking control based on the optimal state estimate until the preset positioning criterion is met; In the second stage, based on the optimal state estimate, a pre-built prediction function controller is used to obtain the torque prediction value, and the optimal control voltage increment is solved in real time in combination with preset soft and hard constraint strategies until the torque prediction value enters and remains in the tolerance neighborhood based on the target torque for a second set time; In the third stage, the control voltage is linearly reduced, and a shutdown operation is performed according to the preset shutdown criterion.

[0015] Furthermore, the expanded-dimensional human-machine coupled system state-space model includes:

[0016] The expanded state vector includes: motor rotation angle, motor angular velocity, tool deflection displacement, tool deflection angular velocity, joint angular displacement, joint angular velocity, torque sensor slow-varying bias, and rotary encoder slow-varying bias;

[0017] The state-space equation of the discrete human-machine coupled system after dimension expansion is expressed as:

[0018] ;

[0019] ;

[0020] in, express The expanded state vector at time step 1. express The expanded state vector at time step 1. This represents the expanded state transition matrix. This represents the expanded input matrix. This represents the expanded output matrix. express Time-based control input, This represents the process noise after dimensional expansion. This represents the expanded observation vector. This indicates the measurement noise after dimensional expansion.

[0021] Furthermore, the process of using the expanded-dimensional human-machine coupled system state-space model to predict the state based on the optimal state estimate from the previous time step and the actual control input at the current time step, and then correcting the predicted value using the current observation vector to obtain the optimal state estimate at the current time step, includes:

[0022] State prediction is performed based on the optimal state estimate from the previous time step and the actual control input at the current time step, expressed by the formula:

[0023] ;

[0024] in, for The state prediction value at time unaware of measurement correction. for The optimal state estimate at time t. express Time-based control input;

[0025] The updated predicted covariance matrix is ​​expressed by the formula:

[0026] ;

[0027] in, For prediction The covariance matrix of the state estimation at time step. for The covariance matrix of the state estimation at time step. This represents the expanded process noise covariance matrix. Process noise in the bias state;

[0028] The Kalman gain is calculated based on the predicted covariance matrix, expressed by the formula:

[0029] ;

[0030] in, for Kalman gain at time step To measure the noise covariance matrix;

[0031] The predicted value is corrected based on the Kalman gain and observation vector at the current time to obtain the optimal state estimate at the current time, which can be expressed by the formula:

[0032] ;

[0033] in, express The optimal state estimate at time t. express The observation vector at time;

[0034] The covariance matrix of the updated state estimate is expressed by the formula:

[0035] ;

[0036] in, express The covariance matrix of the state estimation at time step. Represents the identity matrix.

[0037] Furthermore, the smooth transition of the three stages using a state machine with hysteresis intervals and tolerance neighborhoods includes:

[0038] The hysteresis interval includes unequal trigger thresholds and reset thresholds;

[0039] When switching from the first stage to the second stage, the slope of the torque-angle curve and the average torque within the sliding window are calculated in real time. When the slope of the torque-angle curve and the average torque both reach the corresponding trigger threshold and continue for a first set time, the stage switch is executed. If, during the judgment period based on the first set time, any parameter falls below the corresponding reset threshold, the state machine maintains the first stage and the judgment is reset.

[0040] When switching from the second stage to the third stage, the torque prediction value in the optimal state estimate is monitored in real time. When the torque prediction value enters the tolerance neighborhood and continues for a second set time, the stage switch is executed; if the torque prediction value jumps out of the tolerance neighborhood during the judgment period based on the second set time, the judgment is reset.

[0041] Furthermore, the method of using a proportional-integral controller combined with the optimal state estimate for speed tracking control includes:

[0042] The optimal state estimate is mapped to the estimated motor angular velocity feedback signal;

[0043] Calculate the speed tracking error between the target speed and the estimated motor angular velocity feedback signal;

[0044] The proportional and integral terms are calculated based on the speed tracking error, and the control voltage is generated accordingly.

[0045] The motor speed is maintained at the target speed by using control voltage.

[0046] Furthermore, the step of obtaining the torque prediction value based on the optimal state estimate using a pre-built prediction function controller includes:

[0047] The optimal state estimate at the moment when the in-situ criterion is met in the first stage is used as the initial value of the system state vector in the second stage and input into the predictive function controller.

[0048] The dynamic response sequence of each state variable is obtained using a pre-defined step response model;

[0049] The system state vector at the current moment is obtained based on the initial value of the system state vector, and the instantaneous torque gain, instantaneous deflection gain, and instantaneous deflection velocity gain are calculated in combination with the dynamic response sequence. Then, the predicted torque, deflection, and deflection velocity values ​​at the next moment are calculated.

[0050] Furthermore, the real-time solution of the optimal control voltage increment, combined with a preset soft and hard constraint strategy, includes:

[0051] The optimal input increment is calculated using the closed-form solution of the optimal control quantity, which is expressed by the following formula:

[0052] ;

[0053] in, express The optimal input increment at time t. Indicates the reference torque. Indicates the weighting coefficient. Indicates instantaneous torque gain. express The predicted torque value at any given time;

[0054] Implement soft and hard constraint strategies, wherein the soft and hard constraint strategies include:

[0055] If the optimal input increment causes the torque prediction value at the next moment to exceed the upper limit of the tolerance neighborhood, then the optimal input increment will be projected under hard constraints.

[0056] If the temporary voltage increment exceeds the upper or lower limit of the single-step voltage increment, it will be corrected to the upper or lower limit of the single-step voltage increment.

[0057] Calculate the temporary control quantity. If the temporary control quantity exceeds the upper or lower limit of the control voltage amplitude, then correct it to the upper or lower limit of the control voltage amplitude as close as possible.

[0058] By incorporating the slack variables based on the predicted deflection angle and the predicted deflection velocity into the optimization objective function of the prediction function controller, the corrected objective function is obtained.

[0059] The optimal control voltage increment is calculated using the modified objective function.

[0060] Furthermore, the preset shutdown criteria include:

[0061] Simultaneously satisfy:

[0062] The actual output torque remains within the tolerance neighborhood of the target torque throughout the continuous sampling period;

[0063] The motor angular velocity decays to below the preset angular velocity threshold;

[0064] The rate of change of joint angular displacement reaches below the preset threshold for the rate of change of angular velocity;

[0065] And continue for the third set time.

[0066] Compared with existing technologies, the significant advantages of this invention are as follows: By establishing a state-space model of the human-machine coupled system and combining it with Kalman filtering with slow-varying bias estimation and fixed-delay compensation, this invention achieves high-precision real-time estimation of the system's critical states. Based on this, a three-stage predictive control architecture is adopted. In the tightening stage, the torque change trend is adjusted forward by a predictive function. In the shutoff stage, a linear voltage reduction and multi-condition joint decision strategy is used to effectively suppress torque overshoot caused by system inertia, thereby significantly improving the accuracy and consistency of the final tightening torque and ensuring connection reliability. Simultaneously, the introduction of a state machine with hysteresis and soft and hard constraint strategies enhances the system's adaptability to noise, disturbances, and handheld operating conditions, improving human-machine interaction comfort. The overall control scheme integrates state estimation, rolling optimization, and intelligent decision-making, achieving precise, smooth, and intelligent control of the tightening process. Attached Figure Description

[0067] Figure 1 This is a schematic diagram of the structure of a bolt tightening machine in the prior art;

[0068] Figure 2 This is an integrated control flowchart of the three-stage bolt tightening method based on Kalman filtering and predictive control according to the present invention.

[0069] Figure 3 This is a flowchart of the prediction function control during the Tightening stage in this invention;

[0070] Figure 4 The graph shows a comparison between real data from a bolt tightening machine and the response of the three-stage bolt tightening method and equivalent model based on Kalman filtering and predictive control proposed in this invention.

[0071] Figure 5 This is a comparison diagram of the errors between the three-stage bolt tightening method based on Kalman filtering and predictive control of this invention and its equivalent model.

[0072] Figure 6 This is a comparison chart of the simulated torque of the present invention with that of KF-1stage-MPC and KF-3stage-MPC;

[0073] Figure 7 A comparison chart of the simulated torque of the present invention for different sampling periods. Detailed Implementation

[0074] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0075] This invention provides a three-stage bolt tightening method based on Kalman filtering and predictive control, which can be applied to, for example... Figure 1 The bolt tightening machine shown is configured such that the operator holds the handle 2, and the output of the servo motor 1 is connected to a reducer 3. The reducer 3 reduces the output power of the servo motor 1 to increase torque, matching the torque parameters required for bolt tightening. The output of the reducer 3 is connected to the input of a torque sensor 4, which collects torque data in real time during bolt tightening. The output of the torque sensor 4 is connected to the upper end of a vertically downward-facing bit 5, the lower end of which engages with the bolt to be tightened. The handle 2 is fixedly connected to the non-rotating part of the servo motor 1.

[0076] The control algorithm equipped in this device synchronously receives real-time data on the output torque from the rotary encoder and torque sensor 4 in the servo motor 1. Based on this data, it predicts the torque change trend and precisely controls the operating status of the servo motor 1 in advance. When the device performs bolt tightening operations, after the servo motor 1 is started, its power is processed by the reducer 3 and transmitted to the bit 5 through the torque sensor 4, so that the bit 5 obtains the corresponding speed and torque, driving the bolt into the threaded hole.

[0077] The entire tightening process is divided into three stages. In the Rundown stage (stage 1), the control algorithm controls the speed of servo motor 1 at the set target speed to enable the bolt to be screwed in quickly until the positioning criterion is met. In the Tightening stage (stage 2), the state estimate value output by the Kalman filter is used as input. A predictive function controller is constructed based on the step response model, and the closed-form solution of the optimal control quantity is analyzed. The control quantity is corrected through soft and hard constraint strategies until the torque prediction value enters the preset tolerance neighborhood of the target torque. Then, the process switches to the Shutoff stage (stage 3). When it is determined that the actual torque has entered the tolerance neighborhood of the target torque, the control algorithm controls servo motor 1 to decelerate in advance according to the set rate to prevent overshoot caused by inertia and ensure that the final torque value of the bolt accurately reaches the preset target torque.

[0078] like Figures 2-3 As shown in the figure, this embodiment provides a three-stage bolt tightening method based on Kalman filtering and predictive control. The specific steps are as follows:

[0079] S1. Construct a state-space model of the human-machine coupling system. Based on the coupled dynamics model of the operator of the motor-driven joint tool under the hand-held tightening condition, establish the state-space equation of the human-machine coupling system, and form a discrete state-space expression by discretization through sampling period.

[0080] S1.1. Based on the "motor-transmission-joint-tool / operator" human-machine coupling kinematic model established under hand-held tightening conditions, the state-space equation of the human-machine coupling system is constructed. The state vector of the human-machine coupling system is: .

[0081] The state differential equation of the human-machine coupled dynamics model is:

[0082] ;

[0083] in, For the motor rotation angle, The angular velocity of the motor. This refers to the angular acceleration of the motor. For tool deflection displacement, For the tool's deflection angular velocity, For the tool's deflection acceleration; This refers to joint angular displacement. The joint angular velocity, Joint angular acceleration; This is a reference value for the motor voltage; The moment of inertia of the motor. The total tool moment of inertia, This refers to the moment of inertia of the joint. The torque constant of the motor. For joint stiffness, For operator stiffness; This is the viscous damping coefficient of the motor. This is the transmission damping coefficient. This is the joint damping coefficient. Operator damping coefficient; The total transmission ratio is... For motor resistance, This refers to the operator's lever arm length.

[0084] S1.2. The human-machine coupling system after the sampling period After discretization, its state-space equation is expressed as:

[0085] ;

[0086] ;

[0087] The discretized state transition matrix is ​​as follows:

[0088] ;

[0089] Discretized input matrix: ;

[0090] Discretized output matrix: ;

[0091] in, for The state vector at time t, for Time-based control input, for Time-based process noise, for The observation vector at time t, for Measurement noise at time.

[0092] S2. Implement Kalman filtering with slow-varying bias estimation and fixed time delay compensation, collect motor angle, motor angular velocity and output torque, use the algorithm to update the state using the delayed measurement value, and estimate the slow-varying bias by expanding the state variable to overcome sensor error.

[0093] S2.1. During the actual operation of bolt tightening, the torque sensor and rotary encoder are susceptible to factors such as zero-point drift and ambient temperature fluctuations, leading to a slowly changing systematic error in their measurements. This type of systematic error with a deterministic trend is defined as slow-varying bias. To achieve effective estimation and compensation of the slow-varying bias, thereby improving the system control accuracy, two new state variables, namely the slow-varying bias of the torque sensor and the slow-varying bias of the rotary encoder, are added to the original state variables, completing the dimension expansion design of the state vector.

[0094] The extended-dimensional state vector is represented as:

[0095] ;

[0096] in, For the torque sensor, slow-varying bias is used. This is a slow-change bias for the rotary encoder.

[0097] The bias state change model is as follows:

[0098] ;

[0099] in, for The slowly varying bias state matrix at time t. for The slowly varying bias state matrix at time t. for The small-amplitude process noise at any given time indicates the slow drift characteristic of the bias.

[0100] The expanded discrete state-space equations are expressed as follows:

[0101] ;

[0102] ;

[0103] in, express The expanded state vector at time step 1. express The expanded state vector at time step 1. The observation vector after dimension expansion. For the process noise after dimensional expansion, The measurement noise is after dimension expansion; the state transition matrix after dimension expansion is... , The input matrix is ​​a 2-order identity matrix; the expanded input matrix is... The expanded output matrix is .

[0104] S2.2. The system measurement has a fixed delay of one sampling period, that is, the measurement value obtained at the current moment is the measurement value of the previous moment.

[0105] In the prediction step, the state prediction formula is:

[0106] ;

[0107] in, for The state prediction value at time unaware of measurement correction. for The optimal state estimate at time t. express Time-based control input.

[0108] The updated predicted covariance matrix is ​​expressed by the formula:

[0109] ;

[0110] in, For prediction The covariance matrix of the state estimation at time step. for The covariance matrix of the state estimation at time step. This represents the expanded process noise covariance matrix. This refers to process noise in the bias state.

[0111] In the update step, the Kalman gain is calculated as follows:

[0112] ;

[0113] in, for Kalman gain at time step To measure the noise covariance matrix.

[0114] The state update equation is:

[0115] ;

[0116] in, for The optimal state estimate at time t is used as the input to the subsequent prediction function controller. express The observation vector at time t.

[0117] The covariance matrix update equation for the state estimate is:

[0118] ;

[0119] in, express The covariance matrix of the state estimation at time step. Represents the identity matrix.

[0120] S3. Divide the tightening process into three stages: Rundown, Tightening, and Shutoff, based on the positioning criteria and preset tolerance neighborhood. The stages are then smoothly switched using a state machine with hysteresis.

[0121] S3.1. Utilize a state machine with hysteresis intervals and tolerance neighborhoods to perform smooth switching of the three stages.

[0122] S3.1.1 When switching from the Rundown phase to the Tightening phase, the switching conditions must meet the positioning criteria. The positioning criteria include the threshold transition of the torque-joint angle curve slope and the sliding window statistics of the torque increment, and a hysteresis interval with trigger threshold and reset threshold is set.

[0123] The specific criteria for judging the slope of the torque-angle curve are as follows: the slope of the torque-joint angle curve represents the torque increment caused by a unit change in joint angle. When the bolt transitions from an idle state to a contact state, the bolt's positioning can be determined by detecting the transition in the slope of the torque-joint angle curve.

[0124] The formula for calculating the slope of the torque-joint angle curve is:

[0125] ;

[0126] in, for The slope of the torque-joint angle curve at any given moment. for Output torque at any moment for Output torque at any moment for The joint angle at any moment, for The joint angle at any given moment.

[0127] Real-time computing Time and The joint angle difference at time t is: The criteria for determining changes in joint angles are: , This is the lower limit threshold for angle change. If the joint angle change determination condition is met, the current value is ignored. Slope of the torque-joint angle curve at any moment The calculation.

[0128] The overall criterion for threshold transitions based on the slope of the torque-joint angle curve is:

[0129] ;

[0130] in, The slope trigger threshold, This is the slope growth coefficient, reflecting the magnitude of the increase in the current slope relative to the slope at the previous moment. for The slope of the torque-joint angle curve at any given moment.

[0131] The specific criteria for judging the average torque increment of the sliding window are as follows: the sliding window statistics of torque increment are used to suppress instantaneous fluctuation interference, and the torque increment over a recent period is statistically analyzed through the sliding window. When the bolt transitions from free-spinning to contact, the bolt's positioning can be determined by detecting the transition of the average torque increment of the sliding window.

[0132] Let the size of the sliding window be... The torque sequence within the window is as follows:

[0133] ;

[0134] when At that time, the recursive update window is as follows:

[0135] ;

[0136] in, for The window of time and, for The window of time and...

[0137] when At that time, the average value is calculated using the currently available torque:

[0138] ;

[0139] in, for The average torque within the sliding window at any given time. for Torque at any given moment.

[0140] The formula for calculating the window mean is:

[0141] .

[0142] The threshold condition for the average torque of the sliding window is determined as follows:

[0143] ;

[0144] in, for Torque at any moment The gain threshold represents the amplification factor of the current torque relative to the average torque. for The average torque within the sliding window at any given time. This is set as the minimum torque difference threshold to avoid misjudgments where the multiple is met within a small torque range but the actual increment is insufficient.

[0145] S3.1.2. The state machine must meet the following conditions simultaneously to switch from the Rundown phase to the Tightening phase: the slope of the torque-angle curve and the average torque of the sliding window must both exceed their respective set trigger thresholds, and this judgment result must be maintained for more than a first set time. If any condition falls below its corresponding reset threshold during this process, the state machine will not perform a transition and will maintain its original state.

[0146] S3.2. The specific steps for the state machine with hysteresis to achieve smooth switching between the Tightening and shutdown stages based on the preset tolerance neighborhood are as follows:

[0147] S3.2.1. When switching from the Tightening stage to the shutdown stage, the real-time output torque... Upon first entering the tolerance neighborhood, start the delay timer. If the timer jumps out of the tolerance neighborhood during the timing period, the timer is reset; if Second set time If the state remains within the tolerance neighborhood, the state machine switches to the shutdown phase.

[0148] S3.2.2. During the Tightening stage, when the real-time output torque... Target torque When the preset tolerance neighborhood is reached, it can avoid waiting for the torque to fully reach the target torque. Overshoot caused by immediate shutdown, while ensuring early soft-stop procedure to suppress backlash. Target torque. The tolerance neighborhood set for the center is:

[0149] ;

[0150] Neighborhood Boundary The settings take into account the standard deviation of the torque error estimated by the Kalman filter. System response delay time And the torque fluctuation range required for human-machine comfort:

[0151] ;

[0152] in, For safety reasons, This is the delay compensation coefficient. This is the coefficient for maximum torque variation.

[0153] S4. During the Rundown phase, speed tracking control is executed. A PI controller and Kalman filter are combined to obtain an estimated motor angular velocity through state extraction. Based on the error between the target speed and the estimated value, the controller uses proportional-integral circuitry to collaboratively calculate the control voltage, achieving rapid speed tracking and elimination of steady-state deviations. The system maintains the motor speed at the target value until the predicted torque meets the positioning conditions, then switches to the Tightening phase.

[0154] S4.1. The PI controller and Kalman filter work together by extracting the prior estimated state vector. This is mapped to the estimated motor angular velocity feedback signal required by the PI controller.

[0155] ;

[0156] in, express Estimated motor angular velocity at any given time. Represents the angular velocity state extraction matrix. express The state prediction value at time t is the state vector estimated prior to measurement, which is not corrected by measurement.

[0157] The S4.2.PI controller employs a proportional-integral dual-stage coordinated operation, with the target speed and... The speed tracking error is calculated by taking the estimated motor angular velocity as input at all times.

[0158] ;

[0159] in, for The speed tracking error at any given moment; This is the target rotational speed for the Rundown phase.

[0160] To eliminate static bias, an integral term is introduced to accumulate historical errors, and the integration time compensation is the sampling period. :

[0161] ;

[0162] in, for The integral value of the error at time t.

[0163] S4.3. In a PI controller, the proportional term quickly responds to the current error, while the integral term eliminates steady-state deviation, working together to generate the control input:

[0164] ;

[0165] in, For proportional gain, For integral gain, for Control the output voltage at any time.

[0166] The S4.4.PI controller maintains the motor speed at the target speed until the torque prediction value meets the positioning criterion, at which point the state machine switches to the Tightening stage.

[0167] S5. Execute the predictive function control in the Tightening stage. With the state estimate as input, construct the predictive function controller based on the step response model, deduce the closed-form solution of the optimal control quantity analytically, and use soft and hard constraint strategies to correct and optimize the original control quantity until the torque prediction value enters the target torque preset tolerance neighborhood switching stage.

[0168] The specific steps for constructing the prediction function controller in S5.1 are as follows:

[0169] S5.1.1. Using a step response model, a unit step voltage input is applied to the electric tightening system, and the dynamic response sequence of each state variable is recorded to characterize the system characteristics.

[0170] S5.1.2. Under zero initial conditions, the system's initial state is zero, and a unit step voltage signal is applied to it. The unit step voltage input is defined as:

[0171] That is, in time Previously the voltage was zero, The voltage jumps instantaneously to 1V and remains at that constant value.

[0172] Through iterative calculation, starting from the initial state, the system state vector at each time step is obtained. This includes key states such as torque prediction values, displacements (motor rotation angle, tool deflection displacement, joint angular displacement), and rotational speeds (motor angular velocity, tool deflection angular velocity, joint angular velocity) at various times. The optimal state estimate at the moment when the positioning criterion is met in the first stage is used as the initial value of the system state vector in the second stage and input into the prediction function controller for iteration.

[0173] S5.1.3. The state update equation is:

[0174] ;

[0175] in, For the discretized state matrix, This is for discretizing the input matrix.

[0176] The state vector contains the internal states of the system, and the output response sequence of the corresponding state parameters is as follows:

[0177] Torque step response sequence ;

[0178] Deflection displacement step response sequence ;

[0179] Deflection velocity step response sequence ;

[0180] in, Discretize the output matrix The second line, The system state vector The third element, The system state vector The fourth element.

[0181] S5.1.4. The first non-zero value of the state parameter response sequence reflects the system's instantaneous response capability to the input. Therefore:

[0182] Torque instant gain Used to quantify the instantaneous torque change caused by a unit voltage change; instantaneous gain of deflection. Used to quantify the instantaneous change in deflection angle caused by a unit voltage change; instantaneous gain of deflection velocity. This is used to quantify the instantaneous change in deflection velocity caused by a unit voltage change. Among them, The time step corresponding to the first non-zero value in the response sequence.

[0183] S5.1.5. There is a strong coupling relationship between torque and deflection angle. The two need to be correlated, and the deflection angle needs to be indirectly constrained through torque to obtain the torque-deflection angle proportionality coefficient. The torque sensor's real-time measurement value is converted using a torque-deflection ratio coefficient. Convert to equivalent deflection observations : .

[0184] S5.1.6. The predictive function control algorithm needs to estimate the torque output and corresponding deflection angle for the next step.

[0185] Based on the current actual torque measurement value and input increment Utilizing instantaneous torque gain get Torque prediction at time To obtain the forward prediction of torque:

[0186] ;

[0187] Based on current deflection angle observations and input increment Utilizing the instantaneous gain of the deflection angle get Predicted deflection angle at time 1 To obtain the forward prediction of the deflection angle: .

[0188] Based on the current deflection velocity and input increment Utilizing instantaneous gain of deflection velocity get Predicted deflection velocity at time t. To obtain the forward prediction of the deflection velocity: .

[0189] S5.1.7. The predictive function control algorithm drives the predicted torque to approach the reference torque through an objective function, while limiting the input increment to avoid system oscillation. Its optimization objective function is:

[0190] ;

[0191] in, For reference torque, This is a weighting coefficient used to balance accuracy and smoothness.

[0192] To obtain the optimal input increment For the objective function about Taking the derivative and setting it to zero, we obtain the closed-form solution for the optimal control quantity:

[0193] .

[0194] S6. Perform the Shutoff stage linear voltage reduction and multi-condition shutdown judgment. The control voltage is linearly reduced to the safety threshold at a preset rate. The dynamic response is detected in real time. The machine stops when the torque is stable, the motor angular velocity decays, and the joint angular displacement change rate approaches zero.

[0195] S6.1. During the shutdown phase, the control voltage operates at a preset linear rate. Gradually decrease from the current value to the safety threshold. :

[0196] ;

[0197] in, The settings are comprehensively configured based on system inertia, load characteristics, and human-machine comfort requirements.

[0198] S6.2. During the voltage reduction process, the system continuously monitors key state quantities, namely actual torque, motor angular velocity and joint angular displacement change rate, and checks whether the three shutdown criteria are met simultaneously.

[0199] The three shutdown criteria are torque stability judgment, motor angular velocity decay judgment, and joint angular displacement change rate approaching zero judgment:

[0200] The condition for determining torque stability is: the actual torque is within a continuous range. All samples are within the tolerance neighborhood of the target torque within each sampling period. ,in, .

[0201] The condition for determining the decay of motor angular velocity is: the motor angular velocity decays to near zero speed. ,in The threshold for angular velocity is close to zero.

[0202] The condition for determining when the joint angular displacement rate of change approaches zero is: when the joint angular displacement rate of change approaches zero, it indicates that the bolt has reached the final tightening position. ,in, The threshold for the rate of change of angular velocity is close to zero.

[0203] S6.3. When the above three conditions are met simultaneously and continue for a third set time, the system determines that the tightening is complete, cuts off the control voltage output, and achieves a slow shutdown.

[0204] In another embodiment, a PID feedback control combined with a feedforward compensation algorithm can be used instead of the aforementioned predictive function controller. This scheme is based on the real-time feedback of the output torque from the torque sensor. The PID controller dynamically adjusts the output power of the servo motor, while introducing a feedforward compensation stage to output compensation in advance based on the inertial characteristics of the torque during bolt tightening. When the torque approaches the target value, the PID controller precisely controls the motor to stop and applies a negative torque opposite to the tightening direction through the feedforward compensation strategy to counteract the inertial effect and achieve precise torque control.

[0205] Fuzzy control algorithms can also be used to achieve precise torque control. This algorithm takes the real-time torque value and torque change rate collected by the torque sensor as input, processes them into fuzzy logic, and outputs control commands to the servo motor based on a preset fuzzy rule library (e.g., if the torque deviation is large and the change rate is high, the motor output is adjusted significantly; if the torque is close to the target and the change rate is low, a reverse negative torque is applied). At different stages of bolt tightening, the fuzzy controller can adaptively adjust the motor speed and torque output. Especially during the shutdown stage, it can quickly make decisions and apply a reverse negative torque, effectively avoiding torque overshoot and ensuring the torque accuracy of bolt tightening.

[0206] To verify the validity of this invention patent, simulation experiments were conducted using MATLAB software. The three-stage predictive function control method (KF-3stage-PFC) based on Kalman filtering was verified by comparing it with the single-stage model predictive control method (KF-1stage-MPC) and the three-stage model predictive control method (KF-3stage-MPC) based on Kalman filtering.

[0207] In the engineering implementation of Predictive Function Control (PFC) algorithms, to improve the control performance of bolt tightening systems, the nonlinear dynamic model of the bolt tightening system needs to be equivalently transformed into a first-order linear model that facilitates controller design. To verify the adaptability of different equivalent models, this study selects the traditional first-order model, the optimized first-order model, and the first-order time delay (FOPTD) model as candidate equivalent models. System response data are obtained through step signal excitation experiments, and the dynamic fitting accuracy of each model in the tightening stage is compared and analyzed using the average relative error as the core evaluation index.

[0208] The traditional first-order model simplifies a high-order tightening system to a basic structure containing only gain and inertia elements. Its parameters are determined in a simple and intuitive way, offering ease of calculation but limited accuracy. The optimized first-order model, using traditional parameters as initial values, employs numerical optimization methods to collaboratively adjust parameters throughout the entire response time, minimizing the overall error between the model output and actual data. The first-order time-delay model further introduces a "delay time" parameter into its structure to describe the lag process between the input excitation and the start of the response. By simultaneously optimizing the delay time, gain, and time constant, it minimizes the error throughout the entire time period.

[0209] Actual tightening experiment data shows that during the bolt tightening process based on predictive function control, the transition time from the tightening stage to the shutdown stage is 0.13s. Within this critical transition period (0~0.13s), the equivalent model data is as follows: Figure 4 As shown, the response curve of the high-order tightening system (original system) exhibits a typical dynamic upward process. Comparing the response curves of the three models reveals that the response trajectory of the optimized first-order model is closest to the original system, showing a good fit throughout. The response curves of the traditional first-order model and the first-order time-delay model show visually perceptible deviations from the original system, especially in the initial and middle stages of the response, where tracking lag or amplitude errors are more pronounced. A comparison of the relative errors of the models is provided. Figure 5 As shown, the average relative error of the traditional first-order model is 2.17%, the average relative error of the optimized first-order model is only 0.52%, while the average relative error of the first-order time-delay model is as high as 7.88%. The comparison results show that the optimized first-order model has significantly better dynamic fitting accuracy in the tightening stage than the traditional first-order model and the first-order time-delay model, and can more accurately characterize the dynamic characteristics of the bolt tightening system. Therefore, in the predictive function control algorithm, the optimized first-order equivalent model is selected as the control model for the tightening stage, laying the foundation for subsequent control strategy optimization and control accuracy improvement.

[0210] The system dynamics modeling and Kalman filter design in the method are consistent, with a total simulation time of 0.70s and a sampling period of 0.01s. The target torque is set to 17Nm. Experimental results are as follows: Figure 6 As shown, the steady-state torque value of KF-1stage-MPC is 17.228 Nm, the absolute steady-state error is 0.228525 Nm, and the relative error reaches 1.344%. KF-3stage-PFC and KF-3stage-MPC show the same control effect, with both having a steady-state torque value of 17.165858 Nm, an absolute steady-state error of 0.165858 Nm, and a relative error of 0.9756%.

[0211] Since KF-3stage-PFC and KF-3stage-MPC exhibit consistent error performance, their computational speeds are compared. In the MATLAB software environment, calculations were performed with a sampling period of 0.01s and a simulation duration of 0.70s, totaling 70 sampling points. The reference total computation time for KF-3stage-MPC was 1.55s, with a single-point computation time of 0.022s; the reference total computation time for KF-3stage-PFC was 0.023s, with a single-point computation time of 0.0003s. In actual tightening scenarios, the sampling period is 0.002s. KF-3stage-PFC can be set to a higher sampling period; for example, with a sampling period of 0.001s, the experimental results are as follows... Figure 7 As shown, the steady-state torque value is 16.948404 Nm, the absolute steady-state error is -0.051596 Nm, and the relative error is only 0.3035%.

[0212] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the protection scope of the present invention. Therefore, the protection scope of this invention patent should be determined by the appended claims.

Claims

1. A three-stage bolt tightening method based on Kalman filtering and predictive control, characterized in that, The three-stage bolt tightening method based on Kalman filtering and predictive control includes: A state-space model of the human-machine coupling system is constructed by adding the time-varying bias in the sensor measurement as a new state variable, thus obtaining the expanded-dimensional state-space model of the human-machine coupling system. Collect motor rotation angle, motor angular velocity, and output torque with a fixed delay; Using the expanded state-space model of the human-machine coupled system, state prediction is performed based on the optimal state estimate of the previous time step and the actual control input at the current time step. The predicted value is then corrected using the current observation vector to obtain the optimal state estimate at the current time step. The tightening process is divided into three stages, and a state machine with hysteresis intervals and tolerance neighborhoods is used to smoothly switch between the three stages: In the first stage, a proportional-integral controller is used to perform speed tracking control based on the optimal state estimate until the preset positioning criterion is met; In the second stage, based on the optimal state estimate, a pre-built prediction function controller is used to obtain the torque prediction value, and the optimal control voltage increment is solved in real time in combination with preset soft and hard constraint strategies until the torque prediction value enters and remains in the tolerance neighborhood based on the target torque for a second set time; In the third stage, the control voltage is linearly reduced, and a shutdown operation is performed according to the preset shutdown criterion.

2. The three-stage bolt tightening method based on Kalman filtering and predictive control according to claim 1, characterized in that, The expanded-dimensional state-space model of the human-machine coupling system includes: The expanded state vector includes: motor rotation angle, motor angular velocity, tool deflection displacement, tool deflection angular velocity, joint angular displacement, joint angular velocity, torque sensor slow-varying bias, and rotary encoder slow-varying bias; The state-space equation of the discrete human-machine coupled system after dimension expansion is expressed as: ； ； in, express The expanded state vector at each time step express The expanded state vector at each time step This represents the state transition matrix after dimension expansion. This represents the expanded input matrix. This represents the expanded output matrix. express Time-based control input, This represents the process noise after dimensional expansion. This represents the expanded observation vector. This indicates the measurement noise after dimensional expansion.

3. The three-stage bolt tightening method based on Kalman filtering and predictive control according to claim 2, characterized in that, The process of using the expanded-dimensional state-space model of the human-machine coupled system to predict the state based on the optimal state estimate from the previous time step and the actual control input at the current time step, and then correcting the predicted value using the current observation vector to obtain the optimal state estimate at the current time step, includes: State prediction is performed based on the optimal state estimate from the previous time step and the actual control input at the current time step, expressed by the formula: ； in, for The state prediction value at time without measurement correction. for The optimal state estimate at time t. express Time-based control input; The updated predicted covariance matrix is ​​expressed by the formula: ； in, For prediction The covariance matrix of the state estimation at time step. for The covariance matrix of the state estimation at time step. This represents the expanded process noise covariance matrix. Process noise in the bias state; The Kalman gain is calculated based on the predicted covariance matrix, expressed by the formula: ； in, for Kalman gain at time step To measure the noise covariance matrix; The predicted value is corrected based on the Kalman gain and observation vector at the current time to obtain the optimal state estimate at the current time, which can be expressed by the formula: ； in, express The optimal state estimate at time t. express The observation vector at time; The covariance matrix of the updated state estimate is expressed by the formula: ； in, express The covariance matrix of the state estimation at time step. Represents the identity matrix.

4. The three-stage bolt tightening method based on Kalman filtering and predictive control according to claim 1, characterized in that, The process of using a state machine with hysteresis intervals and tolerance neighborhoods to perform smooth switching across three stages includes: The hysteresis interval includes unequal trigger thresholds and reset thresholds; When switching from the first stage to the second stage, the slope of the torque-angle curve and the average torque within the sliding window are calculated in real time. When the slope of the torque-angle curve and the average torque both reach the corresponding trigger threshold and continue for a first set time, the stage switch is executed. If, during the judgment period based on the first set time, any parameter falls below the corresponding reset threshold, the state machine maintains the first stage and the judgment is reset. When switching from the second stage to the third stage, the torque prediction value in the optimal state estimate is monitored in real time. When the torque prediction value enters the tolerance neighborhood and continues for a second set time, the stage switch is executed; if the torque prediction value jumps out of the tolerance neighborhood during the judgment period based on the second set time, the judgment is reset.

5. The three-stage bolt tightening method based on Kalman filtering and predictive control according to claim 1, characterized in that, The method of using a proportional-integral controller combined with the optimal state estimate for speed tracking control includes: The optimal state estimate is mapped to the estimated motor angular velocity feedback signal; Calculate the speed tracking error between the target speed and the estimated motor angular velocity feedback signal; The proportional and integral terms are calculated based on the speed tracking error, and the control voltage is generated accordingly. The motor speed is maintained at the target speed by using control voltage.

6. The three-stage bolt tightening method based on Kalman filtering and predictive control according to claim 1, characterized in that, The process of obtaining the torque prediction value based on the optimal state estimate using a pre-built prediction function controller includes: The optimal state estimate at the moment when the in-situ criterion is met in the first stage is used as the initial value of the system state vector in the second stage and input into the predictive function controller. The dynamic response sequence of each state variable is obtained using a pre-defined step response model; The system state vector at the current moment is obtained based on the initial value of the system state vector, and the instantaneous torque gain, instantaneous deflection gain, and instantaneous deflection velocity gain are calculated in combination with the dynamic response sequence. Then, the predicted torque, deflection, and deflection velocity values ​​at the next moment are calculated.

7. The three-stage bolt tightening method based on Kalman filtering and predictive control according to claim 1, characterized in that, The method of solving the optimal control voltage increment in real time by combining preset soft and hard constraint strategies includes: The optimal input increment is calculated using the closed-form solution of the optimal control quantity, which is expressed by the following formula: in, express The optimal input increment at time t. Indicates the reference torque. Indicates the weighting coefficient. Indicates instantaneous torque gain. express The predicted torque value at any given time. Implement soft and hard constraint strategies, wherein the soft and hard constraint strategies include: If the optimal input increment causes the torque prediction value at the next moment to exceed the upper limit of the tolerance neighborhood, then the optimal input increment will be projected under hard constraints. If the temporary voltage increment exceeds the upper or lower limit of the single-step voltage increment, it will be corrected to the upper or lower limit of the single-step voltage increment. Calculate the temporary control quantity. If the temporary control quantity exceeds the upper or lower limit of the control voltage amplitude, then correct it to the upper or lower limit of the control voltage amplitude as close as possible. By incorporating the slack variables based on the predicted deflection angle and the predicted deflection velocity into the optimization objective function of the prediction function controller, the corrected objective function is obtained. The optimal control voltage increment is calculated using the modified objective function.

8. The three-stage bolt tightening method based on Kalman filtering and predictive control according to claim 1, characterized in that, The preset shutdown criteria include: Simultaneously satisfy: The actual output torque remains within the tolerance neighborhood of the target torque throughout the continuous sampling period; The motor angular velocity decays to below the preset angular velocity threshold; The rate of change of joint angular displacement reaches below the preset threshold for the rate of change of angular velocity; And continue for the third set time.

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