A time-delay-based multivariable spiral control method for a manipulator considering input saturation

Abstract

The present invention discloses a time-delay-based multivariable spiral control method for a manipulator considering input saturation. First, a cable-driven manipulator model is reasonably selected, and time delay estimation (TDE) is used to estimate and compensate for the residual lumped system dynamics, including the residual linkage dynamics, motor dynamics, and lumped uncertainties of the model parameters. Secondly, an adaptive multivariable spiral sliding mode controller is designed based on the characteristics of the manipulator dynamics model and the control requirements. The parameter uncertainty and unknown external interference are further considered, and the TED estimation is error compensated, which effectively improves the robustness of the control system. Finally, the stability of the closed-loop system is proved based on the Lyapunov stability analysis method.

Description

Technical Field

[0001] The present invention belongs to the technical field of robotic arm control, and in particular relates to a time-delay based multivariable spiral control method for a robotic arm taking input saturation into consideration. Background Art

[0002] In recent years, cable-driven manipulators have gradually become a research hotspot due to their unique characteristics. Compared with traditional industrial manipulators, cable-driven manipulators have smaller movement inertia, better flexibility and higher safety to achieve physical interaction with humans. All of the above characteristics are obtained by moving the drive motor from the joint to the base and transmitting the force through the cable. Due to these advantages, cable-driven manipulators have been widely used in many practical applications. Compared with traditional manipulators, the application of cable-driven technology also brings difficulties to precise control performance. The main difficulties of cable-driven manipulators are low stiffness, high joint flexibility, complex system dynamics and external time-varying disturbances. Therefore, in practical applications, designing a suitable control scheme remains a challenge.

[0003] To improve control performance, time-delay estimation (TDE) schemes have been proposed and studied. TDE primarily utilizes time-delayed system states to estimate the residual dynamics. Since TDE can produce model-free properties, facilitating control, TDE technology has been widely used since its introduction. However, utilizing TDE's delayed signals leads to unavoidable estimation errors—TDE errors—which can degrade control performance. Therefore, additional robust control schemes are often employed to enhance TDE-based nonlinear control, such as SM and TSM control, adaptive methods, fuzzy logic control, and neural network control. As one of the most important control tools, SM and its variants have been extensively combined with TDE schemes, benefiting from their strong robustness and simple form. While excellent results have been achieved using TDE-based SM / TSM control schemes, they remain limited to traditional approaches to addressing chatter.

[0004] During operation, the manipulator is inevitably affected by the external environment, which can interfere with its normal operating conditions. Therefore, achieving convergence of the manipulator within a finite time has been widely studied. Several techniques have been proposed and studied to address the jitter problem of SM control and its variants, such as boundary layer control, continuous arrival law, adaptive control, intelligent control, and high-order SM control. Among them, the adaptive multivariable spiral scheme is a second-order continuous sliding mode control algorithm. When the boundaries are known and there is a smooth matching disturbance with a bounded gradient, a continuous control function is generated, and the sliding variable and its derivative are driven to zero within a finite time. Due to its fast convergence speed and high convergence accuracy, it has been widely studied in academia. Adaptive control can achieve faster convergence speed, provide higher control accuracy and anti-interference ability.

[0005] Furthermore, to ensure the precise and safe operation of the cable-driven manipulator, saturation control requirements cannot be ignored. Under normal control conditions, the system input changes solely based on the set control rate, potentially leading to overload of the input value. Due to the limitations of the manipulator's control structure, exceeding the control force the manipulator can withstand is unsafe. If the manipulator encounters an unexpected event and generates a significant response, this can lead to mechanical system failure, uncontrolled operation, and collisions with external forces, resulting in safety issues and financial losses. Summary of the Invention

[0006] In order to overcome the above-mentioned shortcomings of the existing technology, the present invention takes into account the influence of unknown interference, actuator saturation and strong coupling during the actual operation of the robotic arm, and proposes a new time-delay-based multivariable spiral control method for the robotic arm considering input saturation, thereby realizing limited-time rapid and accurate tracking of the robotic arm posture.

[0007] The technical solution employed in this paper is as follows: First, a reasonable cable-driven manipulator model is selected. Time delay estimation (TDE) is used to estimate and compensate for the residual lumped system dynamics, considering the model parameters, including residual linkage dynamics, motor dynamics, and lumped uncertainties. Second, an adaptive multivariable spiral sliding mode controller is designed based on the characteristics of the manipulator's dynamic model and control requirements. This controller further considers parameter uncertainty and unknown external disturbances, compensating for errors in the TED estimation and effectively improving the robustness of the control system. Furthermore, considering input saturation, this paper also imposes safety constraints on the input quantity to avoid machine necrosis caused by input overload. This strategy does not require accurate estimation of external disturbances, yet still enables the manipulator to accurately track reference commands. The stability of the closed-loop system is demonstrated using Lyapunov's stability analysis method.

[0008] Step 1: Establish a dynamic model to describe the state of the cable manipulator, so as to facilitate the design of a high-precision control method;

[0009] The dynamic model of the n-degree-of-freedom manipulator can be expressed as:

[0010]

[0011]

[0012]

[0013] Where J and d m are the motor inertia and damping matrices, q and θ are the joint and motor position vectors, Respectively represent the first-order derivative and second-order derivative of q and θ, M(q) is the inertia matrix, is the Coriolis / centrifugal matrix, g(q) is the gravitational force, is the friction vector, τ m and τ s are the control torque given by the motor and the joint flexibility torque, d s is the damping matrix, k s is the joint stiffness matrix, τ d represents the lumped unknown uncertainty;

[0014] In order to facilitate the use of the TDE scheme, which is based on the time delay estimation method, (2) is substituted into (1) and the constant parameter is applied. get:

[0015]

[0016] Among them, the expression of f is:

[0017]

[0018] Step 2: Use the TED scheme to estimate the estimated value of the unknown parameter f mentioned above;

[0019] Use the time delay estimation method to estimate the value of f and find its estimated value

[0020]

[0021] Where △t is the delay time. Substituting the kinetic model (4) into (6) yields:

[0022]

[0023] From (6) and (7), it can be seen that the main purpose of the time delay estimation scheme is to estimate the lumped system dynamics using only the time delay values ​​of the control and acceleration signals, and then give a model-free scheme. In engineering applications, τ m (t-△t) can be expressed by τ m The direct time lag is obtained by numerical differentiation to obtain

[0024]

[0025] Where, T≥2△t; in the initial stage, when t≤2△t, q(t) has the actual measured value, and q(t-2△t) will be manually set to zero, which may cause strong fluctuations. Therefore, (8) is used to alleviate the possible strong fluctuations. By setting it in the form of (9), the delay can be estimated and the estimated value can be obtained.

[0026]

[0027] It is widely used in many robust control schemes based on TDE. The simulation part of this paper proves its effectiveness. If no measures are taken, numerical differentiation (9) will significantly amplify the noise effect, thereby reducing the control performance.

[0028] As can be seen from (8) and (9), the current value of the lumped system dynamics is estimated by the TDE scheme using the time-delay system state. Therefore, when large disturbances occur, the estimation error of this method will become larger. However, the adaptive multivariable spiral algorithm method we proposed can reduce the estimation error. The simulation in this paper proves the effectiveness of this method.

[0029] Step 3: Design an adaptive control method with higher control accuracy based on the characteristics of the robotic arm;

[0030] According to the dynamic model of the robotic arm, the attitude controller is designed considering the influence of actuator saturation and external interference, and the attitude tracking error of the robotic arm is defined as:

[0031] e q =qq d (10)

[0032] where q d For the desired posture to be determined, the quadratic derivative of both sides of (10) yields

[0033]

[0034] set up and It's q and q d The second-order derivative of , △d is the external disturbance

[0035]

[0036] Considering the influence of the robot arm input saturation, Equation (12) can be transformed into the following Equation (13):

[0037]

[0038] in is the nonlinear saturation characteristic function The expression of g is as follows:

[0039]

[0040] Among them, τ mimax is the maximum torque value allowed by the actuator, and through formula (14) we can get: when the force generated by the controller |τ mi |greater than τ mimax hour, When the force generated by the controller |τ mi | less than τ mimax hour, Through the function Through the function The output control torque is limited to the range allowed by the actuator.

[0041] Through the above transformation, the control-oriented model of the robotic arm system is obtained:

[0042]

[0043] in

[0044]

[0045] ||ρ||≤D, D>0 and unknown.

[0046] Based on formula (15), the adaptive multivariable spiral algorithm is used to design the attitude controller as shown in formula (17):

[0047]

[0048] in k1>0 is the adaptive gain, which satisfies the adaptive rate shown in equation (18):

[0049]

[0050] α1, β1, ε1 are custom parameters, is the first-order derivative of k1, and the attitude controller (17) is substituted into (15), and the equation (19) is obtained.

[0051]

[0052]

[0053] Theorem 1: For the manipulator system (1) to (3), under the adaptive multivariable spiral sliding mode controller (17) and adaptive gain (18), by selecting appropriate parameters, the posture system can achieve finite time stability, and the joint posture error can converge to a neighborhood near the origin within a finite time.

[0054] Step 4: After designing the control method, use the Lyapunov function to prove the effectiveness of the method;

[0055] In order to prove that the system can achieve finite-time stability under the action of , the following Lyapunov candidate function is selected.

[0056]

[0057] in, choose is a positive definite symmetric matrix;

[0058] Referring to existing articles, we can get formula (21):

[0059]

[0060] Among them, λ max (P1), λ min (P1) is expressed as the maximum and minimum eigenvalues ​​of the matrix

[0061] From formula (21), we can get

[0062]

[0063] in, hour, but

[0064]

[0065] When the adaptive gain k1 increases according to the adaptive law in the formula to meet k1≥2D, e q It can converge to the limited interval ε1 within a limited time. At this time, it can be proved that the present invention can effectively make the robot arm under precise control and avoid the situation where the input torque is too large.

[0066] Compared with the prior art, the present invention has the following beneficial effects:

[0067] (1) The adaptive multivariable spiral algorithm designed in the present invention can achieve accurate and stable control of the system without predicting the upper bound of the disturbance to the system and its derivative.

[0068] (2) For the typical second-order attitude model, an adaptive multivariable spiral sliding mode attitude controller is designed, and the introduction of the adaptive law effectively suppresses the vibration.

[0069] (3) The adaptive multivariable spiral sliding mode controller designed in the present invention has the advantages of fast convergence speed and high convergence accuracy, and is suitable for the requirements of the convergence speed of the posture controller in the fast maneuverability of the robotic arm.

[0070] (4) Avoiding the problem of input saturation. The use of saturation control allows the robot arm input to be maintained within a safe range, providing greater stability. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 This is a comparison chart of the errors between the control method of the present invention and the existing method.

[0072] Figure 2This is a torque comparison diagram between the control method of the present invention and the existing method. DETAILED DESCRIPTION

[0073] The technical solution of the present invention is further described in detail below with reference to specific drawings and implementation steps.

[0074] Step 1: Build a proprietary mathematical model for the cable manipulator to facilitate the design of a high-precision control method

[0075] The dynamic model of an n-degree-of-freedom robotic arm is expressed as:

[0076]

[0077]

[0078]

[0079] Where J and d m are the motor inertia and damping matrices, q and θ are the joint and motor position vectors, Respectively represent the first-order derivative and second-order derivative of q and θ, M(q) is the inertia matrix, is the Coriolis / centrifugal matrix, g(q) is the gravitational force, is the friction vector, τ m and τ s are the control torque given by the motor and the joint flexibility torque, d s is the damping matrix, k s is the joint stiffness matrix. τ d represents the lumped unknown uncertainty.

[0080] Substituting (2) into (1) and applying a constant parameter yields:

[0081]

[0082] Among them, the expression of f is:

[0083]

[0084] It can be simplified to the following formula

[0085]

[0086]

[0087] Step 2: Use the time delay estimation method to estimate the value of f and find its estimated value

[0088]

[0089] Where △t is the delay time. Substituting the comprehensive system dynamics (4) into (6) we can obtain:

[0090]

[0091] From (6) and (7), it can be seen that the main purpose of the time delay estimation scheme is to estimate the lumped system dynamics using only the time delay values ​​of the control and acceleration signals, and then give a model-free scheme. In engineering applications, τ m (t-△t) can be expressed by τ m The direct time lag is obtained by numerical differentiation to obtain

[0092]

[0093] Where, T≥2△t; in the initial stage, when t≤2△t, q(t) has the actual measured value, and q(t-2△t) will be manually set to zero, which may cause strong fluctuations. Therefore, (8) is used to alleviate the possible strong fluctuations. By setting it in the form of (9), the delay can be estimated and the estimated value can be obtained.

[0094]

[0095] It is widely used in many robust control schemes based on TDE. The simulation part of this paper proves its effectiveness. If no measures are taken, numerical differentiation (9) will significantly amplify the noise effect, thereby degrading the control performance. However, it has been shown that this problem can be solved by reducing the gain M or using an additional low-pass filter.

[0096] As can be seen from (8) and (9), the current value of the lumped system dynamics is estimated by the TDE scheme using the time-delay system state. Therefore, when large disturbances occur, the estimation error of this method will become larger. However, the adaptive multivariable spiral algorithm method we proposed can reduce the estimation error. The simulation in this paper proves the effectiveness of this method.

[0097] Step 3: Based on the characteristics of the robotic arm, an adaptive multivariable spiral control method with higher control accuracy is designed;

[0098] According to the dynamic model of the robotic arm, the attitude controller is designed considering the influence of actuator saturation and external interference, and the attitude tracking error of the robotic arm is defined as:

[0099] e q =qq d (10)

[0100] Among them, q d For the desired posture to be determined, the quadratic derivative of both sides of (10) yields

[0101]

[0102] set up and It's q and q d The second-order derivative of , △d is the external disturbance, then

[0103]

[0104] Considering the influence of the robot input saturation, Equation (12) can be transformed into the following Equation (13):

[0105]

[0106] in is the nonlinear saturation characteristic function The expression of g is as follows:

[0107]

[0108] Among them, τ mimax is the maximum torque value allowed by the actuator, and through formula (14) we can get: when the force generated by the controller |τ mi |greater than τ mimax hour, When the force generated by the controller |τ mi | less than τ mimax hour, Through the function Through the function The output control torque is limited to the range allowed by the actuator.

[0109] Through the above transformation, we get the control-oriented model of the robotic arm system

[0110]

[0111]

[0112] in

[0113]

[0114]

[0115] ||ρ||≤D, D>0 and unknown.

[0116] Based on formula (15), the adaptive multivariable spiral algorithm is used to design the attitude controller as shown in formula (17):

[0117]

[0118] k1>0 is the adaptive gain, which satisfies the adaptive rate shown in equation (18):

[0119]

[0120] α1, β1, and ε1 are parameters greater than zero.

[0121] Substituting the attitude control law (17) into (15), we get (19)

[0122]

[0123]

[0124] Theorem 1: For the manipulator system (1) to (2), under the adaptive multivariable spiral sliding mode controller (17) and adaptive gain (18), by selecting appropriate parameters, the posture system can achieve finite time stability, and the joint posture error can converge to a neighborhood near the origin within a finite time.

[0125] Step 4: After designing the control method, use the Lyapunov function to prove the effectiveness of the method.

[0126] In order to prove that the robot arm can be stabilized in a finite time under the action of the control system, the following Lyapunov candidate function is selected.

[0127]

[0128] in, choose is a positive definite symmetric matrix. Referring to existing articles, we can get formula (21)

[0129]

[0130] Among them, λ max (P1), λ min (P1) is expressed as the maximum and minimum eigenvalues ​​of the matrix

[0131] From formula (21), we can get

[0132]

[0133] in, hour, but

[0134]

[0135] When the adaptive gain k1 increases according to the adaptive law in formula (13) to meet k1 ≥ 2D, eq It can converge to the limited interval ε1 within a limited time. At this point, it can be proved that the present invention can effectively make the robot arm under precise control and avoid the situation where the input torque is too large.

[0136] Step 5, use Matlab programming to test the effect of the control rate design;

[0137] The parameters are set as α1=12, β1=2, ε1=0.002.

[0138] in Figure 1 The black solid line in the middle is the adaptive multivariable control method, and the gray solid line is the sliding mode control method. Sliding mode control is a classic method for manipulator control. We compare them from the perspective of error convergence speed and error resistance to external interference. Figure 1 It can be seen that when the initial position of the manipulator under sliding mode control is far from the desired position, its error converges within 4 seconds, while under adaptive multivariable spiral control, the error converges within 1 second, indicating that adaptive multivariable spiral control converges faster. When the control is asymptotically stable, the manipulator under adaptive multivariable spiral control has a smaller error under external disturbances, indicating that the adaptive multivariable spiral control method is more resistant to external disturbances and has higher control accuracy. Figure 2 The figure shows the torque fluctuations of the manipulator under saturation control, as it is subjected to external disturbances. Furthermore, the torque under saturation control always remains within the safe range (the black dashed line represents the safety range we set). This means that saturation control can prevent input overload conditions. In such cases, we can ensure that the control input remains within the range that the manipulator can maintain.

Claims

1. A time-delayed multivariable spiral control method for a manipulator considering input saturation, characterized in that: The steps include: Step 1: Establish a dynamic model to describe the state of the cable manipulator; In step 1, the dynamic model of the n-DOF manipulator can be expressed as; Where J and d m are the motor inertia and damping matrices, q and θ are the joint and motor position vectors, Respectively represent the first-order derivative and second-order derivative of q and θ, M(q) is the inertia matrix, is the Coriolis / centrifugal matrix, g(q) is the gravitational force, is the friction vector, τ m and τ s are the control torque given by the motor and the joint flexibility torque, d s is the damping matrix, k s is the joint stiffness matrix, τ d represents the lumped unknown uncertainty; In order to facilitate the use of the TDE scheme, which is based on the time delay estimation method, (2) is substituted into (1) and the constant parameter is applied. get: Among them, the expression of f is: Step 2: Use the time delay estimation (TDE) scheme to estimate the unknown parameters in the kinetic model; In step 2, the time delay estimation method is used to estimate the value of the unknown parameter f and find its estimated value Where Δt is the delay time. Substituting the kinetic model (4) into (6) yields: From (6) and (7), it can be seen that the purpose of the time delay estimation scheme is to estimate the lumped system dynamics using only the time delay values ​​of the control and acceleration signals, and then give a model-free scheme. In engineering applications, τ m (t-Δt) can be expressed by τ m The direct time lag is obtained by numerical differentiation to obtain Among them, the total time T≥2Δt, in the initial stage, at time t≤2Δt, q(t) has the actual measured value, and q(t-2Δt) will be manually set to zero, which may cause strong fluctuations, so (8) is used to alleviate the possible strong fluctuations; Step 3: Design an adaptive multivariable spiral control method with higher control accuracy based on the characteristics of the robotic arm; According to the dynamic model of the robotic arm, the attitude controller is designed considering the influence of actuator saturation and external interference, and the attitude tracking error of the robotic arm is defined as: e q =q-q d (10) where q d For the desired posture to be determined, the quadratic derivative of both sides of (10) yields set up and It's q and q d The second-order derivative of , Δd is the external disturbance, then Considering the influence of the robot arm input saturation, Equation (12) can be transformed into the following Equation (13): in is the nonlinear saturation characteristic function The expression of g is as follows: Among them, τ mimax is the maximum torque value allowed by the actuator, and through formula (14) we can get: when the force generated by the controller |τ mi |greater than τ mimax hour, When the force generated by the controller |τ mi | less than τ mimax hour, Through the function Through the function The output control torque is limited to the range allowed by the actuator; Through the above transformation, the control-oriented model of the robotic arm system is obtained: in ||ρ||≤D, D>0 and unknown; Based on formula (15), the adaptive multivariable spiral algorithm is used to design the attitude controller as shown in formula (17): in k1>0 is the adaptive gain, which satisfies the adaptive rate shown in equation (18): α1, β1, ε1 are custom parameters, is the first-order derivative of k1, and the attitude controller (17) is substituted into (15), and the equation (19) is obtained.

2. The method for multivariable spiral control of a manipulator based on time delay and considering input saturation according to claim 1, characterized in that: The method further includes step 4, which uses the Lyapunov function to prove the effectiveness of the control method in step 3, and the specific implementation method is as follows; Select the following Lyapunov candidate function: in, choose is a positive definite symmetric matrix; Referring to the existing theory, we can get formula (21): Among them, λ max (P1), λ min (P1) is expressed as the maximum and minimum eigenvalues ​​of the matrix From formula (21), we can get in, hour, but When the adaptive gain k1 increases according to the adaptive law in the formula to meet k1≥2D, e q It can converge to the limited interval ε1 within a finite time. At this time, it can be proved that the control method in step 3 can effectively make the robotic arm under precise control and avoid the situation where the input torque is too large.

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