Dual-loop position control method for universal pneumatic flexible robotic arm based on digital twin

Abstract

A dual-loop position control method for a universal pneumatic flexible robotic arm based on digital twins belongs to the field of industrial intelligent manufacturing technology and robotic arm servo control, comprising: first establishing a virtual body structure of the universal pneumatic flexible robotic arm in digital simulation software according to the structural characteristics of the universal pneumatic flexible robotic arm entity, and only involving kinematic control in the virtual body; when the entity is affected by the disturbances of gravity bias, Coriolis force and coupling, a finite-time expanded state observer is designed to estimate the disturbance, and an integral error controller is used to compensate for the disturbance; a fast integral sliding mode controller is designed to achieve fast and accurate tracking of the desired position; the present invention solves the problem of the universal pneumatic flexible robotic arm being affected by disturbances during movement, and the position and posture of the end point obtained by the digital twin virtual body is not affected by the inaccuracy of D-H modeling and changes in environmental conditions, ultimately ensuring that the universal pneumatic flexible robotic arm based on digital twins can track the spatial position trajectory quickly, accurately and stably.

Description

Technical field:

[0001] The present invention belongs to the field of industrial intelligent manufacturing technology and robotic arm servo control, and specifically relates to a dual-loop position control method for a universal pneumatic flexible robotic arm based on digital twins. Background technology:

[0002] Robotic arms in industrial intelligent manufacturing, as a crucial component of industrial automation, have been widely adopted across multiple industries, including automotive manufacturing, electronics assembly, metal processing, and food and beverage production. The introduction and development of robotic arms have significantly improved production efficiency, product quality, and factory manufacturing capabilities. To meet the demand for compliance and flexibility in intelligent manufacturing, a universal pneumatic flexible robotic arm driven by pneumatic artificial muscles has been designed, capable of grasping and handling. To achieve high-precision grasping and handling, fast and high-precision position control of this universal pneumatic flexible robotic arm is a significant challenge. Currently, in real-world robotic arm position control applications, encoders are typically installed at each joint to measure joint rotation angles to determine the joint position. The end-of-arm pose is then calculated using the DH method. Alternatively, monocular or binocular cameras are used to capture images of the robotic arm, and computer vision algorithms are used to calculate the pose. Alternatively, point cloud data from LiDAR is used to construct an environmental model and the robotic arm pose. With the above methods, the accuracy of the data obtained by the camera and lidar will decrease when environmental conditions (such as occlusion, lighting, and the surface material of the object being measured) change. The accuracy of the manipulator's position will also decrease when the DH model is inaccurate. Therefore, when environmental conditions change or the DH model is inaccurate, fast and high-precision position control of the universal pneumatic flexible manipulator is essential.

[0003] As we all know, when a universal pneumatic flexible robotic arm moves, it will be affected by disturbances such as gravity bias, Coriolis force, and coupling. The above disturbances will have an adverse effect on the servo control of the universal pneumatic flexible robotic arm, making it impossible to achieve high-precision control. At the same time, the universal pneumatic flexible robotic arm is driven by pneumatic artificial muscles, and its slow movement limits its application in certain high-speed and high-precision tasks.

[0004] Digital twins enable interaction and mapping between the physical and digital worlds through digital replicas of physical objects or systems. The core concept of digital twins is to collect real-time data from physical entities through sensors and other devices, transmit this data to digital models, and enable the digital models to dynamically reflect the state and behavior of the physical entities. This allows users to monitor, analyze, predict, and optimize physical entities in a virtual environment, thereby improving efficiency and reducing costs and risks.

[0005] The present invention proposes a dual-loop position control method for a universal pneumatic flexible robotic arm based on digital twins. According to the structural characteristics of the universal pneumatic flexible robotic arm entity, a virtual body structure of the universal pneumatic flexible robotic arm is first established in structural simulation software and digital simulation software, and only kinematic control is involved in the virtual body; when the entity is affected by the disturbances of gravity bias, Coriolis force and coupling, a finite-time expanded state observer is designed to estimate the disturbance, and an integral error controller is used to compensate for the disturbance; a fast integral sliding mode controller is designed to achieve fast and accurate tracking of the desired position; the present invention solves the problem of the universal pneumatic flexible robotic arm being affected by disturbances during movement, and the position and posture of the end point obtained by the digital twin virtual body is not affected by the inaccuracy of DH modeling and changes in environmental conditions, ultimately ensuring that the universal pneumatic flexible robotic arm based on digital twins can track the spatial position trajectory quickly, accurately and stably, effectively solving the above problems. Summary of the invention:

[0006] To achieve the above objectives, the technical solutions adopted by the present invention are as follows:

[0007] The dual-loop position control method for a universal pneumatic flexible robotic arm based on digital twins includes the following steps:

[0008] Step 1: Based on the structural characteristics of the universal pneumatic flexible robotic arm entity, a virtual body structure of the universal pneumatic flexible robotic arm is established in the structural simulation software. The virtual body structure is then imported into the digital simulation software. The angle sensor data of the entity is transmitted to the virtual body through the serial port. The position and posture information of the end points of the virtual body can be obtained in real time. Finally, the position and posture information of the end points of the virtual body are transmitted to the entity through the serial port.

[0009] Step 2: Consider the gravity bias, Coriolis force, and coupling introduced during the movement of the universal pneumatic flexible manipulator as disturbances, and establish a second-order mathematical model of the universal pneumatic flexible manipulator;

[0010] Step 3: Establish a mathematical model of the outer ring of the universal pneumatic flexible manipulator and a mathematical model of the inner ring of the universal pneumatic flexible manipulator;

[0011] Step 4: The difference between the pose information of the virtual end point and the expected pose information of the end point is used as the outer loop pose error. The outer loop pose error is inversely solved into an angle error using the Levenberg-Marquardt method and input into the outer loop. Based on the mathematical model of the outer loop of the universal pneumatic flexible manipulator, an outer loop fast integral sliding mode controller is designed. The output of the controller is used as the expected value of the mathematical model of the inner loop of the universal pneumatic flexible manipulator.

[0012] Step 5: Design an inner loop finite-time extended state observer based on the inner loop mathematical model of the universal pneumatic flexible manipulator to estimate the disturbance of the universal pneumatic flexible manipulator;

[0013] Step 6: Design an inner loop integral error controller based on the mathematical model of the inner loop of the universal pneumatic flexible manipulator and the estimated value of the inner loop finite-time extended state observer for the disturbance;

[0014] Step 7: Use the Lyapunov method to test the convergence of the inner loop finite time extended state observer, the inner loop integral error controller and the outer loop fast integral sliding mode controller.

[0015] Furthermore, the step 1 specifically includes: using the angle sensor data θ of each joint of the entity i (t), where i = 1, 2, ..., 6, thereby performing kinematic control in the virtual body and obtaining the position information and posture information of the end point from the virtual body, 0 T6(t) is the pose information matrix, which is expressed as:

[0016]

[0017] Where: P x (t), P y (t), P z (t) represents the coordinates of the end point of the universal pneumatic flexible manipulator in the global coordinate system X, Y, and Z axes, respectively, and n x (t), n y (t), n z (t) represents the normal axis coordinate of the end point in the global coordinate system X, Y, and Z axes, respectively. x (t), o y (t), o z (t) represents the coordinates of the orientation axis of the end point in the global coordinate system X, Y, and Z axes, respectively. x (t), a y (t), a z (t) represents the coordinates of the approach axis of the end point in the global coordinate system X, Y, and Z axes respectively.

[0018] Furthermore, the step 2 specifically includes:

[0019] The second-order mathematical model of the universal pneumatic flexible robotic arm is:

[0020]

[0021] Where:

[0022] ψ2(u(t),t)=B w (t)20 / πarctan(u(t)),Ω 1,1 =Ω 2,3 =Ω3,3 =diag{b c ,b c}

[0023]

[0024] Where: x i1 (t) = θ i (t) is the angle of each joint of the universal pneumatic flexible robotic arm, is the angular velocity of each joint of the universal pneumatic flexible manipulator, ψ1(X1(t), t) is the state coupling matrix, ψ2(u(t), t) is the control coupling matrix, B w (t) is the coupling matrix, u i (t) is the output of the inner loop integral error controller, f i (t) is the gravitational bias and Coriolis force, ψ1(X1(t),t) and ψ2(u(t),t) are considered as the perturbations W d (t), is the decoupling constant, r is the radius of the joint disk, L0 is the length of the connecting rod, J e represents the moment of inertia of each joint, P0 is the pre-inflation pressure, b is the length of the braided mesh fiber, n is the number of turns of the braided mesh fiber, ζ is the structural compensation parameter, b i1 =(3P0ζr 2 L0) / (πn 2 J e ), X1(t)=[x 11 (t),x 21 (t),…,x 61 (t)] T is the angle matrix, X2(t)=[x 12 (t),x 22 (t),…,x 62 (t)] T is the angular velocity matrix.

[0025] Furthermore, the step three specifically includes:

[0026] The mathematical model of the outer ring of the universal pneumatic flexible robotic arm is:

[0027]

[0028] The mathematical model of the inner ring of the universal pneumatic flexible robotic arm is:

[0029]

[0030] Where: V(t)=[v1(t),v2(t),…,v6(t)] Tis the output of the outer loop fast integral sliding mode controller, G(t)=[g 12 (t),g 22 (t),...,g 62 (t)] T is the inner loop tracking error matrix, X3(t)=W di (t) is the expansion state of the disturbance, W d (t)=[w d1 (t),w d2 (t),…,w d6 (t)] T And W d (t) is continuous and bounded, X3(t)=[x 13 (t),x 23 (t),…,x 63 (t)] T .

[0031] Furthermore, in step 4, the difference between the pose information of the virtual body end point and the pose information of the expected end point is used as the outer loop pose error, and the outer loop pose error is inversely solved into an angle error by the Levenberg-Marquardt method and input into the outer loop, specifically including:

[0032] Pose information of the virtual body end point 0 T6(t) and the pose information of the expected end point 0 T 6des The difference between (t) is taken as the outer ring pose error, and the outer ring pose error is inversely solved into the outer ring angle error dθ by the Levenberg-Marquardt method. i (t) is input to the outer loop fast integral sliding mode controller, and the inverse solution process is as follows using the Levenberg-Marquardt method:

[0033] dθ(k)=Ψ(J T (θ(k))J(θ(k))+χ 2 (k)I6) -1 J T (θ(k))dD(k)

[0034] =Ω(k)

[0035] Where: k is the number of sampling points, dθ(k)=[dθ1(k),dθ2(k),...,dθ6(k)] T is the differential vector of the end point pose, Ω(k) is a nonlinear function used to save dθ(k) at the k sampling point, I6∈R 6×6is the identity matrix, Ψ = diag{ψ1,ψ2,...,ψ6} is the convergence rate parameter matrix, J(θ(k)) is the Jacobian matrix, and χ(k) is a time-varying parameter;

[0036] The update rule of χ(k) is:

[0037]

[0038] According to the outer ring mathematical model of the universal pneumatic flexible manipulator, an outer ring fast integral sliding mode controller is designed, and its output is used as the expected value of the inner ring mathematical model of the universal pneumatic flexible manipulator, which specifically includes:

[0039] The outer loop fast integral sliding mode controller is:

[0040]

[0041]

[0042] The sliding surface is s i (t) is:

[0043]

[0044] Where: v i (t) is the outer loop control law, e i1 (t) is the observation error of the finite-time extended state observer, α and β are two constant parameters greater than zero, m1 and m2 are two constant parameters greater than zero, q1 and q2 are two odd numbers greater than zero and satisfy 0<q1 / q2<1, a∈(0,1) and b∈(0,1) are two exponential terms, λ c is a constant parameter greater than zero.

[0045] Furthermore, the step five specifically includes:

[0046] The inner loop finite-time extended state observer is:

[0047]

[0048] Where: e i1 (t) = z i1 (t)-x i2 (t), e i2 (t) = z i2 (t)-x i3 (t)

[0049]

[0050] Where: sig α (x)=|x| αsgn(x) is the energy function, f1(e i1 (t)) and f2(e i1 (t)) is the designed nonlinear function, l1 and l2 are observer parameters, K1 is an adjustable parameter, z i1 (t) and z i2 (t) are x i2 (t) and x i3 The estimated value of (t), e i1 (t) and e i2 (t) are z i1 (t) and z i2 (t) observation error, a decoupling matrix is ​​given by U d (t)=B0u(t)=[U d1 (t),U d2 (t),…,U d6 (t)] T ;

[0051] Furthermore, the step six specifically includes:

[0052] The inner loop integral error controller is:

[0053]

[0054] Where: Z(t)=[z 12 (t),z 22 (t),...,z 62 (t)] is the perturbation estimation matrix, a1 = 0.5, δ = 0.009, c i and d i are two positive parameters greater than zero, ε i (t) = v i (t)-z i1 (t) is the inner loop angular velocity error, U0(t) is the nonlinear error, and Z(t) is W d (t) is an estimated value.

[0055] Furthermore, the step seven specifically includes:

[0056] The convergence test of the outer loop fast integral sliding mode controller is carried out. The first step is to prove that the outer loop angle error dθ i (t) can converge to the sliding surface s i (t), the Lyapunov function equation is designed as follows:

[0057] For the Lyapunov function equation V reach (t) Derivative:

[0058]

[0059] If there exists a positive number m2 such that m2>max|ε i (t)|, then Established, that is, the outer ring angle error dθ i (t) can converge to the sliding surface s i (t) up;

[0060] The second step is to prove the outer ring angle error dθ i (t) can be along the sliding surface s i (t) converges to zero, and the Lyapunov function equation is designed as follows:

[0061] For the Lyapunov function equation V sm (t) Derivative:

[0062]

[0063] Since α>0, β>0 and dθ i (t)tanh(λ c dθ i (t))>0, so Established, from this we can get the outer ring angle error dθ i (t) can be along the sliding surface s i (t) converges to zero, and the convergence test of the outer loop fast integral sliding mode controller is completed;

[0064] The convergence test of the inner loop finite time extended state observer is carried out to prove that the observation error e i1 (t) and e i2 (t) can converge to zero in a finite time. First, the observation error system of the inner loop finite time extended state observer is given as follows:

[0065]

[0066] The Lyapunov function equation is designed as follows: V ESO (t)=∈ T (t)P∈(t)

[0067] Where: P is a positive definite symmetric matrix, the constructed error ∈ T (t) = [f1(e i1 (t)),e i2 (t)];

[0068] For the Lyapunov function equation V ESO (t) Derivative:

[0069]

[0070] make ξ is a sufficiently small positive constant, the Lyapunov function equation V ESO The derivative of (t) can be written as:

[0071]

[0072] The finite time convergence time is:

[0073] Where t0 is the initial time;

[0074] because And it satisfies the finite time convergence rule, so the observation error e i1 (t) and e i2 (t) can converge to zero in finite time, and the finite time convergence test of the inner loop finite time extended state observer is completed;

[0075] The convergence test of the inner loop integral error controller is carried out to prove that the inner loop angular velocity error ε i (t) can converge to zero. First, the error system of the inner loop integral error controller is given as follows:

[0076]

[0077] Where: g i2 (t) = v i (t)-x i2 (t) is the construction tracking error of the inner loop system;

[0078] In the following proof, fal(ε i (t),a1,δ) is expressed as fal(ε i (t)), the Lyapunov function equation is designed as follows:

[0079] For the Lyapunov function equation V IEC (t) Derivative:

[0080]

[0081] make: And H(t) is bounded, is a positive parameter; it can be concluded that So if there is a small positive number and a large positive number ε0 satisfies: Can make Established; that is, when the parameter c i When it is big enough, Established, that is, the inner ring angular velocity error ε i (t) can converge to zero, and the convergence test of the inner loop integral error controller is completed.

[0082] The beneficial effects of the present invention are:

[0083] 1. The present invention establishes a virtual body through structural simulation software and digital simulation software. Providing entity angle information to the virtual body allows real-time position and posture information of the virtual body's endpoints to be obtained. Because the virtual body and the entity are structured in a mapping relationship, the position and posture information of the virtual body's endpoints can be considered the position and posture information of the entity's endpoints. This eliminates the problem of low accuracy of the obtained position and posture information when environmental conditions change or when DH modeling is inaccurate. In other words, the position and posture information of the virtual body's endpoints is not affected by DH modeling inaccuracies or changing environmental conditions, thereby improving robustness and position and posture acquisition accuracy.

[0084] 2. This invention treats the gravity bias, Coriolis force, and coupling during the motion of a universal pneumatic flexible manipulator as disturbances. It estimates these disturbances using an inner-loop finite-time extended state observer and compensates for them within the inner-loop integral error controller, thereby reducing their impact on the universal pneumatic flexible manipulator and improving control accuracy. Furthermore, the rapid response of the inner-loop integral error controller and the more detailed corrections made by the outer-loop fast integral sliding mode controller based on the inner-loop control improve the overall system response speed.

[0085] 3. The dual closed-loop position control method of this invention is easy to implement in engineering applications. The effectiveness of the designed inner-loop finite-time extended state observer, inner-loop integral error controller, and outer-loop fast integral sliding mode controller was verified using multiple Lyapunov functions. This ensures stable, accurate, and fast control performance. Description of the drawings:

[0086] Figure 1 This is a schematic diagram of the dual-loop position control method for a universal pneumatic flexible robotic arm based on digital twins of the present invention;

[0087] Figure 2 It is a design flow chart of the present invention;

[0088] Figure 3 This is a virtual body diagram of a universal pneumatic flexible robotic arm;

[0089] Figure 4 This is a fixed-point position tracking diagram of the universal pneumatic flexible robotic arm based on digital twins of the present invention;

[0090] Figure 5 is the finite-time extended state observer z of the present invention 21 (t)Signal curve graph. Specific implementation method:

[0091] In order to make the purpose of the present invention more specific and the technical solution more clear, the present invention is described in detail with reference to the following drawings and specific embodiments.

[0092] Figure 1 Shown is a schematic diagram of the control method of the present invention, illustrating the dual-loop position control method of the universal pneumatic flexible robotic arm based on digital twins described in the present invention.

[0093] The following combination Figure 1-5 The control method of the present invention is described in detail, but it is not intended to limit the present invention.

[0094] A dual-loop position control method for a universal pneumatic flexible robotic arm based on digital twins includes the following steps:

[0095] S100. Based on the structural characteristics of the universal pneumatic flexible robotic arm entity, a virtual body structure of the universal pneumatic flexible robotic arm is established in the structural simulation software, and the virtual body structure is imported into the digital simulation software. The angle sensor data of the entity is transmitted to the virtual body via the serial port, so that the position information and posture information of the end point of the virtual body can be obtained in real time. Finally, the position and posture information of the end point of the virtual body is transmitted to the entity via the serial port.

[0096] S200, considering the gravity bias, Coriolis force and coupling introduced during the movement of the universal pneumatic flexible manipulator as disturbances, and establishing a second-order mathematical model of the universal pneumatic flexible manipulator;

[0097] S300, establishing a mathematical model of an outer ring of a universal pneumatic flexible robotic arm and a mathematical model of an inner ring of the universal pneumatic flexible robotic arm;

[0098] S400, the difference between the pose information of the virtual end point and the pose information of the expected end point is used as the outer loop pose error, the outer loop pose error is inversely solved into an angle error using the Levenberg-Marquardt method and input into the outer loop, and an outer loop fast integral sliding mode controller is designed based on the mathematical model of the outer loop of the universal pneumatic flexible manipulator, the output of which is used as the expected value of the mathematical model of the inner loop of the universal pneumatic flexible manipulator;

[0099] S500, designing an inner loop finite-time expansion state observer based on an inner loop mathematical model of the universal pneumatic flexible manipulator to estimate a disturbance of the universal pneumatic flexible manipulator;

[0100] S600. Design an inner loop integral error controller based on the mathematical model of the inner loop of the universal pneumatic flexible manipulator and the estimated value of the inner loop finite time expanded state observer for the disturbance;

[0101] S700, Lyapunov method is used to test the convergence of the inner loop finite time extended state observer, the inner loop integral error controller and the outer loop fast integral sliding mode controller.

[0102] In this embodiment, S100, based on the structural characteristics of the universal pneumatic flexible robotic arm entity, a universal pneumatic flexible robotic arm virtual body structure is established in the structural simulation software, and the virtual body structure is imported into the digital simulation software. The angle sensor data of the entity is transmitted to the virtual body through the serial port, and the position information and posture information of the virtual body end point can be obtained in real time. Finally, the position and posture information of the virtual body end point is transmitted to the entity through the serial port. Specifically, the method includes:

[0103] Using the angle sensor data θ of each joint of the entity i (t), where i = 1, 2, ..., 6, thereby performing kinematic control in the virtual body and obtaining the position information and posture information of the end point from the virtual body, 0 T6(t) is the pose information matrix, which is expressed as:

[0104]

[0105] Where: P x (t), P y (t), P z (t) represents the coordinates of the end point of the universal pneumatic flexible manipulator in the global coordinate system X, Y, and Z axes, respectively, and n x (t), n y (t), n z (t) represents the normal axis coordinate of the end point in the global coordinate system X, Y, and Z axes, respectively. x (t), o y (t), o z (t) represents the coordinates of the orientation axis of the end point in the global coordinate system X, Y, and Z axes, respectively. x (t), a y (t), a z (t) represents the coordinates of the approach axis of the end point in the global coordinate system X, Y, and Z axes, respectively. In this embodiment, S200 considers the gravity bias, Coriolis force, and coupling introduced during the movement of the universal pneumatic flexible manipulator as disturbances, and establishes a second-order mathematical model of the universal pneumatic flexible manipulator, specifically including:

[0106] The second-order mathematical model of the universal pneumatic flexible robotic arm is:

[0107]

[0108] Where:

[0109] ψ2(u(t),t)=B w (t)20 / πarctan(u(t)),Ω 1,1 =Ω2,3 =Ω 3,3 =diag{b c ,b c}

[0110]

[0111]

[0112] Where: x i1 (t) = θ i (t) is the angle of each joint of the universal pneumatic flexible robotic arm, is the angular velocity of each joint of the universal pneumatic flexible manipulator, ψ1(X1(t), t) is the state coupling matrix, ψ2(u(t), t) is the control coupling matrix, B w (t) is the coupling matrix, u i (t) is the output of the inner loop integral error controller, f i (t) is the gravitational bias and Coriolis force, ψ1(X1(t),t) and ψ2(u(t),t) are considered as the perturbations W d (t), is the decoupling constant, r is the radius of the joint disk, L0 is the length of the connecting rod, J e represents the moment of inertia of each joint, P0 is the pre-inflation pressure, b is the length of the braided mesh fiber, n is the number of turns of the braided mesh fiber, ζ is the structural compensation parameter, b i1 =(3P0ζr 2 L0) / (πn 2 J e ), X1(t)=[x 11 (t),x 21 (t),…,x 61 (t)] T is the angle matrix,

[0113] X2(t)=[x 12 (t),x 22 (t),…,x 62 (t)] T is the angular velocity matrix.

[0114] In this embodiment, S300, establishing a mathematical model of an outer ring of a universal pneumatic flexible robotic arm and a mathematical model of an inner ring of the universal pneumatic flexible robotic arm, specifically includes:

[0115] The mathematical model of the outer ring of the universal pneumatic flexible robotic arm is:

[0116]

[0117] The mathematical model of the inner ring of the universal pneumatic flexible robotic arm is:

[0118]

[0119] Where: V(t)=[v1(t),v2(t),…,v6(t)] T is the output of the outer loop fast integral sliding mode controller, G(t)=[g 12 (t),g 22 (t),…,g 62 (t)] T is the inner loop tracking error matrix, X3(t)=W di (t) is the expansion state of the disturbance, W d (t)=[w d1 (t),w d2 (t),…,w d6 (t)] T And W d (t) is continuous and bounded,

[0120] X3(t)=[x 13 (t),x 23 (t),…,x 63 (t)] T .

[0121] In this embodiment, the difference between the pose information of the virtual end point in S400 and the pose information of the expected end point is used as the outer loop pose error. The outer loop pose error is inversely solved into an angle error through the Levenberg-Marquardt method and input into the outer loop. Based on the outer loop mathematical model of the universal pneumatic flexible manipulator, an outer loop fast integral sliding mode controller is designed. Its output is used as the expected value of the inner loop mathematical model of the universal pneumatic flexible manipulator, specifically including:

[0122] Pose information of the virtual body end point 0 T6(t) and the pose information of the expected end point 0 T 6des The difference between (t) is taken as the outer ring pose error, and the outer ring pose error is inversely solved into the outer ring angle error dθ by the Levenberg-Marquardt method. i (t) is input to the outer loop fast integral sliding mode controller, and the inverse solution process is as follows using the Levenberg-Marquardt method:

[0123] dθ(k)=Ψ(J T (θ(k))J(θ(k))+χ 2 (k)I6) -1 J T (θ(k))dD(k)

[0124] =Ω(k)

[0125] Where: k is the number of sampling points, dθ(k)=[dθ1(k),dθ2(k),…,dθ6(k)] T is the differential vector of the end point pose, Ω(k) is a nonlinear function used to save dθ(k) at the k sampling point, I6∈R 6×6 is the identity matrix, Ψ = diag{ψ1, ψ2, …, ψ6} is the convergence rate parameter matrix, J(θ(k)) is the Jacobian matrix, and χ(k) is a time-varying parameter;

[0126] The update rule of χ(k) is:

[0127]

[0128] The outer loop fast integral sliding mode controller is:

[0129]

[0130] The sliding surface is s i (t) is:

[0131]

[0132] Where: v i (t) is the outer loop control law, e i1 (t) is the observation error of the finite-time extended state observer, α and β are two constant parameters greater than zero, m1 and m2 are two constant parameters greater than zero, q1 and q2 are two odd numbers greater than zero and satisfy 0<q1 / q2<1, a∈(0,1) and b∈(0,1) are two exponential terms, λ c is a constant parameter greater than zero.

[0133] In this embodiment, S500, designing an inner loop finite-time extended state observer based on the inner loop mathematical model of the universal pneumatic flexible manipulator to estimate the disturbance of the universal pneumatic flexible manipulator, specifically includes:

[0134] The inner loop finite-time extended state observer is:

[0135]

[0136] Where: e i1 (t) = z i1 (t)-x i2 (t), e i2 (t) = z i2 (t)-x i3 (t)

[0137]

[0138] Where: sig α (x)=|x| α sgn(x) is the energy function, f1(e i1 (t)) and f2(e i1 (t)) is the designed nonlinear function, l1 and l2 are observer parameters, K1 is an adjustable parameter, z i1 (t) and z i2 (t) are x i2 (t) and x i3 The estimated value of (t), e i1 (t) and e i2 (t) are z i1 (t) and z i2 (t) observation error, a decoupling matrix is ​​given by U d (t)=B0u(t)=[U d1 (t),U d2 (t),…,U d6 (t)] T .

[0139] In this embodiment, S600, based on the inner loop mathematical model of the universal pneumatic flexible manipulator and the estimated value of the inner loop finite time extended state observer for the disturbance, designs an inner loop integral error controller, specifically including:

[0140] The inner loop integral error controller is:

[0141]

[0142] Where:

[0143] Where: Z(t)=[z 12 (t),z 22 (t),…,z 62 (t)] is the perturbation estimation matrix, a1 = 0.5, δ = 0.009, c i and d i are two positive parameters greater than zero, ε i (t) = v i (t)-z i1 (t) is the inner loop angular velocity error, U0(t) is the nonlinear error, and Z(t) is W d (t) is an estimated value.

[0144] In this embodiment, S700, using the Lyapunov method to perform convergence testing on the inner-loop finite-time extended state observer, the inner-loop integral error controller, and the outer-loop fast integral sliding mode controller, specifically includes:

[0145] The outer loop fast integral sliding mode controller is tested for convergence. To prove the outer loop angle error dθ i (t) can converge to the sliding surface s i (t), the Lyapunov function equation is designed as follows:

[0146]

[0147] For the Lyapunov function equation V reach (t) Derivative:

[0148]

[0149] Where: Due to s i (t)tanh(λ c s i (t))>0, Established; Order:

[0150] -M=-s i (t)m1|dθ i (t)| a tanh(λ c dθ i (t))

[0151] =Γ(t)-m1|dθ i (t)| a dθ i (t)tanh(λ c dθ i (t))<0

[0152] Where:

[0153] According to the above analysis, we can get:

[0154]

[0155] If there exists a positive number m2 such that m2>max|ε i (t)|, then Established, that is, the outer ring angle error dθ i (t) can converge to the sliding surface s i (t) up;

[0156] Prove that the outer ring angle error dθ i (t) can be along the sliding surface s i (t) converges to zero, and the Lyapunov function equation is designed as follows:

[0157]

[0158] For the Lyapunov function equation Vsm (t) Derivative:

[0159]

[0160] based on And s i (t) = 0, we can get:

[0161]

[0162] Since α>0, β>0 and dθ i (t)tanh(λ c dθ i (t))>0, so Established, from this we can get the outer ring angle error dθ i (t) can be along the sliding surface s i (t) converges to zero, and the convergence test of the outer loop fast integral sliding mode controller is completed;

[0163] The convergence test of the inner loop finite time extended state observer is carried out to prove that the observation error e i1 (t) and e i2 (t) can converge to zero in a finite time. First, the observation error system of the inner loop finite time extended state observer is given as follows:

[0164]

[0165] The Lyapunov function equation is designed as follows:

[0166] V ESO (t)=∈ T (t)P∈(t)

[0167] Where: P is a positive definite symmetric matrix, the constructed error ∈ T (t) = [f1(e i1 (t)),e i2 (t)]; Derivative of the constructed error ∈(t):

[0168]

[0169] Where:

[0170] To complete the following proof, let:

[0171]

[0172] Where: M≥0, and χ are two parameters;

[0173] For the Lyapunov function equation V ESO (t) Derivative:

[0174]

[0175] For the Lyapunov function equation V ESO (t)The following inequality holds:

[0176] Where: ||∈(t)||2 is the 2-norm of ∈(t), λ min {P} is the minimum value of the P eigenvalue;

[0177] make ξ is a sufficiently small positive constant, the Lyapunov function equation V ESO The derivative of (t) can be written as:

[0178]

[0179] The finite time convergence time is:

[0180]

[0181] Where t0 is the initial time;

[0182] because Then the observation error e i1 (t) and e i2 (t) can converge to zero, since the finite convergence time is t ESO , observation error e i1 (t) and e i2 (t) can converge to zero in finite time, and the finite time convergence test of the inner loop finite time extended state observer is completed;

[0183] The convergence test of the inner loop integral error controller is carried out to prove that the inner loop angular velocity error ε i (t) can converge to zero. First, the error system of the inner loop integral error controller is given as follows:

[0184]

[0185] Where: g i2 (t) = v i (t)-x i2 (t) is the construction tracking error of the inner loop system;

[0186] In the following proof, fal(ε i (t),a1,δ) is expressed as fal(ε i (t)), the Lyapunov function equation is designed as follows:

[0187]

[0188] For the Lyapunov function equation V IEC (t) Derivative:

[0189]

[0190]

[0191] make: And H(t) is bounded, is a positive parameter;

[0192] Lyapunov function equation V IEC The derivative of (t) can be written as:

[0193]

[0194] can be expanded to:

[0195]

[0196] From the above formula, we can get So if there is a small positive number and a large positive number ε0 satisfies:

[0197]

[0198] Can make Established;

[0199] Where: ε i (t)∈(-ε0,ε0);

[0200] That is, when the parameter c i When it is big enough, Established, that is, the inner ring angular velocity error ε i (t) can converge to zero, and the inner loop integral error controller completes the convergence test. Therefore, the dual-loop position control method of the universal pneumatic flexible manipulator based on digital twins of the present invention is convergent and effective.

[0201] Example:

[0202] In order to verify the effectiveness of the dual-loop position control method of the universal pneumatic flexible manipulator based on digital twin proposed in this paper, an experimental verification is given to illustrate that the dual closed-loop position control method of the universal pneumatic flexible manipulator based on digital twin is effective, as follows:

[0203] The initial deflection angles of all joints in the physical and virtual universal pneumatic flexible robotic arm are 0°, meaning the physical and virtual bodies are located at the initial coordinates (0m, 0m, 0.865m). Bidirectional communication between the physical and virtual bodies is achieved via serial port transmission. The physical body transmits data from six angle sensors to the virtual body via the serial port. Kinematic control methods are used within the virtual body to input these data into the virtual body, which then outputs the pose information of its endpoints. Finally, this pose information is transmitted back to the physical body via the serial port. The physical universal pneumatic flexible robotic arm consists of connecting rods, universal joints, joint discs, pneumatic artificial muscle actuators, angle sensors, angular velocity sensors, and is equipped with corresponding hardware, an industrial computer, and an air source. The virtual body runs on a dedicated server, where digital simulation software runs the virtual body.

[0204] The control objectives are set as:

[0205] The desired position coordinates are set to (-0.168m, 0.166m, 0.824m). The universal pneumatic flexible robotic arm based on digital twins needs to move quickly and accurately from the initial position coordinates (0m, 0m, 0.865m) to the desired position coordinates.

[0206] When the desired position is given, the position curve output by the universal pneumatic flexible robotic arm based on digital twin is as follows: Figure 4 As shown. The solid lines represent the expected position inputs in the three coordinate directions of -0.168m, 0.166m, and 0.824m, and the dotted lines represent the actual position outputs at the expected positions of -0.168m, 0.166m, and 0.824m respectively. Figure 4 It can be seen that under the input of the desired position in the three coordinate directions, the universal pneumatic flexible robotic arm based on digital twin can quickly and accurately track the desired position.

[0207] When the expected position signal input is -0.168m, 0.166m, and 0.824m, the state estimation signal of the finite time extended state observer is as follows: Figure 5 As shown. 21 (t) represents the deflection angular velocity x of the second joint 22 Estimation curve of (t), e 21 (t) is z 21 (t) Deflection angular velocity x of the second joint 22 (t) is the observation error curve. Figure 5 It can be seen that the finite-time extended state observer can quickly output accurate and stable estimates of the joint deflection angular velocity.

[0208] Based on the disclosure and teachings of the above description, those skilled in the art will be able to make changes and modifications to the above embodiments. Therefore, the present invention is not limited to the above specific embodiments, and any obvious improvements, substitutions, or modifications made by those skilled in the art based on the present invention are within the scope of protection of the present invention.

Claims

1. A dual-loop position control method for a universal pneumatic flexible robotic arm based on digital twins, characterized in that: The following steps are involved: Step 1: Based on the structural characteristics of the universal pneumatic flexible robotic arm entity, a virtual body structure of the universal pneumatic flexible robotic arm is established in the structural simulation software. The virtual body structure is then imported into the digital simulation software. The angle sensor data of the entity is transmitted to the virtual body through the serial port. The position and posture information of the end point of the virtual body can be obtained in real time. Finally, the position and posture information of the end point of the virtual body is transmitted to the entity through the serial port. Step 2: Consider the gravity bias, Coriolis force, and coupling introduced during the movement of the universal pneumatic flexible manipulator as disturbances, and establish a second-order mathematical model of the universal pneumatic flexible manipulator; Step 3: Establish a mathematical model of the outer ring of the universal pneumatic flexible manipulator and a mathematical model of the inner ring of the universal pneumatic flexible manipulator; Step 4: The difference between the pose information of the virtual end point and the expected pose information of the end point is used as the outer loop pose error. The outer loop pose error is inversely solved into an angle error using the Levenberg-Marquardt method and input into the outer loop. Based on the mathematical model of the outer loop of the universal pneumatic flexible manipulator, an outer loop fast integral sliding mode controller is designed. The output of the controller is used as the expected value of the mathematical model of the inner loop of the universal pneumatic flexible manipulator. Step 5: Design an inner loop finite-time extended state observer based on the inner loop mathematical model of the universal pneumatic flexible manipulator to estimate the disturbance of the universal pneumatic flexible manipulator; Step 6: Design an inner loop integral error controller based on the mathematical model of the inner loop of the universal pneumatic flexible manipulator and the estimated value of the inner loop finite-time extended state observer for the disturbance; Step 7: Use the Lyapunov method to test the convergence of the inner loop finite time extended state observer, the inner loop integral error controller and the outer loop fast integral sliding mode controller.

2. The dual-loop position control method for a universal pneumatic flexible robotic arm based on digital twin according to claim 1 is characterized in that: The first step specifically includes: using the angle sensor data of each joint of the entity ,in , thereby performing kinematic control in the virtual body and obtaining the position information of the end point from the virtual body, is the pose information matrix, which is expressed as: ； Where: 、 、 Respectively represent the coordinates of the end point of the universal pneumatic flexible robotic arm in the global coordinate system X, Y, and Z axes, 、 、 Respectively represent the normal axis coordinates of the end point in the global coordinate system X, Y, and Z axes, 、 、 Respectively represent the coordinates of the orientation axis of the end point in the global coordinate system X, Y, and Z axes, 、 、 Represents the coordinates of the approach axis of the end point in the global coordinate system X, Y, and Z axes respectively.

3. The dual-loop position control method for a universal pneumatic flexible robotic arm based on digital twin according to claim 2 is characterized in that: The second step specifically includes: The second-order mathematical model of the universal pneumatic flexible robotic arm is: ； ； Where: , , , , ， ， , , ， ； Where: is the angle of each joint of the universal pneumatic flexible robotic arm, is the angular velocity of each joint of the universal pneumatic flexible robotic arm, is the state coupling matrix, is the control coupling matrix, is the coupling matrix, is the output of the inner loop integral error controller, is the gravitational bias and the Coriolis force, and Considered as a disturbance , is the decoupling constant, is the radius of the articular disc, is the length of the connecting rod, represents the moment of inertia of each joint, is the pre-charge pressure, is the length of the woven mesh fibers, is the number of turns of the braided mesh fibers, is the structural compensation parameter, , is the angle matrix, is the angular velocity matrix.

4. The dual-loop position control method for a universal pneumatic flexible manipulator based on digital twins according to claim 3 is characterized in that: The step three specifically includes: The mathematical model of the outer ring of the universal pneumatic flexible robotic arm is: ； The mathematical model of the inner ring of the universal pneumatic flexible robotic arm is: ； ； Where: is the output of the outer loop fast integral sliding mode controller, is the inner loop tracking error matrix, is the expansion state of the disturbance, and is continuous and bounded, .

5. The dual-loop position control method for a universal pneumatic flexible manipulator based on digital twins according to claim 4 is characterized in that: In step 4, the difference between the pose information of the virtual end point and the pose information of the expected end point is used as the outer loop pose error. The outer loop pose error is inversely solved into an angle error by the Levenberg-Marquardt method and input into the outer loop. Specifically, the following steps are performed: Pose information of the virtual body end point and the pose information of the desired end point The difference is taken as the outer ring pose error, and the outer ring pose error is inversely solved into the outer ring angle error by the Levenberg-Marquardt method. The input is sent to the outer loop fast integral sliding mode controller, and the inverse solution process using the Levenberg-Marquardt method is as follows: ； Where: is the number of sampling points, is the differential vector of the end point pose, is a nonlinear function used to preserve At the sampling point , is the identity matrix, is the convergence rate parameter matrix, is the Jacobian matrix, is a time-varying parameter; In the formula The update rule is: ； According to the outer ring mathematical model of the universal pneumatic flexible manipulator, an outer ring fast integral sliding mode controller is designed, and its output is used as the expected value of the inner ring mathematical model of the universal pneumatic flexible manipulator, which specifically includes: The outer loop fast integral sliding mode controller is: ； The sliding surface is for: ; Where: is the outer loop control law, is the observation error of the finite-time extended state observer, and are two constant parameters greater than zero, and are two constant parameters greater than zero, and are two odd numbers greater than zero and satisfy , and are two exponential terms, is a constant parameter greater than zero.

6. The dual-loop position control method for a universal pneumatic flexible manipulator based on digital twin according to claim 5 is characterized in that: The step five specifically includes: The inner loop finite-time extended state observer is: ， ， Where: , , ， ； Where: is the energy function, and is a nonlinear function of the design, and are the observer parameters, is an adjustable parameter, and They are and The estimated value of and They are and The observation error, a decoupling matrix is ​​given by .

7. The dual-loop position control method for a universal pneumatic flexible manipulator based on digital twins according to claim 6 is characterized in that: The step six specifically includes: The inner loop integral error controller is: ； Where: ; Where: is the perturbation estimation matrix, , , and are two positive parameters greater than zero, is the inner ring angular velocity error, is the nonlinear error, yes estimated value.

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