Adaptive neural network fault-tolerant control method for rope-driven flexible continuous robot

Abstract

The invention discloses an adaptive neural network fault-tolerant control method for a rope-driven flexible continuous robot. The adaptive neural network fault-tolerant control method comprises the following steps: constructing a dynamic model of the rope-driven flexible continuous robot based on a Cosseerat rod theory; a quaternion is introduced to eliminate truncation errors; the tracking error of the working tail end of the robot is obtained, actuator faults are considered, and a state equation of the rope-driven flexible continuous robot is constructed; designing a sliding mode surface, selecting an approaching rate, and enabling a position tracking error to reach the sliding mode surface; approximating unknown dynamic quantities in the kinetic model of the robot by using a radial basis function neural network; constructing a control law and an adaptive parameter updating law; and constructing a Lyapunov function to perform stability analysis on a robot system, adjusting control parameters of a controller, realizing fault-tolerant control, and outputting a trajectory tracking result. According to the method, the offset can be rapidly corrected under the condition that the actuator part of the rope-driven flexible continuous robot breaks down, and trajectory tracking control is effectively achieved.

Description

Technical Field

[0001] The present invention relates to the technical fields of cable-driven flexible continuum robots, Cosserat rods, adaptive neural network control, and fault-tolerant control, and particularly relates to an adaptive neural network fault-tolerant control method for a cable-driven flexible continuum robot. Background Technique

[0002] Cable-driven flexible continuum robots have the characteristics of light weight, low energy consumption, and fast response. Since they are made of flexible materials, they can interact with objects and the environment more naturally and intuitively. This makes them very suitable for applications in intelligent manufacturing, surgical operations, and industrial inspections. In these usage scenarios, often a tiny error can lead to significant damage. Therefore, continuum robots require accurate and reliable control schemes.

[0003] Before designing a dynamic controller, an accurate dynamic model of the cable-driven flexible continuum robot is needed. However, continuum robots are difficult to model due to their flexible and highly nonlinear structures. Whether it is the constant curvature method or the variable curvature method, these kinematic methods ignore the dynamic factors in the control process and assume that the controller can obtain the internal force of the rod. But in most actual control applications of continuum robots, it is difficult to fully meet these conditions. Thus, how to accurately obtain the internal force of the rod in practice is an urgent problem to be solved.

[0004] An adaptive neural network controller designed by Jiangtao (Adaptive neural network control for a soft robotic manipulator) is used to handle the external disturbances of the continuum robot system and the environment and the unmodeled system dynamics, and a neural network is introduced to compensate for these disturbances. However, actuator faults are not considered in this paper. Continuum robots are usually applied in surgical operations or industrial inspections. During intensive work, actuators and sensors are prone to failures. Once a failure occurs, it is very likely to cause closed-loop instability or even serious accidents. To ensure the safety of the control scheme, fault-tolerant design should be considered. Summary of the Invention

[0005] The purpose of the present invention is to solve the above-mentioned defects in the prior art and provide an adaptive neural network fault-tolerant control method for a cable-driven flexible continuum robot.

[0006] The present invention is achieved by at least one of the following technical solutions.

[0007] An adaptive neural network fault-tolerant control method for a cable-driven flexible continuum robot includes the following steps:

[0008] S1. Based on the structure of the cable-driven flexible continuum robot and the Cosserat rod theory, construct the dynamic model of the robot and design the desired reference trajectory.

[0009] S2. Introduce quaternions to eliminate truncation errors; obtain the actual position of the robot's working end, subtract it from the desired reference trajectory in step S1 to obtain the tracking error of the robot's working end position; consider actuator faults and construct the state equation of the cable-driven flexible continuum robot.

[0010] S3. According to the tracking error information of the position, design the sliding surface, select the reaching law, and make the position tracking error reach the sliding surface.

[0011] S4. Use a radial basis function neural network to approximate the unknown dynamic quantities in the dynamic model of the robot.

[0012] S5. Construct the control law and the adaptive parameter update law; construct a Lyapunov function to analyze the stability of the robot system, adjust the control parameters of the controller, achieve fault-tolerant control, and output the trajectory tracking result.

[0013] Furthermore, in step S1, the structure of the cable-driven flexible continuum robot includes an elastic main rod, a plurality of support disks connected in series on the main rod, and four cables that are pairwise parallel and symmetric. The four cables are connected to the main rod through a series of support disks.

[0014] Furthermore, in step S1, using the Cosserat rod theory, the partial derivatives of the position and attitude of the robot's main rod are derived as:

[0015]

[0016] where s ∈ [0, L] is the spatial position variable, i.e., the arc length of the main rod, L is the length of the main rod, t ∈ [0, +∞] is the time variable, p(s, t) is the centerline position of the robot's main rod, R(s, t) is the rotation matrix of the robot's main rod, u(s, t) is the curvature vector of the rod in the local coordinate system, v(s, t) is the rate of change of the rod position with respect to the arc length, q(s, t) is the velocity of the rod in the local coordinate system, and ω(s, t) is the angular velocity of the rod in the local coordinate system;

[0017] where the subscripts s and t represent the first-order derivatives of the variables with respect to space and time, respectively, i.e., p s (s, t) and p t (s, t) are the first-order derivatives of the centerline position of the robot's main rod with respect to space and time, respectively, R s (s, t) and R t (s, t) are the first-order derivatives of the rotation matrix of the robot's main rod with respect to space and time, respectively; the superscript ^ represents the mapping from a three-dimensional vector space to its skew-symmetric matrix. is the skew-symmetric matrix of u(s, t), is the skew-symmetric matrix of ω(s, t); By analyzing the forces on the main rod, the equilibrium differential equation of the main rod can be derived as follows:

[0018]

[0019] where n and m are the internal force and internal moment of the main rod in the global coordinate system, n s , m s are the first-order derivatives of the internal force and internal moment of the main rod in the global coordinate system with respect to space respectively, p tt (s, t) is the second-order derivative of the centerline position of the robot's main rod with respect to time; The forces and moments applied per unit arc length s are defined as f = f e + f td and l = l e + l td , the subscript (·) e represents the external load, (·) td represents the action from the rope, that is, f e is the external load distributed force, l e is the external load distributed moment, f td is the rope distributed force, l td is the rope distributed moment; ρ is the mass density of the robot's main rod, A is the cross-sectional area of the robot's main rod, J is the moment of inertia matrix; ω t (s, t) is the first-order derivative of the angular velocity of the rod in the local coordinate system with respect to time. Using the linear elastic relationship, the constitutive equation of the rod is obtained:

[0020]

[0021] where u s (s, t) is the first-order derivative of the curvature vector in the local coordinate system with respect to space, v s (s, t) is the first-order derivative with respect to space of the rate of change of the rod position with respect to the arc length; The superscript * represents the initial value of the variable, that is, v * (s, t) is the initial value of v(s, t), u(s, t) is the initial value of u(s, t), is the initial value of , is the initial value of u s (s, t), K se is the shear and tensile stiffness matrix, K bt is the bending and torsion stiffness matrix.

[0022] Furthermore, in step S1, the models of the four ropes of the robot are as follows:

[0023] In the global coordinate system, assume that the path of the i-th rope is represented by p i (s, t), satisfying the following relationship:

[0024]

[0025] where represents the offset of the i-th rope from the cross-sectional center of the robot's main rod in the local coordinate system:

[0026]

[0027] where x i (s) and y i (s) are the offset functions of the i-th rope from the cross-sectional center of the robot's main rod on the x-axis and y-axis respectively;

[0028] Assume that the internal force is tangent to the direction of p i (s, t). In the case of neglecting friction and inertia, the distributed force f td and the distributed torque l td are obtained through the following relationship:

[0029]

[0030] where τ i is the tension on the i-th rope, n is the number of ropes, is the skew-symmetric matrix of p is (s, t), is the skew-symmetric matrix of . Taking the first and second partial derivatives of p i (s, t) in space gives:

[0031]

[0032] In the formula, p is (s, t) is the first derivative of p i (s, t) with respect to space, and p iss (s, t) is the second derivative of p i (s, t) with respect to space; is the first derivative of with respect to space, is the second derivative of with respect to space; v(s, t) is the rate of change of the rod position with respect to the arc length, and v s (s, t) is the first derivative of the rate of change of the rod position with respect to the arc length with respect to space; is the skew-symmetric matrix of u(s, t), is the skew-symmetric matrix of u s (s, t).

[0033] Furthermore, in step S2, the quaternion h is introduced as follows:

[0034] h = h 1 + h 2 i x + h 3 j y + h 4 k z (33);

[0035] where h 1 is the real part of the quaternion h, and i x , j y , k z are imaginary units, and h 2 , h 3 , h 4 are the coefficients corresponding to the imaginary units; the first-order derivative h s of the quaternion h with respect to space is:

[0036]

[0037] where u(s, t) = [u 1 u 2 u 3 T , and u 1 u 2 u 3 are the curvature components of u(s, t) on the x, y, and z axes respectively;

[0038] Then the rotation matrix R(s, t) is expressed as:

[0039]

[0040] Furthermore, in step S2, the actual position p(L, t) of the robot end is obtained by a high-speed camera installed around the working environment. The tracking error e between the actual position p x of the cable-driven flexible continuous robot end and the desired reference trajectory x d is:

[0041] e = x d - p x (36);

[0042] where (·) x represents the value of a vector on the x-axis, that is, p x is the value of p(L, t) on the x-axis. The control objective is that the actual position p x of the cable-driven flexible continuous robot end can track the desired reference trajectory x d , that is, the tracking error e finally converges to 0. ​

[0043] Furthermore, for the system on the x-axis, let the first state variable be θ 1 = p x , and the second state variable be The state equation of the following robot system is obtained:

[0044]

[0045] where is the first derivative of θ 1 with respect to time, is the first derivative of θ 2 with respect to time, where p x is the position of the rod end on the x-axis, is the first derivative of p x with respect to time, is the second derivative of p x with respect to time. Substituting into the equilibrium differential equation and constitutive equation of the rod, we have:

[0046]

[0047] where n s is the first derivative of the internal force of the rod with respect to space and time; the subscript (·) x represents the value of a vector on the x-axis, ρ is the mass density of the main rod of the robot, A is the cross-sectional area of the main rod of the robot, f e is the external load distribution force, and f td is the distribution force of the rope;

[0048] To achieve the above-mentioned control objective, since it only considers the tracking problem on the x-axis, only two of the four ropes on the x-axis side need to be controlled. Then, the relationship between the distribution force f td of the rope and the tension of the two ropes is expressed as:

[0049]

[0050] where α i is an intermediate quantity, and α p = [α 1 α 2 is the intermediate quantity of the two ropes on the positive and negative sides of the x-axis. is regarded as the result of the two ropes on both sides of the x-axis being equivalent to a rope that can achieve negative tension. τ 1 , τ 2 are the tensions on the two ropes. Assume that τ equ is the equivalent tension of τ p : τ equ = δ(t)τ c, where δ(t) (0 < δ(t) ≤ 1) is the efficiency factor of the actuator, τ c is the control law.

[0051] Furthermore, in step S3, according to the control objective of the robot system state, define the sliding surface Z 1 and the first derivative of the sliding surface Z 1 with respect to time as:

[0052]

[0053] where e is the tracking error between the actual position p x of the end of the cable-driven flexible continuous robot and the desired reference trajectory x d , is the first derivative of the tracking error with respect to time, is the second derivative of the desired reference trajectory x d with respect to time; ξ is an adjustable parameter and satisfies ξ > 0, α 1 is an intermediate quantity on the positive x-axis side, τ c is the control law; define χ as the quantity that needs to be adaptively estimated online and satisfies Define the reciprocal of χ: And where represents the estimated value of γ, represents the estimation error.

[0054] Furthermore, in step S4, due to the fact that the exact spatial derivative (n s ) of the internal force of the continuous robot cannot be directly obtained under real conditions, a radial basis function neural network is used to approximate the spatial derivative of the internal force of the main rod of the cable-driven flexible continuous robot, that is, n s , and use the Gaussian kernel function: as the activation function of the neural network, where θ = [θ 1 , θ 2 is the input of the neural network; o is the number of input layers of the neural network, and j is the node of the j-th hidden layer; define n sx as the value of n s on the x-axis, and the approximation term of the radial basis function neural network is:

[0055]

[0056] In the formula, h(θ) = [h 1 , h 2 ,..., h n is the output of the Gaussian kernel function, Δ is the approximation error term of the neural network, W *is the ideal weight of the neural network, W *T h(θ) is the exact value; thus, the estimated value and the estimation error of f(θ) can be defined as:

[0057]

[0058]

[0059] where and are the estimated value and the estimation error of f(θ), respectively, is the estimated weight of the neural network, is the estimation error of * W;

[0060] Define the control intermediate quantity where sgn(·) is the sign function, Λ is an adjustable parameter and satisfies:

[0061] Λ≥Δ N ,|Δ|≤Δ N (44);

[0062] where Δ N represents the maximum approximation error of the neural network.

[0063] Furthermore, step S5 includes the following steps:

[0064] S51. Construct the control law and the adaptive parameter update law:

[0065]

[0066] where is the first derivative with respect to time, is the first derivative with respect to time, k>0 and μ>0 are both adjustable parameters;

[0067] S52. Construct the Lyapunov function V as:

[0068]

[0069] Find the first derivative of the Lyapunov function to obtain:

[0070]

[0071] Substitute the control law and the adaptive parameter update law (20) to obtain:

[0072]

[0073] where k is an adjustable positive parameter, obtained by using (19):

[0074]

[0075] where χ > 0 and k > 0, thus:

[0076]

[0077] From the above derivation, V ≥ 0 and and V is bounded when time t → ∞; according to Lyapunov stability theory and Barbalat's lemma, it is obtained that when time t → ∞, V is bounded, while Z 1 → 0, e → 0 and indicating that the cable-driven flexible continuum robot system satisfies asymptotic stability under the action of the adaptive neural network fault-tolerant controller.

[0078] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0079] 1. A radial basis function neural network is used to approximate the spatial derivative of the internal force of the main rod of the cable-driven flexible continuum robot. Therefore, the controller does not need to obtain the internal force of the rod. This is of great significance for applying this control scheme to an actual tendon-driven continuum robot.

[0080] 2. Fault-tolerant control is added to the control scheme. Therefore, the present invention can ensure the safety of the cable-driven flexible continuum robot under actuator faults, and improve the accuracy, robustness and anti-interference ability of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] The drawings described herein are used to provide a further understanding of the present invention, and constitute a part of this application. The schematic embodiments of the present invention and their descriptions are used to explain the present invention, and do not constitute an improper limitation of the present invention. In the drawings:

[0082] Figure 1 is a flowchart of an adaptive neural network fault-tolerant control method for a cable-driven flexible continuum robot disclosed by the present invention;

[0083] Figure 2 is a schematic structural diagram of the cable-driven flexible continuum robot in Embodiment 1 of the present invention;

[0084] Figure 3 is a schematic diagram of the simulation results of the motion trajectory of the end effector of the continuum robot in Embodiment 2 of the present invention under different control schemes;

[0085] Figure 4It is a schematic diagram of the simulation result of the tracking error e on the x-axis of the end effector of the continuous robot in Embodiment 2 of the present invention under different control schemes;

[0086] Figure 5 It is a schematic diagram of the simulation result of the actual tension of the rope and the control input of the controller of the continuous robot in Embodiment 2 of the present invention;

[0087] Figure 6 It is a schematic diagram of the simulation result of the deformation of the main rod of the continuous robot changing with time in the x-z plane in Embodiment 2 of the present invention. Detailed implementation manners

[0088] To make the objectives, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Apparently, the described embodiments are some but not all of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0089] Embodiment 1

[0090] Figure 1 It is a flow chart of an adaptive neural network fault-tolerant control method for a cable-driven flexible continuous robot disclosed in this Embodiment 1, which specifically includes the following steps:

[0091] S1. According to the structure of the cable-driven flexible continuous robot, and based on the Cosserat rod theory, construct the dynamic model of the robot and design the desired reference trajectory;

[0092] S2. Introduce quaternions to eliminate truncation errors; obtain the actual position of the working end of the robot, subtract it from the desired reference trajectory in step S1 to obtain the tracking error of the position of the working end of the robot; considering actuator faults, construct the state equation of the cable-driven flexible continuous robot;

[0093] S3. According to the tracking error information of the position, design a sliding mode surface, select an approach rate, and make the position tracking error reach the sliding mode surface;

[0094] S4. Use a radial basis function neural network to approximate the unknown dynamic quantities in the dynamic model of the robot;

[0095] S5. Construct a control law and an adaptive parameter update law; construct a Lyapunov function to perform stability analysis on the robot system, adjust the control parameters of the controller, achieve fault-tolerant control, and output the trajectory tracking result.

[0096] Specifically, in this embodiment, the specific process of step S1 is as follows:

[0097] S11. The structural schematic diagram of a typical cable-driven flexible continuous robot is as follows. Figure 2 As shown, taking one segment of the multi-segment cable-driven flexible continuous robot as the research object, the cable-driven continuous robot mainly consists of an elastic main rod 1 in the middle, a support disk 2 sleeved on the elastic main rod 1, and four driving cables 3 that are pairwise parallel and symmetric. These cables 3 are connected to the elastic main rod 1 through a series of support disks 2. Among them, the masses of the support disk 2 and the cable 3 can be ignored, and the friction between the support disk 2 and the cable 3 is also ignored.

[0098] In the present invention, the Cosserat rod theory is adopted to model the main rod and the cable of the robot respectively, and then the two models are coupled to obtain the dynamic model of the robot.

[0099] S12. Using the Cosserat rod theory, the partial derivatives of the position and attitude of the robot main rod are derived as follows:

[0100]

[0101] Where s ∈ [0, L] is the spatial position variable, i.e., the arc length of the main rod, L is the length of the main rod, t ∈ [0, +∞] is the time variable, p(s, t) is the centerline position of the robot main rod, R(s, t) is the rotation matrix of the robot main rod, u(s, t) is the curvature vector of the rod in the local coordinate system, v(s, t) is the change rate of the rod position with respect to the arc length, q(s, t) is the velocity of the rod in the local coordinate system, and ω(s, t) is the angular velocity of the rod in the local coordinate system;

[0102] Where the subscripts s and t respectively represent the first-order derivatives of the variables with respect to space and time, that is, p s (s, t) and p t (s, t) are the first-order derivatives of the centerline position of the robot main rod with respect to space and time respectively, R s (s, t) and R t (s, t) are the first-order derivatives of the rotation matrix of the robot main rod with respect to space and time respectively; the superscript ^ represents the mapping from a three-dimensional vector space to its skew-symmetric matrix, is the skew-symmetric matrix of u(s, t), is the skew-symmetric matrix of ω(s, t); performing a force analysis on the main rod, the equilibrium differential equation of the main rod can be derived as:

[0103]

[0104] Where n and m are the internal force and internal moment of the main rod in the global coordinate system respectively, n s , m s are the first-order derivatives of the internal force and internal moment of the main rod in the global coordinate system with respect to space respectively, ptt (s, t) is the second derivative of the centerline position of the robot's main rod with respect to time; the force and moment applied per unit arc length s are defined as f = f e + f td and l = l e + l td , the subscript (·) e represents the external load, (·) td represents the action from the rope, that is, f e is the external load distributed force, l e is the external load distributed moment, f td is the rope's distributed force, l td is the rope's distributed moment; ρ is the mass density of the robot's main rod, A is the cross-sectional area of the robot's main rod, and J is the inertia matrix; ω t (s, t) is the first derivative of the angular velocity of the rod in the local coordinate system with respect to time. The constitutive equation of the rod is obtained using the linear elastic relationship:

[0105]

[0106] where u s (s, t) is the first derivative of the curvature vector in the local coordinate system with respect to space, v s (s, t) is the first derivative of the rate of change of the rod's position with respect to arc length with respect to space; the superscript * represents the initial value of the variable, that is, v * (s, t) is the initial value of v(s, t), u(s, t) is the initial value of u(s, t), is the initial value of v s (s, t), is the initial value of u s (s, t), K se is the shear and tensile stiffness matrix, K bt is the bending and torsion stiffness matrix;

[0107] S13. The models of the four ropes of the robot are as follows:

[0108] In the global coordinate system, assume that the path of the i-th rope is represented as p i (s, t), satisfying the following relationship:

[0109]

[0110] where represents the offset of the i-th rope from the cross-section center of the robot's main rod in the local coordinate system:

[0111]

[0112] where x i(s) and y i (s) are the offset functions of the center of the cross-section of the i-th rope on the x-axis and y-axis respectively from the main rod of the robot.

[0113] Assume that the internal force is tangent to the direction of p i (s,t). Ignoring friction and inertia, the distributed force f of the rope td and the distributed torque l td are obtained through the following relationships:

[0114]

[0115] where τ i is the tension on the i-th rope, n is the number of ropes, is for p is (s,t)'s skew-symmetric matrix, is for 's skew-symmetric matrix. Taking the first and second partial derivatives of p i (s,t) with respect to space gives:

[0116]

[0117] In the formula, p is (s,t) is the first derivative of p i (s,t) with respect to space, and p iss (s,t) is the second derivative of p i (s,t) with respect to space; is for 's first derivative with respect to space, is for 's second derivative with respect to space; v(s,t) is the rate of change of the rod position with respect to the arc length, and v s (s,t) is the first derivative of the rate of change of the rod position with respect to the arc length with respect to space; is the skew-symmetric matrix of u(s,t), is for u s (s,t)'s skew-symmetric matrix.

[0118] The drive motor of the rope of the above-mentioned cable-driven flexible continuous robot is called an actuator. If there is no actuator failure, the input of the actuator is equal to the output. Once an actuator failure occurs, the input and output of the actuator will be inconsistent.

[0119] Specifically, in this embodiment, the specific process of step S2 is as follows:

[0120] S21. Introduce the quaternion h as:

[0121] h = h 1 + h 2 ix +h 3 j y +h 4 k z (58);

[0122] where h 1 is the real part of the quaternion h, i x , j y , k z are the imaginary units, and h 2 , h 3 , h 4 are the coefficients corresponding to the imaginary units; the first-order derivative h s of the quaternion h with respect to space is:

[0123]

[0124] where u(s, t) = [u 1 u 2 u 3 T , and u 1 u 2 u 3 are the curvature components of u(s, t) on the x, y, and z axes respectively;

[0125] then the rotation matrix R(s, t) is expressed as:

[0126]

[0127] S22. The actual position p(L, t) of the robot end is obtained by a high-speed camera installed around the working environment. The tracking error e between the actual position p x of the cable-driven flexible continuous robot end and the desired reference trajectory x d is:

[0128] e = x d - p x (61);

[0129] where (·) x represents the value of a vector on the x-axis, that is, p x is the value of p(L, t) on the x-axis. The control objective is that the actual position p x of the cable-driven flexible continuous robot end can track the desired reference trajectory x d , that is, the tracking error e finally converges to 0.

[0130] ​S23. The rope is considered to be on both sides of the x-z plane. The coordinate origin is located at the fixed end of the rod, the z-axis is parallel to the initial main rod, and the controller only tracks the position of the end effector of the cable-driven flexible continuum robot on the x-axis. For the system on the x-axis, let the first state variable be θ 1 = p x , and the second state variable be The state equation of the following robot system is obtained:

[0131]

[0132] where is the first derivative of θ 1 with respect to time, is the first derivative of θ 2 with respect to time, where p x is the position of the rod end on the x-axis, is the first derivative of p x with respect to time, is the second derivative of p x with respect to time. Substituting the equilibrium differential equation and constitutive equation of the rod, we have:

[0133]

[0134] where n s is the first derivative of the internal force of the rod with respect to space and time; the subscript (·) x represents the value of a vector on the x-axis, ρ is the mass density of the robot's main rod, A is the cross-sectional area of the robot's main rod, f e is the external load distribution force, and f td is the distribution force of the rope;

[0135] To achieve the above control objective, since it only considers the tracking problem on the x-axis, only two of the four ropes on the x-axis side need to be controlled. Then the relationship between the distribution force f td of the rope and the tension of the two ropes is expressed as:

[0136]

[0137] where α i is an intermediate quantity, and α p = [α 1 α 2 is the intermediate quantity of the two ropes on the positive and negative sides of the x-axis. is regarded as the result of the two ropes on both sides of the x-axis being equivalent to a rope that can achieve negative tension. τ 1 , τ 2 are the tensions on the two ropes. Assume that τ equ is the equivalent tension of τ p : τequ = δ(t)τ c , where δ(t) (0 < δ(t) ≤ 1) is the efficiency factor of the actuator, and τ c is the control law;

[0138] Specifically, in this embodiment, the specific process of step S3 is as follows:

[0139] According to the control objective of the robot system state, define the sliding mode surface Z 1 and the first derivative of the sliding mode surface Z 1 with respect to time as:

[0140]

[0141] where e is the tracking error between the actual position p x of the end of the cable-driven flexible continuous robot and the desired reference trajectory x d , is the first derivative of the tracking error with respect to time, is the second derivative of the desired reference trajectory x d with respect to time; ξ is an adjustable parameter and satisfies ξ > 0, α 1 is an intermediate quantity on the positive x-axis side, τ c is the control law; define χ as the quantity to be adaptively estimated online and satisfy Define the reciprocal of χ: And where represents the estimated value of γ, represents the estimation error of;

[0142] Specifically, in this embodiment, the specific process of step S4 is as follows:

[0143] Based on the fact that the exact spatial derivative (n s ) of the internal force of the continuous robot cannot be directly obtained under the actual conditions, use a radial basis function neural network to approximate the spatial derivative of the internal force of the main rod of the cable-driven flexible continuous robot, that is, n s , and use the Gaussian kernel function: as the activation function of the neural network, where θ = [θ 1 , θ 2 is the input of the neural network; o is the number of input layers of the neural network, and j is the node of the j-th hidden layer; define n sx as the value of n s on the x-axis, and the approximation term of the radial basis function neural network is:

[0144]

[0145] where \(h(\theta)=[h 1 ,h 2 ,\cdots,h n \) is the output of the Gaussian kernel function, \(\Delta\) is the approximation error term of the neural network, \(W * \) is the ideal weight of the neural network, \(W *T h(\theta)\) is the exact value; thus, the estimated value and the estimation error of \(f(\theta)\) can be defined as:

[0146]

[0147]

[0148] where and are the estimated value and the estimation error of \(f(\theta)\) respectively, is the estimated weight of the neural network, is the * estimation error of \(W

[0149] Define the control intermediate quantity where \(\text{sgn}(\cdot)\) is the sign function, \(\Lambda\) is an adjustable parameter and satisfies:

[0150] \(\Lambda\geq\Delta N ,|\Delta|\leq\Delta N (69);

[0151] where \(\Delta N represents the maximum approximation error of the neural network;

[0152] Specifically, in this embodiment, the specific process of step S5 is as follows:

[0153] S51. Construct the control law and the adaptive parameter update law:

[0154]

[0155] where is the first derivative with respect to time, is the first derivative with respect to time, \(k > 0\) and \(\mu>0\) are both adjustable parameters;

[0156] S52. Construct the Lyapunov function \(V\) as:

[0157]

[0158] Find the first derivative of the Lyapunov function to obtain:

[0159]

[0160] Substituting the control law and the adaptive parameter update law (20) gives:

[0161]

[0162] where k is an adjustable positive parameter, obtained by using (19):

[0163]

[0164] where χ > 0 and k > 0, thus:

[0165]

[0166] From the above derivation, V ≥ 0 and and V is bounded as time t → ∞; according to Lyapunov stability theory and Barbalat's lemma, it is obtained that as time t → ∞, V is bounded, while Z 1 → 0, e → 0 and indicating that the cable-driven flexible continuous robot system satisfies asymptotic stability under the action of the proposed adaptive neural network fault-tolerant controller;

[0167] Embodiment 2

[0168] Based on the adaptive neural network fault-tolerant control method in Embodiment 1, the Matlab simulation software is used to perform digital simulation on the cable-driven flexible continuous robot system, and at the same time, the sliding mode control and the adaptive neural network sliding mode control are added for comparison. Further verify the effectiveness of the proposed adaptive neural network fault-tolerant control method. Therefore, in this embodiment, based on each step of the adaptive neural network fault-tolerant control method disclosed in Embodiment 1, digital simulation is performed on the cable-driven flexible continuous robot system.

[0169] In this embodiment, the desired reference trajectory is given by x d = 0.04cos(πt / 3) + 0.06m. As a specific embodiment, the adjustable parameters are selected as k = 0.1, ξ = 800, μ = 10, and Λ = 0.2.

[0170] The actuator partial failure fault occurs at t = 11 (s, seconds), and δ(t) is given by:

[0171]

[0172] The initial value of is [10 10.5 10 10.5 10] T .

[0173]

[0174] As Figure 3 shown, after the actuator fails, the adaptive neural network fault-tolerant control method well realizes the trajectory tracking control and can quickly correct the deviation; the adaptive neural network sliding mode control method cannot work effectively; although due to the powerful ability of neural network approximation, the adaptive neural network sliding mode control method resumes tracking after 4 seconds, the adaptive neural network fault-tolerant control method only needs 0.5 seconds to complete. After the partial actuator failure occurs, the sliding mode control method completely loses its accuracy.

[0175] Figure 4 Shows the tracking errors of different control schemes and the adaptive neural network fault-tolerant control method scheme. It can be clearly seen from the tracking error data that the proposed control scheme has stronger robustness and accuracy than the sliding mode control method and the adaptive neural network sliding mode control method scheme.

[0176] The control inputs of different control schemes and the adaptive neural network fault-tolerant control method are as Figure 5 shown. After the partial actuator failure occurs at 11 seconds, the adaptive neural network fault-tolerant control method successfully corrects the fault with the fault-tolerant control scheme. More specifically, Figure 6 shows the change of the deformation of the whole rod over time under the adaptive neural network fault-tolerant control method. It can be observed that the whole rod deforms as expected, and the end effector of the rod successfully tracks the desired trajectory.

[0177] The preferred embodiments of the present invention disclosed above are only used to help illustrate the present invention. The preferred embodiments do not describe all the details in detail, nor limit the invention to the specific embodiments described. Obviously, many modifications and variations can be made according to the content of this specification. These embodiments are selected and specifically described in this specification to better explain the principles and practical applications of the present invention, so that those skilled in the art can well understand and utilize the present invention.

Claims

1. An adaptive neural network fault-tolerant control method for a rope-driven flexible continuous robot, characterized in that: The following steps are involved: S1. According to the structure of the rope-driven flexible continuous robot and based on the Cosserat rod theory, the robot's dynamic model is constructed and the expected reference trajectory is designed; S2, introduce quaternion to eliminate truncation error; obtain the actual position of the robot's working end, and make a difference with the expected reference trajectory in step S1 to obtain the tracking error of the robot's working end position; consider the actuator failure and construct the state equation of the rope-driven flexible continuous robot; S3. Design the sliding surface according to the position tracking error information and select the approach rate so that the position tracking error reaches the sliding surface; S4. Use radial basis function neural network to approximate the unknown dynamic quantities in the robot's dynamic model; S5. Construct control law and adaptive parameter update law; construct Lyapunov function to perform stability analysis on the robot system, adjust the control parameters of the controller, realize fault-tolerant control, and output trajectory tracking results.

2. The adaptive neural network fault-tolerant control method of a rope-driven flexible continuous robot according to claim 1 is characterized in that: In step S1, the structure of the rope-driven flexible continuous robot includes an elastic main rod, a plurality of support plates connected in series through the main rod, and four ropes that are parallel and symmetrical to each other. The four ropes are connected to the main rod through a series of support plates.

3. The adaptive neural network fault-tolerant control method of a rope-driven flexible continuous robot according to claim 1 is characterized in that: In step S1, the partial derivatives of the position and attitude of the robot main rod are derived using the Cosserat rod theory: Where s∈[0,L] is the spatial position variable, i.e., the arc length of the main rod, L is the length of the main rod, t∈[0,+∞] is the time variable, p(s,t) is the centerline position of the robot main rod, R(s,t) is the rotation matrix of the robot main rod, u(s,t) is the curvature vector of the rod in the local coordinate system, v(s,t) is the rate of change of the rod position relative to the arc length, q(s,t) is the velocity of the rod in the local coordinate system, and ω(s,t) is the angular velocity of the rod in the local coordinate system; The subscripts s and t represent the first-order derivatives of the variable with respect to space and time, respectively, that is, p s (s,t) and p t (s, t) are the first-order derivatives of the centerline position of the robot main rod with respect to space and time, respectively, R s (s,t) and R t (s, t) are the first-order derivatives of the robot's main rod rotation matrix with respect to space and time respectively; the superscript ^ represents the mapping from a three-dimensional vector space to its antisymmetric matrix, is the antisymmetric matrix of u(s,t), is the antisymmetric matrix of ω(s,t); the force analysis of the main rod is carried out, and the equilibrium differential equation of the main rod can be derived as follows: Where n and m are the internal force and internal moment of the main rod in the global coordinate system, respectively. s , m s are the first-order derivatives of the internal force and internal moment of the main rod in the global coordinate system with respect to space, p tt (s, t) is the second-order derivative of the centerline position of the robot main rod with respect to time; the force and torque applied per unit arc length s are defined as f = f e +f td and l = l e +l td , subscript (·) e represents external load, (·) td represents the action from the rope, i.e. f e is the external load distribution force, l e is the external load distribution moment, f td is the distributed force of the rope, l td is the distributed torque of the rope; ρ is the mass density of the robot main rod, A is the cross-sectional area of ​​the robot main rod, and J is the moment of inertia matrix; ω t (s, t) is the first-order derivative of the angular velocity of the rod in the local coordinate system with respect to time. The constitutive equation of the rod is obtained using the linear elastic relationship: where u s (s, t) is the first-order derivative of the curvature vector in the local coordinate system with respect to space, v s (s, t) is the first-order derivative of the rate of change of the rod position with respect to the arc length with respect to space; the superscript * indicates the initial value of the variable, i.e., v * (s,t) is the initial value of v(s,t), u(s,t) is the initial value of u(s,t), v s The initial value of (s,t), for u s The initial value of (s, t), K se is the shear and tension stiffness matrix, K bt are the bending and torsional stiffness matrices.

4. The adaptive neural network fault-tolerant control method of a rope-driven flexible continuous robot according to claim 1, characterized in that: In step S1, the models of the four ropes of the robot are as follows: In the global coordinate system, assume that the path of the i-th rope is represented by p i (s, t), satisfying the following relationship: in It represents the offset of the i-th rope from the cross-sectional center of the robot main rod in the local coordinate system: where x i (s) and y i (s) are the offset functions of the cross-sectional center of the i-th rope and the robot main pole on the x-axis and y-axis, respectively; Assume that the internal force is related to p i The direction of (s, t) is tangent. When friction and inertia are neglected, the distributed force f of the rope is td and the distributed moment l td It is obtained through the following relationship: where τ i is the tension on the ith rope, n is the number of ropes, For p is The antisymmetric matrix of (s,t), for The antisymmetric matrix of p in space i Taking the first and second order partial derivatives of (s, t) we can get: Where p is (s,t) is p i The first-order derivative of (s, t) with respect to space, p iss (s,t) is p i The second-order derivative of (s,t) with respect to space; for The first derivative with respect to space is for The second derivative with respect to space; v(s,t) is the rate of change of the rod position with respect to the arc length, v s (s,t) is the first-order derivative of the rate of change of the rod position with respect to the arc length with respect to space; is the antisymmetric matrix of u(s,t), for u s The antisymmetric matrix of (s,t).

5. The adaptive neural network fault-tolerant control method of a rope-driven flexible continuous robot according to claim 1, characterized in that: In step S2, the quaternion h is introduced as: h=h1+h2i x +h3j y +h4k z (8); Where h1 is the real part of the quaternion h, i x ,j y ,k z is the imaginary unit, h2, h3, h4 are the coefficients of the corresponding imaginary unit; the first-order derivative of the quaternion h with respect to space h s for: where u(s,t)=[u1 u2 u3] T , u1 u2 u3 are the curvature components of u(s,t) on the x, y, and z axes respectively; Then the rotation matrix R(s,t) is expressed as:

6. The adaptive neural network fault-tolerant control method of a rope-driven flexible continuous robot according to claim 1, characterized in that: In step S2, the actual position p(L, t) of the robot end is obtained by high-speed cameras installed around the working environment. The actual position p x and the expected reference trajectory x d The tracking error e is: and=x d -p x (11); in(·) x Represents the value of a vector on the x-axis, that is, p x is the value of p(L,t) on the x-axis, and the control target is the actual position p of the end of the rope-driven flexible continuous robot. x Can track the desired reference trajectory x d , that is, the tracking error e eventually converges to 0.

7. According to claim 1, the adaptive neural network fault-tolerant control method for a rope-driven flexible continuous robot is characterized in that: For the system on the x-axis, let the first state variable be θ1=p x , the second state variable is The state equation of the following robot system is obtained: in is the first-order derivative of θ1 with respect to time, is the first-order derivative of θ2 with respect to time, where p x is the position of the end of the rod on the x-axis, For p x The first derivative with respect to time is For p x Substituting the second-order derivative of time into the equilibrium differential equation and constitutive equation of the rod, we have: where n s is the first-order derivative of the internal force of the rod with respect to space and time; subscript (·) x represents the value of a vector on the x-axis, ρ is the mass density of the robot main rod, A is the cross-sectional area of ​​the robot main rod, and f e is the external load distribution force, f td is the distributed force of the rope; In order to achieve the above control objectives, since it only considers the tracking problem on the x-axis, it is only necessary to control the two ropes on the x-axis side of the four ropes, so the distributed force f of the rope is td The relationship with the tension of the two ropes is expressed as: where α i is the intermediate quantity, α p =[α1 α2] is the middle value of the two ropes on the positive and negative sides of the x-axis, The two ropes on both sides of the x-axis are regarded as equivalent to a rope that can achieve negative tension. τ1 and τ2 are the tensions on the two ropes. Assuming τ equ is τ p Equivalent tension: τ equ =δ(t)τ c , where δ(t) (0<δ(t)≤1) is the efficiency factor of the actuator, τ c It is the control law.

8. The adaptive neural network fault-tolerant control method of a rope-driven flexible continuous robot according to claim 1, characterized in that: In step S3, according to the control target of the robot system state, the sliding surface Z1 and the first-order derivative of the sliding surface Z1 in time are defined for: Where e is the actual position p of the end of the rope-driven flexible continuous robot x and the expected reference trajectory x d The tracking error, is the first-order derivative of the tracking error with respect to time, is the expected reference trajectory x d The second-order derivative with respect to time; ξ is an adjustable parameter and satisfies ξ>0, α1 is the intermediate quantity on the positive x-axis side, τ c is the control law; χ is defined as the quantity that needs to be estimated adaptively online and satisfies Define the reciprocal of χ: and in represents the estimated value of γ, express The estimation error.

9. The adaptive neural network fault-tolerant control method of a rope-driven flexible continuous robot according to claim 1, characterized in that: In step S4, the precise spatial derivative of the internal force of the continuous robot cannot be directly obtained based on actual conditions (n s ) is a dynamic quantity, a radial basis function neural network is used to approximate the spatial derivative of the internal force of the main rod of the rope-driven flexible continuous robot, that is, n s , using the Gaussian kernel function: As the activation function of the neural network, where θ = [θ1, θ2] is the input of the neural network; o is the number of input layers of the neural network, j is the node of the jth hidden layer; define n sx n s The value on the x-axis, the approximation of the radial basis function neural network is: Where h(θ)=[h1,h2,...,h n ] is the output of the Gaussian kernel function, Δ is the approximate error term of the neural network, and W * is the ideal weight of the neural network, W *T h(θ) is Exact value; therefore, the estimated value and estimation error of f(θ) can be defined as: in and are the estimated value and estimation error of f(θ), respectively. is the estimated weight of the neural network, It is W * The estimation error of Defining Control Intermediate Quantity where sgn(·) is a sign function, Λ is an adjustable parameter and satisfies: L≥D N ,|Δ|≤Δ N (19); Among them> N Represents the maximum approximation error of the neural network.

10. The adaptive neural network fault-tolerant control method of a rope-driven flexible continuous robot according to claim 1, characterized in that: Step S5 includes the following steps: S51. Construct control law and adaptive parameter update law: in for The first derivative with respect to time is for The first-order derivative with respect to time, k>0 and μ>0 are both adjustable parameters; S52. Construct the Lyapunov function V as: Find the first derivative of the Lyapunov function get: Substituting the control law and adaptive parameter update law (20) into the equation, we obtain: where k is an adjustable positive parameter, obtained by using (19): Where χ>0 and k>0, therefore: Derivation V ≥ 0 and And when time t→∞, V is bounded; According to Lyapunov stability theory and Barbalat lemma, when time t→∞, V is bounded, and Z1→0, e→0 and It is shown that the rope-driven flexible continuous robot system satisfies asymptotic stability under the action of the adaptive neural network fault-tolerant controller.

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