A soft continuum robot finite time robust trajectory tracking control method

Abstract

The present application relates to the field of soft continuum robot control, more particularly to a soft continuum robot finite time robust trajectory tracking control method, step one: based on the segmented constant curvature assumption of the soft continuum robot, considering the system uncertainty including unmodeled dynamics and exogenous disturbance, the soft continuum robot dynamics model is established and converted into a state space model; Step two: based on the nonlinearity of the soft continuum robot state space model, a finite time stability criterion for nonlinear systems is designed; Step three: a disturbance observer of the finite time stability criterion is designed to ensure the finite time convergence of the estimation error; Step four: a non-singular terminal sliding mode controller is designed based on the results of the estimation error; Step five: applied to the trajectory tracking control strategy of the soft continuum robot; accurate trajectory tracking control in the finite time of the soft continuum robot can be achieved.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of soft continuum robot control, and more particularly to a soft continuum robot finite time robust trajectory tracking control method. BACKGROUND

[0002] Soft continuum robots have attracted much attention in recent years due to their unique structure and flexible materials. People hope that they can complete tasks that rigid robots cannot, such as human-robot interaction in dangerous and harsh environments and safe work in uncertain environments. They have the ability to complete tasks that traditional rigid robots cannot achieve.

[0003] However, compared with the limited degrees of freedom and rigid materials of rigid robots, soft continuum robots often have infinite degrees of freedom and nonlinear flexible time-varying characteristics in theory. Therefore, the kinematics and dynamics of soft continuum robots are usually complex and nonlinear, making it difficult to establish an accurate and efficient model, which will result in unmodeled dynamics in the modeling process of soft continuum robots. In addition, in actual engineering applications, there are often external disturbances such as noise, which will further cause uncertainty deviation between the dynamic model of the soft robot and the actual model.

[0004] On the other hand, compared with other methods, the sliding mode control method has strong robustness and can cope with model uncertainty, parameter variation and external disturbance. The goal of sliding mode control is to force the soft robot trajectory to a carefully designed subspace, in which the required tracking performance can be guaranteed according to parameter selection. It needs to be noted that most sliding mode control methods only stabilize the tracking error asymptotically and do not limit the convergence in finite time. With the development of industrial production, traditional asymptotic convergence cannot meet people's needs.

[0005] In this context, based on the above two points of analysis, it is necessary to study soft continuum robots with uncertainty and achieve accurate and efficient finite time trajectory tracking control. SUMMARY

[0006] The purpose of the present application is to provide a soft continuum robot finite time robust trajectory tracking control method, which can solve the problem of system uncertainty caused by both unmodeled dynamics in the dynamics modeling process and external disturbances in actual operation, making it difficult to achieve accurate trajectory tracking control of soft continuum robots in finite time.

[0007] The purpose of the present application is achieved by the following technical solutions:

[0008] A soft continuum robot finite time robust trajectory tracking control method, the method comprising the following steps:

[0009] Step one: Based on the segmented constant curvature assumption of soft continuum robots, considering the system uncertainty including unmodeled dynamics and exogenous disturbances, the dynamics model of soft continuum robots is established and converted into a state space model;

[0010] Step two: Based on the nonlinearity of the state space model of soft continuum robots, a finite time stability criterion for nonlinear systems is designed;

[0011] Step three: A disturbance observer is designed for the finite time stability criterion to ensure the finite time convergence of the estimation error;

[0012] Step four: A non-singular terminal sliding mode controller is designed based on the results of the estimation error, and the trajectory tracking error is converged to the sliding surface in finite time based on the non-singular terminal sliding mode controller;

[0013] Step five: The trajectory tracking control strategy for soft continuum robots is applied;

[0014] The segmented constant curvature assumption of the soft continuum robot is that under the assumption of segmented constant curvature, the soft continuum robot is divided into multiple constant curvature models, where the curvature of each segment is variable in time but constant in space, and each segment is connected smoothly.

[0015] In step one, the dynamics model of the soft continuum robot with n segments including unmodeled dynamics and exogenous disturbances is:

[0016]

[0017] where, q represents the spatial pose angle of the soft continuum robot, R represents the real number field, and represent the first and second time derivatives of the pose angle, respectively, M0(q) represents the inertia matrix, G0(q) simulates the effect of gravity, K0(q) and D0(q) are the stiffness matrix and damping matrix, respectively, A q (q) maps the actual input of force and torque to the configuration space, so that the continuum robot is fully actuated, ΔE represents unmodeled dynamics, and d0 represents exogenous disturbances;

[0018] The dynamics model of the soft continuum robot is converted into a state space model:

[0019]

[0020] Define x1=q, ​respectively, d represents the total uncertainty of the state space model formula (2), and The relationship between the actual input τ and the control input u is defined as M0(q) is a positive definite matrix.

[0021] In step two, the design process of the finite time stability criterion is as follows:

[0022] For any nonlinear system with the initial value f(0)=0, if there exists a Lyapunov function satisfying When the parameters in the formula satisfy the following conditions: 0<m<1, n>1, m+n=2, γ0, γ1, γ2>0, and then the state x of the nonlinear system is finite time convergent to zero, wherein the stable time satisfies t≤T.

[0023] Simplify as:

[0024] Then

[0025] In step three, the design process of the disturbance observer is as follows:

[0026] According to the state space model formula (2), the observation error of the soft continuum robot is defined as:

[0027]

[0028] wherein, is the derivative of the estimated pose angle, is the estimated uncertainty, e1 represents the estimation error of the pose angle derivative, e2 represents the estimation error of the unknown uncertainty, and the design disturbance observer formula is:

[0029]

[0030] wherein, η1, η2, η3, η4, η5, η6, η7, a, b, l, h are observer parameters, and sign represents a sign function.

[0031] Considering the state space model formula (2) and the disturbance observer (5) of the soft continuum robot with uncertainty, when the parameters of the disturbance observer formula (5) satisfy the following conditions and assumptions, the observation error The norm value at any t≥0 satisfies ||e0||≤∈, and the estimation errors e1 and e2 will be respectively in a finite time Te1 and T e2 converges to zero in finite time, where, e0(0), e1(0), e2(0) represent the initial time values of the corresponding state vectors, where the form of function Ψ is defined in formula (3), the assumptions and conditions required for the disturbance observer formula (5) are as follows:

[0032] Assumption one: is bounded and satisfies Condition one: a, b, l, h are all positive odd numbers and satisfy Condition two: matrix satisfies that all its eigenvalues are positive numbers;

[0033] Condition three: observer parameter η1≥∈+δ0, where

[0034] In step four, the design process of the nonsingular terminal sliding mode controller is as follows:

[0035]

[0036] Where the trajectory tracking error of the soft continuum robot is defined as e=x1-x d , x d represents the desired trajectory,

[0037] α, l, h, a1, b1 are sliding surface parameters, where H(e) is a continuous switching function, and its specific form is: a, b, ξ are parameters of the switching function, and in order to ensure the continuity of function H(e), it needs to satisfy The sign variable in the switching function is defined as

[0038] In order to make the trajectory tracking error e reach the sliding surface, the following sliding mode reaching law is designed, and its specific form is:

[0039]

[0040] Where ζ, k, a2, b2, ρ1, ρ2, are all parameters of sliding mode reaching law formula (7);

[0041] Considering the sliding surface formula (6) with sliding mode reaching law formula (7), when the parameters satisfy the following conditions, the sliding surface can converge to zero in finite time Ts, where The form of function Ψ is defined in the finite time stability criterion formula (3);

[0042] Condition one: ζ, k, ρ1, ρ2 are positive numbers, and satisfy 2ρ1k-ρ2>||e|| max ;

[0043] Condition two: l2, h2, p2, q2 are positive odd numbers, and satisfy

[0044] In the step five, considering the soft continuum robot state space model formula (2) with system uncertainty, on the basis of the disturbance observer formula (5), combined with the following form of control input u:

[0045] The trajectory tracking error e converges to the set Π in a finite time T e3 , wherein e = {e|||e||≤ξ} , wherein The form of the function Ψ is defined in formula (3).

[0046] The beneficial effects of the present application are:

[0047] Considering the system uncertainty of the soft continuum robot caused by both unmodeled dynamics in the process of dynamic modeling and exogenous disturbance, and ensuring accurate trajectory tracking in a finite time, the present application can realize accurate trajectory tracking control of the soft continuum robot under uncertainty, can realize accurate tracking of the expected curvature of the curvature of the soft continuum robot, and can realize trajectory tracking in a finite time while ensuring tracking, thereby providing an effective control method for the finite time demand of the soft continuum robot in actual industry

[0048] The non-singular terminal sliding mode controller is used to avoid the problem of singular points, and the sliding mode structure is improved in combination with the finite time stability criterion, thereby improving the control performance; for unmodeled dynamics and exogenous disturbance, a disturbance observer is proposed to ensure that the observation error converges in a finite time, and the observation estimate is used to replace the unknown part, thereby effectively improving the robustness and actual tracking control performance of the soft continuum robot. BRIEF DESCRIPTION OF DRAWINGS

[0049] The present application will be further described in detail below in combination with the drawings and specific implementation methods.

[0050] Figure 1 is a schematic diagram of the soft continuum robot finite time tracking control method of the present application;

[0051] Figure 2is a plot of the curvature trajectory tracking error of the soft continuum robot of the present invention as a function of time;

[0052] Figure 3 is a plot of the curvature trajectory of the soft continuum robot of the present invention as a function of time;

[0053] Figure 4 is a plot of the curvature derivative and estimated curvature derivative of the soft continuum robot of the present invention as a function of time;

[0054] Figure 5 is a plot of the curvature derivative estimation error of the soft continuum robot of the present invention as a function of time;

[0055] Figure 6 is a plot of the uncertainty and estimated uncertainty of the soft continuum robot of the present invention as a function of time;

[0056] Figure 7 is a plot of the uncertainty estimation error of the soft continuum robot of the present invention as a function of time;

[0057] Figure 8 is a plot of the sliding surface of the soft continuum robot of the present invention as a function of time. DETAILED DESCRIPTION

[0058] The present invention is further described in detail below in conjunction with the accompanying drawings.

[0059] As shown in Figures 1 to 8 to solve the technical problem of "it is difficult to achieve accurate trajectory tracking control of a soft continuum robot in a limited time due to the uncertainty of the system caused by both unmodeled dynamics in the process of dynamic modeling and exogenous disturbances in actual operation", the steps and functions of a soft continuum robot finite-time robust trajectory tracking control method are described in detail below;

[0060] Compared with rigid robots, soft continuum robots often have infinite degrees of freedom and nonlinear flexible time-varying characteristics. Therefore, the kinematics and dynamics of soft continuum robots are usually complex and nonlinear, making it difficult to establish an accurate and efficient model, which will lead to unmodeled dynamics in the modeling process of soft continuum robots. In addition, in actual engineering applications, there are often exogenous disturbances such as noise, etc., which will further cause uncertainty deviation between the dynamic model of the soft robot and the actual model. The main current dynamics method is the Euler-Lagrange method based on the assumption of piecewise constant curvature and other variable curvature methods. Under the assumption of piecewise constant curvature, the soft continuum robot is divided into a model of multiple segments of constant curvature, where the curvature of each segment is variable in time but constant in space, and each segment is connected smoothly at the beginning and end;

[0061] A soft continuum robot finite time robust trajectory tracking control method, the method comprising the following steps:

[0062] Step one: based on the segmented constant curvature assumption of the soft continuum robot, considering the system uncertainty including unmodeled dynamics and exogenous disturbances, the soft continuum robot dynamics model is established and converted into a state space model;

[0063] The dynamics model of the soft continuum robot with n segments including unmodeled dynamics and exogenous disturbances is:

[0064]

[0065] wherein, represents the spatial pose angle of the soft continuum robot, represents the real number field, and respectively represent the first and second time derivatives of the pose angle, M0(q) represents the inertia matrix, G0(q) simulates the effect of gravity, K0(q) and D0(q) are respectively the stiffness matrix and the damping matrix, A q (q) will map the actual input of force and torque to the configuration space, so that the continuum robot is fully actuated, ΔE represents the unmodeled dynamics, d0 represents the exogenous disturbance;

[0066] The dynamics model of the soft continuum robot is converted into a state space model:

[0067]

[0068] Define x1=q, respectively represent the pose angle of the soft continuum robot and its first time derivative, d represents the total uncertainty of the state space model formula (2), and In order to facilitate the control design, the relationship between the actual input τ and the control input u is defined as M0(q) is a positive definite matrix;

[0069] Step two: based on the nonlinearity of the state space model of the soft continuum robot, a finite time stability criterion for nonlinear systems is designed;

[0070] The design process of the finite time stability criterion is:

[0071] For any nonlinear system with initial value f(0)=0, if there exists a Lyapunov function satisfying When the parameters in the formula satisfy the following conditions: 0 < m < 1, n > 1, m + n = 2, γ0, γ1, γ2 > 0, and Then the state x of the nonlinear system is finite time convergent, and the stable time satisfies t ≤ T;

[0072]

[0073] Simplify to:

[0074] Then

[0075] Step three: design a disturbance observer of the finite time stability criterion to ensure the finite time convergence of the estimation error;

[0076] The design process of the disturbance observer is as follows:

[0077] According to the state space model formula (2), the observation error of the soft continuum robot is defined as:

[0078]

[0079] Wherein, is the derivative of the estimated pose angle, is the estimated uncertainty, e1 represents the estimation error of the derivative of the pose angle, and e2 represents the estimation error of the unknown uncertainty, and the design formula of the disturbance observer is:

[0080]

[0081] Wherein, η1, η2, η3, η4, η5, η6, η7, a, b, l, h are observer parameters, and sign represents a sign function;

[0082] Considering the state space model formula (2) of the soft continuum robot with uncertainty and the disturbance observer (5), when the parameters of the disturbance observer formula (5) satisfy the following conditions and assumptions, the observation error The norm value at any t ≥ 0 satisfies ||e0|| ≤ ∈, and the estimation errors e1 and e2 will converge to zero in finite time T e1 and T e2 , wherein, e0(0), e1(0), e2(0) respectively represent the initial time values of the corresponding state vectors, wherein the form of the function Ψ is defined in formula (3), and the assumptions and conditions required to be satisfied by the disturbance observer formula (5) are as follows:

[0083] Assumption one: is bounded and satisfies

[0084] Condition one: a, b, l, h are all positive odd numbers and satisfy

[0085] Condition two: the matrix satisfies that all its eigenvalues are positive;

[0086] Condition three: the observer parameter η1≥∈+δ0, where

[0087]

[0088] Step four: design a nonsingular terminal sliding mode controller based on the result of the estimation error, and realize that the trajectory tracking error converges to the sliding surface in finite time based on the nonsingular terminal sliding mode controller;

[0089] The design process of the nonsingular terminal sliding mode controller is as follows:

[0090]

[0091] where the trajectory tracking error of the soft continuum robot is defined as e=x1-x d , x d represents the desired trajectory;

[0092] α, l, h, a1, b1 are sliding surface parameters, where H(e) is a continuous switching function, and its specific form is: a, b, ξ are parameters of the switching function, and in order to ensure the continuity of the function H(e), the following condition needs to be satisfied The sign variable in the switching function is defined as

[0093] In order to make the trajectory tracking error e reach the sliding surface, the following sliding mode reaching law is designed, and its specific form is:

[0094]

[0095] where ζ, k, a2, b2, ρ1, ρ2, are all parameters of the sliding mode reaching law formula (7);

[0096] Considering the sliding surface formula (6) with the sliding mode reaching law formula (7), when the parameters satisfy the following conditions, the sliding surface can converge to zero in finite time Ts, where The form of the function Ψ is defined in the finite time stability criterion formula (3);

[0097] Condition one: ζ, k, p1, p2 are positive numbers, and satisfy 2p1k-p2>||e|| max ;

[0098] Condition two: l2, h2, p2, q2 are positive odd numbers, and satisfy

[0099] Step five: applied to the trajectory tracking control strategy of soft continuum robots;

[0100] Considering the soft continuum robot state space model formula (2) with system uncertainty, on the basis of disturbance observer formula (5), combined with the following form of control input u:

[0101]

[0102] The trajectory tracking error converges to the set Π e3 in finite time T e ={e|||e||≤ξ} within a limited time T where The form of function Ψ is defined in formula (3);

[0103] As Figures 2 to 8 shown, in order to verify and show the high efficiency of the sliding mode controller based on disturbance observer for the finite time tracking control method of soft continuum robots, simulation experiments are carried out on the matlab simulation platform, the dynamics model of soft continuum robot is programmed, and unknown uncertainty disturbance is added to the closed loop feedback, the control law of closed loop system is constructed by estimating the unknown uncertainty through disturbance observer, and the whole system is formulated. Consider a single segment soft continuum robot, the center axis length of the segment is 0.1m, the single point mass is 0.5kg, the stiffness matrix K=0.05N·m and the damping matrix D=0.01N·s·m -1 are selected. The sampling period of simulation experiment is 0.01s, the duration is 20s, and the initial conditions of disturbance observer and control are designed as follows:

[0104] x1(0)=q0=0.5rad, The parameters of the observer and the controller are selected as follows:

[0105] a1=5, b1=5, a=1, l=9, h=7, a=5, b=7, ξ=0.001, ζ=1, k=4, p1=1, p2=0.01, a2=5, b2=5, l2=9, h2=7, p2=5, q2=7, η1=10, η2=81.2409, η3=1650, η4=5, η5=100, η6=2, η7=3;

[0106] The desired trajectory is designed as follows:

[0107]

[0108] The uncertainties of the system are as follows:

[0109]

[0110] The trajectory tracking response curve of the soft continuum robot is shown in Figures 2 to 8 , and the theoretical finite convergence time is T e1 = 0.4s, T e2 = 0.59s, T s = 1.13s, and T e3 = 1.78s. Observing Figure 2 and Figure 3 , the actual convergence time of the soft continuum robot is about 0.7s < T e3 = 1.78s, thus verifying the effectiveness of the finite time stability theory analysis. Observing Figure 4 and Figure 5 , the convergence time of the observer error e1 is about 0.1s < T e1 = 0.4s, and observing Figure 6 , it can be seen that the designed observer can quickly respond and achieve accurate robust tracking in the face of factors such as uncertainty mutations. Similarly, observing Figure 7 , the convergence time of the observer error e2 is about 0.2s < T e2 = 0.59s, and observing Figure 8 , the convergence time of the sliding surface s is about 0.3s < T s = 1.13s. Comparing the theoretical calculation and simulation stability time, it can be seen that the designed control method can guarantee finite time stability, the proposed control method realizes the robust tracking control of the soft continuum robot in finite time, and the effectiveness of the controller is verified.

Claims

1. A finite-time robust trajectory tracking control method for a soft continuum robot, characterized in that: The method includes the following steps: Step 1: Based on the piecewise constant curvature assumption of the soft continuum robot, considering the system uncertainties including unmodeled dynamics and exogenous disturbances, establish the dynamic model of the soft continuum robot and transform it into a state-space model; Step 2: Based on the nonlinearity of the state-space model of the soft continuum robot, design a finite-time stability criterion for the nonlinear system; Step 3: Design a perturbation observer for the finite-time stability criterion to ensure the finite-time convergence of the estimation error; Step 4: Design a non-singular terminal sliding mode controller based on the estimation error, and realize the trajectory tracking error converges to the sliding surface within a finite time based on the non-singular terminal sliding mode controller; Step 5: Application of trajectory tracking control strategy for soft continuum robots; In step two, the design process of the finite-time stability criterion is as follows: For any nonlinear system Given an initial value f(0) = 0, if there exists a Lyapunov function... satisfy 0 when the parameters satisfy the following conditions <m<1,n> 1, m+n=2, γ0, γ1, γ2>0, and Then the state x of the nonlinear system converges to zero in finite time, where the steady-state time satisfies t≤T; Simplified to: but 2. The finite-time robust trajectory tracking control method for a soft continuum robot according to claim 1, characterized in that: The segmented constant curvature assumption of the soft continuum robot is as follows: Under the assumption of segmented constant curvature, the soft continuum robot is divided into a model of multiple segments with constant curvature, wherein the curvature of each segment is variable in time but invariant in space, and each segment is connected end to end and smooth.

3. The finite-time robust trajectory tracking control method for a soft continuum robot according to claim 1, characterized in that: In step one, the dynamic model of the soft continuum robot with n segments, including unmodeled dynamics and exogenous disturbances, is as follows: in, This represents the spatial pose angle of a soft continuum robot. Represents the real number field. and Let represent the first and second time derivatives of the pose angle, respectively, and M0(q) represent the inertia matrix. The Coriolis force and centrifugal force G0(q) are combined to simulate the gravitational effect, where K0(q) and D0(q) are the stiffness matrix and damping matrix, respectively. q (q) The actual inputs containing force and torque Mapping to the configuration space enables the continuum robot to be fully actuated, where ΔE represents the unmodeled dynamics and d0 represents the exogenous perturbation.

4. The finite-time robust trajectory tracking control method for a soft continuum robot according to claim 3, characterized in that: Transform the dynamic model of the soft continuum robot into a state-space model: Define x1 = q, Let represent the pose angle of the soft continuum robot and its first-order time derivative, respectively, and d represent the total uncertainty of the state-space model formula (2). Define the relationship between the actual input τ and the control input u as follows: M0(q) is a pairwise positive definite matrix.

5. The finite-time robust trajectory tracking control method for a soft continuum robot according to claim 4, characterized in that: In step three, the design process of the disturbance observer is as follows: According to the state-space model formula (2), the observation error of a soft continuum robot is defined as: in, The derivative of the estimated pose angle, To represent the uncertainty in the estimation, e1 represents the estimation error of the pose angle derivative, and e2 represents the estimation error of the unknown uncertainty. The formula for designing the perturbation observer is as follows: Where η1, η2, η3, η4, η5, η6, η7, a, b, l, h are observer parameters, and sign represents the sign function.

6. The finite-time robust trajectory tracking control method for a soft continuum robot according to claim 5, characterized in that: Considering the state-space model formula (2) and the perturbation observer (5) of a soft continuum robot with uncertainty, the observation error is as follows: For any norm value of t≥0, ||e0||≤∈, and the estimation errors e1 and e2 will respectively occur in finite time T. e1 and T e2 Converging to zero, where, e0(0), e1(0), and e2(0) represent the initial time values ​​of the corresponding state vectors, where the form of the function Ψ is defined in formula (3). The assumptions and conditions that the perturbation observer formula (5) needs to satisfy are as follows: Assumption 1: It is bounded and satisfies Condition 1: a, b, l, h are all positive odd numbers and satisfy the following conditions: Condition 2: Matrix It satisfies that all of its eigenvalues ​​are positive numbers; Condition 3: Observer parameter η1 ≥ ∈ +δ0, where 7. The finite-time robust trajectory tracking control method for a soft continuum robot according to claim 6, characterized in that: In step four, the design process of the non-singular terminal sliding mode controller is as follows: The trajectory tracking error of a soft continuum robot is defined as e = x1 - x d x d Represents the expected trajectory. α, l, h, a1, b1 are sliding surface parameters, where H(e) is a continuous switching function, specifically in the form of: a, b, and ξ are the parameters of the switching function. To ensure the continuity of the function H(e), the following conditions must be met: The symbolic variable in the switching function is defined as follows:

8. The finite-time robust trajectory tracking control method for a soft continuum robot according to claim 7, characterized in that: To ensure that the trajectory tracking error e reaches the sliding surface, the following sliding surface reaching law is designed, the specific form of which is as follows: Among them, ζ, k, a2, b2, ρ1, ρ2, All of these are parameters of the sliding mode reaching law formula (7); Consider the sliding surface formula (6) with the sliding mode reaching law formula (7). The sliding surface can converge to zero in a finite time Ts when the parameters satisfy the following condition: The form of the function Ψ is defined in the finite-time stability criterion formula (3); Condition 1: ζ, k, ρ1, ρ2 are all positive numbers, and satisfy 2ρ1k - ρ2 > ||e|| max ; Condition 2: l2, h2, p2, q2 are all positive odd numbers, and satisfy the condition...

9. A finite-time robust trajectory tracking control method for a soft continuum robot according to claim 8, characterized in that: In step five, considering the soft continuum robot state-space model formula (2) with system uncertainty, based on the perturbation observer formula (5), the following form of control input u is combined: The trajectory tracking error e in a finite time T e3 Converging inward to set Π e In the case of {e|||e||≤ξ}, where in The form of the function Ψ is defined in formula (3).

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