CN120143624A - A finite-time robust trajectory tracking control method for soft continuum robots - Google Patents
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Abstract
本发明涉及软连续体机器人控制领域，更具体的说是一种软连续体机器人有限时间鲁棒轨迹跟踪控制方法，步骤一：基于软连续体机器人的分段常曲率假设，考虑包括未建模动态和外生扰动的系统不确定性，建立软连续体机器人动力学模型并转化为状态空间模型；步骤二：基于软连续体机器人状态空间模型的非线性，设计针对非线性系统的有限时间稳定准则；步骤三：设计有限时间稳定准则的扰动观测器，保证估计误差的有限时间收敛性；步骤四：基于估计误差的结果设计非奇异终端滑模控制器；步骤五：应用于对软连续体机器人的轨迹跟踪控制策略；可以软连续体机器人有限时间内精确的轨迹跟踪控制。
The invention relates to the field of soft continuum robot control, and more specifically to a finite-time robust trajectory tracking control method for a soft continuum robot. The method comprises the following steps: step one: based on the piecewise constant curvature assumption of the soft continuum robot, taking into account the system uncertainty including unmodeled dynamics and exogenous disturbances, establishing a soft continuum robot dynamics model and converting it into a state space model; step two: based on the nonlinearity of the soft continuum robot state space model, designing a finite-time stability criterion for the nonlinear system; step three: designing a disturbance observer of the finite-time stability criterion to ensure the finite-time convergence of the estimation error; step four: designing a non-singular terminal sliding mode controller based on the result of the estimation error; and step five: applying the method to the trajectory tracking control strategy of the soft continuum robot. The method can realize accurate trajectory tracking control of the soft continuum robot within a finite time.
Description
Technical Field
The invention relates to the field of soft continuum robot control, in particular to a finite time robust track tracking control method of a soft continuum robot.
Background
Soft continuum robots have received attention in recent years due to their unique construction and flexible materials. It is desirable that they accomplish tasks that rigid robots cannot do, such as human interaction in dangerous harsh environments and safety work in uncertain environments. They have the ability to accomplish tasks that traditional rigid robots cannot accomplish.
However, soft continuum robots tend to theoretically have infinite degrees of freedom and nonlinear flexible time-varying properties, as compared to the limited degrees of freedom and rigid materials of rigid robots. Thus, the kinematics and dynamics of soft continuum robots often have complex nonlinearities, resulting in difficulty in building accurate, efficient models, which in turn results in unmodeled dynamics that may exist in the soft continuum robot modeling process, and in addition, in actual engineering applications, exogenous disturbances, such as noise, etc., often exist, which further may cause uncertainty deviations of the dynamic model, such as a soft robot, from the actual model.
On the other hand, compared with other methods, the sliding mode control method has strong robustness and can cope with model uncertainty, parameter change and exogenous interference. The goal of sliding mode control is to force the soft robot trajectory into a carefully designed subspace in which the required tracking performance can be guaranteed according to the parameter selection, it should be noted that most sliding mode control methods only progressively stabilize tracking errors and do not limit convergence over a limited time, whereas conventional progressive convergence has not met the needs of people as industrial production progresses.
In this context, based on the above two-point analysis, it is necessary to study a soft continuum robot with uncertainty and to achieve accurate and efficient finite time trajectory tracking control.
Disclosure of Invention
The invention aims to provide a limited-time robust track tracking control method for a soft continuum robot, which can solve the problem that accurate track tracking control of the soft continuum robot in limited time is difficult to realize due to uncertainty of a system caused by unmodeled dynamics existing in a dynamics modeling process and exogenous disturbance existing in actual operation.
The aim of the invention is achieved by the following technical scheme:
A method for controlling the finite time robust trajectory tracking of a soft continuum robot, the method comprising the steps of:
Based on the piecewise constant curvature assumption of the soft continuum robot, taking system uncertainty comprising unmodeled dynamics and exogenous disturbance into consideration, establishing a dynamics model of the soft continuum robot and converting the model into a state space model;
step two, designing a limited time stability criterion aiming at a nonlinear system based on the nonlinearity of a state space model of the soft continuum robot;
Step three, designing a disturbance observer of a limited time stability criterion, and ensuring limited time convergence of an estimation error;
Designing a nonsingular terminal sliding mode controller based on the result of the estimation error, and converging the track tracking error to a sliding mode surface in a limited time based on the nonsingular terminal sliding mode controller;
fifthly, the method is applied to a track tracking control strategy of the soft continuum robot;
The piecewise constant curvature assumption of the soft continuum robot is that the soft continuum robot is divided into a model with a plurality of sections of constant curvature under the assumption of piecewise constant curvature, wherein the curvature of each section is changeable in time but unchanged in space, and each section is connected end to end and smooth;
in the first step, a dynamic model of the soft continuum robot with n segments including unmodeled dynamics and exogenous disturbance is as follows:
Wherein, Represents the space pose angle of the soft continuum robot,The real number domain is represented by the number,AndRepresenting the first and second time derivatives of the pose angle, respectively, M 0 (q) representing the inertial matrix,The Coriolis force and the centrifugal force G 0 (q) are gathered to simulate the gravity effect, K 0 (q) and D 0 (q) are respectively a rigidity matrix and a damping matrix, A q (q) maps an input tau epsilon R2n containing force and moment to a configuration space, so that the continuum robot is fully actuated, delta E represents unmodeled dynamics, and D 0 represents exogenous disturbance;
Converting the dynamics model of the soft continuum robot into a state space model:
Definition x 1 = q and, Respectively represent the pose angle and the first time derivative of the soft continuum robot, d represents the total uncertainty of the state space model formula (2), and Defining the relation between the actual input tau and the control input u as M 0 (q) is a positive alignment matrix;
In the second step, the design process of the finite time stabilization criterion is as follows:
for any nonlinear system With initial value f (0) =0, if there is a lyapunov functionSatisfy the following requirements0< M <1, n >1, m+n=2, gamma 0,γ1,γ2 >0 when the parameters therein meet the following conditions, andThen the state x of the nonlinear system is converged to zero for a finite time, wherein the settling time satisfies t≤T;
The simplification is as follows:
Then
In the third step, the design process of the disturbance observer is as follows:
defining the observation error of the soft continuum robot according to a state space model formula (2) as follows:
Wherein, To estimate the derivative of the pose angle,For the estimated uncertainty, e 1 represents the estimated error of the pose angular derivative, e 2 represents the estimated error of the unknown uncertainty, and the disturbance observer formula is designed as follows:
wherein η 1,η2,η3,η4,η5,η6,η7, a, b, l, h are observer parameters and sign represents a sign function;
Considering a soft continuum robot state space model equation (2) with uncertainty and a disturbance observer (5), an observation error is observed when parameters of the disturbance observer equation (5) satisfy the following conditions and assumptions The norm value of which is more than or equal to 0 at any T meets the specification of 35 to epsilon, and the estimation errors e 1 and e 2 converge to zero for finite times T e1 and T e2, respectively, where, E 0(0),e1(0),e2 (0) respectively represent the initial time values of the corresponding state vectors, wherein the form of the function ψ 1 is defined in claim 3 (3), the assumptions and conditions that the disturbance observer equation (5) needs to satisfy are as follows:
Assume one: is bounded and meets
Conditions one, a, b, l, h are all positive odd numbers and satisfy a < b, l > h,
Condition two matrixAll characteristic values satisfying the requirement are positive numbers;
Condition three observer parameter η 1≥∈+δ0, wherein δ0>0,η5≥γ;
In the fourth step, the design process of the nonsingular terminal sliding mode controller is as follows:
Wherein the trajectory tracking error of the soft continuum robot is defined as e=x 1-xd,xd representing the desired trajectory, α, l, H, a 1,b1 being the sliding mode plane parameter, wherein H (e) is a continuous switching function, in the specific form: a, b, ζ are parameters of the switching function, which are required to satisfy the continuity of the function H (e) The symbolic variables in the switching function are defined as
In order to enable the track tracking error e to reach the sliding mode surface, the following sliding mode approach law is designed, and the specific form is as follows:
Wherein ζ, k, ρ 1,ρ2, The parameters are parameters of a sliding mode approach law formula (7);
Considering the sliding surface formula (6) with the sliding approach law formula (7), the sliding surface can converge to zero in a finite time Ts when the parameters therein meet the following conditions, wherein The form of the function ψ 2 is defined in the finite time stability criterion equation (3);
Zeta, k and ρ 1,ρ2 are positive numbers and satisfy 2ρ 1k-ρ2>||e||max;
The two conditions, i 2,h2,p2,q2, are positive and odd numbers and meet
In the fifth step, consider a soft continuum robot state space model formula (2) with system uncertainty, and combine the control rate u in the following form on the basis of a disturbance observer formula (5):
the trajectory tracking error e converges to the set n e = { e +.e +.ζ } for a finite time T e3, where Wherein the method comprises the steps ofThe form of the function ψ 3 is defined in equation (3).
The beneficial effects of the invention are as follows:
Considering the system uncertainty of the soft continuum robot caused by both unmodeled dynamics and exogenous interference in the dynamics modeling process and ensuring that accurate track tracking is realized in a limited time, the invention can realize accurate track tracking control on the soft continuum robot under the condition of uncertainty, can realize accurate tracking of the curvature of the soft continuum robot to the expected curvature, and can realize track tracking in the limited time under the premise of ensuring tracking, thereby providing an effective control method for the requirement of the soft continuum robot in the actual industry for the limited time
The non-singular terminal sliding mode controller is adopted to avoid the problem of singular points, meanwhile, the sliding mode structure is improved by combining a limited time stability criterion, so that the control performance is improved, and aiming at unmodeled dynamics and exogenous disturbance, the disturbance observer is provided to ensure that the observation error converges in a limited time, and an unknown part is replaced by an observation estimated value, so that the robustness of the soft continuum robot and the actual tracking control performance are effectively improved.
Drawings
The invention will be described in further detail with reference to the accompanying drawings and detailed description.
FIG. 1 is a schematic diagram of a method for finite time tracking control of a soft continuum robot of the present invention;
FIG. 2 is a graph of the curvature trajectory tracking error of the soft continuum robot of the present invention over time;
FIG. 3 is a graph of the curvature trajectory of the soft continuum robot of the present invention over time;
FIG. 4 is a graph of the change over time of the soft continuum robot curvature derivative and estimated curvature derivative of the present invention;
FIG. 5 is a graph of the change in the soft continuum robot curvature derivative estimation error over time of the present invention;
FIG. 6 is a graph of the change in the soft continuum robot uncertainty and estimated uncertainty over time of the present invention;
FIG. 7 is a graph of the uncertainty estimation error of the soft continuum robot of the present invention over time;
Fig. 8 is a graph showing a change in slip form surface with time of the soft continuum robot of the present invention.
Detailed Description
The invention is described in further detail below with reference to the accompanying drawings.
As shown in fig. 1 to 8, in order to solve the technical problem that "the precise trajectory tracking control of the soft continuum robot within a limited time is difficult to achieve due to the uncertainty of the system caused by both the unmodeled dynamics existing in the dynamics modeling process and the exogenous disturbance existing in the actual operation", the steps and functions of a method for controlling the robust trajectory tracking of the soft continuum robot within a limited time are described in detail below;
Soft continuum robots tend to have infinite degrees of freedom and nonlinear flexible time-varying characteristics as compared to rigid robots. Thus, the kinematics and dynamics of soft continuum robots often have complex nonlinearities, resulting in difficulty in building accurate, efficient models, which in turn results in unmodeled dynamics that may exist in the soft continuum robot modeling process, and in addition, in actual engineering applications, exogenous disturbances, such as noise, etc., often exist, which further may cause uncertainty deviations of the dynamic model, such as a soft robot, from the actual model. The main dynamic method is based on the Euler-Lagrange method under 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, wherein the curvature of each segment is variable in time but is spatially invariant, and each segment is end-to-end and smooth;
A method for controlling the finite time robust trajectory tracking of a soft continuum robot, the method comprising the steps of:
Based on the piecewise constant curvature assumption of the soft continuum robot, taking system uncertainty comprising unmodeled dynamics and exogenous disturbance into consideration, establishing a dynamics model of the soft continuum robot and converting the model into a state space model;
The dynamics model of the soft continuum robot with n segments, including the unmodeled dynamics and exogenous disturbances, is:
Wherein, Represents the space pose angle of the soft continuum robot,The real number domain is represented by the number,AndRepresenting the first and second time derivatives of the pose angle, respectively, M 0 (q) representing the inertial matrix,The Coriolis force and the centrifugal force G 0 (q) are integrated to simulate the gravity effect, K 0 (q) and D 0 (q) are respectively a rigidity matrix and a damping matrix, and A q (q) comprises the input of force and momentMapping to a configuration space such that the continuum robot is fully actuated, Δe represents unmodeled dynamics, d 0 represents exogenous disturbances;
Converting the dynamics model of the soft continuum robot into a state space model:
Definition x 1 = q and, Respectively represent the pose angle and the first time derivative of the soft continuum robot, d represents the total uncertainty of the state space model formula (2), and For the convenience of control design, the relation between the actual input tau and the control input u is defined asM 0 (q) is a positive alignment matrix;
step two, designing a limited time stability criterion aiming at a nonlinear system based on the nonlinearity of a state space model of the soft continuum robot;
The design process of the finite time stabilization criterion is as follows:
for any nonlinear system With initial value f (0) =0, if there is a lyapunov functionSatisfy the following requirements0< M <1, n >1, m+n=2, gamma 0,γ1,γ2 >0 when the parameters therein meet the following conditions, andThen the state x of the nonlinear system is finite time converging, where the settling time satisfies t≤T;
The simplification is as follows:
Then
Step three, designing a disturbance observer of a limited time stability criterion, and ensuring limited time convergence of an estimation error;
The design process of the disturbance observer is as follows:
defining the observation error of the soft continuum robot according to a state space model formula (2) as follows:
Wherein, To estimate the derivative of the pose angle,For the estimated uncertainty, e 1 represents the estimated error of the pose angular derivative, e 2 represents the estimated error of the unknown uncertainty, and the disturbance observer formula is designed as follows:
wherein η 1,η2,η3,η4,η5,η6,η7, a, b, l, h are observer parameters and sign represents a sign function;
Considering a soft continuum robot state space model equation (2) with uncertainty and a disturbance observer (5), an observation error is observed when parameters of the disturbance observer equation (5) satisfy the following conditions and assumptions The norm value of which is more than or equal to 0 at any T meets the specification of 35 to epsilon, and the estimation errors e 1 and e 2 converge to zero for finite times T e1 and T e2, respectively, where, E 0(0),e1(0),e2 (0) respectively represent the initial time values of the corresponding state vectors, wherein the form of the function ψ 1 is defined in claim 3 (3), the assumptions and conditions that the disturbance observer equation (5) needs to satisfy are as follows:
Assume one: is bounded and meets Conditions one, a, b, l, h are all positive odd numbers and satisfy a < b, l > h,Condition two matrixAll characteristic values satisfying the requirement are positive numbers;
Condition three observer parameter η 1≥∈+δ0, wherein δ0>0,η5≥γ;
Designing a nonsingular terminal sliding mode controller based on the result of the estimation error, and converging the track tracking error to a sliding mode surface in a limited time based on the nonsingular terminal sliding mode controller;
The design process of the nonsingular terminal sliding mode controller is as follows:
Wherein the trajectory tracking error of the soft continuum robot is defined as e=x 1-xd,xd representing the desired trajectory, α, l, H, a 1,b1 being the sliding mode plane parameter, wherein H (e) is a continuous switching function, in the specific form: a, b, ζ are parameters of the switching function, which are required to satisfy the continuity of the function H (e) The symbolic variables in the switching function are defined as
In order to enable the track tracking error e to reach the sliding mode surface, the following sliding mode approach law is designed, and the specific form is as follows:
Wherein ζ, k, ρ 1,ρ2, The parameters are parameters of a sliding mode approach law formula (7);
Considering the sliding surface formula (6) with the sliding approach law formula (7), the sliding surface can converge to zero in a finite time Ts when the parameters therein meet the following conditions, wherein The form of the function ψ 2 is defined in the finite time stability criterion equation (3);
Zeta, k and ρ 1,ρ2 are positive numbers and satisfy 2ρ 1k-ρ2>||e||max;
The two conditions, i 2,h2,p2,q2, are positive and odd numbers and meet
Fifthly, the method is applied to a track tracking control strategy of the soft continuum robot;
Considering a soft continuum robot state space model formula (2) with system uncertainty, based on a disturbance observer formula (5), the following form of control rate u is combined:
The trajectory tracking error converges to the set n e = { e ζ over a finite time T e3, where Wherein the method comprises the steps ofThe form of the function ψ 3 is defined in equation (3);
As shown in fig. 2 to 8, in order to verify and demonstrate the effectiveness of the disturbance observer-based sliding mode controller on the finite time tracking control method of the soft continuum robot, a simulation experiment is performed on a matlab simulation platform, an unknown uncertainty disturbance is added to closed loop feedback by programming a dynamic model of the soft continuum robot, a control law of a closed loop system is built by the disturbance observer on the unknown uncertainty estimation, and the whole system is formulated. Considering a single-segment soft continuum robot, the central axis length of the segment is 0.1m, the single-point mass is 0.5kg, and the stiffness matrix k=0.05n·m and the damping matrix d=0.01n·s·m -1 are selected. The sampling period of the simulation experiment is 0.01s, the duration is 20s, and the initial conditions of the disturbance observer and the control are designed as follows:
x1(0)＝q0=0.5rad, the parameters of the observer and the controller are selected as follows:
a1＝5,b1＝5,α＝1,l＝9,h＝7,a＝5,b＝7,ξ＝0.001,ζ＝1,k＝4,ρ1＝1,ρ2＝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;
The desired trajectory is designed as follows:
the uncertainty of the design system is as follows:
Then, as shown in fig. 2 to 8, the trajectory tracking response curves of the soft continuum robot are calculated by parameters to obtain theoretical finite convergence times of T e1＝0.4s,Te2＝0.59s,Ts＝1.13s,Te3 =1.78 s respectively. By observing fig. 2 and 3, the actual convergence time of the soft continuum robot is about 0.7s < t e3 =1.78 s, thus verifying the validity of the finite time stability theory analysis. By observing fig. 4 and 5, it is observed that the convergence time of the observer error e 1 is about 0.1s < t e1 = 0.4s, and observing fig. 6 can result in a fast response and accurate robust tracking of the designed observer in the face of abrupt uncertainty changes. Similarly, the convergence time of the observer error e 2 is about 0.2s < t e2 =0.59 s as observed in fig. 7, and the convergence time of the sliding surface s is about 0.3s < t s =1.13 s as observed in fig. 8. The control method designed can ensure the stability of the limited time through the comparison of theoretical calculation and simulation stability time, the proposed control method realizes the robust tracking control of the soft continuum robot in the limited time, and the effectiveness of the controller is verified.
Claims (10)
1. A finite time robust track tracking control method of a soft continuum robot is characterized by comprising the following steps:
Based on the piecewise constant curvature assumption of the soft continuum robot, taking system uncertainty comprising unmodeled dynamics and exogenous disturbance into consideration, establishing a dynamics model of the soft continuum robot and converting the model into a state space model;
step two, designing a limited time stability criterion aiming at a nonlinear system based on the nonlinearity of a state space model of the soft continuum robot;
Step three, designing a disturbance observer of a limited time stability criterion, and ensuring limited time convergence of an estimation error;
Designing a nonsingular terminal sliding mode controller based on the result of the estimation error, and converging the track tracking error to a sliding mode surface in a limited time based on the nonsingular terminal sliding mode controller;
and fifthly, the method is applied to a track tracking control strategy of the soft continuum robot.
2. The method for finite time robust trajectory tracking control of a soft continuum robot of claim 1, wherein the piecewise constant curvature assumption of the soft continuum robot is that the soft continuum robot is divided into a model of a plurality of segments of constant curvature under the piecewise constant curvature assumption, wherein the curvature of each segment is variable in time but constant in space and each segment is smooth end-to-end.
3. The method for finite time robust trajectory tracking control of a soft continuum robot of claim 1, wherein in step one, the dynamic model of the soft continuum robot with n segments including system uncertainty of unmodeled dynamics and exogenous disturbances is:
Wherein, Represents the space pose angle of the soft continuum robot,The real number domain is represented by the number,AndRepresenting the first and second time derivatives of the pose angle, respectively, M 0 (q) representing the inertial matrix,The coriolis force and centrifugal force G 0 (q) are pooled to simulate the gravitational effect, K 0 (q) and D 0 (q) are stiffness and damping matrices, respectively, a q (q) maps the input τe R2n containing force and moment to configuration space such that the continuum robot is fully actuated, Δe represents unmodeled dynamics, D 0 represents exogenous disturbances.
4. The method for controlling finite time robust trajectory tracking of soft continuum robot of claim 3, wherein the method comprises transforming a dynamic model of the soft continuum robot into a state space model:
Definition x 1 = q and, Respectively represent the pose angle and the first time derivative of the soft continuum robot, d represents the total uncertainty of the state space model formula (2), and Defining the relation between the actual input tau and the control input u as M 0 (q) is a positive definite matrix of the alignment.
5. The method for controlling finite time robust trajectory tracking of soft continuum robot of claim 4, wherein in said step two, the finite time stability criterion is designed as follows:
for any nonlinear system With initial value f (0) =0, if there is a lyapunov functionSatisfy the following requirements0< M <1, n >1, m+n=2, gamma 0,γ1,γ2 >0 when the parameters therein meet the following conditions, andThen the state x of the nonlinear system is converged to zero for a finite time, wherein the settling time satisfies t≤T;
The simplification is as follows:
Then
6. The method for finite time robust trajectory tracking control of a soft continuum robot of claim 5, wherein in step three, the disturbance observer is designed as follows:
defining the observation error of the soft continuum robot according to a state space model formula (2) as follows:
Wherein, To estimate the derivative of the pose angle,For the estimated uncertainty, e 1 represents the estimated error of the pose angular derivative, e 2 represents the estimated error of the unknown uncertainty, and the disturbance observer formula is designed as follows:
Where η 1,η2,η3,η4,η5,η6,η7, a, b, l, h are observer parameters and sign represents a sign function.
7. A finite time robust trajectory tracking control method of a soft continuum robot according to claim 6, wherein the soft continuum robot state space model equation (2) and the disturbance observer (5) with uncertainty are considered, and the observation error is observed when the parameters of the disturbance observer equation (5) satisfy the following conditions and assumptionsThe norm value of which is more than or equal to 0 at any T meets the specification of 35 to epsilon, and the estimation errors e 1 and e 2 converge to zero for finite times T e1 and T e2, respectively, where, E 0(0),e1(0),e2 (0) respectively represent the initial moment values of the corresponding state vectors, wherein the form of the function ψ 1 is defined in claim 3 (3), the assumptions and conditions that the disturbance observer equation (5) needs to satisfy are as follows:
Assume one: is bounded and meets Conditions one, a, b, l, h are all positive odd numbers and satisfy a < b, l > h,
Condition two matrixAll characteristic values satisfying the requirement are positive numbers;
Condition three observer parameter η 1≥∈+δ0, wherein δ0>0,η5≥γ。
8. The method for controlling finite time robust trajectory tracking of soft continuum robot of claim 7, wherein in the fourth step, the design process of the nonsingular terminal sliding mode controller is as follows:
Wherein the trajectory tracking error of the soft continuum robot is defined as e=x 1-xd,xd representing the desired trajectory, α, l, H, a 1,b1 being the sliding mode plane parameter, wherein H (e) is a continuous switching function, in the specific form: a, b, ζ are parameters of the switching function, which are required to satisfy the continuity of the function H (e) The symbolic variables in the switching function are defined as
9. The method for controlling finite time robust trajectory tracking of soft continuum robot of claim 8, wherein the following sliding mode approach law is designed to enable the trajectory tracking error e to reach the sliding mode surface, and the specific form is as follows:
Wherein ζ, k, ρ 1,ρ2, The parameters are parameters of a sliding mode approach law formula (7);
Considering the sliding surface formula (6) with the sliding approach law formula (7), the sliding surface can converge to zero in a finite time Ts when the parameters therein meet the following conditions, wherein The form of the function ψ 2 is defined in the finite time stability criterion equation (3);
Zeta, k and ρ 1,ρ2 are positive numbers and satisfy 2ρ 1k-ρ2>||e||max;
The two conditions, i 2,h2,p2,q2, are positive and odd numbers and meet
10. The method for finite time robust trajectory tracking control of a soft continuum robot according to claim 9, wherein in the fifth step, a soft continuum robot state space model formula (2) with system uncertainty is considered, and the control rate u of the following form is combined on the basis of a disturbance observer formula (5):
The trajectory tracking error e converges to the set n e = { e i ζ } within a finite time T e3, wherein Wherein the method comprises the steps ofThe form of the function ψ 3 is defined in equation (3).
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| CN116277036A (en) * | 2023-05-16 | 2023-06-23 | 湖南工商大学 | A fast fault-tolerant vibration suppression control method for a space robot with flexible base and flexible arm |
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