CN113534666A - Trajectory tracking control method for single-joint robotic arm system under multi-objective constraints - Google Patents

Abstract

The invention provides a track tracking control method of a single-joint mechanical arm system under preset performance, which is based on the current situation that few researches are based on a fuzzy state observer, a fixed time command filter triggered by a self-adaptive event and a barrier Lyapunov function method are combined and applied to a nonlinear system of a single-joint mechanical arm, the track tracking control is researched by taking the nonlinear system with typical repeated motion, such as the single-joint mechanical arm, as an object, and compared with a finite time algorithm, the track tracking control method has higher convergence speed; compared with the general self-adaptive backstepping control, the method can reduce the communication burden and the calculation amount, so that the method has higher engineering practical value for the research of the single-joint mechanical arm.

Description

Trajectory tracking control method for single-joint robotic arm system under multi-objective constraints

technical field

The invention relates to a control method of a single-joint mechanical arm system, in particular to a trajectory tracking control method of a single-joint mechanical arm system under multi-target constraints.

Background technique

The robotic arm is not only an important part of the articulated robot, but also plays an important role in the fields of industry, manufacturing, national defense and military. It can perform production operations in various alternative labor costs and dangerous and complex environments. After years of research It has gradually become practical in various fields, such as: (1) in the civil field, such as etiquette robots providing welcome services to the public, navigation information services, talent shows, etc.; (2) in the industrial field, such as automobile production lines Robotic arms for welding and reinforcing screws, fast brick-and-masonry robots on construction sites, handling and packaging robots in warehouses, assembly robots, etc.; (3) Special fields, such as providing explosive and dangerous work for national defense, military, armed police forces, etc. and so on; (4) aerospace field, such as replacing human beings to pick up and install objects in outer space workstations. With the wide application of multi-joint manipulators in robots, in order to achieve comprehensive optimization of the performance indicators of multi-joint manipulators (controlled systems), the optimal control method of multi-joint manipulators has gradually become the focus of articulated robot design.

Adaptive backstepping control method is an effective algorithm that can deal with nonlinear system control problems, and is mainly used in system tracking control problems. Backstepping is actually a recursive design method from front to back. The virtual control introduced in it is essentially a static compensation idea. The front subsystem must pass the virtual control of the latter subsystem to achieve the purpose of stabilization. In practical systems, there are mostly unknown functions, which can be approximated by fuzzy logic systems or neural networks. At the same time, under the backstepping framework, due to the repeated derivation of the virtual control signal, the problem of "complexity explosion" is generated. This problem is perfectly solved by introducing the dynamic surface control technology. G. Sun et al. Combined with DSC, the influence of uncertain nonlinearity in the system is eliminated. However, this method does not consider the influence of the first-order filtering error; J.A. Farrell et al. further proposed a command filtering technique to reduce the filtering error by constructing an error compensation mechanism However, the above backstepping controller based on command filtering can only achieve asymptotic stability.

Different from the asymptotic control method, the finite-time control method can ensure that the tracking error converges to the equilibrium point quickly. Recently, Y.-X.Li et al. studied the finite-time command filter backstepping for uncertain nonlinear systems. In the above problem, the tuning convergence time is closely related to the initial state, but once the initial state is far from the equilibrium point, the convergence time may be invalid. At present, M. Chen et al. have studied for the first time an adaptive real-time fixed-time tracking algorithm for strictly feedback nonlinear systems. The predicted convergence time is independent of the initial value. A natural question that follows is: how to extend these traditional non-linear systems. Linear control to account for the case of communication load. By introducing an event-triggered control strategy, the communication burden can be effectively alleviated and the waste of unnecessary communication resources can be reduced. W. Yang et al. further solved the fixed-time control problem based on event-triggered control. However, the constraints on the actual system cannot be ignored. influences.

Relevant constraint problems usually arise in engineering examples such as cranes, articulated manipulators, etc. If these constraints are not properly addressed, it may degrade the performance of the system. However, there has not been much research on multi-objective constraint problems. For example: in the material transportation process, the shortest distance and the lowest transportation cost proposed by J.Liu et al. belong to the multi-objective constraints, which can make the basic control method more interesting and challenging, L. Liu et al. Furthermore, an adaptive finite-time control of nonlinear systems with multi-objective constraints is proposed. Finally, the Lyapunov stability theory is used to ensure the stability of the system, so as to realize the trajectory tracking control of a single-joint manipulator.

To sum up, few researches are based on fuzzy state observer, and the combination of adaptive event-triggered fixed-time command filter and obstacle Lyapunov function method is applied to the nonlinear system of single-joint manipulator.

SUMMARY OF THE INVENTION

In view of this, the purpose of the present invention is to propose an adaptive fixed-time command filter tracking control combining state observer and obstacle Lyapunov function for a class of non-strict feedback nonlinear systems with multi-objective constraints and unmeasurable states. strategy, and specifically provides a trajectory tracking control method for a single-joint robotic arm system under multi-objective constraints.

In order to achieve the above purpose, the technical solution adopted in the present invention is: a trajectory tracking control method of a single-joint robotic arm system under multi-objective constraints, comprising the following steps:

Step 1. According to the mathematical model of the single-joint manipulator system, establish the state space model of the single-joint manipulator, and construct the corresponding state observer to estimate the unmeasurable state, and finally carry out the Lyapunov stability analysis with reference to the observation error system ;

Step 2. According to the state space model of the single-joint manipulator system established in step 1, an obstacle function is introduced to solve the multi-objective constraint problem, and the first Lyapunov function is constructed, and the corresponding virtual control law and parameter adaptation law are set. ;

Step 3. According to the state space model of the single-joint manipulator system established in step 1, construct a second Lyapunov function, and set the corresponding virtual control law and parameter adaptation law;

Step 4. According to the state space model of the single-joint robotic arm system established in step 1, construct a third Lyapunov function, and set the corresponding virtual control law and parameter adaptation law;

Step 5. On the basis of the above steps, an event-triggered strategy is introduced to reduce the communication burden, so that the single-joint robotic arm system satisfies the actual fixed time stability condition, that is, the trajectory tracking control of the single-joint robotic arm system is completed.

Further, step 1 specifically includes:

Step 1.1, first, according to the system structure diagram of the single-joint manipulator, the nonlinear mathematical model of the single-joint manipulator is established as:

q分别表示杆的加速度、速度和位置，ν表示电力子系统引起的转矩，u代表着控制输入，D＝1.5kg m²表示机械惯性，B＝1Nms/rad表示在衔接处的粘性摩擦系数，H＝1Ω表示电枢电阻，M＝H表示电枢电感，L＝0.2Nm/A表示反电动势系数；in

q represents the acceleration, velocity and position of the rod respectively, ν represents the torque caused by the power subsystem, u represents the control input, D=1.5kg ^m2 represents the mechanical inertia, B=1Nms/rad represents the viscous friction coefficient at the joint , H=1Ω means armature resistance, M=H means armature inductance, L=0.2Nm/A means back electromotive force coefficient;

x₃＝ν，令单关节机械臂控制系统的输出信号y＝q，则单关节机械臂系统非线性模型可表示为如下形式：Step 1.2, define the system state variable x ₁ =q, the system state

x ₃ =ν, let the output signal of the single-joint manipulator control system y=q, the nonlinear model of the single-joint manipulator system can be expressed as the following form:

内充分光滑的非线性函数，并满足g₁(x₁)≠0，g₂(x₂)≠0和g₃(x₃)≠0；where f ₁ (x)=0, g ₁ (x ₁ )=1, f ₂ (x)=-10sin(x ₁ )-x ₂ , g ₂ (x ₂ )=1, f ₃ (x)=- 0.2x ₂ -x ₃ , g ₃ (x ₃ )=1; f ₁ (x), f ₂ (x), f ₃ (x), g ₁ (x ₁ ), g ₂ (x ₂ ) and g ₃ (x ₃ ) are in the domain of definition

A sufficiently smooth nonlinear function, and satisfy g ₁ (x ₁ )≠0, g ₂ (x ₂ )≠0 and g ₃ (x ₃ )≠0;

Step 1.3, express the nonlinear model of the single-joint manipulator system as the following state space model:

K＝(k₁,k₂,k₃)^T,B_i＝(0,1,0)^T,B＝(0,0,1)^T，C＝(1,0,0)；A是一个严格的Hurwitz矩阵，通过选择合适的K，存在正定矩阵Q＝Q^T＞0，P＝P^T＞0，且满足A^TP+PA＝-Q；In the formula,

K=(k ₁ , k ₂ , k ₃ ) ^T , B _i =(0,1,0) ^T , B=(0,0,1) ^T , C=(1,0,0); A is a Strict Hurwitz matrix, by choosing appropriate K, there is a positive definite matrix Q=Q ^T > 0, P=P ^T > 0, and satisfies A ^T P+PA=-Q;

Step 1.4, set the corresponding state observer as follows:

分别代表着x＝(x₁,x₂,x₃)^T，f_i(x)的估计值；in the formula

respectively represent the estimated values of x=(x ₁ , x ₂ , x ₃ ) ^T , f _i (x);

Based on fuzzy logic rules, we can get:

代表最优权值向量，如果存在

满足

where δ _i represents the minimum approximation error,

represents the optimal weight vector, if there is one

Satisfy

式中δ＝(δ₁,δ₂,δ₃)^T，

Therefore, the observation error can be expressed as

where δ=(δ ₁ ,δ ₂ ,δ ₃ ) ^T ,

Step 1.5, construct the corresponding Lyapunov function as:

Derive it to get:

可得：In view of Young's inequality and fuzzy basis function

Available:

in

可得：Put the above inequality into

Available:

In the formula,

Further, step 2 specifically includes:

Step 2.1, the obstacle function is designed as follows:

m_i(i＝1,...,n)表示加权系数；in,

m _i (i=1,...,n) represents the weighting coefficient;

Step 2.2, define the following coordinate transformation:

z ₁ (t)=ξ-y _d ,

为补偿误差信号，η_i为误差补偿信号；where ξ is the barrier function, _zi is the system state error, y _d is the reference signal,

is the compensation error signal, η _i is the error compensation signal;

的影响：Step 2.3, introduce the following error compensation mechanism to solve the filtering error

Impact:

为一阶滤波器输出信号，α_i代表一阶滤波器的输入信号，β_i＞0是一个时间常数；η_i(0)＝0,χ_j,1＝1(j＝2,...,n)，k_i1＞0,k_i2＞0是设计参数；in

is the output signal of the first-order filter, α _i represents the input signal of the first-order filter, β _i >0 is a time constant; η _i (0)=0, χ _j,1 =1(j=2,... , n), k _i1 > 0, k _i2 > 0 are design parameters;

可得：Introduce the error compensation signal of the above formula

Available:

其中

为参数估计误差，

同时对V₁求导可得：Construct Lyapunov function

in

is the parameter estimation error,

At the same time, taking the derivative _of V1, we get:

并通过杨氏不等式处理可得：Using fuzzy basis functions

And through Young's inequality processing, we can get:

Where τ>0, the above formula can be replaced by:

和参数自适应律

和

其中k₁₁,k₁₂,τ,σ₁,c₁,r₁,

均为正常数；virtual control law

and parameter adaptation law

and

where k ₁₁ ,k ₁₂ ,τ,σ ₁ ,c ₁ ,r ₁ ,

are normal numbers;

Bringing in the virtual control law and the parameter adaptation law, we get:

in

Further, step 3 specifically includes:

Combining the state space model and coordinate transformation of the single-joint robotic arm system in step 2, we can get:

in

The error compensation signal is introduced to solve the influence of filtering error:

Construct the second Lyapunov function that guarantees the stability of the single-joint robotic arm system:

Derive it to get:

及杨氏不等式处理可得：Use fuzzy basis functions

and Young's inequality can be obtained:

Substitute the corresponding formula to get:

和参数自适应律

和

其中k₂₁,k₂₂,τ,σ₂,c₂,r₂,

均为正常数；virtual control law

and parameter adaptation law

and

where k ₂₁ ,k ₂₂ ,τ,σ ₂ ,c ₂ ,r ₂ ,

are normal numbers;

Bringing in the virtual control law and the parameter adaptation law, we get:

in

Further, step 4 specifically includes:

The state space model and coordinate transformation of the single-joint manipulator system combined with the above steps can be obtained:

The error compensation signal is introduced to solve the influence of filtering error:

对其求导得：Construct a third Lyapunov function that guarantees the stability of the single-joint robotic arm system:

Derive it to get:

及杨氏不等式可得：Use the same fuzzy basis functions as above

And Young's inequality can be obtained:

Before setting the actual event-triggered controller u, set the following virtual control law _α4 and parameter adaptation law

Further, step 5 specifically includes:

Define the event trigger error as P(t)=v(t)-u(t)

ρ,μ₁,μ₂均为正常数，并且满足

t_k,k∈z⁺代表输入更新时间；in,

ρ, μ ₁ , μ ₂ are all positive numbers and satisfy

t _k , k∈z ⁺ represents the input update time;

其中

可得：In the interval time [t _k , t _k+1 , based on the event-triggered control strategy, we can obtain |v(t)-u(t)|<τ|u(t)|+μ ₂ , and the controller u is set as

in

Available:

可得

带入可得：Since 0<1+l ₁ (t)τ<1+τ and

Available

Bring in to get:

Compared with the prior art, the beneficial effects of the present invention are: the present invention takes a typical repetitive motion nonlinear system such as a single-joint manipulator as the object to carry out the research on trajectory tracking control, and compared with the finite-time algorithm, it has a faster speed. Convergence speed; compared with the general adaptive backstepping control, it can reduce the communication burden and reduce the amount of calculation, so the research on single-joint manipulators has high engineering practical value.

Description of drawings

Figure 1 is the structure and force analysis diagram of the single-joint robotic arm system;

2 is a schematic flowchart of the trajectory tracking control method of the single-joint robotic arm system under the multi-target constraint of the present invention;

Fig. 3 is the tracking trajectory of the output signal, observation signal and reference signal of the single-joint manipulator;

Figure 4 is a schematic diagram of the tracking error of a single-joint robotic arm.

Detailed ways

In order to make the purposes, 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. Obviously, the described embodiments are the present invention. Part of the embodiments of the invention, but not all of the embodiments, based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative work fall within the protection scope of the present invention.

A trajectory tracking control method for a single-joint robotic arm system under multi-target constraints, as shown in Figure 2, includes the following steps:

Step 1. According to the mathematical model of the single-joint manipulator system, the state space model of the single-joint manipulator is established, and the corresponding state observer is constructed to estimate the unmeasurable state, and finally the Lyapunov stability is carried out with reference to the observation error system. analyze;

Step 2. According to the state space model of the single-joint manipulator system established in step 1, an obstacle function is introduced to solve the multi-objective constraint problem, and the first Lyapunov function is constructed, and the corresponding virtual control law and parameter adaptation law are set. ;

Step 3. According to the state space model of the single-joint manipulator system established in step 1, construct a second Lyapunov function, and set the corresponding virtual control law and parameter adaptation law;

Step 4. According to the state space model of the single-joint robotic arm system established in step 1, construct a third Lyapunov function, and set the corresponding virtual control law and parameter adaptation law;

Step 5. On the basis of the above steps, an event-triggered strategy is introduced to reduce the communication burden, so that the system satisfies the actual fixed time stability condition, that is, the trajectory tracking control of the single-joint robotic arm system is completed.

The technical solutions of each step are described in detail as follows:

Step 1. According to the mathematical model of the single-joint manipulator system, the state space model of the single-joint manipulator is established, and the corresponding state observer is constructed to estimate the unmeasurable state, and finally the Lyapunov stability is carried out with reference to the observation error system. Analysis, specifically:

Step 1.1, first, according to the system structure diagram of the single-joint manipulator, as shown in Figure 1, the nonlinear mathematical model of the single-joint manipulator is established as:

q分别表示杆的加速度，速度和位置，ν表示电力子系统引起的转矩，u代表着控制输入，D＝1.5kg m²表示机械惯性，B＝1Nms/rad表示在衔接处的粘性摩擦系数，H＝1Ω表示电枢电阻，M＝H表示电枢电感，L＝0.2Nm/A表示反电动势系数；in

q represents the acceleration, velocity and position of the rod respectively, ν represents the torque caused by the power subsystem, u represents the control input, D=1.5kg ^m2 represents the mechanical inertia, B=1Nms/rad represents the viscous friction coefficient at the joint , H=1Ω means armature resistance, M=H means armature inductance, L=0.2Nm/A means back electromotive force coefficient;

x₃＝ν令单关节机械臂控制系统的输出信号y＝q，则单关节机械臂系统非线性模型可表示为如下形式Step 1.2, define the system state variable x ₁ =q, the system state

x ₃ =ν Let the output signal of the single-joint manipulator control system y=q, the nonlinear model of the single-joint manipulator system can be expressed as the following form

内充分光滑的非线性函数，并满足g₁(x₁)≠0，g₂(x₂)≠0和g₃(x₃)≠0；where f ₁ (x)=0, g ₁ (x ₁ )=1, f ₂ (x)=-10sin(x ₁ )-x ₂ , g ₂ (x ₂ )=1, f ₃ (x)=- 0.2x ₂ -x ₃ , g ₃ (x ₃ )=1; f ₁ (x), f ₂ (x), f ₃ (x), g ₁ (x ₁ ), g ₂ (x ₂ ) and g ₃ (x ₃ ) are in the domain of definition

A sufficiently smooth nonlinear function, and satisfy g ₁ (x ₁ )≠0, g ₂ (x ₂ )≠0 and g ₃ (x ₃ )≠0;

Step 1.3, consider that some state variables in the system may be unobservable, and a fuzzy state observer needs to be designed to estimate these unobservable states. Therefore, the above system can be expressed as the following state space equation:

in the formula

K=(k ₁ , k ₂ , k ₃ ) ^T , B _i =(0,1,0) ^T , B=(0,0,1) ^T , C=(1,0,0). A is a strict Hurwitz matrix. By choosing a suitable K, there is a positive definite matrix Q=Q ^T > 0, P=P ^T > 0 satisfies A ^T P+PA=-Q

Step 1.4, set the corresponding state observer as follows:

分别代表着x＝(x₁,x₂,x₃)^T，f_i(x)的估计值。in the formula

represent the estimated values of x=(x ₁ , x ₂ , x ₃ ) ^T and f _i (x), respectively.

Based on fuzzy logic rules, we can get:

代表最优权值向量，如果存在

满足

where δ _i represents the minimum approximation error,

represents the optimal weight vector, if there is one

Satisfy

Therefore, the observation error can be expressed as

where δ=(δ ₁ ,δ ₂ ,δ ₃ ) ^T ,

Step 1.5, construct the corresponding Lyapunov function as:

Derive it to get:

可得：In view of Young's inequality and fuzzy basis function

Available:

in

可得：Put the above inequality into

Available:

in the formula

Step 2. According to the state space model of the single-joint manipulator system established in step 1, an obstacle function is introduced to solve the multi-objective constraint problem, and the first Lyapunov function is constructed, and the corresponding virtual control law and parameter adaptation law are set. , including:

Step 2.1, the obstacle function is designed as follows:

m_i(i＝1,...,n)表示加权系数；选择适当的加权系数是为了确保整体目标函数被约束在指定的范围内。由于I是x₁的一个函数，在开放集Ω中，初始值为I(0)是在域中。如果

或

则ξ→∞。简而言之，只要保证ξ是有界的，I也遵循约束条件。in,

m _i (i=1, . . . , n) represent weighting coefficients; appropriate weighting coefficients are selected to ensure that the overall objective function is constrained within the specified range. Since I is a function of x ₁ , in the open set Ω, the initial value of I(0) is in the domain. if

or

Then ξ→∞. In short, I also obeys constraints as long as ξ is guaranteed to be bounded.

Therefore, the problem of satisfying the constraints of the objective function can be transformed into guaranteeing the boundedness of ξ.

Differentiating I can get:

in

可以重写为：Subsequently,

Can be rewritten as:

和

同时可以推论出χ_1,0≠0。如果χ_1,1≠0，那么χ_1,1＝χ_1,0sign(χ_1,0)。in

and

At the same time, it can be deduced that χ _1,0 ≠0. If χ _1,1 ≠0, then χ _1,1 =χ _1,0 sign(χ _1,0 ).

As long as I=x ₁ , the multi-objective constraint problem will be transformed into an output constraint, which is a constraint content commonly studied in engineering.

Step 2.2, define the following coordinate transformation:

z ₁ (t)=ξ-y _d ,

为补偿误差信号，η_i为误差补偿信号。where ξ is the barrier function, _zi is the system state error, y _d is the reference signal,

is the compensation error signal, η _i is the error compensation signal.

来克服现有基于自适应反步法框架中对虚拟控制信号α_i的重复微分问题来减少相应的计算负担。但是已有结果大都忽略了一阶命令滤波器带来的滤波误差

的影响，此时我们引入了如下的误差补偿机制解决滤波误差

的影响，其中

为一阶滤波器输出信号，α_i代表一阶滤波器的输入信号，β_i＞0是一个时间常数。Step 2.3, this application needs to introduce a first-order command filter

To overcome the problem of repeated differentiation of the virtual control signal α _i in the existing framework based on the adaptive backstepping method to reduce the corresponding computational burden. However, most of the existing results ignore the filtering error caused by the first-order command filter.

At this time, we introduce the following error compensation mechanism to solve the filtering error

the impact of which

is the output signal of the first-order filter, α _i represents the input signal of the first-order filter, and β _i > 0 is a time constant.

where η _i (0)=0, χ _j,1 =1 (j=2,...,n), k _i1 >0, k _i2 >0 are design parameters.

可得：Introduce the error compensation signal of the above formula

Available:

其中

为参数估计误差，

同时对V₁求导可得：Construct Lyapunov function

in

is the parameter estimation error,

At the same time, taking the derivative _of V1, we get:

The selection of the Lyapunov function is based on the Lyapunov function selected from similar references:

并通过杨氏不等式处理可得：Using fuzzy basis functions

And through Young's inequality processing, we can get:

Where τ>0, the above formula can be replaced by:

和参数自适应律

和

其中k₁₁,k₁₂,τ,σ₁,c₁,r₁,

均为正常数；virtual control law

and parameter adaptation law

and

where k ₁₁ ,k ₁₂ ,τ,σ ₁ ,c ₁ ,r ₁ ,

are normal numbers;

Bringing in the virtual control law and the parameter adaptation law, we get:

in

Step 3. According to the state space model of the single-joint robotic arm system established in Step 1, construct a second Lyapunov function, and set the corresponding virtual control law and parameter adaptation law, including:

Combine the state space model and coordinate transformation of the single-joint robotic arm system in step 2:

in

The error compensation signal is introduced to solve the influence of filtering error:

Construct the second Lyapunov function that guarantees the stability of the single-joint robotic arm system:

Derive it to get:

及杨氏不等式处理可得：Use the fuzzy basis function as in step 2

and Young's inequality can be obtained:

Substitute the corresponding formula to get:

和参数自适应律

和

其中k₂₁，k₂₂，τ，σ₂，c₂，r₂，

均为正常数；virtual control law

and parameter adaptation law

and

where k ₂₁ , k ₂₂ , τ, σ ₂ , c ₂ , r ₂ ,

are normal numbers;

Bringing in the virtual control law and the parameter adaptation law, we get:

in

Step 4. According to the state space model of the single-joint manipulator system established in step 1, construct a third Lyapunov function, and set the corresponding virtual control law and parameter adaptation law, including:

The state space model and coordinate transformation of the single-joint manipulator system combined with the above steps can be obtained:

The error compensation signal is introduced to solve the influence of filtering error:

Construct a third Lyapunov function that guarantees the stability of the single-joint robotic arm system:

Derive it to get:

及杨氏不等式可得：Use the same fuzzy basis functions as above

And Young's inequality can be obtained:

Before setting the actual event-triggered controller u, the present application sets the following virtual control law _α4 and parameter adaptive law

Step 5. On the basis of the above steps, an event-triggered strategy is introduced to reduce the communication burden, so that the single-joint robotic arm system satisfies the actual fixed time stability condition, that is, the trajectory tracking control of the single-joint robotic arm system is completed, including:

By introducing an event-triggered control strategy based on relative thresholds, the corresponding communication burden and waste of communication resources are reduced.

The following describes the event-triggered control strategy based on relative thresholds in detail:

t _k+1 =inf{t∈R||P(t)|≥τ|u(t)|+μ ₂ }

t_k,k∈z⁺代表输入更新时间。需要注意的是，在时间t∈[t_k,t_k+1)，u可以视作v(t_k)，Define the event trigger error P(t)=v(t)-u(t), 0<τ<1, ρ, μ ₁ , μ ₂ are all positive numbers, and satisfy

t _k , k∈z ⁺ represents the input update time. It should be noted that at time t∈[t _k ,t _k+1 ), u can be regarded as v(t _k ),

Whenever t _k+1 =inf{t∈R||P(t)|≥τ|u(t)|+μ ₂ } is triggered, the moment will be marked as t _k+1 , the actual control input u( t _k+1 ) will be applied to the system. Therefore, we can find the parameters l ₁ (t),l ₂ (t) that satisfy the following equations:

v(t)=(1+l ₁ (t)τ)u+l ₂ (t)μ ₂

Where |l ₁ (t)|≤1, |l ₂ (t)|≤1, so the controller can be obtained:

其中

可得：In the interval time [t _k , t _k+1 , based on the event-triggered control strategy, we can obtain |v(t)-u(t)|<τ|u(t)|+μ ₂ , and the controller u is set as

in

Available:

可得

带入可得：Since 0<1+l ₁ (t)τ<1+τ and

Available

Bring in to get:

可得：Based on Lemma 1:

Available:

where M ₃ =M ₂ +0.557ρ;

definition

Based on Lemma 2:

Based on Lemma 3: H _n ∈ R,i=1,...,n,κ∈[0,1]

(|H ₁ |+…+|H _n |) ^κ ≤|H ₁ | ^κ +…+|H _n | ^κ

in view of

可得：Put the above two inequalities into

Available:

in

definition

Lemma 4: x ₁ , y ₂ represent arbitrary variables, k ₁ , k ₂ , B represent arbitrary constants,

where τ ₁ =0.11;

可得Bring the above formula into

Available

in

和通过以下杨氏不等式相消处理：based on

and are processed by the following Young's inequality cancellation:

可表示为：Lyapunov differential function

can be expressed as:

in the formula

根据

并同上应用引理2和3，上式可转换为：definition

according to

And applying Lemma 2 and 3 as above, the above formula can be transformed into:

满足

分析以下两种情况：At this point, it is assumed that there is an unknown constant

Satisfy

Analyze the following two situations:

Case 1: If

Therefore it is possible to

Case 2: If

set up

Summarizing the above two situations can be obtained:

in

According to Lemma 5: If V(x) is a positive definite function, it also has the following form

In the formula, φ ₁ , φ ₂ , α, β, γ all represent normal numbers, and αγ∈(0,1), βγ∈(1, ∞), ρ＞0 are satisfied at the same time.

Then it can be proved that the origin of the system reaches the actual fixed time stability (compared with asymptotic stability or finite time stability, the advantage of the actual fixed time stability selected in this paper is that the convergence time can be predicted normally without considering the initial conditions).

β＝2,γ＝1更便于实际设计。Consult the existing literature, the parameters are selected as follows

β=2, γ=1 is more convenient for practical design.

Therefore, the single-joint manipulator system can satisfy the actual fixed time stability condition.

The design goal of this application is to design the controller u so that the output signal y can be constrained to track the reference signal y _d within a limited range (k _c1 , k _c2 ) while ensuring that the tracking error z ₁ converges to within a fixed time interval In the small neighborhood range of zero, the calculation amount is effectively reduced and the convergence speed is accelerated; the tracking trajectory of the output signal, observation signal and reference signal of the single-joint manipulator is shown in Figure 3. The schematic diagram of the tracking error of the single-joint manipulator is shown in Figure 4.

This application studies the trajectory tracking control of a typical repetitive motion nonlinear system such as a single-joint robotic arm. Compared with the finite-time algorithm, it has a faster convergence speed; It can reduce the communication burden and reduce the amount of calculation, so the research on the single-joint manipulator has high engineering practical value.

The above description of the disclosed embodiments enables any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims (6)

1. The track tracking control method of the single-joint mechanical arm system under multi-target constraint is characterized by comprising the following steps of:

step 1, establishing a state space model of a single-joint mechanical arm according to a mathematical model of a single-joint mechanical arm system, constructing a corresponding state observer to estimate an unmeasured state, and finally performing luggage Jaconov stability analysis by referring to an observation error system;

step 2, according to the state space model of the single-joint mechanical arm system established in the step 1, introducing a barrier function to solve the multi-target constraint problem, constructing a first Lyapunov function, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law;

step 3, constructing a second Lyapunov function according to the state space model of the single-joint mechanical arm system established in the step 1, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law;

step 4, constructing a third Lyapunov function according to the state space model of the single-joint mechanical arm system established in the step 1, and setting a corresponding virtual control law and a corresponding parameter self-adaptive law;

and 5, introducing an event trigger strategy to reduce the communication burden on the basis of the steps, so that the single-joint mechanical arm system meets the actual fixed time stability condition, and the track tracking control of the single-joint mechanical arm system is completed.

2. The method for controlling the trajectory tracking of the single-joint mechanical arm system under the multi-target constraint according to claim 1, wherein the step 1 specifically comprises:

step 1.1, firstly, according to a system structure diagram of the single-joint mechanical arm, establishing a nonlinear mathematical model of the single-joint mechanical arm as follows:

wherein

q represents the acceleration, velocity and position of the stick, v represents the torque induced by the power subsystem, u represents the control input, and D is 1.5kgm²Represents mechanical inertia, B ═ 1Nms/rad represents a viscous friction coefficient at the joint, H ═ 1 Ω represents armature resistance, M ═ H represents armature inductance, and L ═ 0.2Nm/a represents a back electromotive force coefficient;

step 1.2, define the system state variable x₁Q, system state

x₃And (v) setting the output signal y of the single-joint mechanical arm control system to be q, and then the nonlinear model of the single-joint mechanical arm system can be expressed as follows:

wherein f is₁(x)＝0，g₁(x₁)＝1，f₂(x)＝-10sin(x₁)-x₂，g₂(x₂)＝1，f₃(x)＝-0.2x₂-x₃，g₃(x₃)＝1；

f₁(x)，f₂(x)，f₃(x)，g₁(x₁)，g₂(x₂) And g₃(x₃) Are all in the domain

A non-linear function with fully smooth inner surface and satisfying g₁(x₁)≠0，g₂(x₂) Not equal to 0 and g₃(x₃)≠0；

Step 1.3, representing the nonlinear model of the single-joint mechanical arm system as a state space model as follows:

in the formula,

K＝(k₁,k₂,k₃)^T,B_i＝(0,1,0)^T,B＝(0,0,1)^Tc ═ 1,0, 0; a is a strict Hurwitz matrix, and by choosing the appropriate K, there is a positive definite matrix Q ═ Q^T＞0，P＝P^TIs greater than 0 and satisfies A^TP+PA＝-Q；

Step 1.4, setting the corresponding state observer as follows:

in the formula

Each represents x ═ x₁,x₂,x₃)^T，f_i(x) An estimated value of (d);

based on fuzzy logic rules, we can get:

in the formula of_iWhich represents the minimum approximation error, is,

represents the optimal weight vector, if any

Satisfy the requirement of

The observation error can be expressed as

Wherein δ is (δ)₁,δ₂,δ₃)^T，

Step 1.5, constructing a corresponding Lyapunov function as:

derivation of this can yield:

in view of the Young's inequality and fuzzy basis functions

The following can be obtained:

wherein

Bringing the inequality of the above into

The following can be obtained:

in the formula,

3. the method for controlling the trajectory tracking of the single-joint mechanical arm system under the multi-target constraint according to claim 2, wherein the step 2 specifically comprises:

step 2.1, the barrier function is designed as follows:

k_c1＜I₁(x₁)＜k_c2

k_c1＜I₂(x₁)＜k_c2

k_c1＜I_n(x₁)＜k_c2

wherein,

m_i(i ═ 1, …, n) represents a weighting coefficient;

step 2.2, the following coordinate transformation is defined:

z₁(t)＝ξ-y_d,

where xi is the barrier function, z_iAs systematic state error, y_dAs a reference signal, the reference signal is,

to compensate for error signals, η_iAn error compensation signal;

step 2.3, the following error compensation mechanism is introduced to solve the filtering error

The influence of (a):

wherein

Is the output signal of a first-order filter, alpha_iInput signal, beta, representing a first order filter_i> 0 is a time constant; eta_i(0)＝0,χ_j,1＝1(j＝2,...,n)，k_i1＞0,k_i2> 0 is a design parameter;

introduced into the above formulaError compensation signal of

The following can be obtained:

constructing the Lyapunov function

Wherein

In order to estimate the error for the parameter,

at the same time to V₁The derivation can be:

using fuzzy basis functions

And can be obtained by processing the Young inequality:

where τ > 0, the above formula can be substituted:

law of virtual control

And law of parameter adaptation

And

wherein k is₁₁,k₁₂,τ,σ₁,c₁,r₁,

Are all normal numbers;

the virtual control law and the parameter adaptive law are brought into being available:

wherein

4. The method for controlling trajectory tracking of a single-joint mechanical arm system under multi-target constraint according to claim 3, wherein step 3 specifically comprises:

combining the state space model of the single-joint mechanical arm system in the step 2 and the coordinate transformation, the following can be obtained:

wherein

And introducing an error compensation signal to solve the influence of filtering errors:

constructing a second Lyapunov function for ensuring the stability of the single-joint mechanical arm system:

the derivation of which is:

using fuzzy basis functions

And young inequality treatment can obtain:

replacing the corresponding formula can be:

law of virtual control

And law of parameter adaptation

And

wherein k is₂₁，k₂₂，τ，σ₂，c₂，r₂，

Are all normal numbers;

the virtual control law and the parameter adaptive law are brought into being available:

wherein

5. The method for controlling trajectory tracking of a single-joint mechanical arm system under multi-target constraint according to claim 4, wherein the step 4 specifically comprises:

the state space model and the coordinate transformation of the single-joint mechanical arm system combined with the steps can obtain:

and introducing an error compensation signal to solve the influence of filtering errors:

constructing a third Lyapunov function for ensuring stability of the single-joint mechanical arm system

The derivation of which is:

using fuzzy basis functions as above

And the young inequality can be given as:

before setting the actual event-triggered controller u, the following virtual control law α is set₄And law of parameter adaptation

6. The method for controlling trajectory tracking of a single-joint mechanical arm system under multi-target constraint according to claim 5, wherein the step 5 specifically comprises:

defining an event trigger error as P (t) v (t) u (t)

Wherein,

0＜τ＜1，ρ,μ₁,μ₂are all normal numbers and satisfy

t_k,k∈z⁺Representing an input update time;

at intervals of time

In the method, | v (t) -u (t) | < tau | u (t) | + mu can be obtained based on the event trigger control strategy₂The controller u is set as

Wherein

The following can be obtained:

since 0 < 1+ l₁(t) τ < 1+ τ and

can obtain the product

Bringing into availability:

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