CN112987770B - Anti-saturation finite-time motion control method for walking feet of amphibious crab-imitating multi-foot robot - Google Patents

Abstract

An anti-saturation finite-time motion control method for walking feet of an amphibious crab-imitating multi-legged robot belongs to the technical field of robot control. The method aims to solve the problems of poor precision and low speed of the existing walking foot trajectory tracking control of the crab-imitating multi-foot robot. The method comprises the steps of firstly establishing a robot walking foot dynamics model for the amphibious crab-imitating multi-legged robot, then determining a self-adaptive finite time interference observer based on the robot walking foot dynamics model, processing the influence of input saturation by using an auxiliary system, and finally controlling the robot walking foot movement by using a rapid terminal sliding mode controller based on the self-adaptive finite time interference observer AFTDO under the condition of input saturation. The method is mainly used for controlling the walking feet of the multi-foot robot.

Description

Anti-saturation limited-time motion control method for walking legs of amphibian crab-like multi-legged robot

technical field

The invention relates to a motion control method for a multi-legged robot. It belongs to the field of control technology.

Background technique

In today's world, with the rapid development of science and technology, the demand for the development of marine resources is expanding. Aiming at many tasks in the fields of marine facility inspection, marine resource exploration and data collection, offshore reconnaissance, defense and rescue in complex environments such as offshore seabed and shoals, it is necessary to develop an amphibious crab-like multi-legged robot to complete the above tasks. As shown in Figure 1.

The design and research of the amphibian crab-like multi-legged robot integrates multi-disciplinary and multi-field technologies. The motion control problem in its crawling mode is a key content in the research topic of the amphibian crab-like multi-legged robot, and it is also one of the robot research contents. The reliability of the motion control technology determines the operational efficiency and intelligence level of the amphibious crab-like multi-legged robot, and is also an important prerequisite for the robot to complete specific tasks.

Due to its multi-joint, nonlinear, multi-redundancy and time-varying characteristics, the amphibious crab-like multi-legged robot has a high degree of uncertainty in its kinematics and dynamics. First of all, the robot needs to overcome the model uncertainty caused by its own modeling error and the influence of the disturbance of the ocean current when walking on the seabed during the movement process. At the same time, in the actual control process, the nonlinear characteristics of the actuator system will also This leads to the problem of input saturation. The above problems all put forward higher requirements for the control accuracy and response speed of the robot's walking feet. How to effectively improve the robot's walking foot trajectory tracking control accuracy and speed, and how to ensure that the robot can achieve stable, high-precision and fast control under the condition of actuator saturation are the key issues that need to be solved in the research on the motion control of amphibious crab-like multi-legged robots.

SUMMARY OF THE INVENTION

The invention aims to solve the problems of poor precision and slow speed in the tracking control of the walking foot trajectory of the existing crab-like multi-legged robot.

An anti-saturation limited-time motion control method for a walking foot of an amphibious crab-like multi-legged robot, comprising the following steps:

S1. Establish a robot walking foot dynamics model for the amphibious crab-like multi-legged robot:

为集总不确定性，θ∈R³、

分别表示机器人步行足关节角度、关节角速度、关节角加速度矢量；其中M₀(θ)、

g₀(θ)为模型已知标称部分，分别表示正定惯性矩阵、科氏力和离心力项、重力和浮力项产生的恢复力项，ΔM(θ)、

Δg(θ)对应为建模误差导致的不确定部分，F_s(θ)为机器人受到的地面广义反力项，τ_d为海流扰动，τ为期望控制力/力矩输入，sat(·)为饱和函数；where:

is the lumped uncertainty, θ∈R ³ ,

respectively represent the robot walking foot joint angle, joint angular velocity, and joint angular acceleration vector; where M ₀ (θ),

g ₀ (θ) is the known nominal part of the model, representing the positive definite inertia matrix, Coriolis force and centrifugal force terms, restoring force terms generated by gravity and buoyancy terms, respectively, ΔM(θ),

Δg(θ) corresponds to the uncertain part caused by the modeling error, F _s (θ) is the generalized ground reaction force on the robot, τ _d is the current disturbance, τ is the expected control force/torque input, and sat( ) is saturation function;

S2. Determine the adaptive finite-time disturbance observer based on the walking foot dynamics model of the amphibious crab-like multi-legged robot:

η∈R³为满足以下辅助动力学方程的变量；sliding variable

^η∈R3 is a variable that satisfies the following auxiliary kinetic equation;

为d的估计值，

为滑模项，

为辅助动力学方程的增益；where:

is the estimated value of d,

is the sliding mode term,

is the gain of the auxiliary kinetic equation;

为干扰估计误差，可得：definition

For the interference estimation error, we can get:

The adaptive finite-time disturbance observer is as follows:

ξ为辅助变量，

为α的估计值，

为观测器增益；σ表示对时间t的积分变量；α集总不确定性d的导数边界值；where:

ξ is an auxiliary variable,

is the estimated value of α,

is the observer gain; σ represents the integral variable with respect to time t; α is the derivative boundary value of the lumped uncertainty d;

The adaptive finite-time disturbance observer and disturbance error can be obtained by Eq.:

The adaptive law is:

Among them, γ and δ are normal numbers;

S3. Use the auxiliary system to deal with the influence of input saturation. The auxiliary system is as follows:

In the formula, ζ=(ζ ₁ , ζ ₂ , ζ ₃ ) ^T is the state vector of the auxiliary system, A=diag{a _i } _3×3 , B=[b ₁ ,b ₂ ,b ₃ ] ^T is the parameter matrix and a parameter vector, where a _i > 0, b _i > 0, i=1, 2, 3; sgn(ζ)=(sgn(ζ ₁ ), sgn(ζ ₂ ), sgn(ζ ₃ )) ^T , sgn (·) is the sign function; P=diag{||pi ||} _3×3 , where _pi is the _ith row of the control gain matrix M ₀ ^-1 ;

S4, using the fast terminal sliding mode controller based on the adaptive finite time disturbance observer AFTDO under input saturation to control the motion of the robot's walking foot;

The described fast terminal sliding mode controller based on adaptive finite-time disturbance observer AFTDO under input saturation is as follows:

τ=τ ₀ +τ ₁ +τ ₂

τ ₂ =-M ₀ (θ)(Aζ+B+σ ₀ Psgn(ζ))

分别为关节角位移跟踪误差和角速度跟踪误差；S为全局快速终端滑模面；λ＞0，μ＞0为正对角矩阵；p和q为奇数且p＜q；k₁,k₂＞0为控制参数。Among them, τ ₀ is the equivalent control term, τ ₁ is the auxiliary control term of the disturbance observer, and τ ₂ is the saturation compensation term; e,

are the joint angular displacement tracking error and angular velocity tracking error, respectively; S is the global fast terminal sliding mode surface; λ>0, μ>0 is a positive diagonal matrix; p and q are odd numbers and p<q; k ₁ , k ₂ > 0 is the control parameter.

Further, the joint angular displacement tracking error and angular velocity tracking error are as follows:

_e =θ-θd

Among them, θ _d is the desired angle of the robot walking foot joint.

Further, the described global fast terminal sliding surface is:

where (·) ^p/q represents exponentiation.

如下：Further, the gain of the auxiliary kinetic equation

as follows:

Among them, c ₀ , γ ₀ , γ ₁ , γ ₂ , γ ₃ , and γ ₄ are all positive numbers.

Further, the observer gain

Further, in the described robot walking foot dynamics model, sat(τ)=[sat(τ ₁ ), sat(τ ₂ ), sat(τ ₃ )] ^T , sat( ) is a saturation function:

Among them, i=1, 2, 3, τ _max and τ _min are the maximum control force/torque input and the minimum control force/torque input, respectively.

Further, the process of establishing a robot walking foot dynamics model for the amphibious crab-like multi-legged robot includes the following steps:

First, construct the dynamic model of the robot arm:

分别表示机器人步行足关节角度、关节角速度、关节角加速度矢量；M(θ)＝M₀(θ)+ΔM(θ)，

g(θ)＝g₀(θ)+Δg(θ)；其中M₀(θ)、

g₀(θ)为模型已知标称部分，ΔM(θ)、

Δg(θ)代表建模误差导致的不确定部分，F_s(θ)为机器人受到的地面广义反力项，τ_d为海流扰动；In the formula, θ∈R ³ ,

respectively represent the robot walking foot joint angle, joint angular velocity, and joint angular acceleration vector; M(θ)=M ₀ (θ)+ΔM(θ),

g(θ)=g ₀ (θ)+Δg(θ); where M ₀ (θ),

g ₀ (θ) is the known nominal part of the model, ΔM(θ),

Δg(θ) represents the uncertain part caused by the modeling error, F _s (θ) is the generalized ground reaction force on the robot, and τ _d is the current disturbance;

引入机器人机械臂动力学模型，得到：aggregate uncertainty

Introducing the dynamic model of the robot arm, we get:

The robot walking foot dynamics model is then determined based on input saturation

Beneficial effects:

Aiming at the tracking control problem of the walking foot trajectory of an amphibious crab-like multi-legged robot, the invention comprehensively considers the uncertainty of model parameters and the uncertainty of ocean current disturbance under the condition of input saturation, and proposes an Adaptive Finite Time Disturbance Observer (Adaptive Finite Time Disturbance Observer). , AFTDO) to solve the observation problem of lumped uncertainty caused by current disturbance and modeling error, and adopting a control method that does not depend on the precise dynamic model, an adaptive finite-time disturbance observer-based method considering input saturation is proposed. The global fast terminal sliding mode controller realizes the tracking control of the walking foot trajectory of the amphibious crab-like multi-legged robot.

AFTDO treats the system model parameter uncertainty and ocean current disturbance uncertainty as lumped uncertainties, constructs auxiliary dynamic equations, designs adaptive laws, improves the design of adaptive finite-time disturbance observers, and realizes the detection of lumped uncertainties. Fast and accurate estimation of sex.

The global fast terminal sliding mode control method based on AFTDO under input saturation is based on the adaptive finite time disturbance observer, combined with the advantages of the global fast terminal sliding mode control, which is fast and reaches the sliding mode surface in a limited time, considering the input saturation situation. , construct an input saturation auxiliary system, design a fast terminal sliding mode controller based on an adaptive finite-time disturbance observer under input saturation, and improve the control accuracy and response speed of the robot's walking foot trajectory tracking.

Description of drawings

Figure 1 is a schematic diagram of an amphibious crab-like multi-legged robot;

Figure 2 is a walking foot model of an amphibious crab-like multi-legged robot;

Fig. 3 is the performance curve of the adaptive finite-time disturbance observer, in which Fig. 3(a) is the estimation performance of disturbance d ₁ , Fig. 3(b) is the estimation error of disturbance d ₁ ; Fig. 3(c) is the estimation performance of disturbance d ₂ , Fig. 3(d) is the estimation error of disturbance d ₂ ; Fig. 3(e) is the estimation performance of disturbance d ₃ , and Fig. 3(f) is the estimation error of disturbance d ₃ ;

Figure 4. The response curve of robot walking trajectory tracking control, in which Figure 4(a) is the trajectory tracking performance of joint 1, Figure 4(b) is the trajectory tracking error of joint 1; Figure 4(c) is the trajectory tracking performance of joint 2, Figure 4( d) is the trajectory tracking error of joint 2; Figure 4(e) is the trajectory tracking performance of joint 3, and Figure 4(f) is the trajectory tracking error of joint 3.

Detailed ways

Before describing the specific implementation manner, the parameter definitions of the present invention are first described as follows:

——科氏力和离心力项；g₀(θ)——重力和浮力项产生的恢复力项；θ∈R³、

——机器人步行足关节角度、关节角速度、关节角加速度矢量；e＝θ-θ_d——跟踪误差；τ——控制输入；τ_d——海流扰动；d——包括模型不确定性、地面广义力以及海流扰动的集总不确定性。M ₀ (θ)——positive definite inertia matrix;

- Coriolis force and centrifugal force terms; g ₀ (θ) - restoring force terms generated by gravity and buoyancy terms; θ∈R ³ ,

——joint angle, joint angular velocity, and joint angular acceleration vector of robot walking foot; _e =θ-θd——tracking error; τ——control input; τd——current disturbance; _d ——including model uncertainty, ground Generalized forces and lumped uncertainties in ocean current disturbances.

Specific implementation one:

The limited-time motion control method for walking feet of an amphibious crab-like multi-legged robot described in this embodiment includes the following steps:

S1. Transform the walking foot dynamics model of the amphibious crab-like multi-legged robot based on the walking foot dynamics model of the amphibious crab-like robot:

The walking foot motion model of the amphibian crab-like multi-legged robot is shown in Figure 2. A coordinate system is established for the walking foot. The fixed coordinate system of the walking foot is OX ₀ Y ₀ Z ₀ , and the hip joint coordinate system is OX ₁ Y ₁ Z ₁ . The coordinate system of the hip joint is OX ₂ Y ₂ Z ₂ , the coordinate system of the tibia joint is OX ₃ Y ₃ Z ₃ , and the coordinate system of the end point of the walking foot is OX ₄ Y ₄ Z ₄ . The fixed coordinate system of the walking foot is also the walking foot and the robot body. The coordinate system of the connection.

The dynamic equation of the walking foot of the amphibian crab-like multi-legged robot adopts the dynamic model of the robot arm derived based on the Euler-Lagrange method:

分别表示机器人步行足关节角度、关节角速度、关节角加速度矢量；M(θ)＝M₀(θ)+ΔM(θ)，

g(θ)＝g₀(θ)+Δg(θ)，F_s(θ)＝J_i ^TF_i。其中M₀(θ)、

g₀(θ)为模型已知标称部分，ΔM(θ)、

Δg(θ)代表建模误差导致的不确定部分，F_s(θ)为机器人受到的地面广义反力项，τ_d为海流扰动，J_i ^T为机器人第i条步行足的雅克比转置矩阵；F_i＝[f_ix,f_iy,f_iz]^T为处于支撑相的第i条步行足受到的反力向量。where: θ∈R ³ ,

respectively represent the robot walking foot joint angle, joint angular velocity, and joint angular acceleration vector; M(θ)=M ₀ (θ)+ΔM(θ),

g(θ)=g ₀ (θ)+Δg(θ), F _s (θ)=J _i ^T F _i . where M ₀ (θ),

g ₀ (θ) is the known nominal part of the model, ΔM(θ),

Δg(θ) represents the uncertain part caused by the modeling error, F _s (θ) is the generalized ground reaction force on the robot, τ _d is the current disturbance, and J _i ^T is the Jacobian transpose of the ith walking foot of the robot matrix; F _i =[ _fix , f _iy , f _iz ] ^T is the reaction force vector received by the i-th walking foot in the support phase.

为集总不确定性，代入式(1)得：make

is the lumped uncertainty, substituting into equation (1), we get:

The walking foot of the robot will not be affected by the ground force during the swing process. In addition, the robot will generate model uncertainty and ocean current disturbance due to modeling errors during the movement process. At the same time, the actuator system has nonlinear characteristics of input saturation. , which will cause the system controller to not continue to respond to the system error signal, further making the controller unable to function normally. Therefore, the present invention considers the lumped uncertainty and input saturation of the robot in the current disturbance and modeling error, and establishes the robot walking foot dynamics model as follows:

为集总不确定性，θ∈R³、

分别表示机器人步行足关节角度、关节角速度、关节角加速度矢量；M(θ)＝M₀(θ)+ΔM(θ)，

g(θ)＝g₀(θ)+Δg(θ)，F_s(θ)＝J_i ^TF_i。其中M₀(θ)、

g₀(θ)为模型已知标称部分，ΔM(θ)、

Δg(θ)代表建模误差导致的不确定部分，F_s(θ)为机器人受到的地面广义反力项，τ_d为海流扰动，sat(τ)＝[sat(τ₁),sat(τ₂),sat(τ₃)]^T，τ为设计的期望控制力/力矩输入，sat(·)为饱和函数：where:

is the lumped uncertainty, θ∈R ³ ,

respectively represent the robot walking foot joint angle, joint angular velocity, and joint angular acceleration vector; M(θ)=M ₀ (θ)+ΔM(θ),

g(θ)=g ₀ (θ)+Δg(θ), F _s (θ)=J _i ^T F _i . where M ₀ (θ),

g ₀ (θ) is the known nominal part of the model, ΔM(θ),

Δg(θ) represents the uncertain part caused by modeling error, F _s (θ) is the generalized ground reaction force term received by the robot, τ _d is the current disturbance, sat(τ)=[sat(τ ₁ ), sat(τ ₂ ), sat(τ ₃ )] ^T , τ is the expected control force/torque input of the design, and sat( ) is the saturation function:

S2. Based on the walking foot dynamics model of the amphibious crab-like multi-legged robot, an adaptive finite-time disturbance observer is designed:

并且η∈R³满足以下辅助动力学方程：Define the sliding variable as

And η∈R ³ satisfies the following auxiliary kinetic equation:

为d的估计值，

为滑模项，且标量

为辅助动力学方程(5)的增益。where:

is the estimated value of d,

is the sliding mode term, and the scalar

is the gain of the auxiliary kinetic equation (5).

为干扰估计误差。则由式(2)和式(5)可得：definition

Estimate the error for the interference. Then from formula (2) and formula (5), we can get:

To estimate and compensate for the uncertainty of the lumped parameters, an adaptive finite-time disturbance observer is designed as follows:

ξ为辅助变量，

为α的估计值，

为观测器增益。where:

ξ is an auxiliary variable,

is the estimated value of α,

is the observer gain.

From equation (7) and the definition of interference error, we can get:

For the auxiliary kinetic equation (5) and the adaptive finite-time observer system, the following equation is chosen as the unknown gain:

Then the adaptive law is designed as:

将在有限时间内收敛到包含平衡点

在内的有界区域。Among them: γ, δ are normal numbers, then the estimation error

will converge to the inclusive equilibrium point in finite time

bounded area inside.

S3. Design input saturation auxiliary system:

The system input saturation problem is considered in system (3). For the actual control system, the difference Δτ between the expected control input τ and the actual control input sat(τ) should be small enough, because the control input must satisfy the energy of the system when the control input is saturated. control. Since the disturbance and the system state are bounded, the desired control input is bounded. To satisfy this assumption, the parameter σ ₀ can be large. Define Δτ and assume that it satisfies the condition: ||Δτ||=||τ-sat(τ)||≤σ ₀ , where σ ₀ is a known constant.

Auxiliary systems are designed to handle the effects of input saturation as follows:

In the formula, ζ=(ζ ₁ , ζ ₂ , ζ ₃ ) ^T is the state quantity of the auxiliary system, A=diag{a _i } _3×3 and B=[b ₁ ,b ₂ ,b ₃ ] ^T is the design matrix and design vectors, where a _i > 0, b _i > 0, i=1, 2, 3. In addition, sgn(ζ)=(sgn(ζ ₁ ), sgn(ζ ₂ ), sgn(ζ ₃ )) ^T , sgn(·) is a sign function; P=diag{||pi || _{} 3×3} , where pi is the _ith row of the control gain matrix M ₀ ^-1 . The auxiliary state quantity converges to zero in finite time.

S4. Based on the lumped uncertainty and input saturation of the robot under the current disturbance and modeling error, the fast terminal sliding mode controller based on the adaptive finite-time disturbance observer AFTDO under the input saturation is designed as:

τ=τ ₀ +τ ₁ +τ ₂ (12)

τ ₂ =-M ₀ (θ)(Aζ+B+σ ₀ Psgn(ζ)) (15)

Among them, τ ₀ is the equivalent control item, τ ₁ is the auxiliary control item of the disturbance observer, τ ₂ is the saturation compensation item, and k ₁ , k ₂ >0 are the control parameters.

In order to fully illustrate the innovation of the present invention, the design process and principle of the controller of the present invention are now described as follows:

P1: Using the dynamic model of the robotic arm derived based on the Euler-Lagrangian method to establish the walking foot dynamics equation of the amphibian crab-like multi-legged robot:

分别表示机器人步行足关节角度、关节角速度、关节角加速度矢量；M(θ)＝M₀(θ)+ΔM(θ)，

g(θ)＝g₀(θ)+Δg(θ)，F_s(θ)＝J_i ^TF_i。其中M₀(θ)、

g₀(θ)为模型已知标称部分，ΔM(θ)、

Δg(θ)代表建模误差导致的不确定部分，F_s(θ)为机器人受到的地面广义反力项，τ_d为海流扰动，τ为控制输入。where: θ∈R ³ ,

respectively represent the robot walking foot joint angle, joint angular velocity, and joint angular acceleration vector; M(θ)=M ₀ (θ)+ΔM(θ),

g(θ)=g ₀ (θ)+Δg(θ), F _s (θ)=J _i ^T F _i . where M ₀ (θ),

g ₀ (θ) is the known nominal part of the model, ΔM(θ),

Δg(θ) represents the uncertain part caused by the modeling error, F _s (θ) is the generalized ground reaction force on the robot, τ _d is the current disturbance, and τ is the control input.

为包括模型不确定性、地面广义力以及海流扰动的集总不确定性，代入式(16)得：make

In order to include the model uncertainty, the ground generalized force and the lumped uncertainty of ocean current disturbance, substituting into Equation (16), we get:

in:

ρ=τ _d +F _s (θ) (19)

The above formula has the following properties:

为斜对称矩阵；Property 1: Matrix

is an obliquely symmetric matrix;

Property 2: The matrix M ₀ (θ) is positive definite;

||g₀(θ)||≤g₀成立，c₀、g₀为已知正常数。Property 3: Inequality

||g ₀ (θ)||≤g ₀ holds, and c ₀ and g ₀ are known positive numbers.

||Δg(θ)||≤γ₃；M^-1(θ)存在且有界，M^-1(θ)≤γ₂。Lemma 1: The uncertainty of the system model is unknown but the following bounded conditions are satisfied: ||ΔM(θ)||≤γ ₀ ,

||Δg(θ)||≤γ ₃ ; M ⁻¹ (θ) exists and is bounded, and M ⁻¹ (θ)≤γ ₂ .

Assumption 1: ρ is unknown but bounded, ||ρ||≤γ ₄ .

Lemma 2: Suppose the derivative of d is unknown but bounded, denoted by an unknown constant α.

Among them, γ ₀ , γ ₁ , γ ₂ , γ ₃ , and γ ₄ are known positive numbers.

From the above properties, theorems and assumptions, it can be concluded that:

in:

假设有李雅普诺夫函数V(x)，满足条件(1)V(x)为正定函数，(2)存在任意实数a>0，0＜b＜∞，

以及原点的开放区域U₀∈U，这样

那么系统是有限时间稳定的，且有限收敛时间为

Lemma 3: Consider the following system:

Suppose there is a Lyapunov function V(x), satisfying the conditions (1) V(x) is a positive definite function, (2) there is any real number a>0, 0<b<∞,

and the open area U ₀ ∈ U of the origin, such that

Then the system is stable in finite time, and the finite convergence time is

P2, adaptive finite time disturbance observer:

并且η∈R³满足以下辅助动力学方程：Based on the above assumptions and property conditions of the walking foot dynamics model, an adaptive finite-time disturbance observer is designed, and the sliding variable is defined as

And η∈R ³ satisfies the following auxiliary kinetic equation:

为d的估计值，

为滑模项，且标量

为满足式(17)辅助动力学方程(22)的增益。In the formula: η is a variable that satisfies the above auxiliary equation,

is the estimated value of d,

is the sliding mode term, and the scalar

is the gain of the auxiliary kinetic equation (22) to satisfy equation (17).

为干扰估计误差。则由式(17)和式(22)可得：definition

Estimate the error for the interference. Then from formula (17) and formula (22), we can get:

To estimate and compensate for the uncertainty of the lumped parameters, an adaptive finite-time disturbance observer is designed as follows:

ξ为辅助变量，

为α的估计值，

为观测器增益。σ表示对时间的积分变量，

表示在0到时间t的范围内对g₀(θ)的积分；α集总不确定性d的导数边界值；where:

ξ is an auxiliary variable,

is the estimated value of α,

is the observer gain. σ denotes the integral variable with respect to time,

represents the integral of g ₀ (θ) over the range 0 to time t; the derivative boundary value of α lumped uncertainty d;

From equation (24) and the definition of interference error, we can get:

For the auxiliary kinetic equation (22) and the adaptive finite-time observer system, the following equation is chosen as the unknown gain:

Then the adaptive law is designed as:

将在有限时间内收敛到包含平衡点

在内的有界区域。Among them: γ, δ are normal numbers, then the estimation error

will converge to the inclusive equilibrium point in finite time

bounded area inside.

The proof is as follows:

The first step is to prove that s=0 is reachable in finite time, and the Lyapunov function is taken as:

V ₁ =s ^T M ₀ (θ)s (28)

Derivating the above formula with respect to time gives:

From properties 1 and 3 we get:

From equations (20) and (26), we can get:

内收敛到零，其中V₁(0)为V₁的初始值，此后

始终成立。基于等效变换，将

等效为式(23)中的

即

From the above proof it can be concluded that the sliding variable s can be in finite time

converges to zero, where V ₁ (0) is the initial value of V ₁ and thereafter

always established. Based on the equivalent transformation, the

is equivalent to the formula (23) in

which is

有限时间稳定性，选取李雅普诺夫函数为：The above proves that the state of s=0 is reached in a limited time, and then the interference error needs to be proved

For finite time stability, the Lyapunov function is selected as:

根据式(25)，对V₂关于时间求导：definition

According to Equation ( ₂₅ ), take the derivative of V2 with respect to time:

和自适应律(27)带入上式得：Transform the error equivalently

and the adaptive law (27) into the above equation to get:

Since the following inequality holds:

Substitute into equation (34) to get:

because:

but:

在有限时间内收敛到有界集合

中，其中

干扰误差

的收敛时间为

以上便完成了提出的自适应有限时间干扰观测器的稳定性证明。According to Lemma 3 Lyapunov's finite-time stability theory, the disturbance error

Convergence to a bounded set in finite time

in, of which

interference error

The convergence time of is

The above completes the stability proof of the proposed adaptive finite-time disturbance observer.

P3. Design of global fast terminal sliding mode controller based on AFTDO under input saturation:

Considering the input saturation of the control system, equation (17) can be rewritten as:

Assumption 2: Define Δτ and assume that it satisfies the condition: ||Δτ||=||τ-sat(τ)||≤σ ₀ , where σ ₀ is a known constant.

First, an auxiliary system is designed to deal with the effect of input saturation under assumption 2. The auxiliary system is designed as follows:

The tracking error of joint angular displacement and angular velocity is defined as:

_e =θ-θd (41)

where θ is the actual angle of the robot walking foot joint; θ _d is the expected angle of the robot walking foot joint.

The global fast terminal sliding surface used is:

where ( ) ^p/q represents the power operation, where λ>0, μ>0 is the designed positive diagonal matrix, p and q are odd and p<q, so that the system can quickly reach the equilibrium point in a limited time, and After reaching the sliding mode surface S=0, the error will converge to the equilibrium point e=0 in a finite time, and its convergence time t _s is:

For the defined global fast terminal sliding mode surface, when the error state e is far from the equilibrium point, the linear term λe makes the system rapidly tend to S=0, and when the error state is close to the origin, the convergence rate of the system is mainly determined by the nonlinear term μe ^p/q Therefore, the system state can converge quickly and precisely to the equilibrium point.

Derivation with respect to the defined sliding mode face time gives:

in,

Considering the signal generated by the design auxiliary system, based on the proposed adaptive finite-time disturbance observer, the design input is based on the disturbance observer equation (24), the auxiliary system equation (40), the sliding surface equation (45), and related theorems and assumptions. The fast terminal sliding mode controller based on adaptive finite-time disturbance observer under saturation is:

τ=τ ₀ +τ ₁ +τ ₂ (46)

τ ₂ =-M ₀ (θ)(Aζ+B+σ ₀ Psgn(ζ)) (49)

Among them, τ ₀ is the equivalent control item, τ ₁ is the auxiliary control item of the disturbance observer, and τ ₂ is the saturation compensation item.

其中V(x)表示关于状态x∈R，κ＞0，0＜η＜1的正李雅普诺夫函数，那么对于给定的任意初始条件V(x(0))＝V(0)，函数V(x)可以有限时间内收敛到原点，收敛时间为：

Lemma 4: Given the following first-order nonlinear inequality:

where V(x) represents a positive Lyapunov function with respect to the state x∈R, κ>0, 0<η<1, then for any given initial condition V(x(0))=V(0), the function V(x) can converge to the origin in a finite time, and the convergence time is:

The proof is as follows:

The first step is to prove that the global fast terminal sliding mode controller based on adaptive finite-time disturbance observer is stable. The Lyapunov function is chosen as:

Taking the time derivative of the selected Lyapunov function and substituting it into equation (45), we get:

Substitute the designed control law formula (46) into the above formula to get:

其中χ＞0为一个常数，

为一个正常数使

成立，得到：Among them, (||S ^T sgn(S)|| ₁ ) is the 1-norm, and there is an inequality relationship with the 2-norm (||S||), and select

where χ>0 is a constant,

for a constant

established, get:

以上便完成了快速终端滑模控制器的有限时间稳定性证明。According to Lemma 4, the system can converge in a finite time, and the finite time for convergence is

The above completes the finite-time stability proof of the fast terminal sliding mode controller.

The finite-time convergence of the given auxiliary system after considering input saturation is then demonstrated. The Lyapunov function is chosen as:

According to equation (40), the derivative of _V4 with respect to time is:

According to Lemma 4, the designed auxiliary system (40) can converge in finite time, and the finite convergence time is:

In order to prove the finite-time convergence performance of the global fast terminal sliding mode control method based on adaptive finite-time disturbance observer considering input saturation, it is proved that when the robot controller is designed as equations (47)-(49), the observer and the input are included. The finite time stability of the entire system including saturation.

The Lyapunov function is chosen as:

Derivative with respect to time and according to equations (38), (53), (55), we can get:

in

以上便完成了整个系统的稳定性和有限时间收敛性能的证明。According to Lemma 3, considering the input saturation, the global fast terminal sliding mode controller based on the adaptive finite-time disturbance observer can converge in finite time, and the convergence time is

The above completes the proof of the stability and finite-time convergence performance of the whole system.

Example

δ＝0.1；控制器的参数设置分别为：λ＝diag[5,5,5]，μ＝diag[0.1,0.1,0.1]，k₁＝diag[5,5,5]，k₂＝diag[0.01,0.01,0.01]，p＝3，q＝5；考虑输入饱和情况下设计的辅助系统参数为：A＝diag[5,3,5]，B＝[0.1,0.2,0.3]^T，P＝diag[0.01,0.01,0.02]，σ₀＝100，设置输入饱和界限值为±50Nm；模型参数不确定度为标称系统动力学参数的20％，定义集总不确定性为：d＝[1+0.3sin(0.3t)cos(0.2t),0.5+0.1sin(0.2t)cos(0.1t),0.1+0.1sin(0.2t)]。机器人步行足各关节的角度位置为[θ₁,θ₂,θ₃]＝[30°,45°,90°]，步行足初始位置随机设定。对比分析改进观测器效果以及参考控制算法和改进提出的控制方法的控制性能。In order to verify the control performance of the above joint controller, the dynamic control of the walking foot of the robot is simulated through MATLAB according to the scheme of the specific embodiment 1, and the performance of the designed controller (the present invention) is verified. The parameters of the adaptive finite-time disturbance observer are: c ₀ =50, Θ ₀ =[60,100,120] ^T , γ=1,

δ=0.1; the parameter settings of the controller are: λ=diag[5,5,5], μ=diag[0.1,0.1,0.1], k ₁ =diag[5,5,5], k ₂ =diag [0.01, 0.01, 0.01], p=3, q=5; the auxiliary system parameters designed considering input saturation are: A=diag[5,3,5], B=[0.1,0.2,0.3] ^T , P=diag[0.01,0.01,0.02], σ ₀ =100, set the input saturation limit to ±50Nm; the model parameter uncertainty is 20% of the nominal system dynamic parameters, and the lumped uncertainty is defined as: d =[1+0.3sin(0.3t)cos(0.2t),0.5+0.1sin(0.2t)cos(0.1t),0.1+0.1sin(0.2t)]. The angular position of each joint of the robot walking foot is [θ ₁ , θ ₂ , θ ₃ ]=[30°, 45°, 90°], and the initial position of the walking foot is randomly set. The effect of the improved observer and the control performance of the reference control algorithm and the improved control method are compared and analyzed.

The observation effect of the improved observer and the simulation results of the observation error are shown in Fig. 3 below; Fig. 3 is the performance curve of the adaptive finite-time disturbance observer, in which Fig. 3(a) is the estimated performance of the disturbance d ₁ , and Fig. 3(b) is the The estimation error of disturbance d ₁ ; Fig. 3(c) is the estimation performance of disturbance d ₂ , Fig. 3(d) is the estimation error of disturbance d ₂ ; Fig. 3(e) is the estimation performance of disturbance d ₃ , Fig. 3(f) is the estimation performance of disturbance d ₃ Estimation error.

It can be seen from Fig. 3 that both the reference observer method and the improved observer method can better estimate the observation uncertainty disturbance d, and the final estimation error can keep oscillating around the zero point, and the oscillation range is small; but the improved observer performance The effect is better than the performance of the reference observer, and the improved observer method is better than the reference observer method in terms of estimation speed and estimation error, so the effectiveness of the improved adaptive finite time disturbance observer is verified.

The following verifies the tracking control performance when each joint moves according to the desired trajectory. The displacement curves of the three joints are set as sine curves, and the amplitudes of each joint are different, namely θ ₁ =30sin(t), θ2 _{= 45sin} (t), and θ3 =90sin(t), and the simulation results are shown in 4. Figure 4 shows the control response curve of robot walking trajectory tracking, in which Figure 4(a) is the trajectory tracking performance of joint 1, Figure 4(b) is the trajectory tracking error of joint 1; Figure 4(c) is the trajectory tracking performance of joint 2. 4(d) is the tracking error of joint 2; Fig. 4(e) is the tracking performance of joint 3, and Fig. 4(f) is the tracking error of joint 3.

According to the simulation results in Fig. 4, both controllers have good tracking control performance, but in the initial stage, the improved controller has faster response speed, better convergence performance than the reference controller, and smaller tracking error. , which can better meet the requirements of the robot's walking foot motion joint for rapidity and accuracy, which verifies the effectiveness of the improved controller.

The present invention can also have other various embodiments. Without departing from the spirit and essence of the present invention, those skilled in the art can make various corresponding changes and deformations according to the present invention, but these corresponding changes and deformations are all It should belong to the protection scope of the appended claims of the present invention.

Claims (6)

1. The anti-saturation finite-time motion control method for the walking feet of the amphibious crab-imitating multi-legged robot is characterized by comprising the following steps of:

s1, establishing a robot walking foot dynamics model aiming at the amphibious crab-imitating multi-legged robot:

in the formula:

for lumped uncertainty, θ ∈ R³、

Respectively representing the joint angle, the joint angular velocity and the joint angular acceleration vector of the walking foot of the robot; wherein M is₀(θ)、

g₀(theta) is a known nominal part of the model, and respectively represents a positive definite inertia matrix, a Coriolis force and centrifugal force term, and a restoring force term generated by a gravity and buoyancy term, wherein delta M (theta),

Δ g (θ) corresponds to the uncertainty due to modeling error, F_s(theta) is a ground generalized reaction force term, tau, received by the robot_dFor ocean current disturbance, tau is the expected control force/moment input, sat (-) is the saturation function;

s2, determining a self-adaptive finite time interference observer based on the walking foot dynamics model of the amphibious crab-imitating multi-legged robot:

sliding variable

η∈R³Variables that satisfy the following auxiliary kinetic equations;

in the formula:

is an estimate of the value of d,

in order to form the item of the sliding mode,

is the gain of the auxiliary kinetic equation;

definition of

For interference estimation errors, we can obtain:

the adaptive finite time disturbance observer is as follows:

in the formula:

xi is an auxiliary variable which is a variable,

is an estimated value of a and is,

is the observer gain; σ represents an integral variable over time t; the derivative boundary value of the alpha lumped uncertainty d;

the adaptive finite time disturbance observer and the disturbance error can be used to obtain:

the adaptive law is:

wherein γ and δ are normal numbers;

s3, processing the influence of input saturation by using an auxiliary system, wherein the auxiliary system comprises the following steps:

wherein ζ is (ζ)₁,ζ₂,ζ₃)^TFor the state vector of the auxiliary system, A ═ diag { a }_i}_3×3、B＝[b₁,b₂,b₃]^TIs a parameter matrix and a parameter vector, wherein a_i＞0，b_i＞0，i＝1,2,3；sgn(ζ)＝(sgn(ζ₁),sgn(ζ₂),sgn(ζ₃))^TSgn (·) is a sign function; p ═ diag { | | | P_i||}_3×3Wherein p is_iFor controlling the gain matrix M₀ ^-1Row i of (1);

s4, controlling the walking foot motion of the robot by using a fast terminal sliding mode controller based on an adaptive finite time interference observer (AFTDO) under input saturation;

the fast terminal sliding mode controller based on the adaptive finite time interference observer AFTDO under the input saturation comprises the following steps:

τ＝τ₀+τ₁+τ₂

τ₂＝-M₀(θ)(Aζ+B+σ₀Psgn(ζ))

wherein, tau₀For equivalent control terms, τ₁For auxiliary control terms of disturbance observer, τ₂Is a saturation compensation term; e.

respectively a joint angular displacement tracking error and an angular velocity tracking error; s is a global rapid terminal sliding mode surface; lambda is more than 0, mu is more than 0 and is a positive diagonal matrix; p and q are odd numbers and p < q; k is a radical of₁,k₂The control parameter is more than 0;

the global quick terminal sliding mode surface is as follows:

wherein, (.)^p/qRepresenting an exponentiation.

2. The amphibious crab-imitating multi-legged robot walking foot anti-saturation finite-time motion control method according to claim 1, wherein the joint angular displacement tracking error and the angular velocity tracking error are respectively as follows:

e＝θ-θ_d

wherein, theta_dThe desired angle is the robot walking foot joint.

3. The amphibious crab-imitating multi-legged robot walking foot anti-saturation finite time motion control method according to claim 2, assisting gain of kinetic equation

The following were used:

wherein, c₀、γ₀、γ₁、γ₂、γ₃、γ₄They are all normal numbers.

4. The amphibious crab-imitating multi-legged robot walking foot anti-saturation limited time according to claim 3Inter-motion control method, observer gain

5. The method according to claim 4, wherein sat (τ) ═ sat (τ) in the kinetic model of the walking foot of the robot is [ sat (τ) ]₁),sat(τ₂),sat(τ₃)]^TSat (. cndot.) is the saturation function:

wherein i is 1,2,3, τ_max、τ_minMaximum and minimum control force/torque inputs, respectively.

6. The amphibious crab-imitating multi-legged robot walking foot anti-saturation finite time motion control method according to one of claims 1 to 5, wherein the process of establishing a robot walking foot dynamics model for the amphibious crab-imitating multi-legged robot comprises the following steps:

firstly, a robot mechanical arm dynamic model is constructed:

in the formula, theta ∈ R³、

Respectively representing the joint angle, the joint angular velocity and the joint angular acceleration vector of the walking foot of the robot; m (theta) ═ M₀(θ)+ΔM(θ)，

g(θ)＝g₀(θ) + Δ g (θ); wherein M is₀(θ)、

g₀(θ) is the nominal portion of the model known, Δ M (θ),

Δ g (θ) represents the uncertainty due to modeling error, F_s(theta) is a ground generalized reaction force term, tau, received by the robot_dDisturbance of ocean currents;

will assemble uncertainty

Introducing a mechanical arm dynamic model of the robot to obtain:

determining a robot walking foot dynamics model based on input saturation

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