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

Anti-saturation finite-time motion control method for walking feet of amphibious crab-imitating multi-foot robot Download PDF

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CN112987770B
CN112987770B CN202110217281.6A CN202110217281A CN112987770B CN 112987770 B CN112987770 B CN 112987770B CN 202110217281 A CN202110217281 A CN 202110217281A CN 112987770 B CN112987770 B CN 112987770B
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robot
saturation
walking foot
observer
theta
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CN112987770A (en
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杜雨桐
范金龙
万磊
孙延超
秦洪德
陈欣岩
李凌宇
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Harbin Engineering University
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    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course or altitude of land, water, air, or space vehicles, e.g. automatic pilot
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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 finite-time motion control method for walking feet of amphibious crab-imitating multi-foot robot
Technical Field
The invention relates to a motion control method of a multi-legged robot. Belongs to the technical field of control.
Background
In the world, science and technology are rapidly developed, and the development demand of ocean resources is increasingly expanded. Aiming at a plurality of tasks in the fields of offshore facility inspection, marine resource exploration and data acquisition, offshore investigation, defense, rescue and the like under complex environments such as offshore seabed, shoal and the like, an amphibious crab-like multi-legged robot needs to be developed to meet the operation task requirements. As shown in fig. 1.
The design and research of the amphibious crab-imitating multi-legged robot are crossed and fused with multi-subject and multi-field technologies, the motion control problem in the crawling mode is a key content in the research subject of the amphibious crab-imitating multi-legged robot and is a key technical difficulty in the research content of the robot, the reliability of the motion control technology determines the operation efficiency and the intelligent level of the amphibious crab-imitating multi-legged robot, and the motion control technology is an important premise for ensuring that the robot completes a specific operation task.
The amphibious crab-imitating multi-legged robot has the characteristics of multiple joints, nonlinearity, multiple redundancies, time-varying characteristics and the like, so that a kinematic and dynamic motion model of the amphibious crab-imitating multi-legged robot has high uncertainty. Firstly, the robot needs to overcome the influence caused by factors such as model uncertainty caused by self modeling error and disturbance of ocean current during the walking movement of the seabed in the movement process; meanwhile, the problem of input saturation is also caused by the nonlinear characteristic of an actuator system in the actual control process, and the problems all put higher requirements on the control precision and the response speed of the walking foot of the robot. How to effectively improve the tracking control precision and speed of the walking foot track of the robot and how to ensure that the robot realizes stable high-precision rapid control under the condition of saturated actuator is a key problem to be solved in the motion control research of the amphibious crab-imitating multi-foot robot.
Disclosure of Invention
The invention aims to solve the problems of poor precision and low speed of the existing walking foot trajectory tracking control of the crab-imitating multi-legged robot.
An anti-saturation finite-time motion control method for a walking foot of an amphibious crab-imitating multi-legged robot comprises the following steps:
s1, establishing a robot walking foot dynamics model aiming at the amphibious crab-imitating multi-legged robot:
Figure GDA0003200084050000011
in the formula:
Figure GDA0003200084050000012
for lumped uncertainty, θ ∈ R3、
Figure GDA0003200084050000013
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)、
Figure GDA0003200084050000014
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),
Figure GDA0003200084050000015
Δ g (θ) corresponds to the uncertainty due to modeling error, Fs(theta) is a ground generalized reaction force term, tau, received by the robotdFor 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
Figure GDA0003200084050000021
η∈R3Variables that satisfy the following auxiliary kinetic equations;
Figure GDA0003200084050000022
in the formula:
Figure GDA0003200084050000023
is an estimate of the value of d,
Figure GDA0003200084050000024
in order to form the item of the sliding mode,
Figure GDA0003200084050000025
is the gain of the auxiliary kinetic equation;
definition of
Figure GDA0003200084050000026
For interference estimation errors, we can obtain:
Figure GDA0003200084050000027
the adaptive finite time disturbance observer is as follows:
Figure GDA0003200084050000028
in the formula:
Figure GDA0003200084050000029
xi is an auxiliary variable which is a variable,
Figure GDA00032000840500000210
is an estimated value of a and is,
Figure GDA00032000840500000211
is the observer gain; σ represents an integral variable over time t; the derivative boundary value of the alpha lumped uncertainty d;
the formula self-adaptive finite time disturbance observer and the disturbance error can be obtained:
Figure GDA00032000840500000212
the adaptive law is:
Figure GDA00032000840500000213
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:
Figure GDA00032000840500000214
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 ai＞0，bi＞0，i＝1,2,3；sgn(ζ)＝(sgn(ζ1),sgn(ζ2),sgn(ζ3))TSgn (·) is a sign function; p ═ diag { | | | Pi||}3×3Wherein p isiFor 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
Figure GDA00032000840500000215
Figure GDA00032000840500000216
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.
Figure GDA0003200084050000031
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,k2And > 0 is a control parameter.
Further, the joint angular displacement tracking error and the angular velocity tracking error are respectively as follows:
e＝θ-θd
Figure GDA0003200084050000032
wherein, thetadThe desired angle is the robot walking foot joint.
Further, the global fast terminal sliding mode surface is as follows:
Figure GDA0003200084050000033
wherein, (.)p/qRepresenting an exponentiation.
Further, the gain of the auxiliary kinetic equation
Figure GDA0003200084050000034
The following were used:
Figure GDA0003200084050000035
Figure GDA0003200084050000036
wherein, c0、γ0、γ1、γ2、γ3、γ4They are all normal numbers.
Further, the observer gains
Figure GDA0003200084050000037
Further, sat (τ) ═ sat (τ) in the robot walking foot dynamics model1),sat(τ2),sat(τ3)]TSat (. cndot.) is the saturation function:
Figure GDA0003200084050000038
wherein i is 1,2,3, τmax、τminMaximum and minimum control force/torque inputs, respectively.
Further, the process of establishing the robot walking foot dynamics model for the amphibious crab-like multi-legged robot comprises the following steps:
firstly, a robot mechanical arm dynamic model is constructed:
Figure GDA0003200084050000039
in the formula, theta ∈ R3、
Figure GDA00032000840500000310
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)，
Figure GDA0003200084050000041
g(θ)＝g0(θ) + Δ g (θ); wherein M is0(θ)、
Figure GDA0003200084050000042
g0(θ) is the nominal portion of the model known, Δ M (θ),
Figure GDA0003200084050000043
Δ g (θ) represents the uncertainty due to modeling error, Fs(theta) is a ground generalized reaction force term, tau, received by the robotdDisturbance of ocean currents;
will assemble uncertainty
Figure GDA0003200084050000044
Introducing a mechanical arm dynamic model of the robot to obtain:
Figure GDA0003200084050000045
determining a robot walking foot dynamics model based on input saturation
Figure GDA0003200084050000046
Has the advantages that:
aiming at the problem of tracking and controlling the walking foot trajectory of the amphibious crab-imitating multi-legged robot, model parameter uncertainty and ocean current disturbance uncertainty under the condition of input saturation are comprehensively considered, a self-Adaptive Finite Time Disturbance Observer (AFTDO) is provided to solve the problem of lumped uncertainty observation generated by ocean current disturbance and modeling errors, a control method independent of an accurate dynamic model is adopted, a global fast terminal sliding mode controller based on the self-Adaptive Finite Time disturbance observer and considering input saturation is provided, and the walking foot trajectory tracking and controlling of the amphibious crab-imitating multi-legged robot are realized.
The AFTDO is used for processing uncertainty of parameters of a system model and uncertainty of ocean current disturbance into lumped uncertainty, constructing an auxiliary dynamic equation, designing a self-adaptive law, improving and designing a self-adaptive finite time disturbance observer and realizing rapid and accurate estimation of the lumped uncertainty.
The global fast terminal sliding mode control method based on the AFTDO under the input saturation is characterized in that on the basis of a self-adaptive limited time disturbance observer, the advantages that the global fast terminal sliding mode control is fast and the global fast terminal sliding mode control reaches a sliding mode surface within a limited time are combined, the input saturation condition is considered, an input saturation auxiliary system is constructed, a fast terminal sliding mode controller based on the self-adaptive limited time disturbance observer under the input saturation condition is designed, and the walking foot track tracking control precision and the response speed of the robot are improved.
Drawings
FIG. 1 is a schematic diagram of an amphibious crab-imitating multi-legged robot;
FIG. 2 is a walking foot model of an amphibious crab-imitating multi-foot robot;
FIG. 3 is a graph of adaptive finite time disturbance observer performance, where FIG. 3(a) is the disturbanced1Estimate Performance, FIG. 3(b) is disturbance d1Estimating an error; FIG. 3(c) shows a disturbance d2Estimate Performance, FIG. 3(d) is disturbance d2Estimating an error; FIG. 3(e) shows a disturbance d3Estimate Performance, FIG. 3(f) is disturbance d3Estimating an error;
fig. 4 is a response curve of robot walking trajectory tracking control, in which fig. 4(a) shows joint 1 trajectory tracking performance and fig. 4(b) shows joint 1 trajectory tracking error; fig. 4(c) is the joint 2 trajectory tracking performance, and fig. 4(d) is the joint 2 trajectory tracking error; fig. 4(e) shows the joint 3 trajectory tracking performance, and fig. 4(f) shows the joint 3 trajectory tracking error.
Detailed Description
Before describing the embodiments, the parameter definitions of the present invention will be described below:
M0(θ) — a positive definite inertia matrix;
Figure GDA0003200084050000051
-the terms coriolis force and centrifugal force; g0(θ) -a restoring force term resulting from the gravity and buoyancy terms; theta is formed by R3、
Figure GDA0003200084050000052
-the robot walking foot joint angle, joint angular velocity, joint angular acceleration vector; e-thetad-a tracking error; τ — control input; tau isd-disturbance of the ocean current; d-lumped uncertainties including model uncertainty, ground generalized forces, and ocean current disturbances.
The first embodiment is as follows:
the anti-saturation finite-time motion control method for the walking feet of the amphibious crab-imitating multi-legged robot in the embodiment comprises the following steps of:
s1, carrying out amphibious crab-imitating multi-legged robot walking foot kinetic model transformation based on the amphibious crab-imitating multi-legged robot walking foot kinetic model:
a motion model of the walking foot of the amphibious crab-imitating multi-foot robot is shown in figure 2, and a coordinate system is established for the walking foot, wherein the fixed coordinate system of the walking foot is O-X0Y0Z0Coordinate system of hip joint is O-X1Y1Z1Coordinate system of femoral joint is O-X2Y2Z2The tibia joint coordinate system is O-X3Y3Z3The coordinate system of the end point of the walking foot is O-X4Y4Z4And the walking foot fixing coordinate system is also the coordinate system of the joint of the walking foot and the robot body.
The walking foot dynamics equation of the amphibious crab-imitating multi-legged robot adopts a robot mechanical arm dynamics model deduced based on an Euler-Lagrange method:
Figure GDA0003200084050000053
in the formula: theta is formed by R3、
Figure GDA0003200084050000054
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)，
Figure GDA0003200084050000055
g(θ)＝g0(θ)+Δg(θ)，Fs(θ)＝Ji TFi. Wherein M is0(θ)、
Figure GDA0003200084050000056
g0(θ) is the nominal portion of the model known, Δ M (θ),
Figure GDA0003200084050000057
Δ g (θ) represents the uncertainty due to modeling error, Fs(theta) is a ground generalized reaction force term, tau, received by the robotdFor disturbance of ocean currents, Ji TA Jacobi transpose matrix for the ith walking foot of the robot; fi＝[fix,fiy,fiz]TThe reaction vector received by the ith walking foot in the support phase.
Order to
Figure GDA0003200084050000058
For lumped uncertainty, we have the following formula (1):
Figure GDA0003200084050000059
because the walking foot of the robot is not influenced by the action of ground force in the swinging process, in addition, the robot can generate model uncertainty and ocean current disturbance influence in the moving process due to modeling errors, and meanwhile, because the actuator system has the nonlinear characteristic of input saturation, a system controller can not continuously respond to a system error signal, and the controller can not normally play a role further. Therefore, the invention considers the lumped uncertainty and input saturation condition generated by the robot in ocean current interference and modeling error, and establishes the robot walking foot dynamics model as follows:
Figure GDA0003200084050000061
in the formula:
Figure GDA0003200084050000062
for lumped uncertainty, θ ∈ R3、
Figure GDA0003200084050000063
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)，
Figure GDA0003200084050000064
g(θ)＝g0(θ)+Δg(θ)，Fs(θ)＝Ji TFi. Wherein M is0(θ)、
Figure GDA0003200084050000065
g0(θ) is the nominal portion of the model known, Δ M (θ),
Figure GDA0003200084050000066
Δ g (θ) represents the uncertainty due to modeling error, Fs(theta) is a ground generalized reaction force term, tau, received by the robotdFor ocean current disturbances, sat (τ) ═ sat (τ)1),sat(τ2),sat(τ3)]Tτ is the desired control force/torque input for the design, sat (-) is the saturation function:
Figure GDA0003200084050000067
s2, designing a self-adaptive finite time interference observer based on a walking foot dynamics model of the amphibious crab-imitating multi-legged robot:
defining a sliding variable as
Figure GDA0003200084050000068
And η ∈ R3The following auxiliary kinetic equations are satisfied:
Figure GDA0003200084050000069
in the formula:
Figure GDA00032000840500000610
is an estimate of the value of d,
Figure GDA00032000840500000611
is a sliding mode term and is scalar
Figure GDA00032000840500000612
To assist the gain of equation (5) for kinetics.
Definition of
Figure GDA00032000840500000613
An error is estimated for the interference. Then, the following equations (2) and (5) can be obtained:
Figure GDA00032000840500000614
to estimate and compensate for the uncertainty of the lumped parameters, an adaptive finite time disturbance observer was designed as follows:
Figure GDA00032000840500000615
in the formula:
Figure GDA00032000840500000616
xi is an auxiliary variable which is a variable,
Figure GDA00032000840500000617
is an estimated value of a and is,
Figure GDA00032000840500000618
is the observer gain.
Defined by equation (7) and the interference error:
Figure GDA00032000840500000619
for the auxiliary dynamics equation (5) and the adaptive finite time observer system, the following equation is chosen as the unknown gain:
Figure GDA00032000840500000620
the adaptive law is designed to be:
Figure GDA0003200084050000071
wherein: gamma and delta are normal numbers, then the error is estimated
Figure GDA0003200084050000072
Will converge to the inclusion equilibrium point in a limited time
Figure GDA0003200084050000073
The inner bounded region.
S3, designing an input saturation auxiliary system:
the system (3) takes the system input saturation problem into consideration, and for an actual control system, the difference delta tau between the control input tau and the actual control input sat (tau) is expected to be small enough, because the control input is saturated and the controllability of the system is also met. Since the disturbance and system state are bounded, the required control inputs are bounded. To satisfy this assumption, the parameter σ0May be larger. Δ τ is defined and assumed to satisfy the condition: the | | | delta tau | | | tau-sat (tau) | | is less than or equal to sigma |0Where σ is0Is a known constant.
Design assistance systems to handle the effects of input saturation are as follows:
Figure GDA0003200084050000074
wherein ζ is (ζ)1,ζ2,ζ3)TTo assist the state variables of the system, A ═ diag { a }i}3×3And B ═ B1,b2,b3]TTo design matrices and design vectors, where ai＞0，bi> 0, i ═ 1,2, 3. Further, sgn (ζ) is (sgn (ζ)1),sgn(ζ2),sgn(ζ3))TSgn (·) is a sign function; p ═ diag { | | | Pi||}3×3Wherein p isiFor controlling the gain matrix M0 -1Row i of (2). The auxiliary state quantity converges to zero within a finite time.
S4, designing a fast terminal sliding mode controller based on an Adaptive Finite Time Disturbance Observer (AFTDO) under input saturation based on lumped uncertainty generated by the robot in ocean current disturbance and modeling error and the condition of input saturation, wherein the fast terminal sliding mode controller comprises the following steps:
τ＝τ0+τ1+τ2 (12)
Figure GDA0003200084050000075
Figure GDA0003200084050000076
τ2＝-M0(θ)(Aζ+B+σ0Psgn(ζ)) (15)
wherein tau is0For equivalent control terms, τ1For auxiliary control terms of disturbance observer, τ2For the saturation compensation term, k1,k2And > 0 is a control parameter.
To fully illustrate the inventive innovations, the design process and principles of the inventive controller are now described as follows:
p1: a robot mechanical arm dynamic model deduced based on an Euler-Lagrange method is adopted to establish a walking foot dynamic equation of the amphibious crab-imitating multi-legged robot:
Figure GDA0003200084050000077
in the formula: theta is formed by R3、
Figure GDA0003200084050000078
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)，
Figure GDA0003200084050000081
g(θ)＝g0(θ)+Δg(θ)，Fs(θ)＝Ji TFi. Wherein M is0(θ)、
Figure GDA0003200084050000082
g0(θ) is the nominal portion of the model known, Δ M (θ),
Figure GDA0003200084050000083
Δ g (θ) represents the uncertainty due to modeling error, Fs(theta) is a ground generalized reaction force term, tau, received by the robotdFor ocean current disturbances, τ is the control input.
Order to
Figure GDA0003200084050000084
For lumped uncertainties including model uncertainty, ground generalized forces, and ocean current disturbances, equation (16) is substituted for:
Figure GDA0003200084050000085
wherein:
Figure GDA0003200084050000086
ρ＝τd+Fs(θ) (19)
the above formula has the following properties:
properties 1: matrix array
Figure GDA0003200084050000087
Is an oblique symmetric matrix;
properties 2: matrix M0(θ) positive definite;
properties 3: inequality
Figure GDA0003200084050000088
||g0(θ)||≤g0Is established, c0、g0Known as normal.
Introduction 1: the system model uncertainty is unknown but satisfies the following bounded conditions: gamma is less than or equal to | delta M (theta) |0、
Figure GDA0003200084050000089
||Δg(θ)||≤γ3；M-1(θ) present and bounded, M-1(θ)≤γ2。
Assume that 1: rho is unknown but bounded, | rho | | | is less than or equal toγ4。
2, leading: let d be unknown but bounded, represented by the unknown constant α.
Wherein, γ0、γ1、γ2、γ3、γ4Known as normal.
From the above properties, theorems and assumptions it follows that:
Figure GDA00032000840500000810
wherein:
Figure GDA00032000840500000811
and 3, introduction: consider the following system:
Figure GDA00032000840500000812
assuming a Lyapunov function V (x), satisfying the condition (1) that V (x) is a positive definite function, (2) any real number a exists>0，0＜b＜∞，
Figure GDA00032000840500000813
And an open area U of the origin0E.g. U, such that
Figure GDA00032000840500000814
Then the system is time-limited stable and has a limited convergence time of
Figure GDA0003200084050000091
P2, adaptive finite time disturbance observer:
based on the assumptions and property conditions above the walking foot dynamics model, an adaptive finite time disturbance observer is designed, and a sliding variable is defined as
Figure GDA0003200084050000092
And η ∈ R3The following auxiliary kinetic equations are satisfied:
Figure GDA0003200084050000093
in the formula: η is a variable that satisfies the auxiliary equation described above,
Figure GDA0003200084050000094
is an estimate of the value of d,
Figure GDA0003200084050000095
is a sliding mode term and is scalar
Figure GDA0003200084050000096
To satisfy the gain of the auxiliary kinetic equation (22) of equation (17).
Definition of
Figure GDA0003200084050000097
An error is estimated for the interference. Then, the following equations (17) and (22) can be obtained:
Figure GDA0003200084050000098
to estimate and compensate for the uncertainty of the lumped parameters, an adaptive finite time disturbance observer was designed as follows:
Figure GDA0003200084050000099
in the formula:
Figure GDA00032000840500000910
xi is an auxiliary variable which is a variable,
Figure GDA00032000840500000911
is an estimated value of a and is,
Figure GDA00032000840500000912
is the observer gain. Sigma denotes the integral variation over timeThe amount of the compound (A) is,
Figure GDA00032000840500000913
represents the range of 0 to time t for g0(θ) integration; the derivative boundary value of the alpha lumped uncertainty d;
defined by equation (24) and the interference error:
Figure GDA00032000840500000914
for the auxiliary dynamics equation (22) and the adaptive finite time observer system, the following equation is chosen as the unknown gain:
Figure GDA00032000840500000915
the adaptive law is designed to be:
Figure GDA00032000840500000916
wherein: gamma and delta are normal numbers, then the error is estimated
Figure GDA00032000840500000917
Will converge to the inclusion equilibrium point in a limited time
Figure GDA00032000840500000918
The inner bounded region.
The following was demonstrated:
the first step requires that s-0 is proved to be reachable within a finite time, taking the lyapunov function as:
V1＝sTM0(θ)s (28)
the above equation is derived with respect to time:
Figure GDA0003200084050000101
obtained from properties 1, 3:
Figure GDA0003200084050000102
from formulae (20) and (26):
Figure GDA0003200084050000103
from the above proof, it can be concluded that the sliding variable s can be in a finite time
Figure GDA0003200084050000104
Inner convergence to zero, where V1(0) Is a V1From the initial value of (d), thereafter
Figure GDA0003200084050000105
This is always true. Based on an equivalent transformation, will
Figure GDA0003200084050000106
Equivalent to that in formula (23)
Figure GDA0003200084050000107
Namely, it is
Figure GDA0003200084050000108
The above demonstrates that the state where s ═ 0 is reached within a finite time, and that interference errors need to be demonstrated next
Figure GDA0003200084050000109
The finite time stability is obtained by selecting a Lyapunov function as follows:
Figure GDA00032000840500001010
definition of
Figure GDA00032000840500001011
According to formula (25), for V2With respect to time derivation:
Figure GDA00032000840500001012
equivalently transforming the error
Figure GDA00032000840500001013
And the adaptation law (27) is substituted by the equation:
Figure GDA00032000840500001014
since the following inequality holds:
Figure GDA00032000840500001015
substituting formula (34) to obtain:
Figure GDA0003200084050000111
because:
Figure GDA0003200084050000112
then:
Figure GDA0003200084050000113
according to the theory of 3 Lyapunov finite time stability, interference error
Figure GDA0003200084050000114
Convergence to bounded sets in a finite time
Figure GDA0003200084050000115
In which
Figure GDA0003200084050000116
Interference error
Figure GDA0003200084050000117
Has a convergence time of
Figure GDA0003200084050000118
The above completes the stability certification of the proposed adaptive finite time disturbance observer.
P3, designing a global fast terminal sliding mode controller based on AFTDO under input saturation:
considering the control system input saturation, equation (17) can be rewritten as:
Figure GDA0003200084050000119
assume 2: Δ τ is defined and assumed to satisfy the condition: the | | | delta tau | | | tau-sat (tau) | | is less than or equal to sigma |0Where σ is0Is a known constant.
An auxiliary system is first designed to handle the effects of input saturation under assumption 2, and is designed as follows:
Figure GDA00032000840500001110
the tracking error of the angular displacement and the angular velocity of the joint at the moment is defined as:
e＝θ-θd (41)
Figure GDA00032000840500001111
wherein theta is the actual angle of the joint of the walking foot of the robot; thetadThe desired angle is the robot walking foot joint.
The adopted global fast terminal sliding mode surface is as follows:
Figure GDA00032000840500001112
wherein (·)p/qRepresenting an exponentiation, where λ > 0, μ > 0 is the designed positive diagonal matrix, p and q are odd numbers and p < q, so that the system can reach the equilibrium point quickly within a finite time, and after reaching the sliding mode plane S-0, the error will converge to the equilibrium point e-0 within a finite time, with a convergence time t of 0sComprises the following steps:
Figure GDA0003200084050000121
for a defined global fast terminal sliding mode surface, when an error state e is far away from an equilibrium point, a linear term λ e enables the system to quickly approach S to 0, and when the error state is close to an origin point, the convergence rate of the system is mainly controlled by a nonlinear term μ ep/qThus, the system state can be quickly and accurately converged to the equilibrium point.
The time is derived for a defined sliding mode pair:
Figure GDA0003200084050000122
wherein the content of the first and second substances,
Figure GDA0003200084050000123
considering signals generated by designing an auxiliary system, based on the proposed adaptive finite time disturbance observer, according to a disturbance observer formula (24), an auxiliary system formula (40), a sliding mode surface formula (45) and relevant theorems and assumptions, designing a fast terminal sliding mode controller based on the adaptive finite time disturbance observer under input saturation as follows:
τ＝τ0+τ1+τ2 (46)
Figure GDA0003200084050000124
Figure GDA0003200084050000125
τ2＝-M0(θ)(Aζ+B+σ0Psgn(ζ)) (49)
wherein tau is0For equivalent control terms, τ1For auxiliary control terms of disturbance observer, τ2Is a saturation compensation term.
And (4) introduction: the following first order non-linear inequality is given:
Figure GDA0003200084050000126
where V (x) represents the positive lyapunov function for 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 within a finite time, the convergence time being:
Figure GDA0003200084050000127
the following was demonstrated:
the first step is to prove that the global fast terminal sliding mode controller based on the self-adaptive finite time disturbance observer is stable. The Lyapunov function is chosen to be:
Figure GDA0003200084050000128
and (3) the selected Lyapunov function is derived for time and substituted into the formula (45) to obtain:
Figure GDA0003200084050000131
substituting the designed control law equation (46) into the equation:
Figure GDA0003200084050000132
wherein, (| S)T sgn(S)||1) Is 1 norm, has an inequality relation with 2 norms (| | S | |), and is selected
Figure GDA0003200084050000133
Wherein χ > 0 is a constant,
Figure GDA0003200084050000134
is a normal number
Figure GDA0003200084050000135
If true, we get:
Figure GDA0003200084050000136
according to theorem 4, the system is able to converge for a limited time, the limited time of convergence being
Figure GDA0003200084050000137
Therefore, the proof of the limited time stability of the fast terminal sliding mode controller is completed.
The finite time convergence of the auxiliary system given after input saturation is then demonstrated. The Lyapunov function is chosen to be:
Figure GDA0003200084050000138
according to formula (40), V4The derivative with respect to time is:
Figure GDA0003200084050000139
according to theorem 4, the auxiliary system (40) is designed to converge within a limited time, and the limited convergence time is:
Figure GDA00032000840500001310
in order to prove the finite time convergence performance of the global fast terminal sliding mode control method based on the adaptive finite time disturbance observer considering the input saturation, namely to prove the finite time stability of the whole system including the observer and the input saturation when the robot controller is designed into the formulas (47) - (49).
The Lyapunov function is chosen to be:
Figure GDA0003200084050000141
the time derivative is obtained from equations (38), (53), (55):
Figure GDA0003200084050000142
wherein
Figure GDA0003200084050000143
According to the theorem 3, the global fast terminal sliding mode controller based on the adaptive finite time disturbance observer under the condition of considering input saturation can be converged in finite time, and the convergence time is
Figure GDA0003200084050000144
Figure GDA0003200084050000145
The above completes the verification of the stability and the limited time convergence performance of the whole system.
Examples
In order to verify the control performance of the above joint controller, the dynamic control of the robot walking foot is simulated by MATLAB according to the scheme of the first embodiment, and the performance of the designed controller (the invention) is verified. The parameters of the self-adaptive finite time disturbance observer are respectively as follows: c. C0＝50，Θ0＝[60,100,120]T，γ＝1，
Figure GDA0003200084050000146
δ is 0.1; the parameter settings of the controller are respectively: λ ═ diag [5,5 ]]，μ＝diag[0.1,0.1,0.1]，k1＝diag[5,5,5]，k2＝diag[0.01,0.01,0.01]P is 3, q is 5; the design auxiliary system parameters under the input saturation condition are considered as follows: a ═ diag [5,3,5 ]]，B＝[0.1,0.2,0.3]T，P＝diag[0.01,0.01,0.02]，σ0Setting the input saturation limit value to be +/-50 Nm (100); the model parameter uncertainty is 20% of the nominal system dynamics 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 walking foot of the robot is [ theta ]1,θ2,θ3]＝[30°,45°,90°]The initial position of the walking foot is set randomly. The comparative analysis improves the observer effect as well as the control performance of the reference control algorithm and the proposed control method.
The observation effect of the improved observer and the simulation result of the observation error are shown in the following figure 3; FIG. 3 is a graph of the performance of an adaptive finite time disturbance observer, where d is the disturbance in FIG. 3(a)1Estimate Performance, FIG. 3(b) is disturbance d1Estimating an error; FIG. 3(c) shows a disturbance d2Estimate Performance, FIG. 3(d) is disturbance d2Estimating an error; FIG. 3(e) shows a disturbance d3Estimate Performance, FIG. 3(f) is disturbance d3An error is estimated.
As can be seen from FIG. 3, the reference observer method and the improved observer method can better estimate the observation uncertain disturbance d, and finally, the estimation error can be kept to oscillate near the zero point, and the oscillation range is small; but the improved observer performance effect is better than the reference observer performance, and the improved observer method has better effect than the reference observer method in the aspects of estimation speed and estimation error, so that 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. Setting the displacement curve of three joints as sine curve, the amplitude of each joint is different, and is theta1＝30sin(t)，θ2＝45sin(t)，θ390sin (t), simulation results such as4 is shown in the specification; fig. 4 is a response curve of robot walking trajectory tracking control, in which fig. 4(a) shows joint 1 trajectory tracking performance and fig. 4(b) shows joint 1 trajectory tracking error; fig. 4(c) is the joint 2 trajectory tracking performance, and fig. 4(d) is the joint 2 trajectory tracking error; fig. 4(e) shows the joint 3 trajectory tracking performance, and fig. 4(f) shows the joint 3 trajectory tracking error.
According to the simulation result of fig. 4, both controllers have better tracking control performance, but in the initial stage, the improved controller has higher response speed, better convergence performance than the reference controller, smaller tracking error, and better satisfaction of the requirements of the robot walking foot motion joint on rapidity and precision, and the effectiveness of the improved controller is verified.
The present invention is capable of other embodiments and its several details are capable of modifications in various obvious respects, all without departing from the spirit and scope 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:
Figure FDA0003200084040000011
in the formula:
Figure FDA0003200084040000012
for lumped uncertainty, θ ∈ R3、
Figure FDA0003200084040000013
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)、
Figure FDA0003200084040000014
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),
Figure FDA0003200084040000015
Δ g (θ) corresponds to the uncertainty due to modeling error, Fs(theta) is a ground generalized reaction force term, tau, received by the robotdFor 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
Figure FDA0003200084040000016
η∈R3Variables that satisfy the following auxiliary kinetic equations;
Figure FDA0003200084040000017
in the formula:
Figure FDA0003200084040000018
is an estimate of the value of d,
Figure FDA0003200084040000019
in order to form the item of the sliding mode,
Figure FDA00032000840400000110
is the gain of the auxiliary kinetic equation;
definition of
Figure FDA00032000840400000111
For interference estimation errors, we can obtain:
Figure FDA00032000840400000112
the adaptive finite time disturbance observer is as follows:
Figure FDA00032000840400000113
in the formula:
Figure FDA00032000840400000114
xi is an auxiliary variable which is a variable,
Figure FDA00032000840400000115
is an estimated value of a and is,
Figure FDA00032000840400000116
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:
Figure FDA00032000840400000117
the adaptive law is:
Figure FDA00032000840400000118
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:
Figure FDA00032000840400000119
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 ai＞0，bi＞0，i＝1,2,3；sgn(ζ)＝(sgn(ζ1),sgn(ζ2),sgn(ζ3))TSgn (·) is a sign function; p ═ diag { | | | Pi||}3×3Wherein p isiFor 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
Figure FDA0003200084040000021
Figure FDA0003200084040000022
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.
Figure FDA0003200084040000023
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:
Figure FDA0003200084040000024
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-θd
Figure FDA0003200084040000025
wherein, thetadThe 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
Figure FDA0003200084040000026
The following were used:
Figure FDA0003200084040000027
Figure FDA0003200084040000028
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
Figure FDA0003200084040000031
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:
Figure FDA0003200084040000032
wherein i is 1,2,3, τmax、τminMaximum 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:
Figure FDA0003200084040000033
in the formula, theta ∈ R3、
Figure FDA0003200084040000034
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)，
Figure FDA0003200084040000035
g(θ)＝g0(θ) + Δ g (θ); wherein M is0(θ)、
Figure FDA0003200084040000036
g0(θ) is the nominal portion of the model known, Δ M (θ),
Figure FDA0003200084040000037
Δ g (θ) represents the uncertainty due to modeling error, Fs(theta) is a ground generalized reaction force term, tau, received by the robotdDisturbance of ocean currents;
will assemble uncertainty
Figure FDA0003200084040000038
Introducing a mechanical arm dynamic model of the robot to obtain:
Figure FDA0003200084040000039
determining a robot walking foot dynamics model based on input saturation
Figure FDA00032000840400000310
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