CN108983606A - A kind of Sliding mode self-adaptation control method of mechanical arm system - Google Patents

A kind of Sliding mode self-adaptation control method of mechanical arm system Download PDF

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CN108983606A
CN108983606A CN201810741941.9A CN201810741941A CN108983606A CN 108983606 A CN108983606 A CN 108983606A CN 201810741941 A CN201810741941 A CN 201810741941A CN 108983606 A CN108983606 A CN 108983606A
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mechanical arm
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matrix
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CN108983606B (en
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胡健
李向辉
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Nanjing University of Science and Technology
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Nanjing University of Science and Technology
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance

Abstract

The invention discloses a kind of mechanical arm system Sliding mode self-adaptation control method with progressive tracking performance, belongs to mechanical arm control field.The control method considers the unstructured uncertainties such as the structural uncertainties such as the parameter of mechanical arm system and outer interference, and for structural uncertainties design parameter estimators such as parameters；Continuous robust controller is designed for the upper bound of the unstructured uncertainties such as outer interference；Robust adaptive controller designed by the present invention has good effect to the mechanical arm system for existing simultaneously the unstructured uncertainties such as the structural uncertainties such as parameter and outer interference, and can guarantee mechanical arm system position tracking performance；Sliding mode adaptive controller designed by the present invention is simple and its control output is continuous, applies in practice conducive in engineering.

Description

A kind of Sliding mode self-adaptation control method of mechanical arm system
Technical field
The present invention relates to the Sliding mode self adaptive control sides of mechanical arm control field more particularly to a kind of mechanical arm system Method.
Background technique
Mechanical arm system be a multiple-input and multiple-output, nonlinearity, close coupling complication system.Because of its unique behaviour Make flexibility, be widely used in fields such as industry assembling, safety anti-explosives, as paint-spray robot, spot welding robot, Bomb disposal robot etc..Due to the complexity of mechanical arm system, it is not true that many modelings can be faced with during controller design Qualitative includes structural uncertainty and unstructured uncertainty, these factors can seriously affect controller performance, lead to controller Controlling precision reduces, or even keeps designed controller unstable, to increase the design difficulty of controller.
With being constantly progressive for industrial circle technical level, the control precision of mechanical arm is also being continuously improved.But it passes The control method of system is obviously no longer satisfied the high performance requirements of system, becomes the factor of limit mechanical arm control performance.Closely Nian Lai, with the continuous development of control technology, the various control methods based on modern control theory propose in succession.Wherein sliding formwork control The ratio made in mechanical arm is wide, but its cannot to there are the structural uncertainties such as parameter to estimate in system, when There are the controller of design will be made to seem conservative when the structural uncertainties such as big parameter in system, to make the performance of system Deteriorate.
For the characteristics of Uncertain nonlinear, establishing system mathematic model in mechanical arm system, and design on this basis Mechanical arm system sliding formwork adaptive robust control overcomes system parameter uncertainty and unmodeled uncertainty.
Summary of the invention
The present invention is the uncertain nonlinear problem solved in mechanical arm system control, and then proposes a kind of mechanical arm robust Sliding Mode Adaptive Control method.
The present invention takes that specific step is as follows in order to solve the above problem:
A kind of mechanical arm Sliding mode self-adaptation control method, comprising the following steps:
Step 1, the kinetic model for establishing mechanical arm system, according to Euler-Lagrangian method, a n freedom degree Mechanical arm system kinetic model it is as follows:
Q ∈ R in formula (1)n,Respectively the speed of joint of mechanical arm, angular speed and angle accelerate Degree；H(q)∈Rn×nFor the inertial matrix of mechanical arm system；G(q)∈Rn, τ ∈ RnIt respectively indicates in centripetal section Ao Lili, gravity and input torque: FvFor viscosity friction coefficient, d ∈ RnFor external disturbance vector, including system when dry out It disturbs and is interfered with constant value,
The mechanical arm system has the property that
Property 1:H (q) is a positive definite symmetric matrices, is met:
Wherein, m1And m2∈ R is known bounded positive real number；
Property 2: the differential matrix of mechanical arm inertial matrix and Coriolis matrix meet following skew symmetric matrix relationship:
Property 3: mechanical arm dynamic model is linear relative to one group of physical parameter:
Wherein,It is joint of mechanical arm regression matrix, α=[κ, ρ, ε, μ]TIt is the intrinsic ginseng in manipulator model Number；
The mechanical arm system also meets the following conditions and lemma:
Condition 1: mechanical arm system desired locations instruct qd, its first differentialAnd second-order differentialIt is continuously to have Sector signal；
Condition 2: mechanical arm indeterminate interferes d bounded, i.e.,
||d||≤d0 (5)
Wherein, d0It is a known bounded normal number；
Lemma 1. considers first-order system
Rule reaches finite time stability in above system control as follows:
θ=- θ-λ θ-μ θq/p (7)
Wherein, θ is system state variables, λ and μ for normal number, q > 0 and p > 0 and q and p are odd-integral number, meets q/p < 1；Therefore, convergence time tsAre as follows:
Lemma 2.Have
0≤xtanh(x/a)≤|x| (8)
Lemma 3.Or x >, y >=0 have
x/x+y≤1 (9)；
Step 2, steps are as follows for design mechanical arm sliding formwork adaptive robust control:
Step 2.1, definitionSoqdIt is that system it is expected the position command tracked and this refers to Enable second order micro-, design a sliding-mode surface, limit tracking error, guarantee error convergence to 0,
Wherein Λ is a constant value matrix, and its characteristic value is strictly located at right multiple half-plane, designs virtual reference rail Mark qrInstead of desired trajectory qd,
Therefore,WithIt is replaced with
Definition
Wherein s=[s1,s2,…sn]T,
In order to avoid the shake of sliding formwork control, design control law:
Wherein λ1And μ1It is normal number, q1> 0 and p1> 0 and be all integer odd number, and meet q1/p1< 1,
It is obtained according to formula (1), (14) and (15):
It can be obtained according to formula (16):
Step 2.2, design control law, in conjunction with formula (17) and property 1, the control based on mechanical arm system kinetic model Rule design are as follows:
τ=τa+τs (18)
τs=τs1+τs2 (20)
τs1=-KDs (21)
Wherein, τaFor model feedforward compensation term；τs1For linear feedback item, guarantee the stability of system；τs2For continuous robust , for overcoming the external BOUNDED DISTURBANCES in system, specific design form provides in subsequent designs； KDBe one it is symmetrical just Set matrix is diagonal matrix；WithIt is H (q),G (q) and FvEstimated value；
Step 2.3, design parameter return device and parameter estimator,
Using the parameter update law of parametric regression device are as follows:
Wherein Γ is diagonal adaptive law matrix and Γ > 0, γ are normal number, the adaptive rate of Γ and γ affecting parameters；
Step 2.4 designs continuous robust item according to condition 2, overcomes system interference
τs2=[τs21,τs22,…τs2n]T (24)
Wherein τs2iForm is as follows
In formula (25), kiFor known normal number, ξ (t) meets the following conditions
|ξi(t)|≤δi *
WhereinAnd δiIt is all normal number；
Step 3, the stability for analyzing mechanical arm system, according to the sliding formwork adaptive robust control method designed in step 2, Carrying out stability to system using Lyapunov stability theory proves, obtain system asymptotically stability as a result,
It is as follows to define liapunov function:
WhereinAnd Fv> 0,
To (27) formula derivation, obtain
Formula (16), (18)-(21) are brought into (28) to be obtained according to mechanical arm property 2,3
Adaptive law (22), (23) are brought into formula (29) to obtain
Bring formula (24), (25) into (30)
It can be obtained according to lemma 2
It can be obtained according to lemma 3
Wherein, W is a positive function, while being integrated to formula (33) two sides, is obtained
V (t) ∈ L can be obtained according to formula (34)∞, W ∈ L2, so s,WithIt is all bounded, according to condition 1, it is known that system State q bounded, according to formula (26), system exports τ bounded, so, the equal bounded of the closed signal of system, and available W has Boundary, therefore W congruous continuity, according to Barbalat lemma system asymptotically stability,
S can be obtained according to lemma 1i∈ s is respectively in Finite time tiConverge to 0, tiMoment is as follows
It proves: design liapunov function
(36) derivation is obtained
Formula (15) is brought into (37) to obtain
Wherein λ1And μ1For normal number, q1> 0, q1> 0 and be all odd-integral number,
So in Finite time tiWhen, siConverge to 0.
The beneficial effects of the present invention are: the present invention establishes mechanical arm system using mechanical arm system as research object Kinetic model can accurately track desired locations instruction with the output of its joint position as control target, while consider system The unstructured uncertainties such as the structural uncertainties such as parameter and outer interference, and ginseng is designed for the structural uncertainty of parameter Number estimator bothers the uncertain continuous robust control item of design for outside and guarantees machine in combination with finite-time control The position output of tool arm system can be accurately tracked by desired position command；Mechanical arm system Sliding mode designed by the present invention The control output of self-adaptation control method is smooth continuous, more conducively applies in practice in engineering.Simulation results show it is effectively Property.
Other than objects, features and advantages described above, there are also other objects, features and advantages by the present invention. Below with reference to accompanying drawings, the present invention is described in further detail.
Detailed description of the invention
Fig. 1 is present invention emulation two degrees of freedom mechanical arm structure chart.
Fig. 2 is the signal of mechanical arm system Sliding mode auto-adaptive control theory and flow chart.
Fig. 3 is that system parameter is adaptively schemed under controller action designed by the present invention.
Fig. 4 is controller joint of mechanical arm trace plot designed by the present invention.
Fig. 5 is controller joint of mechanical arm tracking error figure designed by the present invention.
Fig. 6 is PID controller joint of mechanical arm tracking error figure.
Fig. 7 is controller joint control output figure designed by the present invention.
Specific embodiment
With reference to the accompanying drawings of the specification, the present invention is further illustrated.
The present embodiment specifically combines two degrees of freedom mechanical arm (such as Fig. 1) to implement, and the length of robot linkage 1,2 is respectively l1 And l2, quality is respectively m1And m2, and mass center is located at connecting rod half, q1And q2For joint of mechanical arm space joint angles.
Illustrate present embodiment in conjunction with Fig. 1 to Fig. 2, a kind of Sliding mode of mechanical arm system described in present embodiment is adaptive Answering control method, specific step is as follows:
Step 1, the kinetic model for establishing mechanical arm system, according to Euler-Lagrangian method, mechanical arm system Kinetic model is as follows:
Q ∈ R in formula (1)n,The respectively speed of joint of mechanical arm, angular speed and angular acceleration；H (q)∈Rn×nFor the inertial matrix of mechanical arm system；G(q)∈Rn, τ ∈ RnIndicate centripetal Coriolis force, again Power and input torque: FvFor viscosity friction coefficient, d ∈ RnFor external disturbance vector, time-varying interference and constant value including system are dry It disturbs.
There are properties for mechanical arm system:
Property 1:H (q) is a positive definite symmetric matrices, is met:
Wherein, m1And m2∈ R is known bounded positive real number.
Property 2: the differential matrix of mechanical arm inertial matrix and Coriolis matrix meet following skew symmetric matrix relationship:
Property 3: mechanical arm dynamic model is linear relative to one group of physical parameter:
Wherein,It is joint of mechanical arm regression matrix, α=[κ, ρ, ε, μ]TIt is the intrinsic ginseng in manipulator model Number.
In order to which subsequent controllers design and analyze, it is necessary to meet following condition and lemma for mechanical arm system:
Condition 1: mechanical arm system desired locations instruct qd, its first differentialAnd second-order differentialIt is continuously to have Sector signal.
Condition 2: mechanical arm indeterminate interferes d bounded, i.e.,
||d||≤d0 (5)
Wherein, d0It is a known bounded normal number.
Lemma 1. considers first-order system
Rule reaches finite time stability in above system control as follows:
θ=- θ-λ θ-μ θq/p (7)
Wherein, θ is system state variables, λ and μ for normal number, q > 0 and p > 0 and q and p are odd-integral number, meets q/p < 1.Therefore, convergence time tsAre as follows:
Lemma 2.Have
0≤xtanh(x/a)≤|x| (8)
Lemma 3.Or x >, y >=0 have
x/x+y≤1 (9)
Step 2, in conjunction with Fig. 2 design mechanical arm sliding formwork adaptive robust control, steps are as follows:
Step 2.1, definitionSoqdIt is the position command and the instruction second order of system expectation tracking Can be micro-, a sliding-mode surface is designed, tracking error is limited, guarantees error convergence to 0.
Wherein Λ is a constant value matrix, and its characteristic value is strictly located at right multiple half-plane.Then we design void Quasi- reference locus qrInstead of desired trajectory qd。
Therefore,WithIt is replaced with
Definition
Wherein s=[s1,s2,…sn]T。
In order to avoid the shake of sliding formwork control, a kind of control law is designed:
Wherein λ1And μ1It is normal number, q1> 0 and p1> 0 and be all integer odd number, and meet q1/p1< 1.
It is available according to formula (1), (14) and (15):
It can be obtained according to formula (16):
Step 2.2, design control law, in conjunction with formula (17) and property 1, the control law based on Manipulator Dynamic is set It is calculated as:
τ=τa+τs (18)
τs=τs1+τs2 (20)
τs1=-KDs (21)
Wherein, τaFor model feedforward compensation term；τs1For linear feedback item, guarantee the stability of system；τs2For continuous robust , for overcoming the external BOUNDED DISTURBANCES in system, specific design form provides in subsequent designs； KDBe one it is symmetrical just Set matrix is under normal circumstances diagonal matrix；WithIt is H (q),G (q) and FvEstimation Value.
Step 2.3, design parameter return device and parameter estimator
Using the parameter update law of parametric regression device are as follows:
In formula, it is normal number that Γ, which is diagonal adaptive law matrix and Γ > 0, γ, Γ and γ affecting parameters it is adaptive Rate.
Step 2.4 designs a kind of continuous robust item according to condition 2, overcomes system interference
τs2=[τs21, τs22... τs2n]T (24)
Wherein τs2iForm is as follows
In formula (25), kiFor known normal number, ξ (t) meets the following conditions
|ξi(t)|≤δi *
Wherein δiAnd δ *iIt is all normal number.
Step 3, the stability for analyzing mechanical arm system, according to the sliding formwork adaptive robust control side designed in step 2 Method, carrying out stability to system using Lyapunov stability theory proves, obtains the result of system asymptotically stability.
It is as follows to define liapunov function:
WhereinAnd Fv> 0.
To (27) formula derivation, obtain
Formula (16), (18)-(21) are brought into (28) to be obtained according to mechanical arm property 2,3
Adaptive law (22), (23) are brought into formula (29) to obtain
Bring formula (24), (25) into (30)
It can be obtained according to lemma 2
It can be obtained according to lemma 3
Wherein, W is a positive function, while being integrated to formula (33) two sides, is obtained
V (t) ∈ L can be obtained according to formula (34)∞, W ∈ L2, so s,WithIt is all bounded.According to condition 1, it is known that system shape State q bounded.According to formula (26), system exports τ bounded.So the equal bounded of the closed signal of system, and available W bounded, Therefore W congruous continuity.According to Barbalat lemma system asymptotically stability.
S can be obtained according to lemma 1i∈ s is respectively in Finite time tiConverge to 0, tiMoment is as follows
It proves: design liapunov function
(36) derivation is obtained
Formula (15) is brought into (37) to obtain
Wherein λ1And μ1For normal number, q1> 0, q1> 0 and be all odd-integral number.
So in Finite time tiWhen, siConverge to 0.
Embodiment:
Using two degrees of freedom mechanical arm as simulation model, wherein mechanical arm system parameter are as follows: κ=6.7kgm2, ρ=3.4kg m2, ε=3.0kgm2, μ=0, Fv=5Nms/rad, the interference of additionSystem It is expected that the position command of tracking is curve q1d=sin (3.14 × t) (rad) and q2d=sin (3.14 × t) (rad).According to imitative It is as follows that true parameter can obtain manipulator model matrix:
Contrast simulation result: the parameter choosing of the Sliding mode self-adaptation control method of mechanical arm system designed by the present invention It is taken as: Λ=diag [10,10], Γ=diag [5,5,5,5], KD=diag [20,20], γ=80, λ1=5, q1=3, p1 =5, k1=k2=1, ξ1(t)=ξ2(t)=5000/ (1+t2), it meets formula (26), the initial value of mechanical arm parameter be α= [0,0,0,0], Fv=0.PID controller parameter is chosen are as follows: Kp=diag [1000,3000], Ki=0, Kd=diag [500, 1000]。
Fig. 3 be mechanical arm system designed by the present invention Sliding mode self-adaptation control method act on lower system parameter α, FvThe curve that changes over time of estimated value, as can be seen from the figure its estimated value is gradually close to the nominal value of system parameter, And fluctuated in a certain range near the nominal value, so as to which accurately the parameter Estimation of system is come out.
Controller action effect: Fig. 4 is the trace plot of controller mechanical arm doublejointed designed by the present invention, from figure In as can be seen that the present invention designed by controller expectation curve is tracked accurately with good tracking ability.
Fig. 5 and Fig. 6 is the tracking error that controller designed by the present invention and conventional PID controllers act on lower system respectively The correlation curve changed over time, it can be seen from the figure that controller doublejointed tracking error designed by the present invention is much small In the tracking error of conventional PID controllers, the superperformance of designed controller of the invention has been embodied.
Fig. 7 is that the control of the Sliding mode self-adaptation control method of mechanical arm system designed by the present invention inputs at any time The curve of variation is conducive to it can be seen from the figure that the obtained control input signal of the present invention is smooth continuous in engineering reality Middle application.
The foregoing is only a preferred embodiment of the present invention, is not intended to restrict the invention, for the skill of this field For art personnel, the invention may be variously modified and varied.All within the spirits and principles of the present invention, made any to repair Change, equivalent replacement, improvement etc., should all be included in the protection scope of the present invention.

Claims (1)

1. a kind of mechanical arm Sliding mode self-adaptation control method, it is characterised in that the following steps are included:
Step 1, the kinetic model for establishing mechanical arm system, according to Euler-Lagrangian method, the machinery of a n freedom degree Arm system kinetic model is as follows:
Q ∈ R in formula (1)n,The respectively speed of joint of mechanical arm, angular speed and angular acceleration；H(q) ∈Rn×nFor the inertial matrix of mechanical arm system；G(q)∈Rn, τ ∈ RnRespectively indicate centripetal Coriolis force, Gravity and input torque: FvFor viscosity friction coefficient, d ∈ RnFor external disturbance vector, time-varying interference and constant value including system Interference,
The mechanical arm system has the property that
Property 1:H (q) is a positive definite symmetric matrices, is met:
Wherein, m1And m2∈ R is known bounded positive real number；
Property 2: the differential matrix of mechanical arm inertial matrix and Coriolis matrix meet following skew symmetric matrix relationship:
Property 3: mechanical arm dynamic model is linear relative to one group of physical parameter:
Wherein,It is joint of mechanical arm regression matrix, α=[κ, ρ, ε, μ]TIt is the intrinsic parameter in manipulator model；
The mechanical arm system also meets the following conditions and lemma:
Condition 1: mechanical arm system desired locations instruct qd, its first differentialAnd second-order differentialIt is continuous bounded letter Number；
Condition 2: mechanical arm indeterminate interferes d bounded, i.e.,
||d||≤d0 (5)
Wherein, d0It is a known bounded normal number；
Lemma 1. considers first-order system
Rule reaches finite time stability in above system control as follows:
θ=- θ-λ θ-μ θq/p (7)
Wherein, θ is system state variables, λ and μ for normal number, q > 0 and p > 0 and q and p are odd-integral number, meets q/p < 1； Therefore, convergence time tsAre as follows:
Lemma 2.Have
0≤xtanh(x/a)≤|x| (8)
Lemma 3.Or x >, y >=0 have
x/x+y≤1 (9)；
Step 2, steps are as follows for design mechanical arm sliding formwork adaptive robust control:
Step 2.1, definitionSoqdIt is the position command and the instruction second order of system expectation tracking Can be micro-, design a sliding-mode surface, limit tracking error, guarantee error convergence to 0,
Wherein Λ is a constant value matrix, and its characteristic value is strictly located at right multiple half-plane,
Design virtual reference track qrInstead of desired trajectory qd,
Therefore,WithIt is replaced with
Definition
Wherein s=[s1,s2,…sn]T,
In order to avoid the shake of sliding formwork control, design control law:
Wherein λ1And μ1It is normal number, q1> 0 and p1> 0 and be all integer odd number, and meet q1/p1< 1,
It is obtained according to formula (1), (14) and (15):
It can be obtained according to formula (16):
Step 2.2, design control law, in conjunction with formula (17) and property 1, the control law based on mechanical arm system kinetic model is set It is calculated as:
τ=τa+τs (18)
τs=τs1+τs2 (20)
τs1=-KDs (21)
Wherein, τaFor model feedforward compensation term；τs1For linear feedback item；τs2For continuous robust item；KDIt is a symmetric positive definite square Battle array is diagonal matrix；WithIt is H (q),The estimated value of G (q) and Fv；
Step 2.3, design parameter return device and parameter estimator,
Using the parameter update law of parametric regression device are as follows:
Wherein Γ is diagonal adaptive law matrix and Γ > 0, γ are normal number, the adaptive rate of Γ and γ affecting parameters；
Step 2.4 designs continuous robust item according to condition 2, overcomes system interference
τs2=[τs21,τs22,…τs2n]T (24)
Wherein τs2iForm is as follows
In formula (25), kiFor known normal number, ξ (t) meets the following conditions
Wherein δi *And δiIt is all normal number；
Step 3, the stability for analyzing mechanical arm system are utilized according to the sliding formwork adaptive robust control method designed in step 2 Lyapunov stability theory, which carries out stability to system, to be proved, the result of system asymptotically stability is obtained:
It is as follows to define liapunov function:
WhereinAnd Fv> 0,
To (27) formula derivation, obtain
Formula (16), (18)-(21) are brought into (28) to be obtained according to mechanical arm property 2,3
Adaptive law (22), (23) are brought into formula (29) to obtain
Bring formula (24), (25) into (30)
It can be obtained according to lemma 2
It can be obtained according to lemma 3
Wherein, W is a positive function, while being integrated to formula (33) two sides, is obtained
V (t) ∈ L can be obtained according to formula (34)∞, W ∈ L2, so s,WithAll bounded, according to condition 1, it is known that system mode q has Boundary, according to formula (26), system exports τ bounded, and available W bounded, therefore W congruous continuity, according to Barbalat lemma Know system asymptotically stability,
S can be obtained according to lemma 1i∈ s is respectively in Finite time tiConverge to 0, tiMoment is as follows
It proves: design liapunov function
(36) derivation is obtained
Formula (15) is brought into (37) to obtain
Wherein λ1And μ1For normal number, q1> 0, q1> 0 and be all odd-integral number,
So in Finite time tiWhen, siConverge to 0.
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