CN109683478A - Flexible joint mechanical arm fractional order sliding formwork optimal control method - Google Patents

Flexible joint mechanical arm fractional order sliding formwork optimal control method Download PDF

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Publication number
CN109683478A
CN109683478A CN201811568816.9A CN201811568816A CN109683478A CN 109683478 A CN109683478 A CN 109683478A CN 201811568816 A CN201811568816 A CN 201811568816A CN 109683478 A CN109683478 A CN 109683478A
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fractional order
mechanical arm
fractional
formula
joint
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汪允鹤
王继虎
夏正仙
王杰高
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Nanjing Estun Robotics Co Ltd
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Nanjing Estun Robotics Co Ltd
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Priority to PCT/CN2019/086551 priority patent/WO2020124938A1/en
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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
    • G05B13/045—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 using a perturbation signal
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/163—Programme controls characterised by the control loop learning, adaptive, model based, rule based expert control
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/1635—Programme controls characterised by the control loop flexible-arm control

Abstract

The invention discloses a kind of flexible joint mechanical arm fractional order sliding formwork optimal control methods, it buffets present in the joint of mechanical arm integer rank sliding formwork dynamics Controlling flexible and aiming at the problem that Trajectory Tracking Control, in conjunction with the advantages of Sliding mode variable structure control, it is theoretical to introduce fractional calculus, with the fast convergence of fractional order differential operator, imformation memory and heredity feature, the approximation quality for promoting fractional order differential operator is optimized on the basis of conventional fractional rank indirect discretization Oustaloup algorithm, to propose a kind of novel there is strong robustness, the better fractional order Sliding Mode Controller of effect is buffeted in anti-interference and weakening, there is better continuity in the flexibility of joint kinetic control system of mechanical arm, rapidity, robustness and good anti-interference.Finally, the design of fractional order sliding mode variable structure control method is realized.

Description

Flexible joint mechanical arm fractional order sliding formwork optimal control method
Technical field
The present invention relates to a kind of flexible joint mechanical arm control method, specifically a kind of flexible joint mechanical arm fractional order Sliding formwork optimal control method, belongs to robot control field.
Background technique
In terms of the dynamics Controlling of flexible joint mechanical arm and Trajectory Tracking Control, it there is now and become using integer rank sliding formwork The advantages that structure control method is to the strong robust property of parameter time varying and external disturbance, but conventional integer rank System with Sliding Mode Controller is deposited Buffeting larger problem[1].The method that current more commonly used reduction sliding formwork is buffeted has: the positive side method in boundary layer[2]、 Adjusting method based on observer[3,4], the methods of high_order sliding mode control algorithm[5]Buffeting can be weakened to a certain extent, wherein before Two methods do not have the robust property of traditional sliding mode controller, so that system, there are steady-state error, latter algorithm comparison is multiple Miscellaneous, in the lower order system of single order or second order there is the coupling of control output signal and its derivative in control law, be unfavorable for sliding formwork control Make the design of rule.Although fuzzy sliding mode tracking control has strong robust property and do not depend on system model, expert info can be made full use of The advantages that, but there are biggish static differences for Fuzzy control system.
There are still certain disadvantages in existing fractional order sliding-mode control, as Chinese invention patent application document is " a kind of The fractional order sliding-mode control of flexible joint mechanical arm " (publication number CN108181813A) to disclose a kind of flexible joint mechanical The fractional order sliding-mode control of arm system is trembled for present in joint of mechanical arm integer rank sliding formwork dynamics Controlling flexible The problem of vibration and Trajectory Tracking Control, it is theoretical to introduce fractional calculus, and utilize and divide in conjunction with the advantages of Sliding mode variable structure control Fast convergence, imformation memory and the heredity of number rank differential operator, propose a kind of fractional order Sliding Mode Controller, Realize the design of fractional order sliding mode variable structure control method.Using the method for the indirect discretization of tradition in the inventive method, compare There is preferable robust property and noiseproof feature for conventional integer rank Sliding mode variable structure control and can be to a certain degree It is upper weaken sliding formwork movement chattering phenomenon, but the indirect discretization of conventional fractional rank differential operator to control signal low frequency and High band processing has certain defect, therefore controls the higher occasion this method control effect of precision still not in application requirement Enough ideals.
Summary of the invention
Do not consider that flexible joint mechanical arm body construction and outside are negative in existing manipulator motion control system to overcome The uncertainty of disturbance is carried, and the problems such as there are sliding formwork control buffetings, the invention proposes a kind of flexible joint mechanical arms point Number rank sliding formwork optimal control method promotes the approximation accuracy of fractional order differential operator using indirect discretization method is optimized, and answers For in the fractional order sliding-mode control of flexible joint mechanical arm system, can effectively weaken buffeting, the strong robust that sliding formwork moves And noiseproof feature.
In order to solve the above-mentioned technical problem a kind of flexible joint mechanical arm fractional order sliding formwork optimal control that the present invention proposes Method, step include:
Step 1. establishes the dynamics mathematical model of flexible joint mechanical arm servo-system.
The tracking error and sliding-mode surface of step 2. calculating servo-control system.
Using Riemann-Liouville (RL) fractional order integration form of integrable function f (t), it is defined as
In formula,Expression finds a function the α order integral of f (t), and a and t are the bound of operation, and τ is integration variable, Γ () is Gamma function.
Fractional calculusIt is defined as
In formula, Re (α) indicates the real part of α.
Gamma function is defined as
RL type fractional calculus form, has the property that
In formula, m is integer, and N is set of integers.
Step 3. establishes fractional order differential operator discretization filter mathematical model；
This step establishes the mathematical modulo for optimizing indirect discretization algorithm by optimizing to the indirect discretization algorithm of improvement Type.The discretization filter of conventional fractional rank differential operator is to use indirect discretization method in such a way that rational function is cascade In (ωb,ωh) the interior fractional order differential operator s for realizing α order of frequency rangeαApproach, which is
In formula,
0 < α < 1
N is integer, filter order n=2N+1, and the bigger approximation accuracy of n is higher.
Conventional indirect discretization algorithm is improved, be fractional order differential operator fractional order transmission function is approximate, I.e.
In formula, s=j ω (ω is frequency variable, and j is frequency domain symbol) is taken,B and d are Dynamic gene.
In (ωb,ωh) in frequency band, above formula is unfolded, and take first approximation with Taylor's formula, can be obtained
The indirect discretization recursion of K (s) on the right side of above formula is unfolded, approximate formula is then obtained
This step is determined using optimization algorithm optimizing and is filtered by optimizing to the conventional and indirect discretization algorithm of improvement Wave parameter, for above formula is unfolded to increase a problems such as filter brings reduction precision by Taylor's first approximation, optimization Indirect discretization algorithm afterwards can effectively improve approximation quality.
Using the structure for improving indirect discretization algorithm, by fractional calculus sαIt is approximately
sα≈G×Ge
In formula, G is filter, GeFor indirect discretization filter.
The transmission function form of filter G is
And higher accuracy requirement can be reached by extending the order of G, i.e.,
In formula, α1,α2,L,αn+1For the constant coefficient of molecule, β1,β2,L,βn+1For the constant coefficient of denominator.
Using ITAE parameter optimization method performance indicators such as overshoot, regulating time by contrast, filtering is determined after adjusting The parameter of device G then obtains filter G and reaches amplitude-frequency and the optimal propinquity effect of phase frequency in frequency band, realizes score indirectly Rank differential operator sα, avoid loss of significance caused by truncated error.
The fractional order sliding formwork Reaching Law and control amount of step 4. calculating machine arm servo-control system；
Flexible joint mechanical arm joint power model:
In formula, t is time variable,(For motor speed), u (t) is output control amount, K is the stiffness coefficient of flexible joint, and D is flexible joint resistance Buddhist nun's coefficient, J be motor rotary inertia, d (t) be bounded external disturbance amount (be defined as | d (t) |≤dmax, dmaxFor maximum boundary Value).
Fractional orderSliding formworkReaching Law by adjust exponential approach coefficient k, constant speed approaches coefficient ε and differential order α can be with Change speed when system mode reaches sliding-mode surface S (t), expression formula is
DαS (t)=- kS (t)-ε sign (S (t))
In formula, k > 0, ε > 0, sign () is for sign function
According to above step, the control law equation of corresponding fractional order sliding mode controller is designed:
In formula, pose error e (t)=q-qd, qdFor the desired value of joint position, q is the actual value of joint position.
Step 5. updates joint of mechanical arm state parameter；
By the angular transducer being mounted at joint of mechanical arm, collection machinery shoulder joint parameter, and feed back.
The method of the present invention optimizes on the basis of conventional discretization algorithm indirectly, compared to the indirect discretization of tradition It method and improves indirect discretization method there is better propinquity effect in low frequency and high frequency, and there is broader approximate frequency Section.Using the fast convergence, imformation memory and heredity of fractional order differential operator, thus propose with strong robustness, The better fractional order Sliding Mode Controller of effect is buffeted in anti-interference and weakening, can make the flexible joint dynamics of mechanical arm Control system has better continuity, rapidity, robustness and good anti-interference.Finally, realize that optimal fractional order is sliding The design of moding structure control method.
Detailed description of the invention
Fig. 1 is flexible joint mechanical arm fractional order sliding formwork optimal control method flow diagram of the present invention.
Fig. 2 is the flexible joint mechanical arm control structure block diagram of mid-score rank sliding formwork of the present invention.
Fig. 3 is fractional order differential operator frequency response schematic diagram of the present invention.Wherein: upper figure is amplitude-frequency characteristic, and the following figure is phase Frequency characteristic.
Fig. 4 is the joint of mechanical arm fractional order sliding-mode surface track schematic diagram of fractional order sliding formwork control of the present invention.
Fig. 5 is the joint of mechanical arm integer rank sliding-mode surface track schematic diagram of fractional order sliding formwork control of the present invention.
Fig. 6 is the output control amount track schematic diagram of the joint of mechanical arm step response of fractional order sliding formwork control of the present invention.
Fig. 7 is the sliding-mode surface track schematic diagram of the joint of mechanical arm step response of fractional order sliding formwork control of the present invention.
Specific embodiment
In order to be better understood by technical characteristic of the invention, technology contents and its technical effect reached, now in conjunction with Embodiment and attached drawing make more detailed explanation to the method for the present invention.
Embodiment:
As shown in Figure 1, flexible joint mechanical arm fractional order sliding formwork optimal control method proposed by the present invention, step is such as Under:
Step 1. establishes the dynamics mathematical model of flexible joint mechanical arm servo-system；
It is the dynamics mathematical model of calculating machine arm servo-system described in this step, expression formula is
In formula, M (q) is mechanical arm inertia matrix,For centrifugal force and Ge Shili, G (q) is mechanical arm gravity Matrix, τextFor the disturbance of mechanical arm external loading, u is output control amount.
Using the kinetic model for establishing flexible joint mechanical arm for following parameter:
Mechanical arm load rotating inertia: Ja=1.6 × 10-5kg.m2；
Motor rotary inertia: J=2.56 × 10-4kg.m2；
Joint stiffness coefficient: K=1.29Nm/rad；
Damped coefficient: D=3.6 × 102Nm/(rad/s)；
Mechanical arm quality: m=5.3kg；
Mechanical arm mass center: lc=0.15m；
Mechanical arm lengths: l=0.3m.
The tracking error and sliding-mode surface of step 2. calculating servo-control system；
The step will use Riemann-Liouville (RL) fractional order integration form of integrable function f (t), definition For
In formula,Expression finds a function the α order integral of f (t), and a and t are the bound of operation, and τ is integration variable, Γ () is Gamma function.
Fractional calculusIt is defined as
In formula, Re (α) indicates the real part of α.
Gamma function is defined as
RL type fractional calculus form, has the property that
In formula, m is integer, and N is set of integers.
It is the tracking error and fractional order sliding-mode surface of calculating machine arm servo-control system described in this step.
Tracking error expression formula is
E (t)=q-qd (4)
In formula, qdFor desired joint position value, q is the actual value of joint position.
Sliding-mode surface S (t) is designed as
S (t)=λ e (t)+D1-αe(t)+D2-αE (t), 0 < α < 1 (5)
In formula, λ is regulation coefficient, DαIndicate fractional calculus operator, then D1-αE (t) and D2-αE (t) be expressed as asking with 1- α, the 2- α rank differential of track error e (t), expression formula are respectively
According to fractional calculus property, can formula (5) be asked with α rank differential, fractional order sliding-mode surface, which can arrange, is
Step 3. establishes fractional order differential operator discretization filter mathematical model；
The discretization filter of conventional fractional rank differential operator is to be cascaded using indirect discretization method using rational function Mode in (ωb,ωh) the interior realization fractional order differential operator s of frequency rangeαApproach, which is
In formula,
0 < α < 1
N is integer, filter order n=2N+1, and the bigger approximation accuracy of n is higher.
Conventional indirect discretization algorithm is improved, be fractional order differential operator fractional order transmission function is approximate, I.e.
In formula, s=j ω (ω is frequency variable, and j is frequency domain symbol) is taken,B and d are Dynamic gene.
In (ωb,ωh) in frequency band, be unfolded with Taylor's formula, and take first approximation, can be obtained
The indirect discretization recursion of K (s) on the right side of above formula is unfolded, approximate formula is then obtained
This step is determined using optimization algorithm optimizing and is filtered by optimizing to the conventional and indirect discretization algorithm of improvement Wave parameter, for above formula is unfolded to increase a problems such as filter brings reduction precision by Taylor's first approximation, optimization Indirect discretization algorithm afterwards can effectively improve approximation quality.
Using the structure for improving indirect discretization algorithm, by fractional calculus sαIt is approximately
sα≈G×Ge (12)
In formula, G is filter, GeFor indirect discretization filter.
The transmission function form of filter G is
And higher accuracy requirement can be reached by extending the order of G, i.e.,
In formula, α1,α2,L,αn+1For the constant coefficient of molecule, β1,β2,L,βn+1For the constant coefficient of denominator.
Using ITAE parameter optimization method performance indicators such as overshoot, regulating time by contrast, filtering is determined after adjusting The parameter of device G then obtains filter G and reaches amplitude-frequency and the optimal propinquity effect of phase frequency in frequency band, realizes score indirectly Rank differential operator sα, avoid loss of significance caused by truncated error.
According to above-mentioned algorithm, in frequency band (10-3,103) in, N=4 is taken, fractional order differential operator s is calculatedα(α=0.5), The initialization value that filter parameter used in indirect discretization algorithm is parameter optimization is being improved, after parameter optimization, then Obtain the parameter (α of one group of optimal filter G1,α2,α3)=(0.902,1438, -0.0012), (β1,β2,β3)=(0.24, 1438 ,+0.6312), so that filter G reaches the best fit approximation of amplitude-frequency and phase frequency in frequency band, and it is indirect with routine respectively Discretization algorithm improves the comparison that indirect discretization algorithm carries out frequency response, as shown in figure 3, indirectly discrete compared to conventional Change algorithm, optimizing indirect discretization algorithm in low frequency and high band has better propinquity effect, and has broader approximation Frequency band.
The sliding formwork Reaching Law and control amount of step 4. calculating machine arm servo-control system；
It is calculating machine arm servo-control system fractional order sliding formwork Reaching Law and control law described in this step.
According to flexible joint mechanical arm joint power model:
In formula, t is time variable,(For motor speed), u (t) is output control amount,K is the stiffness coefficient of flexible joint, and D is flexible joint damping system Number, J be motor rotary inertia, d (t) be bounded external disturbance amount (be defined as | d (t) |≤dmax, dmaxFor maximum boundary value).
Fractional order Reaching Law is approached coefficient ε and differential order α and can change by adjusting exponential approach coefficient k, constant speed System mode reaches speed when sliding-mode surface, and expression formula is
DαS (t)=- kS (t)-ε sign (S (t)), k > 0, ε > 0 (15)
Designing corresponding control law by formula (14), (15) is
Verified through Lyapunov stability, can obtain control law state will Finite-time convergence in sliding-mode surface S (t)= 0, and Asymptotic Stability.
Step 5. updates joint of mechanical arm state parameter；
To acquire joint parameter by the angular transducer being mounted at joint of mechanical arm described in this step, and feed back.
To sum up, flexible joint mechanical arm fractional order sliding formwork optimal control method proposed by the present invention, as shown in Fig. 2, a kind of Derive fractional order sliding formwork Reaching Law and fractional order Reaching Law based on conventional integer rank sliding formwork Reaching Law, sliding-mode surface, and Optimize the approximation quality for promoting fractional order differential operator on the basis of conventional discretization algorithm indirectly, effect as shown in figure 3, Fractional calculus is had studied with this and Sliding mode variable structure control is incorporated in effect in flexible joint mechanical arm control, and is utilized Lyapunov theory is proved with stability of the fractional order Theory of Stability to whole system.According to the above method, to machinery The single flexible joint of arm uses fractional order Sliding mode variable structure control, optimizes indirect discretization method using following parameter: ωb= 10-3、ωh=103, N=4, α=0.9, obtain fractional order sliding formwork surface curve as shown in figure 4, integer rank (α=1.0) sliding-mode surface Curve as shown in figure 5, under step signal excitation fractional order sliding formwork and integer rank sliding-mode control law correlation curve as shown in fig. 6, Fractional order sliding formwork and integer rank sliding formwork control sliding-mode surface track correlation curve under step signal excitation as shown in fig. 7, by pair Than Fig. 4-7, the fractional order Reaching Law for optimizing indirect discretization algorithm approximate fraction rank differential operator is compared with fractional order sliding-mode surface The chattering phenomenon of sliding formwork movement can effectively be weakened in conventional integer rank sliding formwork control, display has effect under the excitation of step signal Fruit preferably strong robust property and noiseproof feature, and in orbit tracking movement control there is preferable performance boost to make With.
Described above is the flexible joint mechanical arm control method for the optimal fractional order sliding formwork that the present invention provides, it is clear that this Invention is not only to be limited to above-described embodiment, be not partial to essence spirit of the present invention and without departing from involved by substantive content of the present invention It can make various deformations under the premise of range to it to be implemented.

Claims (1)

1. a kind of flexible joint mechanical arm fractional order sliding formwork optimal control method, step include:
Step 1. establishes the dynamics mathematical model of flexible joint mechanical arm servo-system；
The tracking error and sliding-mode surface of step 2. calculating servo-control system
Using Riemann-Liouville (RL) fractional order integration form of integrable function f (t), it is defined as
In formula,Expression finds a function the α order integral of f (t), and a and t are the bound of operation, and τ is integration variable, and Γ () is Gamma function.
Step 3. establishes fractional order differential operator discretization filter mathematical model；
Using the structure for improving indirect discretization algorithm, by fractional calculus sαIt is approximately
sα≈G×Ge
In formula, G is filter, GeFor indirect discretization filter.
The transmission function of filter G is
In formula, α1,α2,L,αn+1For the constant coefficient of molecule, β1,β2,L,βn+1For the constant coefficient of denominator；
The fractional order sliding formwork Reaching Law and control amount of step 4. calculating machine arm servo-control system
Flexible joint mechanical arm joint power model:
In formula, t is time variable,(For motor speed), u (t) is output control amount,K is the stiffness coefficient of flexible joint, and D is flexible joint resistance Buddhist nun's coefficient, J be motor rotary inertia, d (t) be bounded external disturbance amount (be defined as | d (t) |≤dmax, dmaxFor maximum boundary Value)；
Fractional orderSliding formworkReaching Law is approached coefficient ε and differential order α and can change by adjusting exponential approach coefficient k, constant speed System mode reaches speed when sliding-mode surface S (t), and expression formula is
DαS (t)=- kS (t)-ε sign (S (t))
In formula, k > 0, ε > 0, sign () is for sign function
According to above step, the control law equation of corresponding fractional order sliding mode controller is designed:
Step 5. updates joint of mechanical arm state parameter
By the angular transducer being mounted at joint of mechanical arm, collection machinery shoulder joint parameter, and feed back.
CN201811568816.9A 2018-12-21 2018-12-21 Flexible joint mechanical arm fractional order sliding formwork optimal control method Pending CN109683478A (en)

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PCT/CN2019/086551 WO2020124938A1 (en) 2018-12-21 2019-05-13 Fractional-order sliding mode optimization control method for flexible-joint robotic arm

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