CN109878633A - A kind of high-speed catamaran longitudinal direction stabilization method based on Active Disturbance Rejection Control - Google Patents
A kind of high-speed catamaran longitudinal direction stabilization method based on Active Disturbance Rejection Control Download PDFInfo
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Abstract
Subtract the invention belongs to high-speed catamaran longitudinal movement and shake control field, and in particular to a kind of high-speed catamaran longitudinal direction stabilization method based on Active Disturbance Rejection Control.Random seaway model including designing different stage；To catamaran and subtract using STAR-CCM and AQWA software and shake attached body and be designed analysis, establishes the longitudinal movement model of its T-type wing and wave suppression plate, obtain longitudinal motion response heaving height and pitch angular；It is controlled it using the heave of catamaran and pitching as control object using automatic disturbance rejection controller using Active Disturbance Rejection Control system as its control system；It is compensated by closed circuit and is disturbed in the system of the kinematic coupling portion generation of catamaran heave and pitching, static decoupling matrices are designed using Active Disturbance Rejection Control Multivariable Decoupling Control method, decoupling control is carried out to BIBO system, the T-type wing and wave suppression plate are controlled by two controllers respectively, achievees the effect that subtract and shake.The present invention can effectively weaken influence of the wave for catamaran kinetic stability.
Description
Technical field
Subtract the invention belongs to high-speed catamaran longitudinal movement and shake control field, and in particular to is a kind of based on Active Disturbance Rejection Control
High-speed catamaran longitudinal direction stabilization method.
Background technique
Classical control theory and modern control theory are all that controlled device is needed to possess clearly mathematical model, but actual
Control object is unsatisfactory, and the mechanism of system may be very complicated, and scale is very huge, and there are many variable, and parameter is changeable and couples, or is
System has the case where non-linear, uncertainty, time variation, hysteresis quality, and Traditional control theory is difficult to be analyzed with mathematics, and establishes
In accordance with the mathematical model of the characteristics of motion.Will be to non-linear using linearisation when usually modeling, distribution parameter will use lumped parameter,
Time-varying coefficient is needed using constant coefficient, and modeling and actual capabilities can have a tremendous difference, therefore the Traditional control in actually control
Algorithm is difficult to prove effective.Traditional PI D has many advantages in some aspects, but there are following 4 points to be limited to for it: first, extract error
Method and inaccurate；Second, each part " weighted sum " falls flat in some cases；Third, to error
The method for extracting differential is not optimal；There are many side effects for 4th point of feedback.Therefore, we are because when in its advantages
On the basis of studied.Our thinking is to choose appropriate nonlinear element to be freely combined, and is created with this new
Control module is designed controller.Consider in this way, chooses following four angles specifically to solve the above problems: the
One, one " transition " is arranged between controller and system；Second, more accurate when to extract differential signal in systems, choosing
Select one " Nonlinear Tracking Differentiator " of addition；Third chooses more accurate combination for nonlinear element, i.e., and " non-linear group
It closes "；4th, a kind of mode for extracting disturbance is selected, improves its precision, i.e., " extended state observer ".
Automatic disturbance rejection controller has stronger robustness and Model suitability, can disturb system model and not be capable of measuring
External disturbance effect be attributed to total disturbance of system, these are observed with extended state observer and total is disturbed.It will be collected into again
Signal real-time estimation is carried out to it using error feedback controller.The signal of above-mentioned part is passed through into closed circuit to it simultaneously
Compensation.When it carries out real-time estimation to observation signal in selection error feedback control, also make it using special nonlinear effect
Signal accelerates convergence, and the dynamic property of control system is improved with this.
Summary of the invention
The high-speed catamaran longitudinal direction stabilization method based on Active Disturbance Rejection Control that the purpose of the present invention is to provide a kind of, can change
Kind high speed, the stability of wave piercing catamaran longitudinal movement.
A kind of high-speed catamaran longitudinal direction stabilization method based on Active Disturbance Rejection Control, specifically includes the following steps:
Step 1, the random seaway model that different stage is designed according to international meteorological office wave standard；
Step 2, using random seaway model, using STAR-CCM and AQWA implementing hydrodynamic analysis software, to catamaran and
Subtract and shake attached body and be designed analysis, establishes catamaran and subtract the longitudinal movement model for shaking the attached body T-type wing and wave suppression plate, indulged
To motor imagination heaving height and pitch angular；
Step 3 as subtracting for catamaran shakes attached body control system using Active Disturbance Rejection Control system, with the heave of catamaran and
Pitching is controlled it as control object using automatic disturbance rejection controller；
Step 4 itself observes feature for the kinematic coupling portion of catamaran heave and pitching using automatic disturbance rejection controller
Regard that the interior of system is disturbed as, is compensated by closed circuit, and designed using Active Disturbance Rejection Control Multivariable Decoupling Control method
Static decoupling matrices carry out decoupling control to BIBO system, control the T-type wing and pressure wave by two controllers respectively
Plate achievees the effect that subtract and shake.
A kind of high-speed catamaran longitudinal direction stabilization method based on Active Disturbance Rejection Control, random seaway model in step 1 are as follows:
Step 1, the corresponding difference significant wave height H of different stage wave and frequency range according to international meteorological organization prescribed,
The foundation for realizing the random seaway model of different stage is programmed using MATLAB；
Step 2, selection P-M spectrum analog random seaway；
The expression formula of step 3, P-M spectrum are as follows:
In formula: S (ω) is power spectral density, and ω is frequency, and g is acceleration of gravity,For the ariyoshi wave height of wave, most
It is obtained eventually with the random seaway model for meeting with frequency are as follows:
In formula, A=8.1 × 10-3×g2,U is ship's speed, and N is sample number, εi=rand (0,2 π),
The input of step 4, wave module is clock, will be output as the input of module below.
A kind of high-speed catamaran longitudinal direction stabilization method based on Active Disturbance Rejection Control, catamaran described in step 2 are longitudinal
Motion model are as follows:
Step 1, You Dalang BELL'S THEOREM obtain catamaran pitching and heave movement equation group:
In formula: m33It is the quality of catamaran, aijIt is additional mass of the seawater to catamaran, x3It is heave amount, bijIt is damping
Coefficient, cijIt is recovery coefficient, x5It is pitching amount, m55It is the pitching moment of inertia of catamaran
Step 2 obtains catamaran hydrodynamic force coefficient under the different speed of a ship or plane using STAR-CCM and AQWA Platform Analysis；
Step 3 in the marine speed of a ship or plane, Wave Model and meets with frequency according to catamaran, uses MATLAB platform
" ss2tf " function realizes the specific decoupling and solution to catamaran longitudinal movement model.
A kind of high-speed catamaran longitudinal direction stabilization method based on Active Disturbance Rejection Control uses Active Disturbance Rejection Control in step 3
System as catamaran control system the following steps are included:
Step 1 designs the differential tracker (TD) with the T-type wing and the catamaran design automatic disturbance rejection controller of wave suppression plate
And parameter tuning:
Its discrete form are as follows:
In formula, h is integration step, and there are two the factors for influencing the main tracking effect of tracker: speed factor r and integration step
Long h；
Step 2, extended state observer (ESO) design and parameter tuning:
The concrete form of Second Order Eso are as follows:
By formula it is found that system state variables x1(t),x2(t) estimated value is z1(t),z2(t), ESO needs the ginseng adjusted
Number shares 7, respectively β01, β02, β03, a1, a2, δ, b, wherein parameter a1、a2, δ be nonlinear function fal () parameter,
a1, a2Value range between 0 to 1, and the smaller nonlinear function of the two parameters it is linear non-thread sexuality it is stronger,
ESO module is stronger to the control ability of the interference for the system that is observed simultaneously；
Step 3, nonlinear state error Feedback Control Laws (NLSEF) parameter tuning:
The system output that the desired value that TD link is collected into and its differential value and ESO link are collected by NLSEF link
Two status informations carry out making the difference the signal progress nonlinear combination after comparison, obtain control amount u with this0,
When research object is second-order system, following control law is chosen:
There are five the parameters that NLSEF needs to adjust: β1、β2、a1、a2, δ, β1, β2, it is the gain of nonlinear element, works as adjusting
When speed is slow, increase β1, otherwise β can be reduced1；
Step 4 carries out matlab emulation to above-mentioned link, constructs simulink block diagram, chooses different parameters and imitated
True comparison, to verify its performance.
The beneficial effects of the present invention are:
The control method that the present invention designs subtracts for longitudinal direction of the most common high-speed catamaran in marine navigation shakes,
Significantly reduce the hull heaving and pitching in movement.It realizes and Auto-disturbance-rejection Control is used for high-speed trimaran longitudinally subtracts
Function is shaken, suitable for the sail environment of complicated, the high ship's speed of sea situation, can be applied to the stability control of military and civilian multi-hull ship
System aspect, can effectively weaken influence of the wave for catamaran kinetic stability, improve comfort of passenger.
Detailed description of the invention
Fig. 1 is that the band designed in the present invention meets with frequency accidental Wave Model；
Fig. 2 is catamaran longitudinal movement model structure of the present invention；
Fig. 3 is the Control system architecture block diagram that the present invention designs；
Fig. 4 is control method flow chart of the present invention；
Fig. 5 (a)~(f) is the discrete TD modular structure block diagram of automatic disturbance rejection controller；
Fig. 5 (b) is the lower tier model architecture block diagram of the discrete TD module of automatic disturbance rejection controller；
Fig. 5 (c) is automatic disturbance rejection controller ESO modular structure block diagram；
Fig. 5 (d) is the underlying model structural block diagram of automatic disturbance rejection controller ESO module；
Fig. 5 (e) is automatic disturbance rejection controller NLSEF modular structure block diagram；
Fig. 5 (f) is automatic disturbance rejection controller NLSEF module underlying model structural block diagram.
Specific embodiment
The invention will be further described below in conjunction with the accompanying drawings.
It as shown in Fig. 4, is a kind of high-speed catamaran longitudinal direction stabilization method flow chart based on Active Disturbance Rejection Control of the present invention,
Detailed process is as follows:
The foundation of step 1, random seaway model:
The relatively mature P-M of this paper selection technique composes to simulate random seaway.Pass through following emulation experiment energy
Enough know, P-M composes the waveform that can be fitted random seaway well.The expression formula of P-M spectrum are as follows:
In formula: S (ω) is power spectral density, and ω is frequency, and g is acceleration of gravity,For the ariyoshi wave height of wave.
By the thought of random process it is recognised that random seaway wave energy is enough used as a stationary random process to analyze.I
Can use its constant characteristic of statistical value and to carry out qualitative analysis to random seaway by correlation analysis tool.
According to the principle of two dimension waves it is found that it is to have irregular wavelength and width by many that random seaway, which can be regarded,
The wave wave of degree is formed by stacking.Then, under t moment, x0The wave height at place may be expressed as:
In formula, aiFor amplitude, kiFor wave number, ωiFor angular frequency, εiFor initial phase angle.Wind Wave Spectra is distributed as 0 in formula (2)
~∞.It is able to know that by the understanding to wave wave principle, most of energy of wave wave is previously stored in a certain frequency range.Cause
This, the finite harmonic that we can choose under the higher frequency range of energy emulates.It is further simplified the process, fetch bit is set
x0=0.Then formula (2) is rewritten are as follows:
Corresponding slope of wave surface are as follows:
Unit area wave energy:
Wherein ρ is density of sea water, and g is acceleration of gravity, ωi~ωi+ΔωiInterior composition wave energy is:
Wave energy spectrum density S (ω) is defined to make:
ωiWhen → 0, wave energy per unit area are as follows:
Then obtained by above formula:
It is high that above formula substitution (2) is obtained into wave transient wave are as follows:
By equi-energy divided method:
Substitute the above to (10) final model for obtaining random seaway are as follows:
Wherein, A=8.1 × 10-3×g2,N is sample number, εi=rand (0,2 π).
In practical ship's navigation, the experience frequency of trimaran is than complicated many in experiment, so in order to more accurate
Practical wave is simulated, we are artificially introduced experience frequencies omegae,
Period of encounter:
Wherein:K is wave number, and U is ship's speed, and C is velocity of wave, and wave crest relative ship speed is Ce, wherein
Ce=C-U cos β (17)
Then obtain:
When heading sea, β=180 ° are taken, then are had:
Finally obtain with meet with frequency random seaway model it is as follows, random seaway figure is shown in Fig. 1.
The input of wave module is clock, will be output as the input of module below.
Step 2, catamaran longitudinal movement modelling:
It only need to consider the longitudinal movement of catamaran herein, therefore only need to consider pitching and the heave movement of ship.By reaching
Bright BELL'S THEOREM obtains catamaran pitching and heave movement equation group is as follows:
In formula: m33It is the quality of catamaran, aijIt is additional mass of the seawater to catamaran, x3It is heave amount, bijIt is damping
Coefficient, cijIt is recovery coefficient, x5It is pitching amount, m55It is the pitching moment of inertia of catamaran.The heave and pitching that catamaran is subject to
Power mass motion model is as follows.
It tests, is 40 sections in the speed of a ship or plane, meets with frequency weWhen=1.5rad/s, under Pyatyi sea situation, heave that binary is subject to
Power is as follows with pitchmoment:
(23) are decoupled, and by obtaining the heave power of catamaran and pitchmoment is converted into hull after MATLAB calculating
The power and torque of effect itself, as shown in formula (2-26):
To catamaran and it is related subtract to shake attached body and carry out mathematical model build, and pass through UGNX10.0 pairs of Geometric Modeling software
Catamaran and corresponding subtract shake building for attached body progress geometrical model.Wave under driving status is analyzed and modeled, is allowed to
Simulated environment afterwards more can really embody wave environment.Longitudinal movement model analysis to catamaran, establishes catamaran
Longitudinal movement model.And analyze the stress model of the T-type wing and wave suppression plate, obtain its working principle, the system after being
Emulation lays the foundation with modeling.
Step 3, the design of automatic disturbance rejection controller and parameter tuning:
1. tracking-differential (TD) ring layout and parameter tuning:
Its discrete form are as follows:
Wherein, h is integration step.
By formula (25) it is found that there are two the factors of the influence main tracking effect of tracker: speed factor r and integration step h.
After h is determined, influence of the only r to tracking effect is bigger.We input sine and carry out simulink emulation, respectively fast
Tracking waveform when variation sampling step length h and fixed sample time h changes the speed factor r when slow factor r is fixed, which compares, to be ground
Study carefully.The parameter used when simulating, verifying is r0=3, h=0.19, v0=sin (t).
See Fig. 5 (a) TD discrete block, the lower layer model of Fig. 5 (b) TD discrete block.
2. extended state observer (ESO) design and parameter tuning:
The concrete form of Second Order Eso are as follows:
By formula (26), we can read the state variable x to system1(t),x2(t) estimated value z1(t),z2(t)。
Also, we can also read the state variable expanded, with the real-time effect amount z of this acceleration for acting on system3(t)。
ESO needs the parameter adjusted to share 7, respectively β01, β02, β03, a1, a2, δ, b.Wherein parameter a1、a2, δ be nonlinear function
The parameter of fal ().a1, a2Value range between 0 to 1, and linear non-thread of the smaller nonlinear function of the two parameters
Sexuality is stronger, while ESO module is stronger to the control ability of the interference for the system that is observed.
See Fig. 5 (c) ESO module, the underlying model of Fig. 5 (d) ESO module.
3. nonlinear state error Feedback Control Laws (NLSEF) and parameter tuning
The system output that the desired value that TD link is collected into and its differential value and ESO link are collected by NLSEF link
Two status informations make the difference the signal after comparison and carry out nonlinear combination, obtain control amount u with this0。
When research object is second-order system, rule control as follows is chosen:
There are five the parameters that NLSEF needs to adjust: β1、β2、a1、a2,δ.By a upper section it is found that a1、a2, δ generally often take a1
=0.75, a2=1.5, δ=0.08.β1, β2, it is the gain of nonlinear element.β can suitably be increased when adjustment speed is slow1, instead
Can reduce β1.The problem of increase adjustment speed is easy that overshoot is brought to increase, system oscillation, it is in this case, appropriate to increase
Parameter beta2It can provide a system to a degree of inhibition overshoot ability.Reducing adjustment speed then can make transient process slack-off, surpass
Adjust and reduce the proportionality coefficient small, the similar PID of effect is adjusted；β2It is slack-off to increase then transient process, overshoot reduction；β2Reduce then transition
Process is accelerated, and overshoot increases.
See Fig. 5 (e) NLSEF module, Fig. 5 (f) NLSEF module underlying model.
Step 4, the Decoupling design of automatic disturbance rejection controller
Catamaran longitudinal movement can be reduced to heave movement and be coupled to form with pitching.Catamaran longitudinal movement has strong
Coupling individually carries out control to one of them and is likely to generate deleterious effect to an other physical quantity.Therefore
It needs to control catamaran longitudinal movement and shakes attached body being transmitted to controlled subtract after signal decoupling.State observer can be by catamaran liter
Heavy and pitch channel coupling amount regards a kind of disturbance as, and being tracked and being compensated to it independently, to realize solution
Coupling control.
Equipped with mimo system:
Input n dimension output is tieed up for n, wherein the coefficient b of control amountxy(x=1,2, n；Y=1,2, n) be
The function of state variable
Assuming that
Part except title system (5-1) control amountFor dynamic
Coupling unit,Referred to as static coupling unit,Referred to as static coupling matrix.
Remember x=[x1 x2··· xn]T, f=[f1 f2··· fn]T, u=[u1 u2··· un]T, y=[y1
y2··· yn]T, introduce virtual controlling amountSystem governing equation (5-1) just can be changed in this way
It is found that the input/output relation of i-th of access of system is
That is: i-th adjusts the input on access for Ui, export as yi=xi, i=1,2,3, n.Then on every access
Virtual controlling amount and controlled output between be single-input single-output relationship.Then, the virtual controlling amount on every access with
By full decoupled between output.And by a upper chapter to being able to know that in the design and research of controller, act on i-th access
On disturbance summationIt can be obtained in ESO by complete observation.In dominant vector U and system output quantity y
Between be embedded in n automatic disturbance rejection controller using mode in parallel and can be achieved with the decoupling control of multi-variable system.
During this decoupling control, the Dynamic Coupling of systemComponent on individual channelIt is estimated and is compensated as disturbance quantity after being observed by state observer.The static coupling of system
Part does not have excessively high requirement to the estimated accuracy of the static coupling matrix of system, as long as guaranteeing that coupling matrix is reversible, to system
Closed-loop control stability does not influence.It can determine that an invertible matrix is approximate in the approximate extents of matrix.
The specific steps that step 5, the high-speed catamaran longitudinal movement model based on Active Disturbance Rejection Control are realized in MATLAB:
Select the high-speed catamaran speed of a ship or plane for 40 sections, when SSN5 grades of wave ξ (t) are as input, by wave-power/torque
Heaving power F is obtained after resume module3With pitching power F5, as catamaran longitudinal movement model power-heaving/pitching module
Input, obtains heaving x3With pitching x5And their variation accelerationWithBy this four amounts as the defeated of sliding mode controller
Enter, obtains subtracting after the processing based on automatic disturbance rejection controller and shake the attached body T-type wing and wave suppression plate meets the knots modification that the stream angle of attack needs
This knots modification is obtained after subtracting and shaking attached body motion module two kinds as input and subtracts the longitudinal direction shaking attached body and providing for catamaran by α
Power and torque are respectively Ft、FfAnd Mt、Mf, and before being applied to power-heaving/pitching module as feedback quantity, realize that high speed is double
Body ship longitudinal movement stability contorting.
Claims (4)
1. a kind of high-speed catamaran longitudinal direction stabilization method based on Active Disturbance Rejection Control, which is characterized in that specifically includes the following steps:
Step 1, the random seaway model that different stage is designed according to international meteorological office wave standard；
Step 2, using random seaway model, using STAR-CCM and AQWA implementing hydrodynamic analysis software, catamaran and subtracting is shaken
Attached body is designed analysis, establishes catamaran and subtracts the longitudinal movement model for shaking the attached body T-type wing and wave suppression plate, obtains longitudinal fortune
Dynamic response heaving height and pitch angular；
Step 3 as subtracting for catamaran shakes attached body control system using Active Disturbance Rejection Control system, with the heave and pitching of catamaran
As control object, controlled it using automatic disturbance rejection controller；
Catamaran is heaved using itself observation feature of automatic disturbance rejection controller and is regarded as with the kinematic coupling portion of pitching by step 4
It is that the interior of system is disturbed, is compensated by closed circuit, and static using the design of Active Disturbance Rejection Control Multivariable Decoupling Control method
Decoupling matrices carry out decoupling control to BIBO system, control the T-type wing and wave suppression plate by two controllers respectively,
Achieve the effect that subtract and shake.
2. a kind of high-speed catamaran longitudinal direction stabilization method based on Active Disturbance Rejection Control according to claim 1, which is characterized in that
Random seaway model described in step 1 are as follows:
Step 1, the corresponding difference significant wave height H of different stage wave and frequency range according to international meteorological organization prescribed use
The foundation of the random seaway model of different stage is realized in MATLAB programming；
Step 2, selection P-M spectrum analog random seaway；
The expression formula of step 3, P-M spectrum are as follows:
In formula: S (ω) is power spectral density, and ω is frequency, and g is acceleration of gravity,For the ariyoshi wave height of wave,
It finally obtains with the random seaway model for meeting with frequency are as follows:
In formula, A=8.1 × 10-3×g2,U is ship's speed, and N is sample number, εi=rand (0,2 π),
The input of step 4, wave module is clock, will be output as the input of module below.
3. a kind of high-speed catamaran longitudinal direction stabilization method based on Active Disturbance Rejection Control according to claim 1, which is characterized in that
Catamaran longitudinal movement model described in step 2 are as follows:
Step 1, You Dalang BELL'S THEOREM obtain catamaran pitching and heave movement equation group:
In formula: m33It is the quality of catamaran, aijIt is additional mass of the seawater to catamaran, x3It is heave amount, bijIt is damped coefficient,
cijIt is recovery coefficient, x5It is pitching amount, m55It is the pitching moment of inertia of catamaran
Step 2 obtains catamaran hydrodynamic force coefficient under the different speed of a ship or plane using STAR-CCM and AQWA Platform Analysis；
Step 3 in the marine speed of a ship or plane, Wave Model and meets with frequency according to catamaran, uses MATLAB platform " ss2tf " letter
Number realizes the specific decoupling and solution to catamaran longitudinal movement model.
4. a kind of high-speed catamaran longitudinal direction stabilization method based on Active Disturbance Rejection Control according to claim 1, which is characterized in that
Described in step 3 using Active Disturbance Rejection Control system as catamaran control system specifically includes the following steps:
It is step 1, whole to the differential tracker design and parameter of the catamaran design automatic disturbance rejection controller with the T-type wing and wave suppression plate
It is fixed:
Its discrete form are as follows:
In formula, h is integration step, and there are two the factors for influencing the main tracking effect of tracker: speed factor r and integration step h；
Step 2, extended state observer design and parameter tuning:
The concrete form of Second Order Eso are as follows:
By formula it is found that system state variables x1(t),x2(t) estimated value is z1(t),z2(t), extended state observer needs whole
Fixed parameter shares 7, respectively β01, β02, β03, a1, a2, δ, b, wherein parameter a1、a2, δ be nonlinear function fal ()
Parameter, a1, a2Value range between 0 to 1；
Step 3, nonlinear state error Feedback Control Laws parameter tuning:
The desired value and its differential value and expand that differential tracker link is collected by nonlinear state error Feedback Control Laws link
The signal that two status informations of the system output that state observer link is collected into make the difference after comparison carries out non-linear
Combination, obtains control amount u with this0,
When research object is second-order system, following control law is chosen:
There are five the parameters that nonlinear state error Feedback Control Laws need to adjust: β1、β2、a1、a2, δ, β1, β2, it is non-linear loop
The gain of section increases β when adjustment speed is slow1, otherwise reduce β1；
Step 4 carries out matlab emulation to above-mentioned link, constructs simulink block diagram, chooses different parameters and carries out emulation pair
Than to verify its performance.
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| CN111498037A (en) * | 2020-04-05 | 2020-08-07 | 哈尔滨工程大学 | High-speed catamaran longitudinal stabilization method based on variable structure active disturbance rejection control |
| CN111752150A (en) * | 2020-06-12 | 2020-10-09 | 北京理工大学 | Four-wheel cooperative control method for wheeled-foot robot |
| CN112068437A (en) * | 2020-09-17 | 2020-12-11 | 哈尔滨工程大学 | Single-step prediction control anti-rolling method for high-speed multi-hull ship |
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| CN111498037A (en) * | 2020-04-05 | 2020-08-07 | 哈尔滨工程大学 | High-speed catamaran longitudinal stabilization method based on variable structure active disturbance rejection control |
| CN111752150A (en) * | 2020-06-12 | 2020-10-09 | 北京理工大学 | Four-wheel cooperative control method for wheeled-foot robot |
| CN111752150B (en) * | 2020-06-12 | 2021-07-16 | 北京理工大学 | Four-wheel cooperative control method for wheeled-foot robot |
| CN112068437A (en) * | 2020-09-17 | 2020-12-11 | 哈尔滨工程大学 | Single-step prediction control anti-rolling method for high-speed multi-hull ship |
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