CN109343350A - A kind of underwater robot path tracking control method based on Model Predictive Control - Google Patents
A kind of underwater robot path tracking control method based on Model Predictive Control Download PDFInfo
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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
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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
- G05B13/048—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 using a predictor
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- G—PHYSICS
- 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
- G05D1/10—Simultaneous control of position or course in three dimensions
Abstract
The present invention proposes a kind of underwater robot path tracking control method based on Model Predictive Control, belongs to AUV Control technical field.This method obtains discrete time prediction model according to the kinematics and dynamics model of underwater robot first, is then converted to regression model, and be based on least square method of recursion On-line Estimation parameter；Optimization object function and constraint condition are determined to destination path point tracking problem using above-mentioned model, optimum control list entries is sought using projection gradient method, then the optimum control for executing current time to underwater robot inputs, and by continuous iteration, so that underwater robot reaches each destination path point.This method can efficient on-line implement path point tracing control in robot under water, and be capable of handling the constraint of control input, certain robustness can be kept in the case where model has uncertain and external interference.
Description
Technical field
The invention belongs to AUV Control technical field, in particular to a kind of water based on Model Predictive Control
Lower robot path tracking and controlling method.
Background technique
The discovery height of Marine Sciences relies on undersea detection technology and equipment, since marine environment is complicated, condition is extreme, mesh
Preceding main use operation type underwater robot replaces or people is assisted to detect, observe and sample.And for all kinds of underwaters
People guarantees that its movement is controllably a most basic functional requirement, is the premise for realizing every complex job task.Underwater
The dynamics and kinematics model of people is complicated, and with multivariable, non-linear, close coupling, there are inputs or state constraint with timely
The features such as change, and model parameter is uncertain, and there are the external interferences such as ocean current.Complicated model leads to the control of underwater robot
Problem is also extremely complex.With enriching constantly for underwater robot application scenarios, precision, stability of the people to its motion control
It all puts forward higher requirements, the control effect for how improving underwater robot has become important research direction.
Underwater robot path following control task is to control underwater for a space curve pre-planned
People moves according to this paths, does not limit the specific time.In practice, we often do not need underwater robot in strict accordance with one
The smooth curvilinear motion of item, but destination path is portrayed with a series of path points, it is desirable that underwater robot passes through this in order
A little points.Here it is path point tracing controls.From the point of view of the limit, when the path point of setting is close enough, so that it may one approximate
Curve.
In recent years, many scholars study the path point tracking control problem of underwater robot.For underwater machine
The nonlinear model of device people, main control method include Reverse Step Control (Backstepping) method, sliding-mode control etc..
Reverse Step Control is the Control of Nonlinear Systems design method based on Liapunov (Lyapunov) method.It is designed
Thinking is to combine the selection of Lyapunov function with the design of controller, passes through the dynamical equation of the afterbody from system
Starting, the virtual controlling of variation demand and a Lyapunov function to previous stage virtual state, which is calmed to restrain, to be connected,
It is counter step by step to push away, finally obtain a multistage stabilization Feedback Control Laws.It is high that Reverse Step Control design method is suitable for underwater robot
Nonlinear model is spent, is applied in underwater robot path following control by many scholars in recent years, is that the field is most popular
A kind of non-linear control design thinking.It is steady can to guarantee that designed controller has from theoretic for backstepping control method
Qualitative and certain robustness.But its design process is complex, construction and each step Lyapunov function of state variable
" personalization " design that craftsmenship is stronger, often carries out for specific model and control target is chosen, versatility is lacked.And
And this method does not account for control input and there is the case where constraint, i.e., can not directly handle actuator saturation problem.
Sliding-mode control is to design switching law by the diverter surface in definition status space and system mode is received
Diverter surface is held back, to realize a kind of control method of stability contorting target.The design process of sliding formwork control is generally basede on
Lyapunov function, and usually combining adaptive design, thus its stability and robustness have theoretic guarantee.But
The discontinuity of sliding formwork control ratio will lead to " buffeting " phenomenon in practical application, constrain its control performance.And same, the party
Method also can not directly handle actuator saturation problem.
Model Predictive Control Algorithm is a kind of online optimal control algorithm based on model, from 1970s come out with
Come, from the heuristic control algorithm initially applied in process industrial, develops into complete theoretical basis, application
Very extensive a kind of control method.Model Predictive Control Algorithm at each decision moment, by line solver one it is limited when
Open loop optimal control problem in domain obtains control list entries, but only first control amount in sequence is applied in control object.
In subsequent time, system solving optimization problem will determine control amount again, so continue.It is set offline in advance with conventional method
It is different to count control law, Model Predictive Control is that online roll executes, and each moment can all be based on the control of virtual condition solving optimization
Signal processed, therefore inherently closed loop algorithm have certain robustness.
The path point tracking control problem that underwater robot is solved using the frame of Model Predictive Control has three aspects excellent
Gesture.First, Model Predictive Control can optimize the system output of following a period of time, i.e., following one section of motion profile, therefore fit
Adjustment process for path point tracking.Second, Model Predictive Control can explicitly handle the existing constraint of control input, because
This can be avoided the unavoidable actuator saturation problem of other control methods.Third, Model Predictive Control strong robustness are calculated
Method is established on the basis of virtual condition, and the error of model will not be with time integral, and has model feedback correction link, can
Cope with model time variation.
But in Model Predictive Control Algorithm each decision moment require to seek optimum control list entries, this is one
The solution procedure of a nonlinear optimal problem, the iterative step comprising Nonlinear Programming Algorithm calculate complexity, take a long time, answer
When for underwater robot path point tracing control, the problem of control performance and computational efficiency require can not be met simultaneously.Therefore,
The problem of feasibility that algorithm executes online must be taken into consideration, improve algorithm execution efficiency.
Summary of the invention
The purpose of the present invention is the shortcomings to overcome prior art, propose a kind of based on the underwater of Model Predictive Control
Robot path tracking and controlling method.This method can efficient on-line implement path point tracing control in robot under water, and
It is capable of handling the constraint of control input, certain robust can be kept in the case where model has uncertain and external interference
Property.
The present invention proposes a kind of underwater robot path tracking control method based on Model Predictive Control, and feature exists
In, method includes the following steps:
1) prediction model is constructed；Specific step is as follows:
1-1) establish the kinematics and dynamics model of underwater robot；
Kinematics and dynamics model expression under underwater robot continuous time is as follows:
Wherein, η=[x, y, ψ]TIt is the x-axis coordinate of underwater robot in level coordinates system, y-axis coordinate and bow to angle, v
=[u, v, r]TIt is the forward speed of underwater robot, side velocity and yaw rate in underwater robot body coordinate system,
WithRespectively indicate the time-derivative of above-mentioned two variable；
J, M, C, D matrix have following form:
Wherein, d1(u)=Xu-Xu|u||u|,d2=Yv-Yuvu-Yv|v|v,d3=
Yr-Yuru-Yr|r||r|,d4=Nv-Nuvu-Nv|v||v|,d5=Nr-Nuru-Nr|r||r|；M is the quality of underwater robot, IzzIt is
Around the rotary inertia of z-axis,It is mass coefficient, Xu、Xu|u|、Yv、Yuv、Yv|v|、Yr、
Yur、Yr|r|、Nv、Nuv、Nv|v|、Nr、Nur、Nr|r|It is resistance coefficient；
Power and torque on underwater robot three degree of freedom are as follows:
Wherein, ξ is the thrust of propeller, and δ is the yaw angle of rudder；μ=[ξ, δ]TIt is two control inputs, Yuuδ、
NuuδIt is the coefficient of coup；
1-2) determine that the constraint condition of control input, expression formula are as follows:
0≤ξ≤ξmax,-δmax≤δ≤δmax (3)
Wherein ξmaxIt is the maximum thrust of propeller, δmaxIt is the amplitude peak of rudder yaw angle；
It is 1-3) discrete time prediction model by the kinematics and dynamics model conversation of underwater robot；
By formula (1) and (2) discretization, following discrete time prediction model is obtained:
Wherein, Δ t > 0 is the sampling time interval of discretization, i.e. sampling time section [ti,ti+1] length；I is indicated
The ith sample moment；
Formula (4) is denoted as to the discrete time prediction model an of common version:
χ (i+1)=f (χ (i), μ (i)) (5)
Wherein, system state variables are as follows:
χ (i)=[ηT(i),vT(i)]T=[xT(i),yT(i),ψT(i),uT(i),vT(i),rT(i)]T,
Control input are as follows: μ (i)=[ξ (i), δ (i)]T；
2) model feedback corrects；Specific step is as follows:
2-1) derive Partial Linear Models；
It will include that the coupling terms of state variable and control input are considered as independent new variables in formula (5), before each coupling terms
Coefficient individually show, obtain following linear regression model (LRM):
χ (i+1)=Φ (χ (i), μ (i)) Θ (6)
Wherein Φ (χ (i), μ (i)) is i moment state variable and the regression matrix that control input is composed, and is denoted as Φ
(i)；Θ is the regression coefficient vector of all parameter compositions in formula (5) model；
Parameter 2-2) based on least square method of recursion On-line Estimation regression model；
Existed at the i-th moment using the least square method of recursion with forgetting factor to the regression model as shown in formula (6)
Line estimates parameter, and calculation expression is as follows:
Wherein,For the estimated value of i moment parameter Θ, initial valueIt is obtained by off-line identification, P (0)=α2I, 105≤α2
≤1010, forgetting factor 0.95≤λ≤0.995；
3) model determined based on step 1) and step 2), tracks underwater robot destination path point and carries out on-line optimization
It solves；Specific step is as follows:
Optimization object function 3-1) is determined based on line-of-sight navigation strategy；
Assuming that underwater robot current position coordinates are p=(x, y), a series of destination path points set is { p1,…,
pk,pk+1... }, wherein current goal path point is k-th of path point pk=(xk,yk)；
In line-of-sight navigation strategy, underwater robot bow to the deviation between angle and line-of-sight navigation angle be zero, i.e., | | ψ-ψLOS||
→ 0, underwater robot forward speed u is controlled to a stabilized speed ud, i.e., | | u-ud||→0；||ψ-ψLOS| | using anti-remaining
The vector angle form expression of string, i.e. sight line vectorWith underwater robot bow to vectorBetween angle, as shown in formula (8):
The final goal of path point tracking is that underwater robot is made to reach stable state, i.e. ψ=ψLOS, u=udAnd v=r
=0；This corresponding target steady state derives stable state control input from formula (1) are as follows:
μd=[ξd,δd]T=[Xuud-Xu|u|ud|ud|,0]T
Enable current time t0=0, the optimization object function of system is defined as:
Wherein, i indicates the ith sample moment, and Integer N is predicted time sequence length, positive real number aiAnd biIt is balance angle
The weight coefficient of tracking error and speed tracing error, positive real number matrixIt is between two control inputs of balance
Weight coefficient matrix；
It brings formula (8) into formula (9), two of μ (i) will be inputted in optimization object function about system mode χ (i) and control
Divide separately shown as follows:
Wherein,
The following constraint condition that system mode and control input meet:
3-2) judge the value of optimization object function J under current time system mode whether close to zero: if it is, working as
The preceding moment is using stable state control input μdControl as current time inputs, and in sampling time section (t0,t0+ Δ t] in protect
Hold it is constant, subsequently into step 3-7)；If not, thening follow the steps 3-3) carry out specific Optimization Solution；
Optimization object function 3-3) is derived to the partial derivative of control input variable: by state variable { χ (i), i=1 ..., N }
With control input { μ (i), i=0 ..., N-1 } simultaneous as optimized variable, by introducing Lagrange multiplier { λ (i), i=
1 ..., N }, the optimization object function formula (10) comprising equality constraint is converted are as follows:
Partial derivative such as following formula of the J about { μ (0) ..., μ (N-1) }:
By the necessary condition for optimality of state variable { χ (i), i=1 ..., N }, following recurrence formula is obtained:
Therefore, { λ (i), i=1 ..., N } is acquired according to formula (13), then substitutes into formula (12), acquires optimization object function pair
Control the partial derivative of input variable sequence
3-4) determine the initial feasible solution of current time control list entries: in current time t0, to t0What -1 moment was acquired
Optimum control list entries carries out continuation, as initial feasible solution, it may be assumed that
Wherein,For t0The optimum control list entries that -1 moment was acquired,For current t0The initial feasible solution of moment control list entries；In path trace start time, under water
Robot remains static, and the initial feasible solution for controlling list entries at this time is full null sequence；
3-5) iteratively solved using gradient projection method: the iteration since initial feasible solution, in present feasible Xie Chu according to formula
(12) and (13) seek negative gradient direction, determine iteration step length using A meter Huo Armijo rule, then obtained solution is projected to
In the feasible set that inequality defines in formula (11), next feasible solution is obtained；
For first step iteration:
Wherein, s0It is the iteration step length determined by Armijo rule, []+Indicate the projection of formula (11) constraint set；
Above-mentioned iterative process is repeated, until the solution of front and back iteration twice is so that the fall of J is less than the threshold epsilon ∈ of setting
(0,0.01) or iterative steps reach the maximum value M of settingmax, iteration ends obtain optimum control list entries { μ*
(0),…,μ*(N-1)}；
3-6) at current time, control input μ (t)=μ is applied to underwater robot*(0), t=(t0,t0+ Δ t], the control
Amount processed is in sampling time section (t0,t0+ Δ t] in remain unchanged；
Step 2-2 3-7) is returned to using the moment as new current time in subsequent time).
The features of the present invention and the utility model has the advantages that
The present invention proposes a kind of underwater robot path tracking control method based on Model Predictive Control, for the first time non-thread
The Nonlinear Model Predictive Control that can be executed online is realized on the underwater human occupant dynamic model of property, can directly handle control
The existing constraint of system input, and can guarantee stronger robustness in the case where model mismatch.The method overcome before
Model predictive control method can not meet the problem of control performance and computational efficiency requirement simultaneously, improve underwater robot path
The efficiency of search has higher application value.
Method of the invention is broadly divided into three links: building prediction model, model feedback correction, on-line optimization solve.
When constructing prediction model, it is contemplated that underwater robot model nonlinear degree height, drive lacking, this method is directly by its nonlinear model
Type avoids model linearization deviation as prediction model.Presence for model parameter mismatch, time-varying and external interference, this
The model online feedback correction link of invention, improves the robustness of algorithm.Model correction-based, on-line optimization solve link
Line-of-sight navigation strategy is improved first, determines optimization object function, the gradient projection method for being then based on state simultaneous, which is iterated, to be asked
Solution is capable of handling the constraint condition of control input.Meanwhile being calculated based on heuristic initial value and simplifying for omission iterative step, make
It can rapid solving go out optimal control sequence, guarantee the efficiency of on-line implement.Therefore the present invention can be realized simultaneously preferably
Control performance and higher online execution computational efficiency.
Detailed description of the invention
Fig. 1 is the line-of-sight navigation strategy schematic diagram of path point tracing control in the present invention.
Specific embodiment
The present invention proposes a kind of underwater robot path tracking control method based on Model Predictive Control, below with reference to attached
Figure and specific embodiment are further described as follows.
The present invention proposes a kind of underwater robot path tracking control method based on Model Predictive Control, is divided into three rings
Section: building prediction model, model feedback correction, on-line optimization solve；Method includes the following steps:
1) prediction model is constructed；Specific step is as follows:
1-1) establish the kinematics and dynamics model of underwater robot；
Implement the concrete form that this method first has to determine controlled underwater robot kinematics and dynamics model.Underwater machine
The kinematics and dynamics model of device people is the Nonlinear differential eguations that a multivariable intercouples, and this method is implemented to work as
The three-degree-of-freedom motion of preceding widely applied Kongsberg company REMUS Autonomous Underwater Vehicle (AUV) in the horizontal plane is
Example, the kinematics and dynamics model under continuous time are as follows:
Wherein, η=[x, y, ψ]TIt is the x-axis coordinate of underwater robot in fixed level coordinates system, y-axis coordinate and bow
To angle, v=[u, v, r]TIt is the forward speed of underwater robot, side velocity and yaw angle in underwater robot body coordinate system
Speed,WithRespectively indicate the time-derivative of the two variables.J, M, C in REMUS AUV model, D matrix have following shape
Formula:
Wherein, d1(u)=Xu-Xu|u||u|,d2=Yv-Yuvu-Yv|v||v|,d3
=Yr-Yuru-Yr|r||r|,d4=Nv-Nuvu-Nv|v||v|,d5=Nr-Nuru-Nr|r||r|.M is the quality of AUV, IzzIt is around z-axis
Rotary inertia,It is mass coefficient, Xu、Xu|u|、Yv、Yuv、Yv|v|、Yr、Yur、
Yr|r|、Nv、Nuv、Nv|v|、Nr、Nur、Nr|r|It is resistance coefficient.There are certain uncertainties for these coefficients, and may be with outer
The variation of boundary's water environment and have time variation.Therefore this method devises model feedback correction link, will be specific in step 2)
It introduces.
There are two power devices for AUV outfit: 1 tail portion propeller provides preceding to motive force, 1 rudder offer turn
To.According to the power relations of distribution of propeller and rudder, the power and torque on three degree of freedom can be obtained are as follows:
Wherein, ξ is the thrust of propeller, and δ is the yaw angle of rudder.μ=[ξ, δ]TIt is two control inputs, Yuuδ、
NuuδIt is the coefficient of coup (also there is uncertainty).As can be seen that two control inputs can not independent control three degree of freedom
State, therefore the AUV is drive lacking.Formula (1) and formula (2) constitute the kinematics and dynamics of REMUSAUV in the horizontal plane
Model.
1-2) determine the constraint condition of control input；
Often there is saturated phenomenon in the actuator of underwater robot, therefore need clearly to control input when design controller
Constraint condition.In formula (2), according to propeller and the actual mechanical property of rudder, all there is saturation limit in the two actuators
System, that is, control the constraint condition of input are as follows:
0≤ξ≤ξmax,-δmax≤δ≤δmax (3)
Wherein ξmaxIt is the maximum thrust of propeller, δmaxIt is the amplitude peak of rudder yaw angle.
It is 1-3) discrete time prediction model by the kinematics and dynamics model conversation of underwater robot；
In order to avoid model linearization deviation, the present invention directlys adopt the nonlinear model of underwater robot as prediction mould
Formula (1) and (2) discretization are obtained following discrete time prediction model by type:
Wherein, Δ t > 0 is the sampling time interval of discretization, i.e. sampling time section [ti,ti+1] length.I is indicated
The ith sample moment.Δ t takes sufficiently small, 0.1s < Δ t≤1s is generally taken, so that the system mode η (i) in sampling interval
Invariant can be considered as with v (i).Formula (4) can be denoted as the discrete time prediction model an of common version:
χ (i+1)=f (χ (i), μ (i)) (5)
Wherein, system state variables are；
χ (i)=[ηT(i),vT(i)]T=[xT(i),yT(i),ψT(i),uT(i),vT(i),rT(i)]T,
Control input are as follows: μ (i)=[ξ (i), δ (i)]T。
2) model feedback corrects；Specific step is as follows:
2-1) derive Partial Linear Models；
It will include that the coupling terms of state variable and control input are considered as independent new variables in formula (5), before each coupling terms
Coefficient individually show, then obtain following linear regression model (LRM):
χ (i+1)=Φ (χ (i), μ (i)) Θ (6)
Wherein Φ (χ (i), μ (i)) is i moment state variable and the regression matrix that control input is composed, and is denoted as Φ
(i)；Θ is the regression coefficient vector of all parameter compositions in formula (5) model.
Parameter 2-2) based on least square method of recursion On-line Estimation regression model；
Since parameter Θ has uncertain and time variation, at current time, this method will be according to the actual measurement state χ of system
(i) it is corrected with the system information undated parameter of previous moment, implementation model.For the regression model as shown in formula (6), i-th
At the moment, using the least square method of recursion On-line Estimation parameter for having forgetting factor, calculation expression is as follows:
Wherein,For the estimated value of i moment parameter Θ, initial valueIt can be obtained by off-line identification, P (0)=α2I, 105≤
α2≤1010, forgetting factor is usually chosen to 0.95≤λ≤0.995.It is calculatedIt will update into the model of formula (6), use
In the Optimization Solution of step 3)；
3) model determined based on step 1) and step 2), tracks underwater robot destination path point and carries out on-line optimization
It solves；Specific step is as follows:
Optimization object function 3-1) is determined based on line-of-sight navigation strategy (LineofSight)；
Assuming that underwater robot current position coordinates are p=(x, y), a series of destination path points set is { p1,…,
pk,pk+1... }, wherein current goal path point is k-th of path point pk=(xk,yk).P and pkPositional relationship it is as shown in Figure 1.
If AUV is moved to pkFor the center of circle, with rpWhen in the circle of radius, it is considered as AUV and reaches current goal path point.At this point, will
Next destination path point pk+1Line trace is clicked through as new destination path.
In Fig. 1, sight line vectorBe from the position of underwater robot be directed toward destination path point to
Amount.Line-of-sight navigation angle ψLOSIt is sight line vectorDeflection, ψ is the bow of underwater robot to angle.Track cross error is ec,
Sight crossover distance is Δ.
Line-of-sight navigation policy mandates underwater robot bow to the deviation between angle and line-of-sight navigation angle be zero, i.e., | | ψ-ψLOS|
| → 0, while underwater robot forward speed u being required to be controlled to a stabilized speed ud, i.e., | | u-ud||→0.In this method,
||ψ-ψLOS| | it is indicated using the vector angle form of anticosine, i.e. sight line vectorWith AUV bow to vectorBetween angle, as shown in formula (8):
The final goal of path point tracking is that AUV is made to reach stable state (ψ=ψLOS, u=udAnd v=r=0).This
Stable state is exactly that AUV with a constant speed drives towards destination path point straight in fact.This corresponding target steady state, from formula (1)
Derive stable state control input are as follows:
μd=[ξd,δd]T=[Xuud-Xu|u|ud|ud|,0]T
The purpose that on-line optimization solves is to seek a control list entries, and underwater robot is led according to sight
Boat strategy must faster track destination path point.Without loss of generality, current time t is enabled0=0, the optimization object function definition of system
Are as follows:
Wherein, i indicates the ith sample moment after current time, and Integer N is predicted time sequence length, generally takes 5
≤ N≤10, positive real number aiAnd biIt is the weight coefficient for balancing angle error in tracking and speed tracing error, positive real number matrixIt is the weight coefficient matrix balanced between two control inputs.In general, when preceding several after current time
The status tracking effect at quarter is even more important, therefore chooses coefficient ai,bi,RiReduce with the increase of i.
It brings formula (8) into formula (9), two of μ (i) will be inputted in optimization object function about system mode χ (i) and control
Divide separately shown as follows:
Wherein
The following constraint condition that system mode and control input need to meet:
That is equality constraint of the prediction model formula (5) as state variable and control input, controls the amplitude constraint item of input
Part formula (3) is used as inequality constraints.Formula (10) and formula (11) feature the optimization problem for needing to solve.
3-2) judge the value of optimization object function J under current time system mode whether close to zero.If it is, working as
The preceding moment directlys adopt stable state control input μdControl as current time inputs, and in sampling time section (t0,t0+Δt]
It inside remains unchanged, subsequently into step 3-7)；If not, thening follow the steps 3-3) carry out specific Optimization Solution；
Optimization object function 3-3) is derived to the partial derivative of control input variable: by state variable { χ (i), i=1 ..., N }
With control input { μ (i), i=0 ..., N-1 } simultaneous as optimized variable, by introducing Lagrange multiplier { λ (i), i=
1 ..., N }, the optimization object function formula (10) comprising equality constraint is converted are as follows:
Partial derivative such as following formula of the J about { μ (0) ..., μ (N-1) }:
By the necessary condition for optimality of state variable { χ (i), i=1 ..., N }, following recurrence formula can be obtained:
Therefore, { λ (i), i=1 ..., N } is acquired according to formula (13), then substitutes into formula (12), it can be in the hope of optimization aim letter
The partial derivative of several pairs of control input variable sequences
3-4) determine the initial feasible solution of current time control list entries: in current time t0, to t0What -1 moment was acquired
Optimum control list entries carries out continuation, as initial feasible solution, it may be assumed that
Wherein,For t0The optimum control list entries that -1 moment was acquired,For current t0The initial feasible solution of moment control list entries.In path trace start time, under water
Robot remains static, and the initial feasible solution for controlling list entries at this time is full null sequence；
3-5) iteratively solved using gradient projection method: the iteration since initial feasible solution, in present feasible Xie Chu according to formula
(12) and (13) seek negative gradient direction, determine iteration step length using A meter Huo (Armijo) rule, then obtained solution is projected
In the feasible set that inequality defines in formula (11), next feasible solution is obtained, by taking first step iteration as an example:
Wherein
s0It is the iteration step length determined by Armijo rule, []+Indicate the projection of formula (11) constraint set.Armijo rule it is specific
Method are as follows: Yu Xianxuanding initial step length d > 0, reduction factor β and parameter σ meet 0 < β <, 1,0 < σ < 1 respectively.Specifically
Step-length is given by the following formula:
α=βld
Wherein l is the smallest positive integer for setting up following inequality:
Repeat above-mentioned iterative process, until front and back twice iteration solution so that J less than one normal number threshold of fall
Value ε ∈ (0,0.01) or iterative steps reach the maximum value M of settingmax(generally take 20≤Mmax≤ 50), iteration ends obtain
To optimum control list entries { μ*(0),…,μ*(N-1)}；
3-6) at current time, control input μ (t)=μ is applied to the actuator of AUV*(0), t=(t0,t0+ Δ t], it should
Control amount is in sampling time section (t0,t0+ Δ t] in remain unchanged；
Step 2-2 3-7) is returned to using the moment as new current time in subsequent time).
Claims (1)
1. a kind of underwater robot path tracking control method based on Model Predictive Control, which is characterized in that this method includes
Following steps:
1) prediction model is constructed；Specific step is as follows:
1-1) establish the kinematics and dynamics model of underwater robot；
Kinematics and dynamics model expression under underwater robot continuous time is as follows:
Wherein, η=[x, y, ψ]TIt is the x-axis coordinate of underwater robot in level coordinates system, y-axis coordinate and bow to angle, v=[u,
v,r]TIt is the forward speed of underwater robot, side velocity and yaw rate in underwater robot body coordinate system,WithPoint
The time-derivative of above-mentioned two variable is not indicated；
J, M, C, D matrix have following form:
Wherein, d1(u)=Xu-Xu|u||u|,d2=Yv-Yuvu-Yv|v||v|,d3
=Yr-Yuru-Yr|r||r|,d4=Nv-Nuvu-Nv|v||v|,d5=Nr-Nuru-Nr|r||r|；M is the quality of underwater robot, Izz
It is the rotary inertia around z-axis,It is mass coefficient, Xu、Xu|u|、Yv、Yuv、Yv|v|、Yr、
Yur、Yr|r|、Nv、Nuv、Nv|v|、Nr、Nur、Nr|r|It is resistance coefficient；
Power and torque on underwater robot three degree of freedom are as follows:
Wherein, ξ is the thrust of propeller, and δ is the yaw angle of rudder；μ=[ξ, δ]TIt is two control inputs, Yuuδ、NuuδIt is coupling
Collaboration number；
1-2) determine that the constraint condition of control input, expression formula are as follows:
0≤ξ≤ξmax,-δmax≤δ≤δmax (3)
Wherein ξmaxIt is the maximum thrust of propeller, δmaxIt is the amplitude peak of rudder yaw angle；
It is 1-3) discrete time prediction model by the kinematics and dynamics model conversation of underwater robot；
By formula (1) and (2) discretization, following discrete time prediction model is obtained:
Wherein, Δ t > 0 is the sampling time interval of discretization, i.e. sampling time section [ti,ti+1] length；I is indicated i-th
Sampling instant；
Formula (4) is denoted as to the discrete time prediction model an of common version:
χ (i+1)=f (χ (i), μ (i)) (5)
Wherein, system state variables are as follows:
χ (i)=[ηT(i),vT(i)]T=[xT(i),yT(i),ψT(i),uT(i),vT(i),rT(i)]T,
Control input are as follows: μ (i)=[ξ (i), δ (i)]T；
2) model feedback corrects；Specific step is as follows:
2-1) derive Partial Linear Models；
It will include that state variable and the coupling terms of control input are considered as independent new variables in formula (5), before each coupling terms is
Number individually shows, and obtains following linear regression model (LRM):
χ (i+1)=Φ (χ (i), μ (i)) Θ (6)
Wherein Φ (χ (i), μ (i)) is i moment state variable and the regression matrix that control input is composed, and is denoted as Φ (i)；Θ
It is the regression coefficient vector of all parameter compositions in formula (5) model；
Parameter 2-2) based on least square method of recursion On-line Estimation regression model；
The regression model as shown in formula (6) is estimated at the i-th moment using the least square method of recursion with forgetting factor online
Parameter is counted, calculation expression is as follows:
Wherein,For the estimated value of i moment parameter Θ, initial valueIt is obtained by off-line identification, P (0)=α2I, 105≤α2≤
1010, forgetting factor 0.95≤λ≤0.995；
3) model determined based on step 1) and step 2) is tracked progress on-line optimization to underwater robot destination path point and asked
Solution；Specific step is as follows:
Optimization object function 3-1) is determined based on line-of-sight navigation strategy；
Assuming that underwater robot current position coordinates are p=(x, y), a series of destination path points set is { p1,…,pk,pk +1... }, wherein current goal path point is k-th of path point pk=(xk,yk)；
In line-of-sight navigation strategy, underwater robot bow to the deviation between angle and line-of-sight navigation angle be zero, i.e., | | ψ-ψLOS| | → 0,
Underwater robot forward speed u is controlled to a stabilized speed ud, i.e., | | u-ud||→0；||ψ-ψLOS| | using anticosine
The expression of vector angle form, i.e. sight line vectorWith underwater robot bow to vector
Between angle, as shown in formula (8):
The final goal of path point tracking is that underwater robot is made to reach stable state, i.e. ψ=ψLOS, u=udAnd v=r=0；
This corresponding target steady state derives stable state control input from formula (1) are as follows:
μd=[ξd,δd]T=[Xuud-Xu|u|ud|ud|,0]T
Enable current time t0=0, the optimization object function of system is defined as:
Wherein, i indicates the ith sample moment, and Integer N is predicted time sequence length, positive real number aiAnd biIt is balance angleonly tracking
The weight coefficient of error and speed tracing error, positive real number matrix Ri=diag (ri 1,ri 2) it is between two control inputs of balance
Weight coefficient matrix；
It brings formula (8) into formula (9), will divide in optimization object function about system mode χ (i) and two parts of control input μ (i)
It opens and is expressed as follows:
Wherein,
L (μ (i))=ri 1(ξ(i)-ξd)2+ri 2δ2(i)
The following constraint condition that system mode and control input meet:
3-2) judge the value of optimization object function J under current time system mode whether close to zero: if it is, when current
It carves using stable state control input μdControl as current time inputs, and in sampling time section (t0,t0+ Δ t] in keep not
Become, subsequently into step 3-7)；If not, thening follow the steps 3-3) carry out specific Optimization Solution；
Optimization object function 3-3) is derived to the partial derivative of control input variable: by state variable { χ (i), i=1 ..., N } and control
System input { μ (i), i=0 ..., N-1 } simultaneous be used as optimized variable, by introducing Lagrange multiplier λ (i), i=1 ...,
N }, the optimization object function formula (10) comprising equality constraint is converted are as follows:
Partial derivative such as following formula of the J about { μ (0) ..., μ (N-1) }:
By the necessary condition for optimality of state variable { χ (i), i=1 ..., N }, following recurrence formula is obtained:
Therefore, { λ (i), i=1 ..., N } is acquired according to formula (13), then substitutes into formula (12), acquires optimization object function to control
The partial derivative of input variable sequence
3-4) determine the initial feasible solution of current time control list entries: in current time t0, to t0- 1 moment was acquired optimal
It controls list entries and carries out continuation, as initial feasible solution, it may be assumed that
Wherein,For t0The optimum control list entries that -1 moment was acquired,For current t0The initial feasible solution of moment control list entries；In path trace start time, under water
Robot remains static, and the initial feasible solution for controlling list entries at this time is full null sequence；
3-5) iteratively solved using gradient projection method: the iteration since initial feasible solution, in present feasible Xie Chu according to formula (12)
(13) negative gradient direction is sought, iteration step length is determined using A meter Huo Armijo rule, then obtained solution is projected into formula
(11) in the feasible set that inequality defines in, next feasible solution is obtained；
For first step iteration:
Wherein, s0It is
The iteration step length determined by Armijo rule, []+Indicate the projection of formula (11) constraint set；
Repeat above-mentioned iterative process, until front and back twice iteration solution so that the fall of J be less than setting threshold epsilon ∈ (0,
0.01) or iterative steps reach the maximum value M of settingmax, iteration ends obtain optimum control list entries { μ*(0),…,
μ*(N-1)}；
3-6) at current time, control input μ (t)=μ is applied to underwater robot*(0), t=(t0,t0+ Δ t], the control amount
In sampling time section (t0,t0+ Δ t] in remain unchanged；
Step 2-2 3-7) is returned to using the moment as new current time in subsequent time).
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