CN105174061B - Double pendulum crane length of a game optimal trajectory planning method based on pseudo- spectrometry - Google Patents

Abstract

A kind of double pendulum crane length of a game optimal trajectory planning method based on pseudo- spectrometry.Solving non-linear double pendulum bridge type crane system and automatically controlling problem, there is method good chassis positioning performance is eliminated with two-stage hunting of load.Line translation is entered to system kinematics model first, to facilitate ensuing analysis.Consider including the various constraints including two-stage pivot angle and machine speed and acceleration higher limit afterwards, construct corresponding optimization problem.Subsequently, the optimization problem of the belt restraining is converted into using Gauss puppet spectrometry and is easier to the nonlinear programming problem of solution and solves, obtain time optimal chassis track.The present invention is processed and is converted to complicated time optimal problem using the thought of pseudo- spectrometry, reduces the difficulty of solution；Meanwhile, this method can obtain the optimum result of length of a game, can be greatly enhanced the work efficiency of crane system.Emulation with test result indicate that, the present invention can obtain good control effect, with good actual application value.

Description

Double pendulum crane length of a game optimal trajectory planning method based on pseudo- spectrometry

Technical field

The invention belongs to the technical field that non-linear lack of driven electric system is automatically controlled, more particularly to a kind of based on puppet The double pendulum crane length of a game optimal trajectory planning method of spectrometry.

Background technology

In industrial processes, it is carrying load to desired position, including overhead crane, rotary cranes, tower Formula crane, marine hoist have a very wide range of applications in interior all kinds of crane systems.In order to simplify the machinery knot of crane system Structure, often not directly control load, but the motion by chassis, drag indirectly and be loaded to target location.This structure band Come as a result, the control input dimension of crane system be less than degree of freedom dimension to be controlled.System with the characteristic is so-called Under-actuated systems^[1].Full drive system is compared, due to the presence of drive lacking characteristic, under-actuated systems are often more difficult to control, and In actual production, crane system is often by veteran operative.In the event of maloperation, load may be caused violent Swing causes collision, or even security incident occurs.Therefore, the research of crane system autocontrol method is had realistic meaning with Extensively using value, has obtained the concern of numerous scholars.

For bridge type crane system, main control targe includes two aspects, i.e., chassis fast and accurately positioning with The suppression of hunting of load and elimination.However, these two aspects be typically it is conflicting, i.e., too fast trolley movement often lead to compared with Big hunting of load.Therefore, while realizing that the control targe of these two aspects has higher difficulty.In order to obtain preferably control Effect, current Chinese scholars have been proposed for many crane system autocontrol methods.Tuan etc. proposes anti-based on part The control method of linearization^[2-3], the control algorithm design of bridge type crane system can be simplified.In document [4], in [5], Singhose etc. is controlled to crane system using the thought of input shaper, effectively can suppress to load Residual oscillations.For place Uncertain external interference is managed, research worker controls crane system using sliding Mode Algorithm^[6-7], it is possible to obtain good robustness. Hu Zhou etc. proposes a kind of nonlinear transformations fused controlling method^[8], the problem of saturation can be input into processing controller, it is right to realize The high performance control of crane system.Document [9], [10] propose the control strategy based on energy with passivity, can obtain preferably Effect.In addition, in recent years, including genetic algorithm^[11], fuzzy control^[12]Deng some intelligent control methods equally in crane Control field has certain application.

It is well known that the hunting of load of crane system is caused by the acceleration and deceleration motion of chassis, in trolley movement and load pendulum Stronger coupling be there is between dynamic.Based on this, a suitable track can be planned for chassis, when chassis is transported according to the track When dynamic, you can realize to its quick pinpoint target.Simultaneously, it is contemplated that pivot angle suppresses and the requirement for eliminating, and advises in track During drawing, by analysing in depth and rationally using the coupled relation between trolley movement and hunting of load, one can be cooked up Bar has the chassis track of the pendulum ability that disappears.The double goal that chassis is quickly accurately positioned the pendulum that disappears with load can so be completed.Base In the thought, research worker has been presented for many crane method for planning track^[13-17].For example, in document [13], Uchiyama Etc. a kind of open loop control strategy for jib crane is proposed, in order to eliminate Residual oscillations, they are the horizontal movement of cantilever A S types track is planned.Document [14] then proposes a kind of method for planning track based on phase plane analysis, and it can be preferable Ground suppresses hunting of load, and eliminates Residual oscillations.

For crane system, although above-mentioned various methods can obtain preferable control effect in the ideal case, but These methods are assumed that the quality of suspension hook can be ignored and load to regard particle as, and the hunting of load of crane system is considered as list Oscillator system.If the quality of suspension hook is larger, it is impossible to ignore, or load shapes are very big, it is impossible to be simply viewed as particle, in such case Under, the swing of crane system will present double pendulum phenomenon, i.e. suspension hook carries out one-level swing around chassis, while load is sent out around suspension hook Raw two grades of swings.Now, above-mentioned single pendulum overhead crane control method cannot obtain gratifying control performance, only exist pole at present The crane system control strategy of double pendulum effect is considered less.Natural frequency of the document [18-20] by analysis double pendulum crane system, will The method of input shaper successfully expands to double pendulum crane system.Sun Ning etc. is considering the constraint of system pivot angle, machine speed constraint On the premise of etc. a series of constraints, it is proposed that a kind of double pendulum crane optimal trajectory planning method theoretical based on differential flat^[21]。 A kind of energy of the Guo Wei equalitys by analysis crane system, it is proposed that double pendulum overhead crane control strategy based on passivity^[22]。

Although above-mentioned control strategy can realize the control to double pendulum crane system, they cannot realize length of a game most Excellent control effect, i.e., cannot ensure the maximization of crane system operational efficiency.Therefore need to design suitable control method, with Improve the work efficiency of crane system.

The content of the invention

Present invention aim to address existing double pendulum effect bridge type crane system method for planning track above shortcomings, A kind of double pendulum crane length of a game optimal trajectory planning method based on pseudo- spectrometry is provided.

This invention address that by the kinematics model of bridge type crane system of the analysis with double pendulum effect, it is proposed that a kind of Based on the double pendulum crane length of a game optimal trajectory planning method of pseudo- spectrometry, the optimum trolley rail of a length of a game has been obtained Mark, while completing chassis and being accurately positioned, realizes the quick suppression of two-stage hunting of load and eliminates, and be applied to reality Border crane platform is tested, and can be greatly enhanced the work efficiency of crane system.

What the present invention was provided is included based on the double pendulum crane length of a game optimal trajectory planning method of pseudo- spectrometry：

1st, profile constraints are analyzed and corresponding optimization problem is constructed

The control targe of analysis crane system, it is considered to including including two-stage pivot angle and machine speed and acceleration higher limit Various constraints, draw the following optimization problem with the shipping time as cost function：

Wherein, x (t) represents the position of chassis, x_fRepresent the target location of chassis, the t express times in bracket, after variable Face (t) represents that the variable is the variable with regard to the time, for simplicity's sake, (t) in most of variable, T tables is omitted in formula Show the total time for completing to transport, min represents minimum, connects the constraints for needing to consider is represented behind s.t.；Respectively Represent first derivative and second dervative of chassis position x (t) with regard to the time, i.e. machine speed and acceleration；v_max,a_maxGeneration respectively Chassis maximal rate and peak acceleration that table is allowed；θ₁(t),θ₂T () represents one-level and two grades of pivot angles respectively,Represent one-level and two grades of angular velocity；θ_1max,θ_2maxRepresent that the one-level allowed during transporting most is put on two grades Angle, ω_1max,ω_2maxRepresent allowed one-level and two grades of maximum angular rates.

2nd, acceleration driving model is set up and is converted with optimization problem

Analysis obtains following speed-up degree drive system model with using double pendulum bridge type crane system：

Wherein, ζ represents the total state vector of system, is defined as follows：

Wherein, x (t),Chassis Position And Velocity, θ are represented respectively₁(t),Represent one-level pivot angle and angular velocity, θ₂ (t),Two grades of pivot angles and angular velocity are represented, the subscript T of bracket represents the transposition computing of matrix；What u (t) represented the system is System input, and For chassis acceleration；F (ζ), h (ζ) are represented with the total state vector ζ of system as independent variable Function, obtained by crane system kinematical equation, concrete form is shown in (7)；Total state for system is vectorial with regard to the time Derivative.

Using above-mentioned acceleration drive system model, former optimization problem is converted into following form：

Wherein, ζ represents the total state vector of system, and u (t) represents the system input of the system, and For Chassis acceleration；T express times in bracket, behind variable, (t) represents that the variable is the variable with regard to the time, is simple and clear rising See, (t) in most of variable is omitted in formula, and T represents the total time for completing to transport, and min represents minimum, connects behind s.t. Represent the constraints for needing to consider；The subscript T of vector represents the transposition computing of matrix；x_fRepresent given chassis target position Put,Represent machine speed, v_max,a_maxAllowed chassis maximal rate and peak acceleration are represented respectively；θ₁(t),θ₂ T () represents one-level and two grades of pivot angles respectively,Represent one-level and two grades of angular velocity；θ_1max,θ_2maxRepresent transport process The one-level of middle permission and two grades of maximum pendulum angles, ω_1max,ω_2maxRepresent allowed one-level and two grades of maximum angular rates.

3rd, the trajectory planning based on Gauss puppet spectrometry

The optimization problem in the 2nd step is processed and solved using the thought of Gauss puppet spectrometry, comprised the following steps that：

3.1st, first with lagrange-interpolation, select at Legendre-Gauss (Legendre-Gauss, LG) point Discrete system state trajectory and input trajectory, by discrete loci and Lagrange interpolation polynomial, represent corresponding near Like locus model.

3.2nd, then, the locus model after pairing approximation carries out derivation, by the derivative Lagrange of system mode Multinomial derivative is represented.

3.3rd, subsequently, using discrete locus model and its derivative, original system kinematics model is converted into a series of Polynomial equation；Using Gauss integration, the boundary condition in the 2nd step in optimization problem is equally expressed as the shape of polynomial equation Formula.

3.4th, last, time optimal trajectory planning problem is converted into a kind of Non-Linear Programming with Algebraic Constraint and asks Topic, obtains global optimum's time and optimal trajectory by solving.

4th, track following

By code-disc or laser sensor, test desk truck position and rate signal x (t),Using obtained by the 3.4th step Chassis time optimal reference locus to be tracked and corresponding speed trajectory, selection percentage differential (proportional- Derivative, PD) controller is as follows：

Wherein, driving force of F (t) roles of delegate on chassis, x_r(t),Represent with reference to deformation trace respectively and Speed trajectory, k_p,k_dIt is the positive control gain for needing adjustment.Using the controller, corresponding real-time control can be calculated Signal, drives trolley movement, completes control targe.

The theoretical foundation and derivation of the inventive method

1st, profile constraints are analyzed and corresponding optimization problem is constructed

Overhead crane with double pendulum effect, its kinematics model are as follows：

Wherein, m₁,m₂The quality of suspension hook and load is represented respectively, and M represents the quality of chassis；X (t) represents chassis displacement,Represent second dervatives of the x (t) with regard to the time, i.e. chassis acceleration；T express times, behind variable, (t) represents that the variable is With regard to the variable of time, for simplicity's sake, (t) in most of variable is omitted in formula；θ₁(t),θ₂(t) represent one-level with Two grades of pivot angles (pivot angle of suspension hook pivot angle and load around suspension hook),For corresponding angular velocity,Accelerate for angle Degree；l₁Represent the length of lifting rope, l₂For equivalent rope length, that is, load the distance between barycenter and suspension hook barycenter；G is acceleration of gravity.

For formula (1), (2), both sides are with divided by (m respectively₁+m₂)l₁With m₂l₂, and abbreviation obtains：

Formula (3), (4) describe chassis displacement x (t) and system two-stage pivot angle θ₁(t),θ₂Coupled relation between (t), i.e., Impact of the motion of chassis to hunting of load.By analysing in depth the coupled relation, a chassis with the pendulum ability that disappears is planned Track, is the basis of the present invention.

For deadline optimal trajectory planning, it is contemplated that target of the crane system in real work, physical constraint and peace Quan Xing, the present invention are systematically considered profile constraints following aspects^[21]:

1) to realize quick and accurate chassis positioning, chassis is from initial position x₀Setting in motion, the elapsed time, T reached mesh Cursor position x_f, and the machine speed of start time and finish time, acceleration are 0, i.e.,

Wherein, T represents the time required for transport process；For initial position, without loss of generality, x is chosen here₀=0.

2) in view of the output bounded of real electrical machinery, during transport, the speed of chassis, acceleration all should be maintained at suitable In the range of, i.e.,

Wherein, v_max,a_maxThe chassis maximal rate and acceleration of permission are represented respectively,Represent machine speed With the absolute value of acceleration.

3) it is the process that ensures at the end of transport directly to carry out load next step, chassis should after reaching target location Without Residual oscillations and angular velocity is also 0, i.e.,

4) it is to avoid due to the collision that causes of violent swing for loading, during transport, pivot angle and angle that two-stage swings In allowed limits, i.e., speed should all keep

Wherein, θ_1max,θ_1maxThe maximum angle that one-level is allowed with two grades of pivot angles is represented respectively；ω_1max,ω_2maxRepresenting allows One-level and two grades of maximum angular rates.

To sum up, construct following optimization problem:

Wherein, min represents minimum, connects and represent the constraints for needing to consider behind s.t..Next, will be by pseudo- spectrometry The optimization problem is solved, and a time optimal track is cooked up for chassis.

2nd, acceleration driving model is set up and is converted with optimization problem

For convenience of follow-up trajectory planning, first double pendulum crane system model and optimization problem (5) are carried out turning here Change.System total state vector ζ (t) is defined for this as follows：

Wherein, x (t),Chassis Position And Velocity, θ are represented respectively₁(t),Represent one-level pivot angle and angular velocity, θ₂ (t),Two grades of pivot angles and angular velocity are represented, the subscript T of bracket represents the transposition computing of matrix.According to formula (3), (4), by platform Input of the acceleration of car as system.Now, kinematics model is converted into following form：

Wherein,Represent derivatives of the ζ (t) with regard to the time；U (t) is chassis accelerationF (ζ), h (ζ) represent with regard to The auxiliary function of ζ (t), concrete form are as follows：

Wherein, for convenience of describing, following auxiliary variable A, B, C, D are defined：

In above formula, following reduced form has been used：

S₁=sin θ₁,S₂=sin θ₂,C₁=cos θ₁,C₂=cos θ₂,

S_1-2=sin (θ₁-θ₂),C_1-2=cos (θ₁-θ₂).

Using acceleration drive system model (6) of gained, former optimization problem (5) is being converted into：

Wherein, ζ represents the total state vector of system, and u (t) represents the system input of the system, and For Chassis acceleration；T express times in bracket, behind variable, (t) represents that the variable is the variable with regard to the time, is simple and clear rising See, (t) in most of variable is omitted in formula, and T represents the total time for completing to transport, and min represents minimum, connects behind s.t. Represent the constraints for needing to consider；The subscript T of vector represents the transposition computing of matrix；x_fRepresent given chassis target position Put,Represent machine speed, v_max,a_maxAllowed chassis maximal rate and peak acceleration are represented respectively；θ₁(t),θ₂ T () represents one-level and two grades of pivot angles respectively,Represent one-level and two grades of angular velocity；θ_1max,θ_2maxRepresent transport process The one-level of middle permission and two grades of maximum pendulum angles, ω_1max,ω_2maxRepresent allowed one-level and two grades of maximum angular rates.

Next, the optimization problem will be solved by pseudo- spectrometry, and a time optimal track is cooked up for chassis.

3rd, the trajectory planning based on Gauss puppet spectrometry

To adapt to the requirement of Gauss puppet spectrometry, it is necessary first to utilize coordinate transform, by track corresponding time interval by t ∈ On interval τ ∈ [- 1,1], i.e., [0, T] is transformed into

Here τ represents the auxiliary variable of similar time.Subsequently choose K Legendre-Gauss (Legendre-Gauss, LG) point { τ₁,τ₂,...,τ_K∈ (- 1,1) composition point ranges.Here τ₁,τ₂,...,τ_KSelected LG points, subscript table are represented Show the serial number 1,2 of the point ..., K；K is the number of the LG points of selection.The selection of LG points is multinomial by the Legendre for solving K ranks The zero point of formula is obtained.Meanwhile, τ₀=-1 first place for being added to point range, system state amount to be planned and input quantity discrete representation into Following form：

ζ(τ₀),ζ(τ₁),ζ(τ₂),...,ζ(τ_K),

u(τ₀),u(τ₁),u(τ₂),...,u(τ_K),

Using the K+1 node, K+1 Lagrange interpolation polynomial is constructed, concrete form is as follows：

Wherein,Represent the Lagrangian differential polynomial of serial number i, i ∈ { 0,1 ..., K }；Here, τ ∈ [- 1, 1]；The company of expression takes advantage of symbol, i.e., from the beginning of the item of sequence number j=0, take the item of j=K always, and during skip j=i's .Using the value of system state amount and input quantity at formula (11) and LG points, the quantity of state track of system and input quantity rail Mark is gone out by the approximate table of manner below：

Wherein, ζ (τ_i),u(τ_i) τ=τ is represented respectively_iThe system state amount and input quantity at place；Represent summation sign, Start to be added to the item of i=K from the item of sequence number i=0.To formula (12) derivation, and using in formula (11) interpolating function it is concrete Form, calculates and abbreviation, and the derivative for obtaining quantity of state track is as follows：

Wherein,Represent τ=τ_kThe state trajectory derivative value at place；Represent τ=τ_kPlace, lagrange polynomialDerivative value, concrete form is as follows：

It is worth using the track at formula (13), (14) and LG points, the differential equation in optimization problem (9) is constrainedCarry out discretization to process with approximation, concrete outcome is as follows：

Wherein, k ∈ { 0,1,2 ..., K }.The form of (15) with Algebraic Constraint formula.Next, the side in optimization problem Boundary's constraint is also required to the form for changing into Algebraic Constraint, wherein the boundary constraint of 0 moment directly rewrites as follows:

ζ (0)=[0 0000 0]^T.

To represent the boundary constraint of transport process finish time, τ is defined_K+1=1.From formula (10), τ_K+1=1 corresponds to Transport finish time t=T.Using Gauss integration, the edge-restraint condition is expressed as：

Wherein, ζ (τ₀) i.e. above-mentioned system initial state vector；w_kRepresent k-th Legendre weights (Legendre Weight), occurrence is tried to achieve in the lump when LG points are solved.

To sum up, in optimization problem, all of constraint can be gone out by the form table of Algebraic Constraint, and based on this, former optimization is asked Topic changes into a kind of nonlinear programming problem with Algebraic Constraint, specific as follows shown：

min T

s.t.

ζ (0)=[0 0000 0]^T,

ζ(τ)-χ≤0,-ζ(τ)-χ≤0,

u(τ)-a_max≤0,-u(τ)-a_max≤0

Wherein, vectorial χ is defined as follows：

χ=[∞ v_max θ_1max ω_1max θ_2max ω_2max]^T

Wherein, ∞ represents infinity.For above-mentioned constrained nonlinear programming, the planning of continuous quadratic type is selected here Method (sequential quadratic programming, SQP) is solved, and obtains following time optimal state vector sequence Row：

ζ(τ₀),ζ(τ₁),ζ(τ₂),...,ζ(τ_K),ζ(τ_K+1),

Above formula is time-discrete optimum state sequence vector.Take first two (chassis displacement and chassis speed of each vector Degree), row interpolation of going forward side by side obtains the optimum chassis displacement of corresponding length of a game and speed trajectory.

4th, track following

By code-disc or laser sensor, test desk truck position and rate signal x (t),Using treating obtained by the 3rd step Tracking chassis time optimal reference locus and corresponding speed trajectory, selection percentage differential (proportional- Derivative, PD) controller is as follows：

Wherein, driving force of F (t) roles of delegate on chassis, x_r(t),Represent with reference to deformation trace respectively and Speed trajectory, k_p,k_dIt is the positive control gain for needing adjustment.Using the controller, corresponding real-time control can be calculated Signal, drives trolley movement, completes control targe.

The advantages of the present invention

The present invention is for the overhead crane with double pendulum effect, it is proposed that when a kind of double pendulum crane based on pseudo- spectrometry is global Between optimal trajectory planning method.Specifically, the kinematics model of crane system is converted into into a kind of acceleration first and drives mould Type, and it is based on this model, it is considered to various constraints, construct the optimization problem of belt restraining；Subsequently, using Gauss puppet spectrometry to gained Optimization problem is processed, and is translated into the nonlinear programming problem of more convenient solution.On this basis, you can obtain the time Optimum chassis track.This method for planning track proposed by the present invention except consider disappear pendulum target in addition to, can also be very square Just the actual physics constraints such as pivot angle constraint, angular speed constraint, machine speed constraint, acceleration constraint are processed.With existing method Except for the difference that, method proposed by the present invention can obtain length of a game's optimal solution, drastically increase the work efficiency of crane system. Finally, by emulation and experiment, demonstrate effectiveness of the invention.

Description of the drawings：

Fig. 1 represents Trajectory Planning result 1 (chassis displacement and rate curve) in the present invention；

Fig. 2 represents Trajectory Planning result 2 (two-stage pivot angle and angular velocity curve) in the present invention；

Fig. 3 represents trajectory planning experimental result in the present invention；

Fig. 4 represents liner quadratic regulator device experimental result；

Fig. 5 representative polynomial trajectory planning experimental results.

Specific embodiment：

Embodiment 1：

The control targe of analysis crane system, it is considered to including including two-stage pivot angle and machine speed and acceleration higher limit Various constraints, obtain the following optimization problem with the shipping time as cost function：

Here, the target location for selecting chassis is x_f=0.6m, profile constraints are as follows：

θ_1max=θ_2max=2deg, v_max=0.3m/s, ω_1max=ω_2max=5deg/s, a_max=0.15m/s²

2nd, acceleration driving model is set up and is converted with optimization problem

Analysis sets up following speed-up degree drive system model with using double pendulum bridge type crane system：

As the model expression is excessively complicated, repeat no more here, only provide the respective physical ginseng of crane system Number, it is as follows：

M=6.5kg, m₁=2.003kg, m₂=0.559kg, g=9.8m/s²,l₁=0.53m, l₂=0.4m.

Then optimization problem is converted into following form：

3rd, the trajectory planning based on Gauss puppet spectrometry

To realize the method for planning track based on pseudo- spectrometry proposed by the invention, used here as GPOPS software tools Case^[23]And SNOPT workboxes^[24]The offline optimization problem solved in the 2nd step simultaneously obtains corresponding time optimal track.Wherein, Choose Legendre-Gauss point parameter K=750.Specific result is referring to emulation experiment description section.

4th, emulation experiment effect description

4.1st, simulation result

The feasibility of trajectory planning algorithm is proposed for the checking present invention, is carried out in MATLAB/Simulink environment first Numerical simulation.Simulation process is divided into two steps.The first step, is that chassis plans a time optimal reference locus according to this method； Second step, it is assumed that chassis is run according to the reference locus, obtains the track of chassis and pivot angle.

The result of emulation is as shown in accompanying drawing 1, accompanying drawing 2.In accompanying drawing 1, dotted line represents chassis target location, and dotted line represents platform Vehicle speed is constrained, and solid line represents simulation result.In accompanying drawing 2, dotted line represents angle restriction, and dotted line represents angular speed constraint, real Line represents simulation result.From accompanying drawing 1 as can be seen that when chassis is run along reference locus, chassis is quickly and accurately restrained To target location x_f=0.6m；Meanwhile, in whole process, the speed of chassis meets set constraint.Can from accompanying drawing 2 Go out, during chassis is transported, two-stage pivot angle is respectively less than given binding occurrence 2deg；Meanwhile, angular velocity is also in institute's restriction range；And At the end of transport, there are no Residual oscillations in two-stage pivot angle, i.e., the target of the pendulum that quickly disappears also is capable of achieving.

4.2nd, experimental result

By code-disc or laser sensor, test desk truck position and rate signal x (t),Using treating obtained by the 3rd step Tracking chassis time optimal reference locus and corresponding speed trajectory, selection percentage differential (proportional- Derivative, PD) controller is as follows：

Wherein, driving force of F (t) roles of delegate on chassis, x_r(t),Represent with reference to deformation trace respectively and Speed trajectory, k_p,k_dIt is the positive control gain for needing adjustment.Using the controller, corresponding real-time control can be calculated Signal, drives trolley movement, completes control targe.

In an experiment, the tracking control unit of selection controls gain and is：

k_p=750, k_d=150

Experimental result is as shown in Figure 3.Wherein, dotted line represents pivot angle constraint, and solid line represents actual trolley movement rail Mark and pivot angle track.It can be seen that under the effect of PD control device, chassis can preferably track the reference locus, Realize the control targe of quick accurate chassis positioning.Two-stage pivot angle is held in given scope during whole transport, And at the end of transporting, almost no Residual oscillations.The experiment show present invention can realize preferable effect.

Further to embody effectiveness of the invention, as a comparison, this give the optimal trajectory planning of document [21] Method, and the experimental result of linearquadratic regulator (linear quadratic regulator, LQR) method.Wherein, The constraint of method for planning track in document [21] chooses consistent with the constraint of institute's extracting method of the present invention；And for LQR methods, its control Device expression formula processed is as follows：

Wherein, driving force of F (t) roles of delegate on chassis；x(t),Represent the chassis position of measurement in real time and speed Degree；x_fI.e. given target location, is set to x_f=0.6m；θ₁(t),Represent one-level pivot angle and its angular velocity；θ₂(t), Represent two grades of pivot angles and its angular velocity, k₁,k₂,k₃,k₄,k₅,k₆Corresponding control gain is, concrete value sees below.Meanwhile, The method cost function is chosen as follows：

Wherein, X is defined as follows

E (t) represents chassis position error, e (t)=x (t)-x_f；The selection of matrix Q, R is as follows：

Q=diag { 200,1,200,1,200,1 }, R=0.05

It is calculated controller gain as follows：

k₁=63.2456, k₂=50.7765, k₃=-129.3086, k₄=-6.9634, k₅=19.9137, k₆=- 6.7856.

Method in LQR methods and document [21], experimental result is as shown in accompanying drawing 4, accompanying drawing 5.Wherein, accompanying drawing 5 is utilization The experimental result of method in document [21], solid line represent experimental result, and dotted line represents pivot angle constraint.

As can be seen that when chassis tracks planned optimal trajectory and runs, completing transport process only needs from accompanying drawing 3 Within in 4.095s, and whole process, two-stage swings and is held in given constraint 2deg, substantially without Residual oscillations when transport is completed. And for document [21] institute extracting method, completing transport process needs 5.445s；For LQR methods, transport process needs are completed 7.425s.Meanwhile, caused by LQR methods, one-level swing maximum pendulum angle reaches 6.5deg, and two grades swing maximum pendulum angles and reach 11.5deg, much larger than the pivot angle of institute's extracting method of the present invention.In summary, the method designed by the present invention can realize chassis It is accurately positioned and quick elimination that system two-stage swings, obtains good control performance.

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Claims (1)

1. a kind of double pendulum crane length of a game optimal trajectory planning method based on pseudo- spectrometry, it is characterised in that the method includes：

1st, profile constraints are analyzed and corresponding optimization problem is constructed

The control targe of analysis crane system, it is considered to including various including two-stage pivot angle and machine speed and acceleration higher limit Constraint, draws the following optimization problem with the shipping time as cost function：

$\begin{array}{c}\min T\\ s.t.x\left(0\right)=\stackrel{\·}{x}\left(0\right)=\stackrel{\·\·}{x}\left(0\right)=0,\\ x\left(T\right)=x_f,\stackrel{\·}{x}\left(T\right)=\stackrel{\·\·}{x}\left(T\right)=0,\\ \left|\stackrel{\·}{x}\left(t\right)\right|\≤v_{\max },\left|\stackrel{\·\·}{x}\left(t\right)\right|\≤a_{\max },\\ {\mathrm{\θ}}_1\left(0\right)={\stackrel{\·}{\mathrm{\θ}}}_1\left(0\right)=0,{\mathrm{\θ}}_1\left(T\right)={\stackrel{\·}{\mathrm{\θ}}}_1\left(T\right)=0,\\ {\mathrm{\θ}}_2\left(0\right)={\stackrel{\·}{\mathrm{\θ}}}_2\left(0\right)=0,{\mathrm{\θ}}_2\left(T\right)={\stackrel{\·}{\mathrm{\θ}}}_2\left(T\right)=0,\\ \left|{\mathrm{\θ}}_1\left(t\right)\right|\≤{\mathrm{\θ}}_{1\max },\left|{\mathrm{\θ}}_2\left(t\right)\right|\≤{\mathrm{\θ}}_{2\max },\\ \left|{\stackrel{\·}{\mathrm{\θ}}}_1\left(t\right)\right|\≤{\mathrm{\ω}}_{1\max },\left|{\stackrel{\·}{\mathrm{\θ}}}_2\left(t\right)\right|\≤{\mathrm{\ω}}_{2\max },\end{array}---\left(5\right)$

Wherein, x (t) represents the position of chassis, x_fRepresent the target location of chassis, the t express times in bracket, behind variable (t) Represent that the variable is the variable with regard to the time, for simplicity's sake, (t) in most of variable is omitted in formula, T is represented and completed The total time of transport, min represent minimum, connect and represent the constraints for needing to consider behind s.t.；Chassis is represented respectively First derivative and second dervative of position x (t) with regard to the time, i.e. machine speed and acceleration；v_max,a_maxRepresenting respectively is allowed Chassis maximal rate and peak acceleration；θ₁(t),θ₂T () represents one-level and two grades of pivot angles respectively,Represent one-level With two grades of angular velocity；θ_1max,θ_2maxRepresent the one-level and two grades of maximum pendulum angles allowed during transporting, ω_1max,ω_2maxRepresent institute The one-level of permission and two grades of maximum angular rates；

2nd, acceleration driving model is set up and is converted with optimization problem

Analysis obtains following speed-up degree drive system model with using double pendulum bridge type crane system：

$\stackrel{\·}{\mathrm{\ζ}}=f\left(\mathrm{\ζ}\right)+h\left(\mathrm{\ζ}\right)u,---\left(6\right)$

Wherein, ζ represents the total state vector of system, is defined as follows：

$\mathrm{\ζ}={\left[\begin{array}{cccccc}x& \stackrel{\·}{x}& {\mathrm{\θ}}_1& {\stackrel{\·}{\mathrm{\θ}}}_1& {\mathrm{\θ}}_2& {\stackrel{\·}{\mathrm{\θ}}}_2\end{array}\right]}^T$

Wherein, x (t),Chassis Position And Velocity, θ are represented respectively₁(t),Represent one-level pivot angle and angular velocity, θ₂(t),Two grades of pivot angles and angular velocity are represented, the subscript T of bracket represents the transposition computing of matrix；The system that u (t) represents the system is defeated Enter, and For chassis acceleration；F (ζ), h (ζ) represent total state vector letters of the ζ as independent variable with system Number, is obtained by crane system kinematical equation, and concrete form is shown in (7)；For the vectorial derivative with regard to the time of total state of system；

$f\left(\mathrm{\ζ}\right)={\left[\begin{array}{cccccc}\stackrel{\·}{x}& 0& {\stackrel{\·}{\mathrm{\θ}}}_1& A& {\stackrel{\·}{\mathrm{\θ}}}_2& B\end{array}\right]}^T,h\left(\mathrm{\ζ}\right)={\left[\begin{array}{cccccc}0& 1& 0& C& 0& D\end{array}\right]}^T---\left(7\right)$

Wherein, for convenience of describing, following auxiliary variable A, B, C, D are defined：

$\begin{array}{c}A=-\frac{m_2{C}_{1-2}}{l_1(m_1+m_2)-m_2{l}_1{C}_{1-2}^2}\[l_1{S}_{1-2}{\stackrel{\·}{\mathrm{\θ}}}_1^2+\frac{m_2{l}_2}{m_1+m_2}{S}_{1-2}{C}_{1-2}{\stackrel{\·}{\mathrm{\θ}}}_2^2-g(S_2-S_1{C}_{1-2})\]\\ -\frac{1}{l_1}{\mathrm{gS}}_1-\frac{m_2{l}_2}{l_1(m_1+m_2)}{S}_{1-2}{\stackrel{\·}{\mathrm{\θ}}}_2^2\\ B=\frac{m_1+m_2}{l_2(m_1+m_2)-m_2{l}_2{C}_{1-2}^2}\[\frac{m_2{l}_2}{m_1+m_2}{S}_{1-2}{C}_{1-2}{\stackrel{\·}{\mathrm{\θ}}}_2^2+l_1{S}_{1-2}{\stackrel{\·}{\mathrm{\θ}}}_1^2-g(S_2-S_1{C}_{1-2})\],\\ C=\frac{m_2{C}_{1-2}}{l_1(m_1+m_2)-m_2{l}_1{C}_{1-2}^2}\[C_2-C_1{C}_{1-2})-\frac{1}{l_1}{C}_1,\\ D=-\frac{m_1+m_2}{l_2(m_1+m_2)-m_2{l}_2{C}_{1-2}^2}(C_2-C_1{C}_{1-2}),\end{array}---\left(8\right)$

(8), in formula, following reduced form has been used：

$\begin{array}{c}{S}_1={\mathrm{sin\θ}}_1,S_2={\mathrm{sin\θ}}_2,C_1={\mathrm{cos\θ}}_1,C_2={\mathrm{cos\θ}}_2,\\ S_{1-2}=sin({\mathrm{\θ}}_1-{\mathrm{\θ}}_2),C_{1-2}=\cos ({\mathrm{\θ}}_1-{\mathrm{\θ}}_2).\end{array};$

Using above-mentioned acceleration drive system model, former optimization problem is converted into following form：

$\begin{array}{c}\min T\\ s.t.\stackrel{\·}{\mathrm{\ζ}}=f\left(\mathrm{\ζ}\right)+h\left(\mathrm{\ζ}\right)u,\\ \mathrm{\ζ}\left(0\right)={\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\end{array}\right]}^T,\\ \mathrm{\ζ}\left(T\right)={\left[\begin{array}{cccccc}{x}_f& 0& 0& 0& 0& 0\end{array}\right]}^T,\\ \left|\stackrel{\·}{x}\left(t\right)\right|\≤v_{\max },\left|u\left(t\right)\right|\≤a_{\max },\\ \left|{\mathrm{\θ}}_1\left(t\right)\right|\≤{\mathrm{\θ}}_{1\max },\left|{\mathrm{\θ}}_2\left(t\right)\right|\≤{\mathrm{\θ}}_{2\max },\\ \left|{\stackrel{\·}{\mathrm{\θ}}}_1\left(t\right)\right|\≤{\mathrm{\ω}}_{1\max },\left|{\stackrel{\·}{\mathrm{\θ}}}_2\left(t\right)\right|\≤{\mathrm{\ω}}_{2\max }\end{array}---\left(9\right)$

Wherein, ζ represents the total state vector of system, and u (t) represents the system input of the system, and For chassis Acceleration；The subscript T of vector represents the transposition computing of matrix；

3rd, the trajectory planning based on Gauss puppet spectrometry

The optimization problem in the 2nd step is processed and solved using the thought of Gauss puppet spectrometry, comprised the following steps that：

3.1st, first with lagrange-interpolation, select Legendre-Gauss (Legendre-Gauss, LG) point at from Scattered system mode track and input trajectory, by discrete loci and Lagrange interpolation polynomial, represent corresponding approximate rail Mark model；

3.2nd, then, the locus model after pairing approximation carries out derivation, and the derivative of system mode is multinomial with Lagrange Formula derivative is represented；

3.3rd, subsequently, using discrete locus model and its derivative, original system kinematics model is converted into a series of multinomial Formula equation；Using Gauss integration, the boundary condition in the 2nd step in optimization problem is equally expressed as the form of polynomial equation；

3.4th, last, time optimal trajectory planning problem is converted into a kind of nonlinear programming problem with Algebraic Constraint, Global optimum's time and optimal trajectory are obtained by solving；

4th, track following

By code-disc or laser sensor, test desk truck position and rate signal x (t),Using to be tracked obtained by the 3.4th step Chassis time optimal reference locus and corresponding speed trajectory, selection percentage differential (proportional-derivative, PD) controller is as follows：

$F\left(t\right)=-k_p\left(x\right(t)-x_r(t\left)\right)-k_d\left(\stackrel{\·}{x}\right(t)-{\stackrel{\·}{x}}_r(t\left)\right)---\left(16\right)$

Wherein, driving force of F (t) roles of delegate on chassis, x_r(t),Represented with reference to deformation trace and speed rail respectively Mark, k_p,k_dIt is the positive control gain for needing adjustment；Using the controller, corresponding real-time control signal can be calculated, Trolley movement is driven, control targe is completed.