CN104192713A - Time-optimal bridge crane track planning method based on differential flatness and B-spline - Google Patents

Abstract

The invention relates to a time-optimal bridge crane track planning method based on differential flatness and a B-spline, and aims at solving the automatic control problem of a nonlinear bridge crane. The method is good in performance of locating a trolley and eliminating the swinging of a load. The method comprises the following steps of proposing system flatness output which is used for processing a coupling relation between the motion of the trolley and the swinging of the load; converting track parameters to a B-spline curve form with to-be-optimized parameters by utilizing the continuity and smoothness of the B-spline curve; obtaining a time-optimal track by utilizing a polygonal optimization algorithm by considering constraints such as the swinging angle of the load and the acceleration of the trolley. The track planning is carried out by utilizing the differential flatness characteristics of a bridge crane system, so that the analysis on the complicated coupling characteristics of the system can be avoided; moreover, the selected track form has an analytic expression; meanwhile, the time-optimal track is obtained by utilizing the optimization algorithm. The simulation and experimental result shows that the control effect is good, and the real application value is good.

Description

The time optimal traverse crane method for planning track of and B batten smooth based on differential

Technical field

The invention belongs to the technical field that non-linear lack of driven electric system is automatically controlled, particularly relate to a kind of time optimal traverse crane method for planning track of and B batten smooth based on differential.

Background technology

Bridge type crane system is a kind of system being widely used in industrial process, it have non-linear with owe drive performance.What is called is owed drive system, refers to that the control inputs number of system is less than degree of freedom in system dimension.Compare full drive system, owe drive system have simple in structure, the advantage such as low consumes energy; But meanwhile, the difficulty of this system control also will be higher than full drive system.Specific to traverse crane, the motion of load is that the lifting rope by being fixed on chassis drives, and cannot directly control, and this has also just increased the difficulty of traverse crane control.

The target of bridge type crane system control comprises two aspects, the one, and chassis location fast and accurately, the 2nd, the quick elimination of hunting of load.For this control target, conventional control method can be divided into two large classes.The first kind is that calm/adjusting is controlled, and comprises linear Feedback Control [1], input shaper [2-3], based on the nonlinear Control of energy [4-5], fuzzy control [6]deng.Equations of The Second Kind is that trajectory planning is controlled with tracking.Particularly, first cook up a level and smooth track for chassis, design subsequently tracking control unit and make chassis follow the tracks of this track, to complete the control to bridge type crane system.The thought of trajectory planning is widely used in a lot of fields, as mechanical arm [7], mobile robot etc.But for bridge type crane system, due to the drive performance of owing of system, very large to the planning difficulty of chassis track, the method for a lot of trajectory plannings has only provided Tracking Feedback Control device, and has ignored the planning process of track.In fact, the trolley movement of bridge type crane system and hunting of load have close coupling characteristic, for completing chassis trajectory planning, need to carry out deep analysis to this coupling behavior.Based on this, a lot of scholars are studied method for planning track.Lee [8]propose a kind of traverse crane method for planning track of system, but cannot assess in advance the maximum pendulum angle of load.Blajer and Ko odziejczyk [9]be studied keeping away barrier problem, and utilize batten equation to carry out matching to specified point, design corresponding load path.The people such as Uchiyama [10]by the cross motion of jib crane being cooked up to S type track, and in planning process, consider the remaining elimination that swings, proposed a kind of simple open-loop control method.Ma Bojun etc. [11]propose a kind of S type track of chassis according to engineering experience, and designed a kind of adaptive tracing controller this track is followed the tracks of.But this track is due to the analysis lacking in planning process system, coupled characteristic, the elimination cannot proof load swinging.Sun Ning etc. have proposed a series of method for planning track [12-14], comprise iterative learning method, phase plane analysis method, online method for planning track etc., all can realize the accurate location of chassis and the pendulum that disappears fast of load.But meanwhile, above-mentioned method for planning track is not all considered time optimal problem, has limited the efficiency of transporting of bridge type crane system.

Summary of the invention

The object of the invention is to solve existing bridge type crane system method for planning track above shortcomings, a kind of time optimal traverse crane method for planning track of and B batten smooth based on differential is provided.

The present invention is devoted to by analyzing the dynamic coupling relation between chassis and load, a kind of time optimal traverse crane method for planning track of and B batten smooth based on differential has been proposed, obtain a time optimal chassis track, when improving chassis positioning precision, realize the quick elimination of hunting of load, and be applied to actual crane platform and test, improve the work efficiency of system.

And time optimal traverse crane method for planning track B batten smooth based on differential provided by the invention comprise:

1st, the structure smooth output of bridge type crane system the constraint of definite track

According to Bridge System kinematical equation, analysis obtains the smooth output of system and is load level attitude ; The smooth system of differential [15]have character as described below, i.e. all quantity of states and output in system, can represent with the smooth output of system and limited order derivative thereof;

According to above-mentioned character, the constraint of chassis track all can be converted into smooth output constraint, obtain concrete optimization problem:

(16)

Wherein, represent the target location of chassis, the expression time, represent the total time of transport, represent minimum, represent to need the constraint condition of consideration; represent respectively load level attitude about first derivative and the second derivative of time, represent about the time order derivative, represent the order of differentiate; represent acceleration due to gravity, represent the length of lifting rope, represent acceptable maximum load pivot angle, for the intermediate parameters in computation process.

2nd, load path parametrization

Selection B-spline curves represent , concrete form is:

(18)

Wherein, represent individual B spline base function; for the order of B spline base function, get here ; for predetermined B batten sequence node parameter, be controlling point corresponding to individual B spline base function; The sequence node concrete form that these B-spline curves are corresponding is:

(17)

Wherein represent individual node; Meanwhile, for arithmetic progression.

3rd, trajectory parameters optimization

Design controlling point sequence has following form:

(22)

Can find out, it is right to need be optimized and solve; Select polynomial function to represent the controlling point in this section, concrete form is:

(23)

Wherein, the SQL that representative is introduced, the independent variable of representative function, for the independent variable parameter of introducing, undetermined parameter in representative function; According to track described in the 1st step constraint, solve, utilize numerical method, can obtain optimum trajectory parameters and corresponding time optimal load path equation ; Time optimal chassis reference locus to be asked be

(36)

Wherein, representative about the second derivative of time.

4th, track following

By real-time test desk truck position , speed , consider the to be tracked optimum reference locus that planning obtains , select proportional-derivative(PD, ratio-differential) and tracking control unit is as follows:

(37)

Wherein, representative imposes on the real-time control effort of chassis, for reference velocity track, it is the positive ride gain that needs adjustment; Utilize controller (37), calculate online corresponding real-time control signal, drive crane work, realize and control target.

the theoretical foundation of the inventive method and derivation

1st, the structure smooth output of bridge type crane system the constraint of definite track

The kinetic model of two dimension bridge type crane system is expressed as follows:

(1)

Wherein, represent respectively the quality of chassis and load; represent chassis position, represent chassis acceleration/accel; the expression time, after variable represent that this variable is the variable about the time, for simplicity's sake, omits in most of variable in formula ; for load pivot angle, for cireular frequency, for angular acceleration; for lifting rope length; for acceleration due to gravity; for motor offers the propulsive effort of chassis.The horizontal displacement of load be expressed as:

(2)

To the second formula in (1), both sides with divided by :

(3)

Generally, in crane system operational process, load pivot angle requires can not be too large, meet little angle approximate condition, and this constraint also can be considered in track planning process.Utilize little angle approximate, to formula (2), (3) linearization, can obtain:

(4)

(5)

Ask two order derivatives to obtain formula (4) about the time:

(6)

By formula (5), (6), can be calculated:

(7)

Can find out system state amount from formula (7) all can be by load level attitude and limited order derivative represents, this two dimension bridge type crane system is that differential is smooth [15], and smooth output is .According to the character of smooth system, all can be converted into constraint and the planning to the smooth output of system to the constraint of chassis and load pivot angle and planning.

For completing the smooth output of system planning, consider the track constraint of following several aspects:

(1) load position constraint at the whole story

This process of transporting should ensure the reference position from 0 moment load , be transported to the target location in moment .Without loss of generality, choose reference position here and be 0.This constraint can be expressed as:

(8)

(2) load path smoothness constraint

For guaranteeing the trackability of track, selected track should can be led for four times about the time, and the all-order derivative in moment at the whole story all should be 0.This constraint is:

(9)

Wherein, represent derivative order.

(3) transport efficiency constraints

Whole transporting in process, the horizontal velocity of load should keep non-negative,

(10)

(4) load pivot angle constraint

For ensureing to transport the safety in process, load pivot angle should remain in certain scope,

(11)

(5) chassis acceleration/accel constraint

For avoiding controller saturation problem, chassis acceleration/accel should keep within the specific limits,

(12)

Analysis mode (8) ~ (12) are known, and formula (8) ~ (10) are for directly right constraint.Utilize formula (7), formula (11), (12) can be converted into:

(13)

(14)

Wherein, for the parameter of introducing in computation process, expression is as follows:

(15)

To sum up can obtain, time optimization problem is:

(16)

2nd, load path parametrization

For solving (16), need to select suitable track pair carry out parametrization.Here select B-spline curves as parametrization track.According to formula (9), select the order of B-spline curves to be , retrain with proof load smooth trajectory.Designing the sequence node that these B-spline curves are corresponding is:

(17)

Wherein, parameter be used for adjusting the number of B-spline curves node.Choose this sequence stage casing for arithmetic progression, tolerance is .Design as follows:

(18)

Wherein, represent the controlling point sequence of B-spline curves, represent the sequence number of B spline base function, represent individual the B spline base function on rank.B-spline curves have following character [16]:

character 1: the derivative of B-spline curves equally also has the form of B-spline curves, has identical sequence node between them, exists definite functional relation between the sequence of controlling point simultaneously.

character 2: B-spline curves have closure property about controlling point sequence, specific as follows: suppose to exist the form of B-spline curves to be if,

Set up, must have

Introduce array representative respectively 1 ~ 4 order derivative controlling point sequence, specific as follows:

(19)

Utilize character 1, can be calculated:

(20)

From formula (17), (20), when time, corresponding ; Meanwhile, correspondence b spline base function be defined in sequence node upper, in this sequence, each node all equates, B spline base function perseverance is in this case 0.Meanwhile, when also can this thing happens while getting other numerical value, but can not exert an influence to the calculating of B-spline curves.Therefore,, for this type of controlling point, only need ignore.

According to formula (16), (17), (18), (20) and character 2, by calculating, can obtain the time optimal problem after abbreviation:

(21)

3rd, trajectory parameters optimization

Here suppose B-spline curves sequence node parameter, determine.Known according to formula (21), the controlling point sequence of B-spline curves has following form:

(22)

Therefrom can find out to only have the stage casing of this sequence need to determine.According to actual crane operation experience, the load path of expectation should be Central Symmetry curve, consider the character of B-spline curves, controlling point sequence also be centrosymmetric.Therefore, only need be with representing first half section sequence, and second half section sequence can utilize symmetry principle to obtain.Analysis can obtain, and Central Symmetry point is center-point , and numerical value is .According to the concrete form of constraint condition in formula (21), for ensureing the alerting ability of optimizing process, we select 5 order polynomial function representations simultaneously .According to above-mentioned analysis, can be expressed as:

(23)

Wherein, the SQL that representative is introduced, the independent variable of representative function, for the independent variable parameter of introducing, undetermined parameter in representative function.According to what design form is known, centered by symmetrical curve and symmetric points coordinate be ., can be obtained by formula (23) meanwhile, when time, with second derivative opposite number each other.For ensureing overall continuity, need to meet .The above analysis convolution (20), (23), the constraint in optimization problem (21) can be converted into:

(24)

For the inequality constrain in formula (24), the inequality constrain in the corresponding formula that is (21).But due to only represent in a part, and controlling point before also needs consideration, known by analyzing, here span should be , agreement simultaneously, when time, .

Bring formula (23) into formula (24), and arrange abbreviation, can obtain:

(25)

After in analysis mode (25), 5 formulas are known, have three unknown parameters , two equality constraint and three inequality constrains.For simplification problem, choice for use represent ,

(26)

Bring formula (26) in (25) the 4th formula, can obtain:

(27)

Wherein, be the parameter of introducing, concrete form is:

Due to value be discrete point, can be by traveling through all values to solve formula (27).Because symbol uncertain, need the discussion of classifying here:

1) when time, have

2) when time, have

For different , correspondence also different.So, choose and will affect in ergodic process value.If occurred when traversal situation, can by select other avoid this situation.

To sum up, can obtain effective span be:

(28)

Next, consider the 5th and the 6th formula in (25).Bring formula (23) abbreviation into, can obtain:

(29)

Wherein, for the parameter of introducing, concrete form is:

(30)

(31)

Utilize , abbreviation formula (29) can obtain:

(32)

Solve formula (32) and just can obtain optimum parameter , obtain the optimum time simultaneously .Here use Numerical Methods Solve formula (32).Particularly, according to trying to achieve above effective span (28), therefrom select abundant some composition ordered series of numbers , here represent the sequence number of ordered series of numbers middle term.For each , bring into , obtain corresponding .Traversal , can obtain maxim and minimum value.Then,, according to the maxim of gained and minimum value, by asking its absolute value, can obtain the maxim of absolute value, is expressed as follows:

(33)

By formula (32), (33), can obtain:

(34)

Abbreviation formula (34) can obtain:

(35)

Wherein, concrete form is:

The solution of inequality group (32) is:

Wherein,

By to ordered series of numbers travel through, we can obtain maxim, .Analyze known, to be asked optimal time be , this value is corresponding simultaneously be optimum parameter .According to this parameter, can obtain optimum load path , then utilize formula (7), can calculate optimum chassis track for:

(36)

4th, track following

By real-time test desk truck position , speed , consider the to be tracked optimum reference locus that planning obtains , select PD tracking control unit as follows:

(37)

Wherein, representative imposes on the real-time control effort of chassis, for reference velocity track, it is the positive ride gain that needs adjustment; Utilize controller (37), calculate online corresponding real-time control signal, drive crane work, realize and control target.

advantage of the present invention and beneficial effect

The present invention proposes a kind of time optimal traverse crane method for planning track of and B batten smooth based on differential.First the present invention by analyzing the kinematics characteristic of bridge type crane system, has determined that system has the characteristic that differential is smooth, has proposed the smooth output of system simultaneously.Next utilize the feature of smooth output, will the trajectory planning problem of chassis be converted into the trajectory planning problem to smooth output, also avoided the complex analyses to system, coupled quantity of state simultaneously.Subsequently, the continuity based on B-spline curves and flatness, turn to load path parameter the B-spline curves with parameter to be optimized.Subsequently, utilize a kind of optimized algorithm based on polynomial function, consider various track constraints, solve time optimal trajectory parameters.Simulation and experiment result shows, simplicity of design of the present invention is directly perceived, and can realize the accurate location of bridge type crane system chassis and the pendulum that disappears fast of load.

brief description of the drawings:

Fig. 1 trajectory planning algorithm simulating result.

Fig. 2 trajectory planning algorithm experimental result.

detailed description of the invention:

Embodiment 1:

1st, the structure smooth output of bridge type crane system the constraint of definite track

According to Bridge System kinematical equation, analysis obtains the smooth output of system and is load level attitude ; The smooth system of differential has character as described below, i.e. all quantity of states and output in system, can represent with the smooth output of system and limited order derivative thereof;

According to above-mentioned character, the constraint of chassis track all can be converted into smooth output constraint, obtain concrete optimization problem:

(16)

Here, the system parameter of selection is as follows:

2nd, load path parametrization

Selection B-spline curves represent , concrete form is:

(18)

The sequence node concrete form that these B-spline curves are corresponding is:

(17)

Wherein for arithmetic progression, and .The concrete numerical value at node and controlling point will be determined below.

3rd, trajectory parameters optimization

Design controlling point sequence has following form:

(22)

Can find out, it is right to need be optimized and solve; Select polynomial function to represent the controlling point in this section, concrete form is:

(23)

Wherein, the SQL that representative is introduced, the independent variable of representative function, for the independent variable parameter of introducing, represent the undetermined parameter in SQL.According to the constraint of track described in the 1st step, bring the system parameter that first two steps are chosen into, solve, utilize numerical method, can obtain optimized parameter corresponding in formula (23) and be respectively:

Meanwhile, also obtaining the optimum time is .According to the parameter of trying to achieve, the sequence node that can obtain correspondence in equation of locus (18) is:

Controlling point sequence is:

With controlling point sequence, just can obtain corresponding B-spline curves according to sequence node above [16], namely ; Corresponding time optimal chassis reference locus be

(36)

Wherein, representative about the second derivative of time.

4th, emulation experiment effect is described

4.1st, simulation result

For verifying that the present invention proposes the feasibility of trajectory planning algorithm, first carries out numerical simulation in MATLAB/Simulink environment.Simulation process can be divided into two steps.The first step is a time optimal reference locus of chassis planning according to above-mentioned algorithm; Second step, supposes that chassis, according to this reference locus operation, obtains the track of chassis and load.

The result of emulation as shown in Figure 1, as can be seen from the figure, in the time that chassis moves along reference locus, chassis fast and accurately converge to target location.Meanwhile, in process is transported in whole load, load pivot angle is all less than given binding occurrence , and in the time transporting end, there is not remaining swing, the target of the pendulum that disappears fast also can realize.

4.2nd, experimental result

By real-time test desk truck position , speed , consider the to be tracked optimum reference locus that planning obtains , select PD tracking control unit as follows:

(37)

Wherein, representative imposes on the real-time control effort of chassis, for reference velocity track, it is the positive ride gain that needs adjustment; Utilize controller (37), calculate online corresponding real-time control signal, drive crane work, realize and control target.

In experiment, the tracking control unit ride gain of choosing is:

Experimental result as shown in Figure 2.Wherein, dotted line represents chassis reference locus to be tracked , solid line represents actual trolley movement track.As can be seen from the figure, under PD controller action, chassis can be followed the tracks of this reference locus preferably, realizes fast the accurately control target of chassis location.Simultaneously load pivot angle is after transporting end, and also rapid convergence to 0 very, and there is no remaining swing has been realized the load pendulum that disappears fast.Experimental result shows, this method for planning track has good control effect.

bibliography

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[16]C. de Boor, A Practical Guide to Splines, New York: Springer-Verlag, 1978.

Claims (1)

1. a time optimal traverse crane method for planning track for and B batten smooth based on differential, is characterized in that the method comprises:

1st, the structure smooth output of bridge type crane system the constraint of definite track

According to bridge type crane system kinematical equation, analysis obtains the smooth output of system and is load level attitude ; Crane system has following character, i.e. all quantity of states and output in system, can both represent with the smooth output of system and limited order derivative thereof; According to this character, the constraint of chassis track all can be converted into smooth output constraint, obtain concrete optimization problem:

(16)

Wherein, represent the target location of chassis, the expression time, represent the total time of transport, represent minimum, represent to need the constraint condition of consideration; represent respectively load level attitude about first derivative and the second derivative of time, represent about the time order derivative, represent the order of differentiate; represent acceleration due to gravity, represent the length of lifting rope, represent acceptable maximum load pivot angle, for the intermediate parameters in computation process;

2nd, load path parametrization

Selection B-spline curves represent , concrete form is:

(18)

Wherein, represent individual B spline base function; for the order of B spline base function, get here ; for predetermined B batten sequence node parameter, be controlling point corresponding to individual B spline base function; The sequence node concrete form that these B-spline curves are corresponding is:

(17)

Wherein represent individual node; Meanwhile, for arithmetic progression;

3rd, trajectory parameters optimization

Design controlling point sequence has following form:

(22)

Can find out, it is right to need be optimized and solve; Select polynomial function to represent the controlling point in this section, concrete form is:

(23)

Wherein, the SQL that representative is introduced, the independent variable of representative function, for the independent variable parameter of introducing, represent the undetermined parameter in SQL; According to track described in the 1st step constraint, solve, utilize numerical method, can obtain optimum trajectory parameters and corresponding time optimal load path equation ; Time optimal chassis reference locus to be asked be

(36)

Wherein, representative about the second derivative of time;

4th, track following

By real-time test desk truck position , speed , consider the optimum reference locus of chassis to be tracked that planning obtains , select proportional-derivative tracking control unit as follows:

(37)

Wherein, representative imposes on the real-time control effort of chassis, for reference velocity track, it is the positive ride gain that needs adjustment; Utilize controller (37), calculate online corresponding real-time control signal, drive crane work, realize and control target.