CN105174061A - Double-pendulum crane global time optimal trajectory planning method based on pseudo-spectral method - Google Patents

Abstract

Disclosed is a double-pendulum crane global time optimal trajectory planning method based on a pseudo-spectral method. The purpose of automatic control over a nonlinear double-pendulum bridge crane system is achieved. The method has the good functions of trolley positioning and two-stage load pendulum elimination. The method comprises the steps that firstly, a system kinematic model is transformed so as to facilitate following analysis; then, corresponding optimization problems are constructed by considering various constraints including two-stage pendulum angles and trolley speed and acceleration upper limit values; and then, the Gaussian pseudo-spectral method is used for transforming the optimization problems with the constraints into nonlinear programming problems easy to solve to be solved to obtain a trolley trajectory with the optimal time. The thought of the pseudo-spectral method is utilized for handling and transforming the complex time optimal problem, the solving difficulty is lowered; and meanwhile, the result with the optimal global time can be obtained through the method, and the working efficiency of a crane system can be greatly improved. Simulation and experiment results show that the good control effect can be obtained and the good actual application value is achieved.

Description

Based on the double pendulum crane length of a game optimal trajectory planning method of pseudo-spectrometry

Technical field

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

Background technology

In industrial process, for carrying load is to desired position, comprises all kinds of crane systems of traverse crane, rotary cranes, tower crane, marine hoist, have a very wide range of applications.In order to simplify the physical construction of crane system, often direct control load, but by the motion of chassis, indirectly drag and be loaded to target location.It is that this structure is brought as a result, the control inputs dimension of crane system is less than degree of freedom dimension to be controlled.There is the system of this characteristic and so-called under-actuated systems ^[1].Compare full drive system, due to the existence of drive lacking characteristic, under-actuated systems is more difficult control often, and in actual production, crane system is often by veteran operative.If generation maloperation, load may be caused acutely to swing and to cause collision, even safety misadventure occur.Therefore, to the research of crane system autocontrol method, there is realistic meaning and widespread use is worth, obtain the concern of numerous scholars.

For bridge type crane system, main control objectives comprises two aspects, i.e. chassis location fast and accurately and the suppression of hunting of load and elimination.But these two aspects is normally conflicting, namely too fast trolley movement often causes larger hunting of load.Therefore, the control objectives simultaneously realizing these two aspects has higher difficulty.In order to obtain good control effects, current Chinese scholars has proposed a lot of crane system autocontrol method.Tuan etc. propose the control method based on Partial feedback linearization ^[2-3], the control algorithm design of bridge type crane system can be simplified.In document [4], in [5], Singhose etc. utilize the thought of input shaper to control crane system, can effectively suppress load Residual oscillations.For processing uncertain external interference, researchist utilizes sliding Mode Algorithm to control crane system ^[6-7], good robustness can be obtained.Hu Zhou etc. propose a kind of nonlinear transformations fused controlling method ^[8], saturated problem can be inputted by processing controller, realize the high performance control to crane system.Document [9], [10] propose the control policy based on energy and passivity, can obtain good effect.In addition, in recent years, genetic algorithm is comprised ^[11], fuzzy control ^[12]certain application is had in overhead crane control field equally Deng some intelligent control methods.

As everyone knows, the hunting of load of crane system is caused by the acceleration and deceleration motion of chassis, there is stronger being coupled between trolley movement with hunting of load.Based on this, a suitable track can be planned for chassis, when chassis is according to this orbiting motion, can realize its quick pinpoint target.Meanwhile, consider that pivot angle suppresses and the requirement eliminated, in track planning process, by the coupled relation analysed in depth and between Appropriate application trolley movement and hunting of load, the chassis track that has the pendulum ability that disappears can be cooked up.Accurately location and load disappear the double goal put fast can to complete chassis like this.Based on this thought, researchist has proposed a lot of crane method for planning track ^[13-17].Such as, in document [13], Uchiyama etc. propose a kind of open loop control strategy for jib crane, and in order to eliminate Residual oscillations, they are that the horizontal motion of cantilever has planned a S type track.Document [14] then proposes a kind of method for planning track based on phase plane analysis, and it can suppress hunting of load preferably, and eliminates Residual oscillations.

For crane system, although above-mentioned various method can obtain good control effects in the ideal case, these methods all suppose that the quality of suspension hook can be ignored and load can regard particle as, and the hunting of load of crane system is considered as single pendulum system.If the quality of suspension hook is comparatively large, cannot ignore, or load shapes is very large, simply can not regard particle as, in this case, the swing of crane system will present double pendulum phenomenon, and namely suspension hook carries out one-level swing around chassis, and load simultaneously swings around suspension hook generation secondary.Now, above-mentioned single pendulum overhead crane control method cannot obtain gratifying controller performance, only there is few crane system control policy considering double pendulum effect at present.The method of input shaper, by analyzing the natural frequency of double pendulum crane system, is successfully expanded to double pendulum crane system by document [18-20].Sun Ning etc., under the prerequisite considering a series of constraints such as the constraint of system pivot angle, machine speed constraint, propose a kind of double pendulum crane optimal trajectory planning method based on differential flat theory ^[21].Guo Wei equality, by analyzing the energy of crane system, proposes a kind of double pendulum overhead crane control strategy based on passivity ^[22].

Although above-mentioned control policy can realize the control to double pendulum crane system, they all cannot realize the control effects of length of a game's optimum, namely cannot ensure the maximization of crane system operating efficiency.Therefore need to design suitable control method, to improve the work efficiency of crane system.

Summary of the invention

The object of the invention is to solve 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.

The present invention is devoted to by analyzing the kinematics model with the bridge type crane system of double pendulum effect, propose a kind of double pendulum crane length of a game optimal trajectory planning method based on pseudo-spectrometry, obtain the chassis track of length of a game's optimum, complete chassis pinpoint while, achieve quick suppression and the elimination of two-stage hunting of load, and be applied to actual crane platform and test, greatly can improve the work efficiency of crane system.

Double pendulum crane length of a game optimal trajectory planning method based on pseudo-spectrometry provided by the invention comprises:

1st, analyze profile constraints and construct corresponding optimization problem

Analyze the control objectives of crane system, consider the multiple constraint comprising two-stage pivot angle and machine speed and acceleration/accel higher limit, show that following take haulage time as the optimization problem of 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, t in bracket represents the time, after variable, (t) represents that this variable is the variable about the time, for simplicity's sake, (t) in most of variable is omitted in formula, T has represented the total time of transport, and min represents minimum, connects the constraint condition representing and need to consider after s.t.; represent that chassis position x (t) is about the first derivative of time and second derivative, i.e. machine speed and acceleration/accel respectively; v _max, a _maxrepresent the chassis maximum speed allowed and peak acceleration respectively; θ ₁(t), θ ₂t () represents one-level and secondary pivot angle respectively, represent one-level and secondary cireular frequency; θ _1max, θ _2maxrepresent the one-level and secondary maximum pendulum angle that allow in transport process, ω _1max, ω _2maxrepresent the one-level and secondary maximum angular rate that allow.

2nd, acceleration/accel driving model Establishment and optimization problem transforms

Analyze and utilize double pendulum bridge type crane system, obtaining following acceleration/accel drive system model:

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

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), represent chassis Position And Velocity respectively, θ ₁(t), represent one-level pivot angle and cireular frequency, θ ₂(t), represent secondary pivot angle and cireular frequency, the subscript T of bracket represents transpose of a matrix computing; U (t) represents the system input of this system, and for chassis acceleration/accel; The function that it is independent variable that f (ζ), h (ζ) all represent with the total state of system vector ζ, obtained by crane system kinematical equation, concrete form is shown in (7); the vectorial derivative about the time of total state for system.

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

Utilize above-mentioned acceleration/accel 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 this system, and for chassis acceleration/accel; T in bracket represents the time, after variable, (t) represents that this variable is the variable about the time, for simplicity's sake, (t) in most of variable is omitted in formula, T has represented the total time of transport, min represents minimum, connects the constraint condition representing and need to consider after s.t.; The subscript T of vector represents transpose of a matrix computing; x _frepresent given chassis target location, represent machine speed, v _max, a _maxrepresent the chassis maximum speed allowed and peak acceleration respectively; θ ₁(t), θ ₂t () represents one-level and secondary pivot angle respectively, represent one-level and secondary cireular frequency; θ _1max, θ _2maxrepresent the one-level and secondary maximum pendulum angle that allow in transport process, ω _1max, ω _2maxrepresent the one-level and secondary maximum angular rate that allow.

3rd, based on the trajectory planning of the pseudo-spectrometry of Gauss

Utilize the thought of the pseudo-spectrometry of Gauss that the optimization problem in the 2nd step is carried out to process and solved, concrete steps are as follows:

3.1st, first lagrange-interpolation is utilized, select Legendre-Gauss (Legendre-Gauss, LG) the discrete system state trajectory at some place and input trajectory, by discrete loci and Lagrange interpolation polynomial, represent corresponding approximate trajectories model.

3.2nd, then, carry out differentiate by the locus model after pairing approximation, the derivative of state of the system lagrange polynomial derivative is represented.

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

3.4th, last, namely time optimal trajectory planning problem is converted into a kind of nonlinear programming problem with Algebraic Constraint, namely obtains global optimum's time and optimal trajectory by solving.

4th, track following

By code-disc or laser sensor, test desk truck position and speed signal x (t), utilize the speed trajectory of the 3.4th step gained chassis time optimal to be tracked reference locus and correspondence, 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, the propulsive effort of F (t) role of delegate on chassis, x _r(t), represent respectively with reference to deformation trace and speed trajectory, k _p, k _dit is the positive ride gain needing adjustment.Utilize this controller, corresponding real-time control signal can be calculated, drive trolley movement, complete control objectives.

The theoretical foundation of the inventive method and derivation

1st, analyze profile constraints and construct corresponding optimization problem

Have the traverse crane of double pendulum effect, its kinematics model is as follows:

$\begin{array}{c}\left(m_1+m_2\right)l_1{\mathrm{cos\θ}}_1\stackrel{\·\·}{x}+\left(m_1+m_2\right)l_1^2{\stackrel{\·\·}{\mathrm{\θ}}}_1+m_2{l}_1{l}_2\cos \left({\mathrm{\θ}}_1-{\mathrm{\θ}}_2\right){\stackrel{\·\·}{\mathrm{\θ}}}_2\\ +m_2{l}_1{l}_2\sin \left({\mathrm{\θ}}_1-{\mathrm{\θ}}_2\right){\stackrel{\·}{\mathrm{\θ}}}_2^2\sin \left(m_1-m_2\right){\mathrm{gl}}_1{\mathrm{sin\θ}}_1=0,\end{array}---\left(1\right)$

$\begin{array}{c}{m}_2{l}_2{\mathrm{cos\θ}}_2\stackrel{\·\·}{x}+m_2{l}_1{l}_2\cos \left({\mathrm{\θ}}_1-{\mathrm{\θ}}_2\right){\stackrel{\·\·}{\mathrm{\θ}}}_1\\ +m_2{l}_2^2{\stackrel{\·\·}{\mathrm{\θ}}}_2-m_2{l}_1{l}_2{\stackrel{\·}{\mathrm{\θ}}}_1^2\sin \left({\mathrm{\θ}}_1-{\mathrm{\θ}}_2\right)+m_2{\mathrm{gl}}_2{\mathrm{sin\θ}}_2=0\end{array}---\left(2\right)$

Wherein, m ₁, m ₂represent the quality of suspension hook and load respectively, M represents the quality of chassis; X (t) represents chassis displacement, represent the second derivative of x (t) about the time, i.e. chassis acceleration/accel; T represents the time, and after variable, (t) represents that this variable is the variable about the time, for simplicity's sake, omits (t) in most of variable in formula; θ ₁(t), θ ₂t () represents one-level and secondary pivot angle (suspension hook pivot angle and load are around the pivot angle of suspension hook), for corresponding cireular frequency, for angular acceleration; l ₁represent the length of lifting rope, l ₂for equivalence rope is long, the distance namely between load barycenter and suspension hook barycenter; G is acceleration due to gravity.

For formula (1), (2), both sides are same divided by (m respectively ₁+ m ₂) l ₁with m ₂l ₂, and abbreviation obtains:

${\mathrm{cos\θ}}_1\stackrel{\·\·}{x}+l_1\stackrel{\·\·}{\mathrm{\θ}}+\frac{m_2{l}_2}{m_1+m_2}cos({\mathrm{\θ}}_1-{\mathrm{\θ}}_2)\stackrel{\·\·}{\mathrm{\θ}}+\frac{m_2{l}_2}{m_1+m_2}sin({\mathrm{\θ}}_1-{\mathrm{\θ}}_2){\stackrel{\·}{\mathrm{\θ}}}_2^2+g{\mathrm{sin\θ}}_1=0,---\left(3\right)$

${\mathrm{cos\θ}}_2\stackrel{\·\·}{x}+l_{1}cos({\mathrm{\θ}}_1-{\mathrm{\θ}}_2)\stackrel{\·\·}{\mathrm{\θ}}+l_2\stackrel{\·\·}{\mathrm{\θ}}-l_1{\stackrel{\·}{\mathrm{\θ}}}_1^{2}sin({\mathrm{\θ}}_1-{\mathrm{\θ}}_2)+g{\mathrm{sin\θ}}_2=0---\left(4\right)$

Formula (3), (4) describe chassis displacement x (t) and system two-stage pivot angle θ ₁(t), θ ₂t the coupled relation between (), namely the motion of chassis is on the impact of hunting of load.By analysing in depth this coupled relation, planning a chassis track with the pendulum ability that disappears, is basis of the present invention.

For deadline optimal trajectory planning, consider the target of crane system when real work, physical constraint and safety, the present invention will systematically consider the profile constraints of following several aspect ^[21]:

1) for realizing accurate chassis location fast, chassis is from initial position x ₀setting in motion, elapsed time T reaches target location x _f, and the machine speed of start time and finish time, acceleration/accel are 0, namely

$x\left(0\right)=\stackrel{\·}{x}\left(0\right)=\stackrel{\·\·}{x}\left(0\right)=0,x\left(T\right)=x_f,\stackrel{\·}{x}\left(T\right)=\stackrel{\·\·}{x}\left(T\right)=0,$

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

2) consider the output bounded of real electrical machinery, in transport process, the speed of chassis, acceleration/accel all should remain in suitable scope, namely

$\left|\stackrel{\·}{x}\left(t\right)\right|\≤v_{max},\left|\stackrel{\·\·}{x}\left(t\right)\right|\≤a_{max},$

Wherein, v _max, a _maxrepresent chassis maximum speed and the acceleration/accel of permission respectively, represent the absolute value of machine speed and acceleration/accel.

3) directly can carry out next step process for ensureing to load at the end of transporting, chassis reaches behind target location should without Residual oscillations and cireular frequency is also 0, namely

${\mathrm{\θ}}_1\left(0\right)={\stackrel{\·}{\mathrm{\θ}}}_1\left(0\right)=0,{\mathrm{\θ}}_1\left(T\right)={\stackrel{\·}{\mathrm{\θ}}}_1\left(T\right)=0,{\mathrm{\θ}}_2\left(0\right)={\stackrel{\·}{\mathrm{\θ}}}_2\left(0\right)=0,{\mathrm{\θ}}_2\left(T\right)={\stackrel{\·}{\mathrm{\θ}}}_2\left(T\right)=0.$

4) be the collision that the violent swing avoided due to load causes, in transport process, the pivot angle that two-stage swings and cireular frequency all should keep in allowed limits, namely

$\left|{\mathrm{\θ}}_1\left(t\right)\right|\≤{\mathrm{\θ}}_{1max},\left|{\mathrm{\θ}}_2\left(t\right)\right|\≤{\mathrm{\θ}}_{2max},\left|{\stackrel{\·}{\mathrm{\θ}}}_1\left(t\right)\right|\≤{\mathrm{\ω}}_{1max},\left|{\stackrel{\·}{\mathrm{\θ}}}_2\left(t\right)\right|\≤{\mathrm{\ω}}_{2max}.$

Wherein, θ _1max, θ _1maxrepresent the maximum angle that one-level and secondary pivot angle allow respectively; ω _1max, ω _2maxthe one-level that representative allows and secondary maximum angular rate.

To sum up, following optimization problem is constructed:

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

Wherein, min represents minimum, connects the constraint condition representing and need to consider after s.t..Next, this optimization problem will be solved by pseudo-spectrometry, and cook up a time optimal track for chassis.

2nd, acceleration/accel driving model Establishment and optimization problem transforms

For convenience of follow-up trajectory planning, first double pendulum crane system model and optimization problem (5) are transformed here.Define system total state vector ζ (t) is as follows for this reason:

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

Wherein, x (t), represent chassis Position And Velocity respectively, θ ₁(t), represent one-level pivot angle and cireular frequency, θ ₂(t), represent secondary pivot angle and cireular frequency, the subscript T of bracket represents transpose of a matrix computing.According to formula (3), (4), using the input of the acceleration/accel of chassis as system.Now, kinematics model is converted into following form:

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

Wherein, represent the derivative of ζ (t) about the time; U (t) is chassis acceleration/accel f (ζ), h (ζ) represent the auxiliary function about ζ (t), and concrete form is as follows:

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

Wherein, for convenience of describing, following subsidiary variable A is defined, B, C, D:

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

In above formula, employ following reduced form:

S ₁＝sinθ ₁,S ₂＝sinθ ₂,C ₁＝cosθ ₁,C ₂＝cosθ ₂,

S _1-2＝sin(θ ₁-θ ₂),C _1-2＝cos(θ ₁-θ ₂).

Utilize the acceleration/accel drive system model (6) of gained, former optimization problem (5) is to be converted into:

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

Wherein, ζ represents the total state vector of system, and u (t) represents the system input of this system, and for chassis acceleration/accel; T in bracket represents the time, after variable, (t) represents that this variable is the variable about the time, for simplicity's sake, (t) in most of variable is omitted in formula, T has represented the total time of transport, min represents minimum, connects the constraint condition representing and need to consider after s.t.; The subscript T of vector represents transpose of a matrix computing; x _frepresent given chassis target location, represent machine speed, v _max, a _maxrepresent the chassis maximum speed allowed and peak acceleration respectively; θ ₁(t), θ ₂t () represents one-level and secondary pivot angle respectively, represent one-level and secondary cireular frequency; θ _1max, θ _2maxrepresent the one-level and secondary maximum pendulum angle that allow in transport process, ω _1max, ω _2maxrepresent the one-level and secondary maximum angular rate that allow.

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

3rd, based on the trajectory planning of the pseudo-spectrometry of Gauss

For adapting to the requirement of the pseudo-spectrometry of Gauss, first needing to utilize coordinate transform, time interval corresponding for track being transformed on interval τ ∈ [-1,1] by t ∈ [0, T], namely

$\mathrm{\τ}=\frac{2t}{T}-1---\left(10\right)$

Here τ representation class is like the subsidiary variable of time.Choose K Legendre-Gauss (Legendre-Gauss, LG) point { τ subsequently ₁, τ ₂..., τ _k∈ (-1,1) forms point range.Here τ ₁, τ ₂..., τ _knamely represent selected LG point, subscript represents that the sequence number of this point is 1,2 ..., K; K is the number of the LG point selected.The zero point choosing the Legendre polynomial by solving K rank of LG point obtains.Meanwhile, τ ₀=-1 first place being added to point range, system state amount to be planned and input discrete sheet are shown as following form:

ζ(τ ₀),ζ(τ ₁),ζ(τ ₂),...,ζ(τ _K),

u(τ ₀),u(τ ₁),u(τ ₂),...,u(τ _K),

Utilize this K+1 node, construct K+1 Lagrange interpolation polynomial, concrete form is as follows:

Wherein, represent the Lagrangian fit differential polynomial that sequence number is i, i ∈ 0,1 ..., K}; Here, τ ∈ [-1,1]; represent to connect and take advantage of symbol, namely from the item of sequence number j=0, take the item of j=K always, and in process, skip the item of j=i.Utilize the system state amount at formula (11) and LG point place and the value of input, the quantity of state track of system and input track are shown out by mode is below approximate:

Wherein, ζ (τ _i), u (τ _i) represent τ=τ respectively _ithe system state amount at place and input; represent summation sign, from the item of sequence number i=0, be namely added to the item of i=K.To formula (12) differentiate, and utilize the concrete form of the middle interpolating function of formula (11), calculate and abbreviation, the derivative obtaining quantity of state track is as follows:

Wherein, represent τ=τ _kthe state trajectory derivative value at place; represent τ=τ _kplace, lagrange polynomial derivative value, concrete form is as follows:

Utilize the track value at formula (13), (14) and LG point place, to the differential equation constraint in optimization problem (9) carry out discretization and approximation process, concrete outcome is as follows:

Wherein, k ∈ 0,1,2 ..., K}.Formula (15) has the form of Algebraic Constraint.Next, the boundary condition constraint in optimization problem also needs the form changing into Algebraic Constraint, and wherein the boundary constraint of 0 moment is directly rewritten as follows:

ζ(0)＝[000000] ^T.

For representing the boundary constraint of transport process finish time, definition τ _k+1=1.From formula (10), τ _k+1=1 is corresponding transport finish time t=T.Utilize Gauss integration, this edge-restraint condition is expressed as:

$\mathrm{\ζ}\left({\mathrm{\τ}}_{K+1}\right)=\mathrm{\ζ}\left({\mathrm{\τ}}_0\right)+\frac{2}{T}\underset{k=1}{\overset{K}{\Σ}}{w}_k\[f\left(\mathrm{\ζ}\right({\mathrm{\τ}}_k\left)\right)+h\left(\mathrm{\ζ}\right({\mathrm{\τ}}_k\left)\right)u\left({\mathrm{\τ}}_k\right)\]$

Wherein, ζ (τ ₀) namely above-mentioned system initial state vector; w _krepresent kth Legendre's weights (Legendreweight), occurrence is tried to achieve in the lump when solving LG point.

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

minT

s.t.

ζ(0)＝[000000] ^T,

$\mathrm{\ζ}\left({\mathrm{\τ}}_0\right)+\frac{2}{T}\underset{k=1}{\overset{K}{\Σ}}{w}_k\[f\left(\mathrm{\ζ}\right({\mathrm{\τ}}_k\left)\right)+h\left(\mathrm{\ζ}\right({\mathrm{\τ}}_k\left)\right)u\left({\mathrm{\τ}}_k\right)\]={\left[\begin{array}{cccccc}{x}_f& 0& 0& 0& 0& 0\end{array}\right]}^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 _maxθ _1maxω _1maxθ _2maxω _2max] ^T

Wherein, ∞ represents infinity.For above-mentioned constrained nonlinear programming, select continuous quadratic type planing method (sequentialquadraticprogramming, SQP) to solve here, obtain following time optimal state vector sequence:

ζ(τ ₀),ζ(τ ₁),ζ(τ ₂),...,ζ(τ _K),ζ(τ _K+1),

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

4th, track following

By code-disc or laser sensor, test desk truck position and speed signal x (t), utilize the speed trajectory of the 3rd step gained chassis time optimal to be tracked reference locus and correspondence, 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, the propulsive effort of F (t) role of delegate on chassis, x _r(t), represent respectively with reference to deformation trace and speed trajectory, k _p, k _dit is the positive ride gain needing adjustment.Utilize this controller, corresponding real-time control signal can be calculated, drive trolley movement, complete control objectives.

Advantage of the present invention and beneficial effect

The present invention is directed to the traverse crane with double pendulum effect, propose a kind of double pendulum crane length of a game optimal trajectory planning method based on pseudo-spectrometry.Specifically, first the kinematics model of crane system is converted into a kind of acceleration/accel driving model, and based on this model, considers various constraint, construct the optimization problem of belt restraining; Subsequently, utilize the pseudo-spectrometry of Gauss to process gained optimization problem, be translated into the more convenient nonlinear programming problem solved.On this basis, time optimal chassis track can be obtained.This method for planning track that the present invention proposes disappears except consideration except the target of putting, and can also process the actual physics constraints such as pivot angle constraint, angular speed constraint, machine speed constraint, acceleration/accel constraint easily.The method proposed unlike, the present invention with existing method can obtain length of a game's optimal solution, drastically increases the work efficiency of crane system.Finally, by simulation and experiment, demonstrate validity of the present invention.

Accompanying drawing illustrates:

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

Fig. 2 represents Trajectory Planning result 2 (two-stage pivot angle and cireular frequency 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 result.

Detailed description of the invention:

Embodiment 1:

Analyze the control objectives of crane system, consider the multiple constraint comprising two-stage pivot angle and machine speed and acceleration/accel higher limit, obtaining following take haulage time as the optimization problem of cost function:

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

Here, the target location selecting chassis is x _f=0.6m, profile constraints is as follows:

θ _1max＝θ _2max＝2deg,v _max＝0.3m/s,ω _1max＝ω _2max＝5deg/s,a _max＝0.15m/s ²

2nd, acceleration/accel driving model Establishment and optimization problem transforms

Analyze and utilize double pendulum bridge type crane system, setting up following acceleration/accel drive system model:

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

Because this model expression is too complicated, repeat no more here, only provide the respective physical parameter of crane system, as follows:

M＝6.5kg,m ₁＝2.003kg,m ₂＝0.559kg,g＝9.8m/s ²,l ₁＝0.53m,l ₂＝0.4m.

Then optimization problem is converted into following form:

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

3rd, based on the trajectory planning of the pseudo-spectrometry of Gauss

For realizing the method for planning track based on pseudo-spectrometry proposed by the invention, use GPOPS software toolkit here ^[23]and SNOPT tool box ^[24]off-line solves the optimization problem in the 2nd step and obtains corresponding time optimal track.Wherein, Legendre-Gauss point parameter K=750 is chosen.Concrete result describes part see emulation experiment.

4th, emulation experiment effect describes

4.1st, simulation result

For checking the present invention proposes the feasibility of trajectory planning algorithm, first in MATLAB/Simulink environment, carry out numerical simulation.Simulation process is divided into two steps.The first step is that a time optimal reference locus planned by chassis according to this method; Second step, assuming that chassis runs according to this 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, represented by dotted arrows chassis target location, dotted line represents machine speed constraint, and solid line represents simulation result.In accompanying drawing 2, represented by dotted arrows angle restriction, dotted line represents angular speed constraint, and solid line represents simulation result.As can be seen from accompanying drawing 1, when chassis runs along reference locus, chassis converges to target location x quickly and accurately _f=0.6m; Meanwhile, in whole process, the speed of chassis meets set constraint.As can be seen from accompanying drawing 2, in chassis transport process, two-stage pivot angle is all less than given binding occurrence 2deg; Meanwhile, cireular frequency is also in institute's restriction range; And at the end of transport, two-stage pivot angle does not all exist Residual oscillations, the target of the pendulum that namely disappears fast also can realize.

4.2nd, experimental result

By code-disc or laser sensor, test desk truck position and speed signal x (t), utilize the speed trajectory of the 3rd step gained chassis time optimal to be tracked reference locus and correspondence, 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, the propulsive effort of F (t) role of delegate on chassis, x _r(t), represent respectively with reference to deformation trace and speed trajectory, k _p, k _dit is the positive ride gain needing adjustment.Utilize this controller, corresponding real-time control signal can be calculated, drive trolley movement, complete control objectives.

In an experiment, the tracking control unit ride gain chosen is:

k _p＝750,k _d＝150

Experimental result as shown in Figure 3.Wherein, dotted line represents pivot angle constraint, and solid line represents actual trolley movement track and pivot angle track.As can be seen from the figure, under PD controller action, chassis can follow the tracks of this reference locus preferably, realizes the control objectives of accurately chassis location fast.Two-stage pivot angle all remains in given scope in whole transport process, and at the end of transporting, does not almost have Residual oscillations.The present invention of this experiment show can realize good effect.

For embodying validity of the present invention further, as a comparison, this give the optimal trajectory planning method of document [21], and the experimental result of linearquadratic regulator (linearquadraticregulator, LQR) method.Wherein, the constraint of method for planning track is chosen consistent with the constraint of institute of the present invention extracting method in document [21]; And for LQR method, its controller expression formula is as follows:

$F\left(t\right)=-k_1\left(x\right(t)-x_f)-k_2\stackrel{\·}{x}\left(t\right)-k_3{\mathrm{\θ}}_1\left(t\right)-k_4{\stackrel{\·}{\mathrm{\θ}}}_1\left(t\right)-k_5{\mathrm{\θ}}_2\left(t\right)-k_6{\stackrel{\·}{\mathrm{\θ}}}_2\left(t\right),$

Wherein, the propulsive effort of F (t) role of delegate on chassis; X (t), represent the chassis Position And Velocity measured in real time; x _fnamely given target location, is set to x _f=0.6m; θ ₁(t), represent one-level pivot angle and cireular frequency thereof; θ ₂(t), represent secondary pivot angle and cireular frequency thereof, k ₁, k ₂, k ₃, k ₄, k ₅, k ₆be corresponding ride gain, concrete value sees below.Meanwhile, the method cost function is chosen as follows:

$J={\∫}_0^{\mathrm{\∞}}(X^{T}QX+{\mathrm{RF}}^2)dt,$

Wherein, X is defined as follows

$X={\left[\begin{array}{cccccc}e\left(t\right)& \stackrel{\·}{x}\left(t\right)& {\mathrm{\θ}}_1\left(t\right)& {\stackrel{\·}{\mathrm{\θ}}}_1\left(t\right)& {\mathrm{\θ}}_2\left(t\right)& {\stackrel{\·}{\mathrm{\θ}}}_2\left(t\right)\end{array}\right]}^T,$

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

Calculate controller gain as follows:

k ₁＝63.2456,k ₂＝50.7765,k ₃＝-129.3086,k ₄＝-6.9634,k ₅＝19.9137,k ₆＝-6.7856.

LQR method and the middle method of document [21], experimental result is as shown in accompanying drawing 4, accompanying drawing 5.Wherein, accompanying drawing 5 is for utilizing the experimental result of method in document [21], and solid line represents experimental result, and dotted line represents pivot angle constraint.

As can be seen from accompanying drawing 3, when chassis follow the tracks of institute plan that optimal trajectory runs time, complete transport process and only need 4.095s, and in whole process two-stage swing all remain on given constraint 2deg within, substantially without Residual oscillations when having transported.And for document [21] institute extracting method, complete and transport process need 5.445s; For LQR method, complete and transport process need 7.425s.Meanwhile, the one-level that LQR method causes swings maximum pendulum angle and reaches 6.5deg, and secondary swings maximum pendulum angle and reaches 11.5deg, much larger than the pivot angle of institute of the present invention extracting method.In summary, the method designed by the present invention can realize the accurate location of chassis and the quick elimination of system two-stage swing, obtains good controller performance.

Bibliography

[1]LiuY,YuH.Asurveyofunderactuatedmechanicalsystems.IETControlTheory&Applications.2013,7(7):921-935

[2]TuanLA,LeeSG,DangVH,MoonS,KimB.Partialfeedbacklinearizationcontrolofathree-dimensionaloverheadcrane.InternationalJournalofControl,AutomationandSystems,2013,11(4):718-727

[3]TuanLA,KimGH,KimMY,LeeSG.Partialfeedbacklinearizationcontrolofoverheadcraneswithvaryingcablelengths.InternationalJournalofPrecisionEngineeringandManufacturing,2012,13(4):501-507

[4]SinghoseW,KimD,KenisonM.Inputshapingcontrolofdouble-pendulumbridgecraneoscillations.ASMEJournalofDynamicSystems,Measurement,andControl,2008,130(3):034504.1-034504.7

[5]BlackburnD,SinghoseW,KitchenJ,PatrangenaruV,LawrenceJ,KamoiT,TauraA.Commandshapingfornonlinearcranedynamics.JournalofVibrationandControl,2010,16(4):477-501

[6] Wang Wei, Yi Jianqiang, Zhao Dongbin, Liu Diantong. the classification sliding-mode control of bridge type crane system. automation journal, 2004,30 (5): 784-788

[7]XiZ,HeskethT.Discretetimeintegralslidingmodecontrolforoverheadcranewithuncertainties.IETControlTheory&Applications,2010,4(10):2071-2081

[8] Hu Zhou, Wang Zhisheng, Zhen Ziyang. the drive lacking crane nonlinear transformations fused controlling that tape input is saturated. automation journal, 2014,40 (7): 1522-1527

[9]SunN,FangY,ZhangX.Energycouplingoutputfeedbackcontrolof4-DOFunderactuatedcraneswithsaturatedinputs.Automatica,2013,49(5):1318-1325

[10]SunN,FangY.Newenergyanalyticalresultsfortheregulationofunderactuatedoverheadcranes:Anend-effectormotion-basedapproach.IEEETransactionsonIndustrialElectronics,2012,59(12):4723-4734

[11]NakazonoK,OhnishiK,KinjoH,YamamotoT.Loadswingsuppressionforrotarycranesystemusingdirectgradientdescentcontrolleroptimizedbygeneticalgorithm.TransactionsoftheInstituteofSystems,ControlandInformationEngineers,2011,22(8):303-310

[12]ZhaoY,GaoH.Fuzzy-model-basedcontrolofanoverheadcranewithinputdelayandactuatorsaturation.IEEETransactionsonFuzzySystems,2012,20(1):181-186

[13]UchiyamaN,OuyangH,SanoS.Simplerotarycranedynamicsmodelingandopen-loopcontrolforresidualloadswaysuppressionbyonlyhorizontalboommotion.Mechatronics,2013,23(8):1223-1236

[14]SunN,FangY,ZhangX,YuanY.Transportationtask-orientedtrajectoryplanningforunderactuatedoverheadcranesusinggeometricanalysis.IETControlTheory&Applications,2012,6(10):1410-1423

[15] Sun Ning, Fang Yongchun, Wang Pengcheng, Zhang Xuebo. the design of drive lacking three-dimensional bridge type crane system adaptive Gaussian filtering device. automation journal, 2010,36 (9): 1287-1294

[16]SunN,FangY,ZhangX,MaB.Anovelkinematiccoupling-basedtrajectoryplanningmethodforoverheadcranes.IEEE/ASMETransactionsonMechatronics,2012,20(1):241-248

[17]WangPeng-Cheng,FangYong-Chun,JiangZi-Ya.Adirectswingconstraint-basedtrajectoryplanningmethodforunderactuatedoverheadcranes.ActaAutomaticaSinica,2014,40(11):2414-2419

[18]VaughanJ,KimD,SinghoseW.Controloftowercraneswithdouble-pendulumpayloaddynamics.IEEETransactionsonControlSystemsTechnology,2010,18(6):1345-1358

[19]SinghoseW,KimD.Manipulationwithtowercranesexhibitingdouble-pendulumoscillations.In:Proceedingsof2007IEEEInternationalConferenceonRoboticsandAutomation.Roma,Italy:IEEE,2007.4550-4555

[20]MasoudZ,AlhazzaK,Abu-NadaE,MajeedM.Ahybridcommand-shaperfordouble-pendulumoverheadcranes.JournalofVibrationandControl,2014,20(1):24-37

[21] Sun Ning, Fang Yongchun, money Yu wise man. with the double pendulum effect crane trajectory planning of state constraint. control theory and application, 2014,31 (7): 974-980

[22] Guo Weiping, Liu Diantong. secondary pendulum-type crane system is dynamically and based on passive control. Journal of System Simulation, 2008,20 (18): 4945-4948

[23]GargD,PattersonM,HagerW,RaoA,BensonD,HuntingtonG.Aunifiedframeworkforthenumericalsolutionofoptimalcontrolproblemsusingpseudospectralmethods.Automatica,2010,46(11):1843-1851

[24]GillP,MurrayW,SaundersM.SNOPT:AnSQPalgorithmforlarge-scaleconstrainedoptimization.SIAMReview,2005,47:99-131。

Claims (1)

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

1st, analyze profile constraints and construct corresponding optimization problem

Analyze the control objectives of crane system, consider the multiple constraint comprising two-stage pivot angle and machine speed and acceleration/accel higher limit, show that following take haulage time as the optimization problem of 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, t in bracket represents the time, after variable, (t) represents that this variable is the variable about the time, for simplicity's sake, (t) in most of variable is omitted in formula, T has represented the total time of transport, and min represents minimum, connects the constraint condition representing and need to consider after s.t.; represent that chassis position x (t) is about the first derivative of time and second derivative, i.e. machine speed and acceleration/accel respectively; v _max, a _maxrepresent the chassis maximum speed allowed and peak acceleration respectively; θ ₁(t), θ ₂t () represents one-level and secondary pivot angle respectively, represent one-level and secondary cireular frequency; θ _1max, θ _2maxrepresent the one-level and secondary maximum pendulum angle that allow in transport process, ω _1max, ω _2maxrepresent the one-level and secondary maximum angular rate that allow;

2nd, acceleration/accel driving model Establishment and optimization problem transforms

Analyze and utilize double pendulum bridge type crane system, obtaining following acceleration/accel drive system model:

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

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), represent chassis Position And Velocity respectively, θ ₁(t), represent one-level pivot angle and cireular frequency, θ ₂(t), represent secondary pivot angle and cireular frequency, the subscript T of bracket represents transpose of a matrix computing; U (t) represents the system input of this system, and for chassis acceleration/accel; The function that it is independent variable that f (ζ), h (ζ) all represent with the total state of system vector ζ, obtained by crane system kinematical equation, concrete form is shown in (7); the vectorial derivative about the time of total state for system;

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

Utilize above-mentioned acceleration/accel 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 this system, and for chassis acceleration/accel; The subscript T of vector represents transpose of a matrix computing;

3rd, based on the trajectory planning of the pseudo-spectrometry of Gauss

Utilize the thought of the pseudo-spectrometry of Gauss that the optimization problem in the 2nd step is carried out to process and solved, concrete steps are as follows:

3.1st, first lagrange-interpolation is utilized, select Legendre-Gauss (Legendre-Gauss, LG) the discrete system state trajectory at some place and input trajectory, by discrete loci and Lagrange interpolation polynomial, represent corresponding approximate trajectories model;

3.2nd, then, carry out differentiate by the locus model after pairing approximation, the derivative of state of the system lagrange polynomial derivative is represented;

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

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

4th, track following

By code-disc or laser sensor, test desk truck position and speed signal x (t), utilize the speed trajectory of the 3.4th step gained chassis time optimal to be tracked reference locus and correspondence, 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, the propulsive effort of F (t) role of delegate on chassis, x _r(t), represent respectively with reference to deformation trace and speed trajectory, k _p, k _dit is the positive ride gain needing adjustment; Utilize this controller, corresponding real-time control signal can be calculated, drive trolley movement, complete control objectives.