CN111704038B - Bridge crane path planning method considering obstacle avoidance - Google Patents

Bridge crane path planning method considering obstacle avoidance Download PDF

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CN111704038B
CN111704038B CN202010669502.9A CN202010669502A CN111704038B CN 111704038 B CN111704038 B CN 111704038B CN 202010669502 A CN202010669502 A CN 202010669502A CN 111704038 B CN111704038 B CN 111704038B
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bridge crane
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theta
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CN111704038A (en
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王昕炜
刘洁
董献洲
彭海军
张盛
陈飙松
李云鹏
吕琛
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Dalian University of Technology
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B66—HOISTING; LIFTING; HAULING
    • B66C—CRANES; LOAD-ENGAGING ELEMENTS OR DEVICES FOR CRANES, CAPSTANS, WINCHES, OR TACKLES
    • B66C17/00—Overhead travelling cranes comprising one or more substantially horizontal girders the ends of which are directly supported by wheels or rollers running on tracks carried by spaced supports
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B66—HOISTING; LIFTING; HAULING
    • B66C—CRANES; LOAD-ENGAGING ELEMENTS OR DEVICES FOR CRANES, CAPSTANS, WINCHES, OR TACKLES
    • B66C13/00—Other constructional features or details
    • B66C13/18—Control systems or devices
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B66—HOISTING; LIFTING; HAULING
    • B66C—CRANES; LOAD-ENGAGING ELEMENTS OR DEVICES FOR CRANES, CAPSTANS, WINCHES, OR TACKLES
    • B66C13/00—Other constructional features or details
    • B66C13/18—Control systems or devices
    • B66C13/22—Control systems or devices for electric drives
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B66—HOISTING; LIFTING; HAULING
    • B66C—CRANES; LOAD-ENGAGING ELEMENTS OR DEVICES FOR CRANES, CAPSTANS, WINCHES, OR TACKLES
    • B66C13/00—Other constructional features or details
    • B66C13/18—Control systems or devices
    • B66C13/48—Automatic control of crane drives for producing a single or repeated working cycle; Programme control

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  • Mechanical Engineering (AREA)
  • Automation & Control Theory (AREA)
  • Control And Safety Of Cranes (AREA)

Abstract

A bridge crane path planning method considering obstacle avoidance belongs to the field of mechanical automation. The path planning method comprises the following steps: firstly, a controlled dynamic model of the bridge crane is established according to relevant parameters of the bridge crane. Second, a control saturation constraint is established. Third, a state quantity constraint is established. Fourthly, establishing obstacle avoidance conditions. Fifthly, determining the boundary condition of the dispatching task. And sixthly, establishing an energy-time optimal control problem and solving the energy-time optimal control problem according to a dynamic equation, boundary conditions and constraint conditions of the bridge crane. The invention provides a simple obstacle avoidance condition in a mathematical form, and can ensure the stability and efficiency of a calculation process. Meanwhile, time and energy consumption in the dispatching process are comprehensively considered, and the method has important significance in actual control tasks.

Description

Bridge crane path planning method considering obstacle avoidance
Technical Field
The invention belongs to the field of mechanical automation, and relates to a bridge crane path planning method considering obstacle avoidance.
Background
The bridge crane is common equipment for material allocation and transportation in engineering scenes such as factories, wharfs, construction sites and the like. Bridge cranes are usually operated by experienced drivers, and manual operation is inevitably disturbed by factors such as line-of-sight disturbances, fatigue operation, etc., resulting in potential operational risks. Therefore, the automated operation of bridge cranes has been extensively studied. According to whether feedback exists in the design of the controller, the control strategy of the bridge crane is divided into open-loop control (such as an input shaping method, a path planning method, a filtering method and the like) and closed-loop control (such as model prediction control, sliding mode control, feedback linearization and the like). In the practical application process, if a feasible path is planned by using an open-loop method at first, and then a closed-loop strategy is used for path tracking, the safety level and robustness of the control strategy are greatly improved. Under the control strategy again, the path planning technology is a key link influencing the control effect.
The existing bridge Crane traffic path Planning technique usually considers only the shortest traffic Time or the best energy consumption (e.g. Zhang X, Fan Y, Sun N. minimum-Time traffic Planning for the following executed Overhead road Systems with State and Control constraints IEEE Transactions on Industrial Electronics,2014,61(12): 6915. lou 6925 and Chen H, Yang P, Y.A Time optical traffic Planning Method for Overhead road with Obstate Availance. IEEE/ASME International Conference on Advanced Intelligent Control. If the time consumption and energy consumption of dispatching can be comprehensively considered in the path planning link, the production unit can be selected in a targeted manner according to the specific task condition. In addition, cargo or product accumulation often occurs in real-world production or construction environments, creating potential obstacles to bridge crane path planning (e.g., Yang J S, Huang M L, Chien W F, Tsai M H. application of Machine Vision to Collision Avoidence Control of the Overhead road and International Conference electronics, 2015 and Hara Y, Noda Y. operation access system for Overhead road and local traffic in Overhead road and International Conference, Man, and CyberCyberics, 2016). In the dispatching path planning link, if the existence of the constraint is considered, a huge threat can be generated on the safety of dispatching loop regulation, but the existence of the constraint is rarely considered in the existing research.
Disclosure of Invention
In order to solve the technical problem, the dispatching path planning of the bridge crane in the obstacle environment is carried out under the framework of nonlinear optimal control. A conservative obstacle avoidance condition is proposed using a mathematically simple inequality, thereby simplifying the complexity of the optimal control problem. In the optimal control problem column, limiting factors such as control saturation, speed constraint, swing angle constraint, obstacle avoidance conditions and the like are comprehensively considered, and a time-energy optimal performance index is adopted.
In order to achieve the purpose, the invention adopts the technical scheme that:
a path planning method for a bridge crane considering obstacle avoidance comprises the following steps that firstly, a controlled dynamic equation of the bridge crane is established according to relevant parameters of the bridge crane. Second, a control saturation constraint is established. Third, a state quantity constraint is established. Fourthly, establishing obstacle avoidance conditions. Fifthly, determining the boundary condition of the dispatching task. And sixthly, establishing an energy-time optimal control problem and solving the energy-time optimal control problem according to a dynamic equation, boundary conditions and constraint conditions of the bridge crane. The calculation flow chart of the invention is shown in fig. 1, and comprises the following steps:
step 1: establishing controlled dynamic equation of bridge crane according to related parameters of bridge crane
The bridge crane is shown in figure 2, wherein X represents the displacement of the trolley along the X axis, Y represents the displacement of the cart along the Y axis, and thetaxAnd thetayFor indicating the swing angle of the hoist rope. The length of the lifting rope is denoted as l. m isp、mtAnd mgRespectively representing the mass of the heavy object, the small vehicle and the big vehicle. FxRepresenting a control force applied to the trolley in the direction of the X-axis, FyRepresenting the control force applied to the cart in the Y-axis direction. Note vxAnd vySpeed, omega, of small and large vehicles, respectivelyxAnd omegayAre each thetaxAnd thetayThe corresponding acceleration. For convenience, let us rememberx≡sinθx，Sy≡sinθy，Cx≡cosθxAnd Cy≡cosθy. The acceleration of gravity is g. The generalized coordinate of the selection system is q ═ x, y, θx,θy]TThen the controlled dynamics equation of the system is:
Figure GDA0002960940040000021
wherein t is a time variable,
Figure GDA0002960940040000022
and u ═ Fx,Fy]TRespectively, a state variable and a control variable of the system. The matrices M, V, G and F represent the inertia matrix, coriolis force matrix, gravity correlation matrix, and control force matrix of the system, respectively, and the corresponding expressions are:
Figure GDA0002960940040000023
Figure GDA0002960940040000024
G＝[0,0,mpglSxCy,mpglCxSy]T (4)
F＝[Fx,Fy,0,0]T (5)
step 2: establishing control saturation constraints
In the control process, FxAnd FyRespectively is Fx,maxAnd Fy,maxThen the control saturation constraint is expressed as:
|Fx|≤Fx,max,|Fy|≤Fy,max (6)
and step 3: establishing state variable constraints
In order to ensure that the system can realize rapid parking under the emergency condition, the speed of the small car and the big car is ensuredIs within a safe range. The upper speed limits of the small car and the big car are respectively recorded as vx,maxAnd vy,maxThen the corresponding constraint is expressed as:
|vx|≤vx,max,|vy|≤vy,max (7)
in addition, the swing angle in the process of allocation and transportation is ensured to be within a safe range. Note thetaxAnd thetayRespective upper limit is θx,maxAnd thetay,maxThen the corresponding constraint is expressed as:
|θx|≤θx,max,|θy|≤θy,max (8)
and 4, step 4: establishing obstacle avoidance conditions
K obstacles placed on the ground are recorded in the environment. The ground can use a plane Pground:{(x,y,z)|z＝-z0},(z0> 0) is used. For convenience, assume that each obstacle is a cylinder and the radius for the weight is rpIs shown as a ball. For the obstacle K (K ═ 1,2, …, K), let its radius be robs,kHeight of hk(hk＜z0) The top surface of the substrate is located on a plane Ptop,k:{(x,y,z)|z＝-z0+hkThe circle centers of the bottom and top surfaces are respectively recorded as
Figure GDA0002960940040000031
And
Figure GDA0002960940040000032
in order to avoid the obstacle, the lifting rope and the heavy object are not required to collide with any obstacle.
According to fig. 3, the coordinates of the cart are G (x, y,0) and the coordinates of the center of the sphere of the characteristic sphere of the weight are P (x + LC)ySx,y+LSy,-LCyCx). The point G and the point P are related to the plane PgroundRespectively is G*(x,y,-z0) And P*(x+LCySx,y+LSy,-z0). For any point A and point B in space, the symbols are used
Figure GDA0002960940040000033
Representing the euclidean distance between two points. The obstacle avoidance condition between the weight and the obstacle is expressed as:
Figure GDA0002960940040000034
and according to point Oobs,kAnd point P*Equation (9) is further expressed as:
Figure GDA0002960940040000041
according to FIG. 3, the line segment GP and the plane P are recordedtop,kIs point SkThen point SkThe coordinates are expressed as:
Sk(x+(z0-hk)tanθx,y+(z0-hk)secθxtanθy,-z0+hk),k＝1,2,...,K (11)
therefore, the obstacle avoidance condition between the lifting rope and the obstacle is expressed as:
Figure GDA0002960940040000042
and according to the point
Figure GDA0002960940040000043
And point SkEquation (12) may be further expressed as:
Figure GDA0002960940040000044
the 2K inequalities in the formula (10) and the formula (13) accurately describe the obstacle avoidance condition, however, it can be found that the numerous trigonometric functions involved therein make the finally constructed optimal control problem have extremely high nonlinearity, which is not favorable for the stable and fast solution of the problem. Therefore, considering establishing obstacle avoidance conditions favorable for solving, the following are specific:
due to thetaxAnd thetayGiven the maximum allowable value of (c), segment G*P*Subject to the following inequality constraints
Figure GDA0002960940040000045
Defining a function with respect to the variable α as
Figure GDA0002960940040000046
According to fig. 4, the obstacle avoidance condition for the obstacle can be described by using the following inequality
Figure GDA0002960940040000047
Wherein α > 1 is a given safety factor. According to point Oobs,kAnd point G*Equation (15) may be further expressed as:
Figure GDA0002960940040000048
it can be found that compared with the formula (10) and the formula (13), the obstacle avoidance condition in the formula (16) is more conservative and mathematically simpler, and the number of the constraint conditions is reduced by half, which facilitates stable and fast solution of the optimal control problem.
And 5: determining boundary conditions for a dispatch task
Recording the starting time of the dispatching task as tsThe initial state corresponding to the bridge crane is xs. Recording the expected terminal state of the bridge crane in the dispatching task as xf。
Step 6: according to a dynamic equation, boundary conditions and constraint conditions of the bridge crane, establishing an energy-time optimal control problem and solving the problem:
constructing the following optimal control problem with optimal time-energy as a performance index according to the dynamic equation of the bridge crane established in the step 1, the constraint conditions established in the steps 2 to 4 and the task boundary conditions determined in the step 5:
Figure GDA0002960940040000051
wherein, tfThe time when the dispatching task is completed; positive definite matrix
Figure GDA0002960940040000052
A weight matrix which is a control variable;
Figure GDA0002960940040000053
is the weight coefficient of the transit time relative to the energy consumption. The control variable obtained by solving the problem is recorded as
Figure GDA0002960940040000054
Then
Figure GDA0002960940040000055
And
Figure GDA0002960940040000056
corresponding to optimal control inputs acting on the cart and cart, respectively.
The invention has the beneficial effects that: based on a dynamic equation of the bridge crane, the environmental obstacle in the hoisting process is considered, the obstacle avoidance condition is described in a conservative and mathematically simple mode, and finally the hoisting path of the bridge crane is solved under the framework of an energy-time hybrid optimal control problem, so that the method has important practical value on the operation safety and efficiency of the bridge crane in a complex operation environment. The method disclosed by the invention has strong operability and feasibility and is convenient for practical application.
Drawings
FIG. 1 is a flow chart of the calculation of the present invention.
Fig. 2 is a schematic structural diagram of a bridge crane in consideration of the present invention.
Fig. 3 is a spatial configuration of a bridge crane and an obstacle in consideration of the present invention.
Fig. 4 is a schematic plan geometric view corresponding to the obstacle avoidance condition of the structure of the present invention.
FIG. 5 is a three-dimensional view of a trajectory in an embodiment of the present invention.
FIG. 6 is a top view of a track in an embodiment of the invention.
FIG. 7 is a control input for the cart in an embodiment of the present invention.
FIG. 8 is a control input for the cart in an embodiment of the present invention.
Detailed Description
The present invention is further illustrated by the following specific examples.
The mass of a trolley of the bridge crane is recorded as mt20kg, cart height z03m, the mass of the cart is mg60kg, and the length of the lifting rope is 1.5 m; mass mp15kg, the radius of the characteristic sphere is rp0.4 m; the acceleration of gravity is h ═ 9.8m/s2。
In the path planning link, the upper limit of the control variable is set to be Fx,max30N and Fy,max30N; the upper limit of the speed variable is set to vx,max0.8m/s and vy,max0.8 m/s; the upper limit of the swing angle is set to theta x,max3 deg. and thetay,max＝3deg。
Assuming that K ═ 2 obstacles exist in the environment, relevant parameters of the two obstacles are shown in table 1; and when the obstacle avoidance condition is established, selecting the safety factor alpha as 1.5.
TABLE 1 obstacle parameter information
Figure GDA0002960940040000061
Setting the initial time of the dispatching task as ts0s, corresponding to an initial state of xs＝[0,0,0,0,0,0,0,0]T(ii) a The expected terminal state is xf＝[5,4.5,0,0,0,0,0,0]T。
When an optimal control problem is constructed, a weight matrix of a control variable is recorded as R ═ diag (1,1), and a weight coefficient of the tuning time to the energy consumption is recorded as ω 1000.
A bridge crane path planning method considering obstacle avoidance comprises the following steps:
step 1: establishing controlled dynamic equation of bridge crane according to related parameters
Selecting a state space of
Figure GDA0002960940040000062
The controlled variable is u ═ Fx,Fy]TThen the controlled dynamics equation of the system is
Figure GDA0002960940040000063
Wherein the expressions of the matrices M, V, G and F are respectively
Figure GDA0002960940040000064
Figure GDA0002960940040000071
G＝[0,0,15×1.5×gSxCy,15×1.5×gCxSy]T (21)
F＝[Fx,Fy,0,0]T (22)
Step 2: establishing control saturation constraints
The control saturation constraint is expressed as:
|Fx|≤30,|Fy|≤30 (23)
and step 3: establishing state variable constraints
The constraint condition corresponding to the speed variable is expressed as
|vx|≤0.8,|vy|≤0.8 (24)
The constraint condition corresponding to the swing angle is expressed as
|θx|≤3,|θy|≤3 (25)
And 4, step 4: establishing obstacle avoidance conditions
For the first obstacle, the corresponding obstacle avoidance condition is expressed as
Figure GDA0002960940040000072
For the second obstacle, the corresponding obstacle avoidance condition is expressed as
Figure GDA0002960940040000073
And 5: determining boundary conditions for a dispatch task
Recording the starting time of the dispatching task as t s0, the bridge crane corresponds to an initial state xs＝[0,0,0,0,0,0,0,0]T. Recording the expected terminal state of the bridge crane in the dispatching task as xf＝[5,4.5,0,0,0,0,0,0]T。
Step 6: establishing an energy-time optimal control problem and solving the problem according to a dynamic equation, boundary conditions and constraint conditions of the bridge crane
Constructing the following optimal control problem with optimal time-energy as a performance index according to the kinetic equation established in the step 1, the constraint conditions established in the steps 2 to 4 and the task boundary conditions determined in the step 5
Figure GDA0002960940040000081
According to the steps, the optimal terminal time obtained by calculation is tf11.9688s, performance index J1.51376 × 104. The three-dimensional view and the top view of the calculated track of the bridge crane are respectively as the figure5 and 6, the control inputs for the cart and cart are shown in figures 7 and 8, respectively.
The above-mentioned embodiments only express the embodiments of the present invention, but not should be understood as the limitation of the scope of the invention patent, it should be noted that, for those skilled in the art, many variations and modifications can be made without departing from the concept of the present invention, and these all fall into the protection scope of the present invention.

Claims (1)

1. A bridge crane path planning method considering obstacle avoidance is characterized in that a bridge crane consists of a cart, a trolley, a lifting rope and a heavy object, and comprises the following steps:
step 1: establishing controlled dynamic equation of bridge crane according to related parameters of bridge crane
In the bridge crane, X represents the displacement of the trolley along the X axis, Y represents the displacement of the trolley along the Y axis, and thetaxAnd thetayUsed for showing the swing angle of the lifting rope; the length of the lifting rope is recorded as l; m isp、mtAnd mgRespectively representing the mass of the heavy object, the small vehicle and the big vehicle; fxRepresenting a control force applied to the trolley in the direction of the X-axis, FyRepresenting the control force applied to the cart in the Y-axis direction; note vxAnd vySpeed, omega, of small and large vehicles, respectivelyxAnd omegayAre each thetaxAnd thetayA corresponding acceleration; and remember Sx≡sinθx，Sy≡sinθy，Cx≡cosθxAnd Cy≡cosθy(ii) a The gravity acceleration is g; the generalized coordinate of the selection system is q ═ x, y, θx,θy]TThen the controlled dynamics equation of the system is:
Figure FDA0002581752250000011
wherein t is a time variable,
Figure FDA0002581752250000012
and u ═ Fx,Fy]TRespectively a state variable and a control variable of the system; the matrices M, V, G and F represent the inertia matrix, coriolis force matrix, gravity correlation matrix, and control force matrix of the system, respectively, and the corresponding expressions are:
Figure FDA0002581752250000013
Figure FDA0002581752250000014
G＝[0,0,mpglSxCy,mpglCxSy]T (4)
F＝[Fx,Fy,0,0]T (5)
step 2: establishing control saturation constraints
In the control process, FxAnd FyRespectively is Fx,maxAnd Fy,maxThen the control saturation constraint is expressed as:
|Fx|≤Fx,max,|Fy|≤Fy,max (6)
and step 3: establishing state variable constraints
The speed of the small car and the big car is required to be within a safe range, so that the system can realize rapid parking under an emergency condition, and the upper speed limits of the small car and the big car are respectively recorded as vx,maxAnd vy,maxThen the corresponding constraint is expressed as:
|vx|≤vx,max,|vy|≤vy,max (7)
in addition, the swing angle is required to be within a safe range in the dispatching process; note thetaxAnd thetayRespective upper limit is θx,maxAnd thetay,maxThen the corresponding constraint is expressed as:
|θx|≤θx,max,|θy|≤θy,max (8)
and 4, step 4: establishing obstacle avoidance conditions
Recording K barriers placed on the ground in the environment; the ground adopts a plane Pground:{(x,y,z)|z＝-z0},(z0> 0) represents; assuming that each obstacle is a cylinder and the weight is represented by a characteristic sphere; for the obstacle K (K ═ 1,2, …, K), let its radius be robs,kHeight of hk(hk＜z0) The top surface of the substrate is located on a plane Ptop,k:{(x,y,z)|z＝-z0+hkThe circle centers of the bottom and top surfaces are marked as Oobs,k(xobs,k,yobs,k,-z0) And
Figure FDA0002581752250000021
in order to avoid obstacles, the lifting rope and the heavy object are not required to collide with any obstacle;
the coordinate of the trolley is G (x, y,0), and the sphere center coordinate of the characteristic sphere of the weight is P (x + LC)ySx,y+LSy,-LCyCx) (ii) a The point G and the point P are related to the plane PgroundRespectively is G*(x,y,-z0) And P*(x+LCySx,y+LSy,-z0)；
For any point A and point B in space, the symbols are used
Figure FDA0002581752250000022
Representing the euclidean distance between two points; the obstacle avoidance condition between the weight and the obstacle is expressed as:
Figure FDA0002581752250000023
and according to point Oobs,kAnd point P*Equation (9) is further expressed as:
Figure FDA0002581752250000024
note the segment GP and the plane Ptop,kIs point SkThen point SkThe coordinates are expressed as:
Sk(x+(z0-hk)tanθx,y+(z0-hk)secθxtanθy,-z0+hk),k＝1,2,...,K (11)
therefore, the obstacle avoidance condition between the lifting rope and the obstacle is expressed as:
Figure FDA0002581752250000025
and according to the point
Figure FDA0002581752250000026
And point SkEquation (12) is further expressed as:
Figure FDA0002581752250000031
based on the formula (10) and the formula (13), establishing obstacle avoidance conditions beneficial to solving, specifically as follows:
since theta is already givenxAnd thetayIs the maximum allowable value of (c), then line segment G*P*Subject to the following inequality constraints:
Figure FDA0002581752250000032
defining a function with respect to the variable α as
Figure FDA0002581752250000033
The obstacle avoidance condition of the obstacle is described by using the following inequality:
Figure FDA0002581752250000034
wherein α > 1 is a given safety factor; according to point Oobs,kAnd point G*Equation (15) is further expressed as:
Figure FDA0002581752250000035
and 5: determining boundary conditions for a dispatch task
Recording the starting time of the dispatching task as tsThe initial state corresponding to the bridge crane is xs(ii) a Recording the expected terminal state of the bridge crane in the dispatching task as xf；
Step 6: establishing an energy-time optimal control problem and solving the energy-time optimal control problem according to the dynamic equation of the bridge crane established in the step 1, the constraint conditions established in the steps 2 to 4 and the task boundary conditions determined in the step 5, and constructing the following optimal control problem with optimal time-energy as a performance index:
Figure FDA0002581752250000036
wherein, tfThe time when the dispatching task is completed; positive definite matrix
Figure FDA0002581752250000041
A weight matrix which is a control variable;
Figure FDA0002581752250000042
a weight coefficient of the transfer time relative to the energy consumption; the control variable obtained by solving the problem is recorded as
Figure FDA0002581752250000043
Then
Figure FDA0002581752250000044
And
Figure FDA0002581752250000045
corresponding to optimal control inputs acting on the cart and cart, respectively.
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