TÊN ĐỀ TÀI: THIẾT KẾ BỘ ĐIỀU KHIỂN PHI TUYẾN TỐI ƯU CHO CẦU TRỤC HOẠT ĐỘNG TRONG KHÔNG GIAN 3D NONLINEAR OPTIMAL CONTROL OF A 3D OVERHEAD CRANE NHIỆM VỤ VÀ NỘI DUNG: - Xây dựng mô hìn
INTRODUCTION
Categorizing and working principle of cranes
Cranes can be categorized into four types: mobile cranes, tower cranes and jib cranes and overhead cranes.
A mobile crane (as shown in Fig 1.1.) is a self-propelled vehicle that is designed solely or principally for lifting objects using a boom with lifting gear which only carries loads that are necessary for its own propulsion or equipment
Trucks fitted with crane apparatus, such as a knuckle boom crane (a crane fitted to a vehicle to lift goods on and off the vehicle)
(Image from http://www.braggcrane.com/index.php/bragg_crane_service)
2 A number of researches have been focus on dynamics and control of mobile cranes such as Qian et al [1] presented the dynamics and trajectory tracking control of cooperative multiple mobile cranes Compared with a single mobile crane, cooperative cable parallel manipulators for multiple mobile cranes (CPMMC) are more complex in configuration, which has the characters of both series and parallel manipulators Sun et al [2] investigated a new method for dynamic calculation of mobile cranes In this method, the flexible multibody model of the structure will be coupled with the model of the drive system Reddy et al [3] reports the work done towards applying configuration space (C-space) and search concepts to develop a tool to identify lift paths that satisfies the planning requirements, etc
Tower cranes (as shown in Fig 1.2) are a common fixture at any major construction site They are pretty hard to miss, they often rise hundreds of feet into the air, and can reach out just as far The construction crew uses the tower crane to lift steel, concrete, large tools like acetylene torches and generators, and a wide variety of other building materials
A number of researches have been focus on dynamics and control of tower cranes such as Schlott et al [4], a tower crane is modeled as a multibody system which describes the fundamental dynamic behavior of the crane structure using only few degrees of freedom
(Image from http://www.craneticket.com)
3 Carmona et al [5], introduced a new configuration of a hammerhead crane
Instead of a single cable, it is introduced a second hoisting cable It is shown a mathematical model that relates both hoisting cables and is set the corresponding linear feedback control law that coordinate both cable Hosseini, M., et al [6], proposed a method to select the suitable type of crane and locate the best place for crane erection based on a minimum radius for requested crane and minimum cost, etc
A jib crane (as shown in Fig 1.3), is not restricted to a fixed location and can be mounted on a movable chassis for use in military operation or other temporary work sites Such cranes often focus more on mobility rather than lifting capacity
Outriggers are often used in mobile jib cranes to keep them stable under load On the other hand, the stationary cranes are generally anchored in place
In the modern jib cranes, robust metal cable is wrapped around the jib strut ends, with the end of the hoist often connected to an electromagnet or hook, while the other end is connected to a metallic winch When you activate the winch, the pulley delivers a lifting force that is equal to the force applied by the metallic winch The hoist can move inwards or outwards along the jib length, providing even better flexibility of motion
(Image from http://www.mmengineers.com/jib_crans.html)
4 A number of researches have been focus on dynamics and control of jib cranes such as Kumada et al [7] used adaptive control for jib crane with nonlinear uncertainties Doỗi et al [8] focuses on control of oscillations using modeling and simulations Dynamic parameters investigated are: acceleration, angular velocity, forces and torque in main parts of crane, and influence of load swinging
Meshcheryakov et al [9] approached to the adaptive control system design on the basis of fuzzy logic The main idea is to create the operator’s actions model for the jib crane operations automation, etc
Overhead crane (as shown in Fig 1.4) is a kind of bridge type crane whose crane span structure travels on the carrier track Crane bridge travels longitudinally on the track laying on both sides of the elevated rail and the lifting trolley travels on the track laying on the bridge, which becomes a rectangular working range so that the room under the bridge can be made full use to lift articles and avoid blocking from ground equipment.
Overhead crane is widely used in the warehouse, workplace, port and open storage place indoors or outdoors Overhead cranes can be classified into ordinary overhead crane, simple overhead girder crane and metallurgical special overhead crane Ordinary overhead crane generally consists of lifting trolley, bridge traveling mechanism and bridge metal mechanism Lifting mechanism is composed with lifting mechanism, trolley traveling mechanism and trolley frame
(Image from http://www.mhi.org/fundamentals/cranes)
5 Objectives of the overhead crane system: bring cargo from place to another place as fast as possible and suppress the vibration of the cargo during the crane working
In this thesis, I will focus on 3-D overhead crane system and solve some objectives as follows:(i) Analysing and constructing a dynamic model for 3-D overhead crane system with five DOFs (i.e., trolley and girder positions, rope length, and two sway angles) (ii) Designing a nonlinear optimal control based on this dynamic-model to suppress the vibration of payload, drive the trolley to desired position as fast as possible and (iii) Minimizing the control efforts.
Modeling for overhead crane system
We have 2 methods to modeling the 3-D overhead crane system: Ordinary differential equations (ODEs) and Partial differential equations (PDEs)
In ODEs system (as shown in Fig 1.5), we assumed that the payload is a lumped mass model, neglect affection of elasticity of cable and inertial payload This model is used popularly to modeling crane systems in research and practice
A number of researches have been focus on ODEs model such as: Elling et al
[10], Chin et al [11], Abdel-Rahman et al [12], Oguamanam et al [13], etc
In PDEs system (as shown in Fig 1.6), we assumed that:
The cable is completely flexible and nonstretching
Transversal and angular displacements are small
The acceleration of the mass of the payload is negligible with respect to the gravitational acceleration g
A number of researches have been focus on PDEs model such as D’Andréa- Novel et al [14] used feedback stabilization of a hybrid PDE-ODE system to apply to an overhead crane D’Andréa-Novel et al [15] used PDEs to prove the exponential stabilization of an overhead crane with flexible cable, Abdel-Rahman et al [16] recommend appropriate models and control for various crane application and suggest
7 direction for further work He et al [17] used PDEs to construct an adaptive control of a flexible crane system with the boundary output constraint
As mentioned above, using PDEs model has more obstacles because control theory for PDEs has not completed and we cannot apply the control theory for ODEs
In this thesis, I used nonlinear ODEs model to construct the dynamic model for 3-D overhead crane system This model will be used to design the nonlinear optimal control.
Control methods for overhead crane system in practice
There has two control methods for overhead crane system: automatic and semi-automatic control
1.3.1 Semi-automatic overhead crane systems
Repetitive tasks that are performed in an unchanging sequence of steps but still require the attention of an operator can be partly automated without difficulty
Processes are performed according to a defined sequence of steps, with the semi- automatic crane remaining completely capable of manual operation at all times and without loss of flexibility
The semi-automatic crane control system allows only logically correct operation of the crane, thereby supporting its operator, who can concentrate on his or her tasks and the product while the process chain is monitored by the control system
Fig 1.7 Semi-automatic crane systems
(Image from https://www.altmann-foerdertechnik.de.html)
8 Wherever it is impossible to completely block off the hazard area because people are present there, a semi-automatic crane is the right choice The control system allows only the next correct step, for instance moving to a target position along a defined route, and requires constant approval by the operator Without this approval, the crane stops and transitions to a safe system status If the operator moves the semi-automatic crane into a collision zone during manual operation, it will stop automatically
Benefits of semi-automatic cranes:
Collision avoidance with obstacles (e.g shelves, machinery, fixtures) stored in the control system
Reduction of damage caused by collisions
A suitable choice of sensors makes automation of the following crane tasks possible:
Movement to a nearly unlimited number of positions with millimetre precision
Picking up and setting down loads, load swaying prevention
Transfer of attached load or other process variables to a higher-level control system
(Image from http://www.centrocranes.com)
9 We make two types of automated cranes In the semi-automatic systems, various features assist the operator, but allow more manual control With a fully- automated crane, the operator makes the settings, and the crane automatically takes care of repetitive or difficult actions This is especially useful in demanding and hazardous environments
Automated overhead cranes can reduce labor costs, track inventory, optimize storage, reduce damage, increase productivity and reduce the capital expense associated with forklift systems Some common applications include steel coil and paper roll handling, waste-to-energy, shipbuilding, container handling, food production, metals processing and general manufacturing.
Literature review
3-D overhead cranes are widely used in manufacturing and maintenance applications because of its efficiency and saving downtime in transport operations
In an overhead crane, a trolley travels in horizontal beam, which is called a bridge
This bridge moves on another horizontal beam A hoisting motor that lifts/lowers the payload is assembled to the trolley It is obvious that the degree of freedom of the overhead crane is five (i.e., trolley movement, bridge movement, hoisting, and two sway angles of the payload) However, there are only three actuators, which drive bridge, trolley, and hoist motions Therefore, it is shown that overhead cranes are unactuated systems, in which sway motions are not directly driven by actuators
The first priority in crane operation is eliminating hazard generated by sway motions of the payload This problem can be solve by move the trolley and/or the bridge with low speeds In contrast to the safety issue, a common demand is to reduce processing time in cranes, which can be achieved by increasing speeds of a trolley and a bridge However, high speeds of the trolley and the bridge may result in large residual sway of the payload and consequently, the processing time because the crane has to wait until the sway motions of the payload are completely suppressed Moreover, it should be noted that the crane also needs to be driven to desired positions Therefore, prompt sway suppression and precise position control for overhead crane are simultaneously required
10 In Viet Nam, there have many papers about overhead crane were presented such as [18-20] Ngo [18], the author approached to anti-swing control of container cranes with friction compensate The friction always exist in crane system and we cannot know the coefficient exactly It makes difficult for control of the overhead crane The author introduces a method to estimate the coefficient of the friction and constructs a nonlinear controller to drive the trolley to desire position and anti-swing the payload Nguyen and Ngo [19] investigated an adaptive fuzzy sliding mode control scheme for container crane but the controller is construct based on 2D-model
Nguyen et al [20] investigated a vision anti-sway control system for a container crane, the author uses vision system to derive the value of the two sway angles and design the controller based on Lyapunov method to anti-swing sway angles But there have never seen investigated about optimal control for overhead cranes system
In the world, the mathematical models for 3-D overhead cranes are presented in [21-35] The pioneering work on 3-D overhead cranes goes back to the work [21], in which the nonlinear model of 3-D overhead crane was introduced Most of the researches on 3-D overhead cranes are developed based on the Euler-Lagrange equation The 3-D crane models are usually complexity because of the nonlinear coupling between variables of motions For example, the motion of the payload affects to the motions of bridge and the trolley and vice versa Since the complexity of the nonlinear models of 3-D overhead cranes yield difficulties in control design, a number of researches [21, 23-30, 34] did not consider hoisting motion, i.e., the overhead crane only possesses 4-DOFs It should be noted that the hoisting motion generates the variation of the rope length, which significantly influences to the sway motions of the payload [22, 31, 34] It is obvious that the researches focused on 5- DOF overhead cranes (i.e., the hoisting motion is included) which are rare In this research, a nonlinear model of a 5-DOF overhead crane is employed to develop a control algorithm
Because the intricateness of the nonlinear models of overhead cranes system, many authors exploited linearization technique to simplify mathematical models [22]
In the work [22] based on linearized crane dynamics, a fuzzy logic anti-swing
11 controller was presented However, these methods based on the linearization technique of overhead cranes may lose the sufficient accuracy of information about position of the bridge and trolley, especially the sway angles because the linear models were approximately the nonlinear models To improve the control performance, control design based on nonlinear models was investigated Chwa [26] developed a nonlinear behavior of the system resulting from the payload variation
Ngo and Hong [29] considered an overhead crane in moving platform, where the sliding mode control algorithm was employed to suppress the sway motion Sun et al
[30] proposed an energy coupling-based feedback control scheme for an overhead crane under control input constrains Zhang et al [34] also investigated an overhead crane with input constrains, but Lyapunov method was employed to derive the control law It is obvious that the control algorithms based on the nonlinear models improves the control performance of the overhead cranes Therefore, in this research, nonlinear model is used to develop the proposed control algorithm
As mentioned above, the time for loading/unloading process of cranes needs to be reduced while sway angles is required to be small The control problem of the cranes can be considered as optimization problem that minimizes time and state variables with dynamic constrains Optimal control has been used to develop control laws for cranes [36-47] Karihaloo and Parbery [36] proposed an optimal control scheme for a gantry crane representing by a linear model of the crane, where the minimization of control efforts within given transport time was considered However, the suppression of sway motion was not included in control design Auernig and Troger [38] developed a control algorithm that optimizes transport time of a container crane It should be noted that since a container crane operates in 2D space, its dynamics are simpler than dynamics of a 3-D overhead crane [37-40] introduced a minimum time control algorithm for an overhead crane, where the overhead crane only moves in 2D space, and a linear model was used to represent the crane dynamics
Optimal tracking control [45] proposed a novel time-optimal off-line trajectory planning method, together with a tracking controller, is proposed for a two- dimensional (2D) underactuated overhead crane, optimal sliding mode control [47]
12 proposed a quadratic performance index is given and an optimal nonlinear switching manifold It is recognized that although the optimal control is fully suitable solution to control design of 3-D overhead crane, this control method has not been focused by researchers The possibility of a reason is the complexity of searching an optimal control input, where nonlinear differential equations need to be solved However, the use of computation method provided by Matlab software can make the computation more efficient In this thesis, an optimal control problem that is able to minimize sway motions together with control efforts is developed, where a nonlinear model representing 3-D overhead crane with varying rope length is considered The effectiveness of the proposed control law is illustrated by numerical simulations and experiment results.
Objectives of the thesis
a In practice b In experimental model
Fig 1.9 Simultaneous mechanism of pulley
Constructing the dynamic model for the 3-D overhead crane systems (ODEs) with the motions:
Translational motion in X -, Y -directions of the trolley and in Z-direction of the payload (hoist up and hoist down)
Vibration of the payload has two components: and
13 Three motions are driven by three actuators AC servo Mitsubishi The vibration of the payload is suppressed by combine two motions in X -, Y - directions of the trolley
In practice, the overhead crane systems is also has another motion: rotate around Z- axis But this motion is eliminated by simultaneous mechanism of pulley on hoisting compartment (as shown in Fig 1.9) Therefore, we can neglect this motion in constructive dynamic model of 3-D overhead crane system
1.5.2 Nonlinear optimal control design for 3-D overhead crane system
The nonlinear optimal control is constructed to ensure three objectives of the 3-D overhead crane system such as:
1 Driving the trolley from initial to desired position as fast as possible
2 The vibration of the payload had to be suppressed as fast as possible
3 Optimal trajectory of three motion in X-, Y-, Z-directions to minimize time and control efforts of the system.
Organization of the thesis
The thesis begins with the preliminaries in Chapter 2 that introduce about preliminaries such as generalized coordinate, Euler -Lagrange equations and a nonlinear optimal control
Chapter 3 presented how to construct the model for crane system to operate in 3-D with five motion: trolley and girder position, rope length and two sway angles
Chapter 4 considered the modeling that is constructed in Chapter 3 to construct a nonlinear optimal control for a 3-D overhead crane system to ensure three objectives: (1) Driving the trolley from initial to desired position as fast as possible (2) The vibration of the payload had to suppress as fast as possible (3) Optimal trajectory of three motion in X-, Y-, Z-directions to minimize time and control efforts of the system
Chapter 5 implemented simulations in Matlab software and experiments in the model in practice at Control and Automation Laboratory to compare with simulation results and conclude about the effectiveness of the controller
PRELIMINARIES
Generalized coordinate
Generalized coordinate is used to describe the structure of the system If the system has N particles, each particle needs three coordinates to describe its position in space Therefore, it needs 3N coordinates to describe N particles and it is said that the system has 3 degrees (DOFs) If coordinates are linked to each other by i formulas, the degree of the system is 3N-i Following [48, pp 12 -13], we obtain:
Generalized coordinate is defined as follows:
The generalized coordinates of the system is defined as follows:
( ) rr q,q , (2.2) where qand qare independent function, respectively
Generalized forces can be obtained from the computation of the virtual work, δW, of the applied forces
The virtual work of the forces, F, acting on the particle Pi (x,y,z) ( i =1, , n), is given by:
F r (2.3) where r i is defined as follows :
From Eq (2.3) and Eq (2.4), we have:
Closed-loop systems have the potential energy (V), the potential force (F) is calculate as follows: i i i i i
In closed-loop systems, generalized force is opposite in sign with partial derivative of potential energy of the systems follow as generalized coordinate.
Euler Lagrange equations
Considering the system that has N particles with n (DOFs) In general, the particle Pi is denote by vector r i , which is the function of generalized coordinate and time Following [48, pp 13 -17], we obtain:
( , , , , ) ( , ) i i q q q t n i t r r r q , (2.8) where q and t are independent variables, respectively Velocity of particle Pi is defined as follows:
( , ) i i t v v q,q , (2.9) where q( ,q q 1 2 , ,q n )is generalized velocity We have:
Besides, small displacement is defined as follows:
(2.11) where is used for small displacement Implement partial derivative v i of particle Pi followed generalized velocityq k , from Eq (2.10), we have:
Substituting r i from Eq (2.10), we have:
Compared Eq (2.13) and Eq (2.14), we obtained: d d i i k k r r t q q
Applied D’Alembert principle for particle Pi of the system, which has N particles: i i 0
Summing virtual work of small displacementr i of particle Pi:
Applied D’Alembert principle, we assume that the applied force is inertia force
At equilibrium position, virtual work of the applied forces is equal zero, we prove this problem as below:
, (2.18) where Q i is the generalized force
Combine Eq (2.12) and Eq (2.15), we obtained:
, (2.21) where kinetic energy of the system 2
Because q j is independent with each other, formula in (2.21) is zero, we obtained: d d j j j
The potential energy can be rewritten:
where V is the potential energy of the system Eq (2.23) becomes: d d j j j
Because the potential energy and the velocity are independent, we have:
We can rewritten as follows: d 0 ( 1, 2, , ), d j j
With non-potential system, applied force Q j includes substituting force and non-substituting force: j j 0 j
The Lagrange equation is written as follows: d 1, 2, , d j j j
Nonlinear optimal control theory
We considered the nonlinear system dynamics [49, pp 184-325] that are described by state equation as follows:
( )t a( ( ), ( ), )t t t x x u , (2.29) where ( )x t ∈ n is the states of the system, ( )u t ∈ m is the control law and cost function J is defined as follows:
19 Optimal control law is the law that we had to find out u * (t) during defined time [t 0 ;T] to J obtain minimum cost Final state function is defined as follows:
So that the performance can be expressed as:
Because x(t o ), t o are fixed, the optimization does not affect the ( ( ), )h x t o t o term, so we need consider only the function:
Clean this up by defining the Hamiltonian:
To proceed, we note that integrating by parts we get:
So we can rewrite the variation as: x u
So necessary conditions for J( )u 0 are that for t[0, ]T
Hamilton function:H( , , )x u t L( , , )x u t λ T f( , , )x u t (2.41) Final state constraint: ( ( ), )x T T 0 (2.42)
CHAPTER 3: PROBLEM FORMULATION FOR A 3-D OVERHEAD
In this chapter, I will represent how to construct dynamic model for 3-D overhead crane system by Euler-Lagrange method This system includes five generalized degree (DOFs): trolley and girder position, rope length and two swing angles The system uses three-actuators to lead three motions in X -, Y -directions and motion of rope length (in Z-direction)
The 3-D overhead crane system has three motions: (1) The trolley travels in horizontal beam, which is called a bridge (Y-direction) is driven by the first AC servo motor, (2) This bridge moves on another horizontal beam (X-direction), are driven by the second AC servo motor A hoisting system, which is assembled to the trolley is driven by the third AC servo motor to hoist up or hoist down the payload in Z- direction When the trolley motions, the swing angle of the payload fluctuates around equilibrium with angle ( )t (this angle has two components ( ) and ( )t t )
Fig 3.1 Components of the 3-D overhead crane system
Fig 3.2 Sway angle of the 3-D overhead crane system
Table 3.1: Parameters of the 3-D overhead crane system
Symbol Unit Description m tr kg Mass of trolley m g kg Total mass of girders m p kg Mass of payload ( ) x t m Trolley position in X-direction
( ) t deg Swing angle of the payload in OXYZ Angle ( ) t has two components ( ) t and ( ) t as shows in Fig
F N Control force applied to the trolley
F l N Control force for the hoisting f cx N Friction force between the girders in X-direction f cy N Friction force between the trolley and girder in Y- direction
23 As shown in Fig 3.1, the girder, the trolley, and the payload position vectors are given as follows:
( ) ( ) 0 ( ) 0 0 ( ) ( )sin ( ) cos ( ) ( ) ( )sin ( ) ( ) cos ( )sin ( ) tr g p x t y t x t x t l t t t y t l t t l t t t
(3.1) where x(t) and y(t) are the trolley position in X- and Y-directions, respectively
Let q(t) 5 be the generalized coordinate vector defined as follows: q ( ) t x t ( ) y t ( ) l t ( ) ( ) t ( ) t T (3.2) The forces applied to the system are given by:
The friction forces in the X-direction and Y-direction respective given as follows: f t cx ( ) c x t x ( ), f t cy ( ) c y t y ( ), (3.4) where c x and c y are the viscous friction coefficients in X- and Y-directions, respectively
The total kinetic energy K and the potential energy P of the crane system are given as:
( ) ( ) cos ( ) ( ) ( )sin ( )sin ( ) ( ) ( ) ( ) p p p p p tr g p tr p p p p
(3.5) Pm gl t p ( ) 1 cos ( ) cos ( ) t t (3.6) Using Euler-Lagrange equation [50, pp 423-435], the equations of motion are derived as follows:
Substituting Eqs (3.5) and (3.6), the Eq (3.7) can be written:
( ) ( ) sin ( ) cos ( ) ( ) ( ) cos ( ) cos ( ) ( ) ( )sin ( )sin ( ) ( 2 cos ( ) cos ( ) ( ) ( ) 2 sin ( )sin ( ) ( ) ( ) 2 ( ) cos ( )sin ( ) ( ) ( ) ( )sin ( ) cos ( tr g p p p p p p p p m m m x t m t t l t m l t t t t m l t t t m t t l t t m t t l t t m l t t t t t m l t t t
( )sin ( ) ( ) sin ( ) cos ( ) ( ) sin ( ) ( ) ( ) ( ) cos ( ) ( )
2 cos ( ) cos ( ) ( ) cos ( ) cos ( ) ( ) ( ) cos ( ) ( ) 2 ( ) cos ( )sin ( ) ( ) ( )
2 ( ) cos ( ) ( ) ( ) ( )sin ( ) cos ( ) 0 ( )sin ( )sin ( ) ( ) ( ) cos ( ) ( p p l p p p p p
(3.8) The dynamic equations (3.8), we can be rewritten as:
M(q)q + C(q,q)q + G(q) = u, (3.9) whereM(q) 5x5 is inertia matrix of the crane system and C(q) 5x5 represent the centripetal Coriolis and G(q) 5 is the gravity term, defined by:
0 0 cos ( ) cos ( ) ( ) sin ( )sin ( ) ( ) cos ( ) cos ( ) ( ) ( )sin ( ) cos ( ) ( ) ( ) cos ( )sin ( ) ( ) sin ( )sin ( ) ( ) ( )sin ( ) cos ( ) ( ) ( ) cos ( ) p p p p p p p p
26 ( ) 0 0 m g p (1 cos ( ) cos ( ))t t m gl p sin ( ) cos ( )t t m gl p cos ( )sin ( )t t T
Based on the structure of M(q) andC(q, q) given by Eq (3.9), it should be noted that the following skew-symmetric relationship is satisfied
2 (3.10) where M(q) can be upper and lower bounded by the following inequality: n 1 2 T Μ(q) n 2 2 , 5 (3.11) where n 1 and n 2 ∈ are positive bounded constants
In derivation of the dynamic model of the crane, the following assumptions are made:
(i) The payload and the trolley are connected by a mass-rigid link
(ii) The mass of the trolley and the length of the connecting rod are known
(iii) The friction of forces f cx and f cy are assumed known, where the viscous friction coefficients c x and c y
In this chapter, I presented how to construct the motion equations of the 3-D overhead crane system The formulas will be used to construct the nonlinear optimal control for the 3-D overhead crane system in next Chapter
DESIGN A NONLINEAR OPTIMAL CONTROL
In this chapter, I will represent how to design a nonlinear optimal control for the 3-D overhead crane system to satisfy three objectives:
Driving the crane system to desired positions
Suppressing the vibrations of the payload
Optimal trajectories of the system to minimize time and control efforts of the system
A control law is proposed to drive the crane to the desired position and to suppress the sway angles simultaneously For convenience, a new generalized coordinate vector is defined as follows: q T (q q p , s ), (4.1) where
The equations of motion of the overhead crane (3.19) can be rewritten as follows:
( ) 0 pp ps p pp ps p p p sp ss s sp ss s s
0 0 0 0 x pp ps pp y ps sp ss sp ss m m m m c C C C m m m c C C m m m C C m m C m m m C
To achieve the control objectives, we given desired signals q q q d , d , d (which are assumed to be bounded), the control law is designed to ensure the asymptotical convergence of q to q d
28 The error signals are defined as:
, where x y l d , d , , , d d d are defined trajectories of x y l, , , , , respectively
To obtain three-objectives of the system, we had to minimize the cost function which is displayed as follows:
k u , k e ∈ 5x5 are the real symmetric positive definite weighting matrix ofe, u
The initial and final conditions are given as follows
The limitation of the actuators are derived as follow:
, where τ , τ , τ x y l are the moment of three AC motors in X-,Y-and Z-directions
29 From Eq (3.9), we define state equation of the system as follows:
With λ( )t λ 1( )t λ 2( )t T , based on Eq (2.41) we define the Hamiltonian:
From Eq (2.40), the costate equations are defined as follows:
From Eq (2.43), we can derived the control law as follows:
Solve Eqs (4.7)-(4.8), we can obtain: λ 1 ( )t t.k q e ( d q 1 )c 1 (4.9) d 2 1 ( ) 1 2 ( ) t t dt
Eq (4.10) can rewritten as follows:
32 d = - + e + e + e + e + e dt d = - + e + e + e + e + e dt d = - + e + e + e + e l + e dt d = - + e + e + e + e + e dt d = - + e dt
C q q , where c x , c y are the viscous friction coefficients in X-,Y-directions
We derive that the initial of the λ λ 1 , 2 as follows:
λ λ (4.19) Now impose the boundary conditions:
Eq (4.20) can rewritten as follows:
34 The Eq (4.21) demonstrated that the boundary conditions are always truth with the parameters of the system change because each element of the Eq (4.20) is equal zero at t=T
In closing, we designed the nonlinear optimal control for 3-D overhead crane system
NUMERICAL SIMULATIONS AND EXPERIMENTS
Numerical simulations
To verify the feasibility of the control algorithm, we perform simulation the proposed nonlinear optimal control algorithm in Matlab software with the following parameters as follows:
The initial state of the system is chosen as:
The desired position of the system is chosen as:
To derive the control gain k e , we implement as follows: when k e11 is increased, the velocity of the trolley in X-direction will increase and the trolley will obtain the desired position fast, but the sway angles are also increased We do not hope it occurs, because the vibrations of the system are eliminated by the combination of the motion of the trolley in X-, Y- directions, matrix k e with k e22 is increasing to the value to decrease in the vibrations of the two sway angles of the system to minimum value and the trolley reaches the desired position in Y-direction fast By simulations, the optimal nonlinear control law is turned until a best performance is achieved, we derive the optimal control gain k e as shown in Eq (5.1)
The energy of the system is restricted by the constraints of the AC servo motor Therefore, to minimize the control effort of the system and had to ensure the trolley
36 and payload reach to the desired position as fast as possible We had to choose the suitable k u By simulations, the optimal nonlinear control law is turned until a best performance is achieved, we derive the optimal control gain k u as shown in Eq (5.2)
Fig 5.1 Simulation results with the mass of the payload changes
Fig 5.1 compares the simulation results with the mass of the payload changes
(m p =1kg, m=1.5kg, m=3kg, m=4kg and m=5.5kg) First, the payload is hoisted to desired position, after that the trolley reached to Y-desired position Finally, the trolley reached to X-desired position Because the length of the cable affects to vibration of the payload during the motion, the length of the cable is the shortest, vibration of the payload is minimum, the trolley reached to desire position x d = 0.8m after 5s duration, y d = 0.5m after 2.9s, the rope will hoist up the payload from initial height l(0)=0.8m to zero position after t=1.6s duration will all payloads The position of the trolley and payload at this time are different but the difference is too small
Therefore, the responses of the trolley in X-, Y-, Z-directions are coincident During trolley is moving, the maximum vibration amplitudes of the sway angles are
|θ max(t)|=2.85deg, | ϕ max (t)|= 3.4deg After t=7.1s, the vibrations almost eliminated
Fig 5.2 The control efforts of the system in simulation
Fig 5.2 shown that the values of the control efforts after deriving the optimal control gain k u The maximum control efforts in X-, Y-, Z-directions are 6.5N, 6.5N and 3.2 N, respectively.
Introducing experiment system
In practice, an overhead crane system at the port is very big, include three motions: motion of the trolley in Y-direction, motion of the trolley and the girder in
X-direction and hoist up of the payload In this thesis, I will do experiments on model that simulates this system in port
In this model, a trolley travels in horizontal beam, which is called a bridge
This bridge moves on another horizontal beam A hoisting motor which is assembled to the trolley to hoist down and hoist up the payload as shown in Fig 5.4 The payload is link directly to the cable
The experiment model is described in Fig 5.3, the motion of the trolley in X- and Y-directions are driven by 2 AC motors servo Mitsubishi HC-MF-43(400W)
(Driver AC MR-J2-40A and hoisting cable is driven by AC motor servo Mitsubishi HC-MFS-23(200W) (Driver AC MR-J2S-20A) Two high resolution encoders Sony Magnescale RE90C-2048C are installed to measure two sway angles , during the
3-D overhead crane operating (as shown in Fig 5.5) The central controller includes one PC (Core i5, Ram 4GB), which is installed a card SMC-4DF-PCI of CONTEC company to receive the data from encoders (Encoders of three AC motors in X-,Y-,
Z-directions and two encoders Sony) and controls three AC motors
AC servo of X direction (HC-MF-43)
AC servo of hoisting motion
Controller (CPU, Ram 4GB, card SMC-4DF-PCI)
AC servo of Y direction (HC-MF-43)
39 The control program runs in Windows 7 environment with C sharp software (Visual studio ultimate version 2010) In the control program, the sampling time was set 0.01 second After every the sampling time, the PC will receive the signals from five encoders to derive the positions and velocities of the trolley, payload and the vibration of the two sway angles, C sharp program will calculate the value of λ λ 1 , 2 Therefore, we can derive the control law u p and the PC will transmit these values to the AC motors in X-, Y-, Z-directions to drive the bridge, trolley, payload This process is repeat until the trolley and the payload reach to the desired positions
Fig 5.5 Measurements of the sway angles using high resolution encoders
To derive the sway angles of the payload, I installed measurements of the sway angles system below the trolley, which is shown in Fig 5.5 In this system, the cable is installed through two U-bars, which are orthogonal The centroid of two U-bar will cross together at the point, where is also the output of the cylinder The origin is the cross-point of two U bars
In the system, sway angles need to be measure are A A A 1 0 3 and A A A 2 0 3 , which are shown simply ( )t and( )t With model is designed as shown in Fig 5.6, we will measure ( )t andA A A 1 0 4 '( ) t ( ) t is derived as follows:
arctan(cos * tan ') (5.1)The measurements of the swayangles system are linked with the trolley The variation of the payload is measured by two encoders of Sony company (EC90-2048 pulses/circle), which is installed in U-bar axis The positions and velocities of the trolley in X- and Y-directions and Z-direction are determined by the feedback signals from encoders of the AC motors (8192 pulses/ circle for AC servo HC-MF-43 and 131072 pulses/circle for AC servo HC-MFS-23)
Fig 5.7 Control interface in C# environment
The signals of the encoders will be feedback to PC through card PCI These signals will be process and calculate the applied forces for three AC motors in each direction The results will transmit to Card PCI After that, card PCI will generated the suitable pulses for drivers to control three AC motors Control interface is implemented in PC and this progress is implemented by programing (C# Visual studio 2010) with the library, which is provided by CONTEC company In these progresses, the feedback signals will be stored in text files to draw graph after program finished
Fig 5.8 The mass of the payload from 1kg to 5.5kg.
Experiments
To illustrate the control performance, we perform experiments the proposed nonlinear optimal control in Chapter 4 with the following parameters:
The initial state of the system is chosen as:
The desired position of the system is chosen as:
Fig 5.9 Experimental comparison between simulation and experiment, when m p =1kg
Fig 5.9 compares the performance between simulation result and experiment result with the mass of the payload is m p =1kg As shown in Fig 5.9, first the rope will hoist up the payload from initial height l(0)=0.8m to zero position after t=1.6s (simulation) and 1.7s (experiment) duration After that, the trolley is driven to desired position in Y-direction within 2.9s (simulation) and 3.7s (experiment) duration, is driven to desired position in X-direction within 5s (simulation) and 6.4s (experiment) duration During the trolley is moving, the maximum vibration amplitudes of the sway angles are |θ max(t)| = 2.840deg (simulation) and |θ max(t)|=1.756deg (experiment),
|ϕ max(t)|=2.790deg (simulation) and |ϕ max(t)|=3.867deg (experiment) After ts, the vibrations almost eliminated
Fig 5.10 Experimental comparison between simulation and experiment, when m p =1.5kg
Fig 5.10 compares the performance between simulation result and experiment result with the mass of the payload is m p =1.5kg As shown in Fig 5.10, first the rope will hoist up the payload from initial height l(0)=0.8m to zero position after t =1.6s (simulation) and 1.7s (experiment) duration After that, the trolley is driven to desired position in Y-direction within 2.9s (simulation) and 3.7s (experiment) duration, is driven to desired position in X-direction within 5s (simulation) and 6.6s (experiment) duration During the trolley is moving, the maximum vibration amplitudes of the sway angles are |θ max(t)|=2.840deg (simulation) and |θ max(t)|=1.934deg (experiment),
|ϕ max(t)|=2.790deg (simulation) and |ϕ max(t)|=3.867deg (experiment) After t 3s, the vibrations almost eliminated
Fig 5.11 Experimental comparison between simulation and experiment, when m p =3kg
Fig 5.11 compares the performance between simulation result and experiment result with the mass of the payload is m p =3kg As shown in Fig 5.11, first the rope will hoist up the payload from initial height l(0)=0.8m to zero position after t =1.6s (simulation) and 1.7s (experiment) duration After that, the trolley is driven to desired position in Y-direction within 2.9s (simulation) and 3.8s (experiment) duration, is driven to desired position in X-direction within 5s (simulation) and 6.6s (experiment) duration During the trolley is moving, the maximum vibration amplitudes of the sway angles are |θ max(t)|=2.840deg (simulation) and |θ max(t)|=1.934deg (experiment),
|ϕ max(t)|=2.790deg (simulation) and |ϕ max(t)|=3.164deg (experiment) After t.5s, the vibrations almost eliminated
Fig 5.12 Experimental comparison between simulation and experiment, when m p =4kg
Fig 5.12 compares the performance between simulation result and experiment result with the mass of the payload is m p =4kg As shown in Fig 5.12, first the rope will hoist up the payload from initial height l(0)=0.8m to zero position after t =1.6s (simulation) and 1.7s (experiment) duration After that, the trolley is driven to desired position in Y-direction within 2.9s (simulation) and 3.8s (experiment) duration, is driven to desired position in X-direction within 5s (simulation) and 6.6s (experiment) duration During the trolley is moving, the maximum vibration amplitudes of the sway angles are |θ max(t)|=2.838deg (simulation) and |θ max(t)|=1.934deg (experiment),
|ϕ max(t)|=2.790deg (simulation) and |ϕ max(t)|=3.164deg (experiment) After t.2s, the vibrations almost eliminated
Fig 5.13 Experimental comparison between simulation and experiment, when m p =5.5kg
Fig 5.13 compares the performance between simulation result and experiment result with the mass of the payload is m p =1kg As shown in Fig 5.13, first the rope will hoist up the payload from initial height l(0)=0.8m to zero position after t =1.6s (simulation) and 1.7s (experiment) duration After that, the trolley is driven to desired position in Y-direction within 2.9s (simulation) and 3.7s (experiment) duration, is driven to desired position in X-direction within 5s (simulation) and 6.6s (experiment) duration During the trolley is moving, the maximum vibration amplitudes of the sway angles are |θ max(t)|=2.838deg (simulation) and |θ max(t)|=2.109deg (experiment),
|ϕ max(t)|=2.790deg (simulation) and |ϕ max(t)|= 3.691deg (experiment) After t.8s, the vibrations almost eliminated
Table 5.1 Summarizing the simulation and experiment results with the mass of the payload changes
The mass of the payload (kg)
Time(s) Maximum amplitudes of the sway angles (deg) To x desire
Fig 5.14 Position of the trolley in X-direction with the mass of the payload changes
Fig 5.15 Position of the trolley in Y-direction with the mass of the payload changes
Fig 5.16 Position of the payload in Z-direction with the mass of the payload changes
Fig 5.17 Sway angle theta (θ) with the mass of the payload changes
Fig 5.18.Sway angle phi (ϕ) with the mass of the payload changes
Figs 5.14 - 5.18 compares the performance between the best results in simulation with payload mass m p =1kg and experiment results with the mass of the payload changes The trolley is driven to desired position in X-direction within 6.6s duration, in Y-direction within 3.8s and the payload is hoist up within 1.7s with both payloads The difference among the payloads is the time to eliminate the vibration of two sways angle Combine with table 5.1, we can see that the time to eliminate completely the vibration increased when the mass of payload increased (10.2s with m p =1kg, 12.3s with m p =1.5kg, 16.5s with m p =3kg, 17.2s with m p =4kg, 18.8s with m p =5.5kg) It should be noted that the author uses ordinary differential equations (ODE) model to modeling the system and considered the payload as lumped mass model, the control law does not consider the vibration of rope and the inertial payload is neglected.
Concluding remarks
In this chapter, I displayed the experiment results with the mass of the payload changes to verify the effectiveness of the nonlinear optimal control law proposed in
CONCLUSIONS
Figs 5.9-5.18 shown that the vibrations of the payload obtain the maximum amplitudes at the initial period because the acceleration of the AC motors The responses of the system and the time to eliminate completely the vibrations in experiments are not faster than the results in simulations because of several reasons such as the vibrations of the system in acceleration process and deceleration process, the stiffness of the system and the vibration of the rope, inertial payload, etc Because the simulation results is approximated with the experiment results, we can conclude that the controller can operate in the condition that the mass of the payload changes
In this thesis, a nonlinear model of a 3-D overhead crane system was developed, where the Euler-Lagrange equations was employed Based on this model, a nonlinear optimal control is designed to guarantee three objectives: (1) Driving trolley from initial to desired position (2) Suppressing the sway angle of the payload during system operating (3) Optimizing control efforts applied to the crane system
The effectiveness of the proposed control algorithm was verified by numerical simulations and experiment results The proposed control design are considered as a viable solution for overhead cranes in practice
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