TÊN ĐỀ TÀI: - Tiếng Việt: Điều khiển cầu trục hoạt động trong không gian 3 chiều sử dụng adaptive command shaping - Tiếng Anh: Adaptive command shaping control of a 3d overhead crane I
INTRODUCTION
General review of cranes and some factual applications
Cranes are equipment which are used to lifting and transfer heavy weight objects in working fields, stores, seaport, etc Nowadays, the demands for lifting, loading and unloading of goods, materials, equipment whose loads exceed human capacity are very large, so the cranes are now widely used
According to the working condition, cranes are built in many different types like mobile cranes, tower cranes, floating cranes, overhead cranes, etc
Mobile cranes can be moved within a site or even in public traffic, giving them great flexibility They can be either mobile wheeled, truck-mounted, track-mounted (Figure 1.1), etc This is the most standard and versatile type of crane used in reality today
(Images from http://www.palfinger-sany.com/en/psv/products/truck-crane/stc750 )
(Images from https://denverinfill.com/blog/2016/12/tower-crane-census-winter-
2016.html ) The tower crane is a modern form of a balance crane When fixed to the ground, tower cranes will often give the best combination of height and lifting capacity and are also used when constructing tall buildings (Figure 1.2) They are used mostly in the construction of high buildings
Floating cranes are mostly used in offshore construction or purpose They are specialized in the lifting of heavy loads like bridge sections, offshore-rig construction, etc They are also called crane ship, crane vessel They can also be used to load or unload ships or lift sunken ships from the water (Figure 1.3)
(Images from https://www.breakbulk.com/events/breakbulk-china-
The overhead cranes (also referred to as bridge cranes) are cranes with a lifting system like hoist or open winch, which can move along the rails on fixed frame (Figure 1.4) They are of very robust construction with a high level of standardization, making them very modular and adaptable to any need, high reliability components and available for a wide range of applications They are usually used in stores, factories, seaports, etc In seaports, because they are used to move the containers, they also called container crane
Cranes often use steel cable to lift up the cargos Consequently, when a crane with heavy cargo move, the cargo oscillates and creates a sway angle with vertical line In reality, this kind of oscillation is harmful for equipment or even man around
Moreover, time is wasted for waiting this oscillation stops Thus, the sway angle has to be minimized as small as possible In conclusion, control a crane in reality need to satisfy two basic goals: (1) Reach to the desired point as soon as possible; (2) Minimize the sway angle of the cargo lifted
This thesis focus on studying 3-D overhead crane with a particular method to control it as follows: (i)Develop a dynamic model of 3-D overhead crane (ii)Design an adaptive command shaping control for controlling 3-D overhead crane
(iii)Examining the sway angle in comparing with bared shaping method
(Images from http://www.crane-tec.com)
Operation of overhead crane
Overhead cranes are very adaptive with working condition of small space like
Chapter 1: Introduction factory, store, etc Moreover, with simple structure, it can be easily install to many working fields These make overhead cranes are really popular in modern industry and reality life
Figure 1.5 describes a basic 3-D overhead crane It consists a lifting-lowering system which is assembled on a trolley (4) to lift the payload on the hook (1) by cable (3) Hook (1) is connected with cable (3) by a set of pulleys (2) Trolley can move along the rails (5), which is called bridge or girder Bridge (5) can moves along the based frame structure (6) The trolley and girder movements are perpendicular
Hence, the hook/payload can be move in 3-D space There are three actuators acting on the girder, the trolley movements and the pay out, pay in the cable
Table 1.1 Basic structure of 3-D Overhead Crane
(4): Lifting-lowering system/Trolley (5): Rails/bridge/Girder (6): Based frame structure
(Image from http://www.konecranesusa.com/underhung-overhead-cranes) In modern industry, overhead cranes are widely used So that, they need to be increase productivity and the reliability, and also the cheaper cost The speed of
Chapter 1: Introduction process decides the productivity and the economic efficiency However, increasing speed affect on reliability of the crane Moreover, it leads to increasing the payload’s oscillation and takes more waste time Therefore, to archive both goals, increasing speed of the crane and minimizing the sway angle, automatic control method has to be applied in the working process.
Modeling of an overhead crane
There are two methods for modeling overhead cranes, which have been using:
Ordinary differential equations (ODEs) and partial differential equations (PDEs) [1]
Figure 1.6 present a diagram of an ODEs model In ODEs model, payload is assumed as lumped mass model, elasticity of the cable is also ignored d
This model is commonly used in researching overhead cranes, when mass of the payload is extremely bigger than mass of the cable In other words, mass of the cable
Chapter 1: Introduction is neglected A number of researches have been recorded in using ODEs model to develop, typically as: Elling et al [2], Chin et al [3], Abdel-Rahman et al [4], Oguamanam et al [5], etc
In PDEs system (as shown in Figure 1.7), the overhead cranes are considered with following hypothesis:
The cable is completely flexible and non-stretching
Transversal and angular displacements are small
The acceleration of the mass of the payload is negligible with respect to the gravitational acceleration g
Thus, the effects of the elasticity and mass of cable has to be involved The dynamic equations of cable elements are constructed by wave equations Until now, because of its complexity, PDE models only exist in 2-D like D’Andréa-Novel et al
[6] which applied feedback stabilization of a hybrid PDE-ODE system to an overhead crane D’Andréa-Novel et al [7] used PDEs to prove the exponential stabilization of an overhead crane with flexible cable, Abdel-Rahman et al [8] recommend appropriate models and control for various crane application and suggest direction for further work He et al [9] used PDEs to construct an adaptive control of a flexible crane system with the boundary output constraint
PDEs model is more effective when considering to the mass of the cable
However, applying PDEs model is more complicated In addition, theories for PDEs model has not been enough yet Consequently, using PDEs has to face more obstacle than ODEs model
In this research, an ODEs model was employed to construct a dynamic model of a 3-D overhead crane based on Euler-Lagrange equation with generalized coordinates After that, this model is linearization to compute the natural frequency, damping ratio of a 3-D overhead crane as well as applying adaptive input command shaping control to 3-D overhead crane
Control of 3-D overhead cranes
It is recognizable that 3-D overhead crane has five degrees of freedom: trolley movement, bridge movement, lifting-lowering movement, and two sway angles of the payload It nevertheless has three actuators to perform movement of the trolley, the bridge and hoist motion Thus, overhead cranes are underactuated systems, with the sway angles are driven indirectly
As a consequence of commonly used, many control methods for minimizing sway angles of overhead cranes have been researched, analyzed and applied in reality
They can be categorized into three main groups:
Open-loop control method: based on the theory of oscillation synthesis and the particular specifications of the overhead crane, reverse-phase oscillations are created appropriately to manipulate the cranes Consequently, the sway angles of crane can be eliminated when it reaches to the desired position Singer et al [10] presented a ZV input shaper to control gantry crane followed this method Singhose et al [11] examined the effects on hoisting on the input shaping control of the gantry cranes,
Chapter 1: Introduction considered ZV shaper and ZVD shaper and shows how effectively this method is
Nguyen et al [12], Liu and Cheng [13] also used this method to plan the suitable trajectory for eliminating the sway angles Tho and Nguyen [14] applied input shaping in control 2-D overhead crane
The advantages of this method are uncomplicated, not difficult to design, applied in reality Because it does not need the feedback signals of the sway angles
And thus, the equipment of this system is simplified On the contrary, the response of this control method is so sensitive with the variation in natural frequency of the crane In addition, if there are unpredictable impacts make change to the sway angle of the crane, this controller can not eliminate
Closed-loop control methods: all feedback signals are used to compute the control signals for the overhead cranes Many closed-loop controllers have been reached and experimented Nguyen [15], Kim [16] applied state feedback to control sway angles of the crane Cheng and Chen [17], Park et al [18] made it simplified by using state feedback linearization method Butler et al [19], Qian et al [20] accessed another method to control non-linear model of overhead crane by using adaptive method Bartolini et al [21] designed a sliding mode controller for controlling 2-D overhead crane model Optimal controller was also used by Nguyen and Vu [22]
Although there are many different types of closed-loop controller were designed, they all need feedback sway angles tracked Hence, this method requires more devices to monitor the sway angles It increases the cost of control system But the advantage is they can adapt quite well when disturbance presents because all the errors are considered to compute the control signal
Furthermore, in company with the development intelligent control, Lee and Cho [23], Dragan et al [24] used fuzzy logic theory in controlling overhead crane
This controller is easy to design and adaptive to disturbance But its effectiveness depends on the experience of designers It also need full feedback parameters as same as closed-loop controllers In addition, as other fuzzy controllers, it can not be demonstrated clearly the stable of this controller
This thesis will focus on the open-loop control method, which depends on
Chapter 1: Introduction the natural frequency of the system This method is only satisfied when the natural frequency of the system unchanged So that the controller of this method need to be improved when the natural frequency change.
Objectives of the thesis
This thesis is focus on analyzing and improving the open-loop controller for 3- D overhead crane As mentioned above, open-loop controller is contingent on natural oscillation specifications of the system, especially natural frequency of the system
Feedback signals of the sway angles are not necessary for designing this controller
So that, it is easy to design However, if the natural frequency changes through the operating process, responses of the system are not satisfied the desired goals Under these circumstances, adaptive input shaping is designed to adapt with the change of system natural frequency
Dynamic model of 3-D overhead crane is constructed by using Euler-Lagrange equations The motions of this system consist:
Translational motions of trolley, bridge and hoisting of the payload These motions are driven by three actuators which controlled by appropriate
Oscillation of the payload, which is composed of two sway angles θ, φ
Based on this dynamic model, natural frequency of overhead crane is determined It is analyzed to exam what happen if it changes through out the control process
1.5.2.Applied input shaping in overhead crane control
Command shaping is described and simulation LQR controller is employed to track the desired input after using command shaping The responses of this method are compared with the responses of the same system without using command shaping input to experience the effectiveness of shaping method Furthermore, response of system according as natural frequency changes are considered to survey the effectiveness of this controller
1.5.3.Improve input shaping by adaptive input shaping
In real operating process, the height of the payload can be required changing for
Chapter 1: Introduction particular purposes In these cases, the natural frequency of the system also changes
As a result, the command shaping input with initial natural frequency is no longer accordant with updated system Therefore, adaptive command shaping input method is employed This controller improves the response of the system even if the natural frequency changes.
Organization of the thesis
After the introduction of this thesis which presented in Chapter 1, Chapter 2 describes two basic fundamental theories: (i) Euler-Lagrange equation with generalized coordinate; (ii) input command shaping method in eliminating oscillation of second-order system
Based on the Euler-Lagrange equation, dynamic ODEs model of overhead crane which is constructed in Chapter 3 This model is then linearized to computer the oscillation characteristics of overhead crane, which are used to apply input command shaping method in Chapter 4
Chapter 4 presents how to apply input command shaping method in eliminating oscillation of second-order system to overhead controller Effectiveness of input command shaping is then simulated by Matlab ® software Its response is compare to non-input shaper and a nonlinear optimal controller of Nguyen and Nguyen [25] to clarify how effective it is Even though input shaper can work well, but when characteristic oscillations change, the sway angles increase To overcome this issue, adaptive input shaping method is introduced in Chapter 5 and also simulated by
Matlab ® software to experience its efficacy
FUNDAMENTAL THEORY
Generalized coordinate [25]
Generalized coordinates are commonly used to provide the minimum number of independent coordinates that define the configuration of a system, which simplifies the formulation of Euler-Lagrange's equations of motion For a system has N particles, each position vector of a particle needs three coordinates to describe its position in Cartesian coordinates:
If this system has C holonomic constraint:
Then the number of independent coordinates is n = 3N − C It is said that this system has n degrees of freedom (DOFs) This number is used as the parameter quantity of generalized coordinate Let q is the generalized coordinate, then: n
The position vector r i of particle i th is a function of all the r i generalized coordinates and time: r = (q(t),t) i (2.4)
The corresponding time derivatives of q k are called the generalized velocities:
The velocity vector v k is the total derivative of r k with respect to time:
Because of that, with holonomic constraints, v k is generally depends on the generalized velocities and generalized coordinates
Generalized forces can be obtained from the computation of the virtual work,
Chapter 2: Fundamental theories δW, of the applied forces [27] Let F i , i=1: N are the applied forces, the virtual work of F i , acting on the particles P i , i=1: N, is given by:
Where δr i is the virtual displacement of the particle Pi Inferring from (2.4):
Then the virtual work for the system of particles (2.7) becomes:
So that, the virtual work of a system of particles can be written in the form:
And Q j , j=1: N are the generalized forces associated with the generalized coordinates
In 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 [27]
Now consider a system of particles: j=1: N p Let r (k) denote the position of k (th)
Chapter 2: Fundamental theories particle relative to an inertial reference frame Let f (k) be the force acting on the mas m k of the k (th) particle From Newton’s Second Law:
Let δr (k) be the virtual displacement that is compatible with all constraints on the motion of the k th particle Then:
Let divide f (k) into two elements:
, 1, , k k k f F R k N p (2.17) where R (k) , k=1:N P are the forces due to the constraints, and F (k) , k=1:N P are other forces acting on the k (th) , k=1:N P particles Then, because the constraint forces R (k) do not bring out the motion of system, so R (k) δr (k) =0, equation (2.16) becomes:
(2.18) for all δr (k) For holonomic dynamical system with n DOFs, all displacements r (k) can be expressed in terms of n independent generalized coordinates, follow (2.3) and (2.4):
From (2.7), (2.11), the virtual work of this system is:
Interchange the order of summation (2.22):
In addition, the kinetic energy of the system is:
Take partial derivative of (2.24) by q j :
The velocity of a particle depends on ( , , ) q q t , or:
Differentiation (2.26) with respect to time t:
On the other hand, take differentiation of (2.6) follow t:
Differentiation (2.30) with respect to time t:
Because δq j in (2.33) are independent, so that: j 0, j j d T T
Equation (2.34) are Euler-Lagrange’s equations for systems with holonomic constrains, whether or not the forces are conservative If all forces are conservative, then Q j V q j , then (2.34) becomes:
Equation (2.37) are Euler-Lagrange’s equations for conservative systems with holonomic constrains With non-potential system, applied force Q i includes substituting force and non-substituting force: j j 0 j
The Euler-Lagrange equation is written as follows:
Input shaping (IS) method
As a first step to understanding how to generate commands that move systems
Chapter 2: Fundamental theories without vibration, it is helpful to start with the simplest such command Singh and Singhose [29] explained this concept so clearly Firstly, giving the system an impulse will cause it to move and vibrate also However, if we apply a second impulse to the system, we can cancel the vibration induced by the first impulse when the second impulse has opposite phase and appropriate amplitude Figure 2.1 [29] was used to described this concept
Figure 2.1 Diagram of input shaping concept
To calculate the exact amplitude and time location respectively of the impulses, a standard second order system is considered
Denote G is a second order system, ω and ξare natural frequency and damping ratio of G respectively The time response of G to an impulse, whose amplitude and time location are A i and t i , is given as:
Based on the superposition law for linear system, the response of system G to a series of n impulses is given as follows:
The magnitude of y Σ (t) in (2.43) is obtained as follow:
The vibration percentage of magnitude response between the impulse sequence and a unity impulse which is applied at t = 0 is given as:
Noted that the ratio V(ω,ξ) represents the vibration percentage between input shaping to that without input shaping The input command shaping impulses can be obtained by letting V(ω,ξ) =0, which yields in:
Without lost of generality, assume there has two impulses in the sequence and the first impulse is applied at t = 0 (i.e t 1 = 0), hence:
The minimum time of t 2 is chosen as:
Real systems cannot be moved around with impulses, so we need to convert the properties of the impulse sequence to a usable command This can be done by convolution product The impulse sequence is convolved with any desired command signal The result of convolution product is then used as the command to the system
If the impulse sequence causes no vibration, then the result of “convolution product” will also cause no vibration
So that to preserve the response of original command, following constraint should be hold:
Solution of (2.52) and (2.53) is two impulses input which called ZV input The first impulse locates at the beginning of the process input, it is showed as follow:
If three impulses or four impulses are chosen for applying input command shaping method, the solutions are call ZVD and ZVDD input shaping respectively
This thesis focus on dealing with the response of 3-D crane when the natural frequencies change Hence, the input command shaper which is chosen is ZV shaping.
OVERHEAD CRANE DYNAMICS MODEL
Overhead crane modeling
Overhead crane model in Figure 1.5 can be described as principle diagram in Figure 3.1 This system as three actuators, which provide three forces: (1) F l is the force acting on the payload to lift it up; (2) F x acts on the girder/bridge to move the girder translationally along the X-axis, and also affect to the movement of trolley and the payload; (3) F y makes the trolley and also the payload translate along Y-axis
Figure 3.1 3-D Overhead crank principle diagram
While the trolley and the girder move, the payload oscillates back and forth the
Chapter 3: Overhead crane dynamic model equilibrium point Let γ is the angle from Z-axis to the cable direction To analyze this angle, divide it into two elements: θ and φ which are showed in Figure 3.2
Dynamic ODEs model of 3-D overhead crane system is constructed by Euler- Lagrange method Mass of the cable is ignored, there are three moving components: the girder, the trolley and the payload There positions are:
3 Payload position: r p [ x lsin cos y lsin lcos cos ]
Figure 3.2 Sway angles of the payload
To make the equations are easy to write, let letter “s” represents for “sin operation”, letter “c” represents for “cos operation”
Let q(t) ∈ R 5 be the generalized coordinate vector defined as follows: q(t)= [ x(t) y(t) (t) (t) l(t)] T (3.1) The forces applied to the system are given by:
Chapter 3: Overhead crane dynamic model
.x; x y bx bx f b f b 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:
Using Euler-Lagrange equation as follows: i i i d L L dt q q f
Where: L=K-P, for each generalized coordinate of q, 5 equations can be inferred as below:
Chapter 3: Overhead crane dynamic model
Chapter 3: Overhead crane dynamic model
These dynamic equations can be rewritten as:
Chapter 3: Overhead crane dynamic model
With five dynamic equations can be written as set of equations (3.11) below:
It can be inferred from (3.11) and (3.12) that:
Chapter 3: Overhead crane dynamic model
System linearizing and oscillation specifications
To applied input command shaping method, system must be linearized around equilibrium point of the payload, where θ, φ≈0, the force around the operating point is (0, 0, -m p g):
The equations (3.16) can be rewrite as 3.17
Now, the natural frequency and damping ratio of linearized 3-D crane system are compute by the set of equations (3.17) The payload oscillates around equilibrium point which is similar to the single pendulum To compute the natural frequency of the payload, damping of girder and trolley are ignored, assume the friction applied on the payload swing motions are àx, ày Equations 1 st , 2 nd , 3 rd , 4 th become (3.18)
Chapter 3: Overhead crane dynamic model
Inferring from 3 rd , 4 th of (3.18):
Replace (3.19) to 1 st , 2 nd of (3.18):
Natural frequencies ω nx, ω ny and damping ratios ξ nx , ξ ny can be inferred from equations (3.21) as below:
Chapter 3: Overhead crane dynamic model
The residual vibration frequencies of payload are indicated by:
At the end of this chapter, the natural frequencies and damping ratios of linearized 3-D crane are indicated in (3.22) These results are essential for designing a controller which uses input command shaper turning This method will be introduced in following chapter.
INPUT SHAPING METHOD
ZV Input shaping with 3-D overhead crane
Follow the results of (3.22), (3.23) the natural frequencies of this system depend on the length of the cable and the mass of components which can be change through the control process For checking the response of 3-D crane system, l and m p are kept unchanged in computing shaping impulses To applied the LQR controller, the linearized dynamic model is used to find the state space function form The state space function will be constructed in 10-variables for further use
Let u l mg u l , adding damping ratios à x , à y of trolley oscillation, 3-D dynamic overhead crane described in (3.17) is rewritten as follow:
Solve this set of equations with the variables:
Chapter 4: Input shaping method in 3-D overhead crane
This set of equations can be described in 10-variables state space function:
Chapter 4: Input shaping method in 3-D overhead crane
This section only cares for the effectiveness of input shaping method with 3-D overhead crane The natural frequency of the system has to be hold as a constant This leads to keep the length of the cable unchanging through the control process
To track the trajectory of 3-D crane, LQR method should be used However, the errors of the sway angles are not considered in this controller due to the desire of analyzing the effect of input shaping method
Figures 4.1 shows the control schematic of 3-D crane system using only LQR tracking controller and using Input Shaper without input shaping method Figure 4.2 presents the combination of LQR tracking controller and input command shaping method In this solution, the desire inputs are turned by input command shaper before computing the error input for the LQR tracking controller They are used for tracking
Chapter 4: Input shaping method in 3-D overhead crane the positions of the trolley and the gilder Noted that in both methods: l=const
(Gilder-trolley Position) 3-D Crane System
Figure 4.1 LQR Position Tracking Control Applied on 3-D Crane
Figure 4.2 Input Shaping applied on 3-D crane control
The LQR cost function is described in (4.6), with u(t) stands for control signals, x(t) for positions
These both methods are now simulated in Matlab ® for inspecting the differences As mention before, the purpose of this section is analyzing the effectiveness of input command shaping method, so that the matrix R will be chosen as (4.7) randomly in which the coefficients that involve with sway angles are ignored and let be zeroes
The Q matrix is chosen (4.8) In which, the coefficients in relative with sway
Chapter 4: Input shaping method in 3-D overhead crane angles are ignored, and set to be zero
The parameters of the 3-D overhead crane are used in simulation follow real system at Control and Automation Laboratory, Mechanical Department, Ho Chi Minh city university of Technology, Viet Nam They are measured and pointed as (4.9)
Figure 4.3 Positions Response of 3-D overhead crane on X-Axis
Chapter 4: Input shaping method in 3-D overhead crane
Figure 4.3 and Figure 4.4 present the position responses of this system in both cases: with and without input shaper Two dash-line are desire position and response of system without using input shaper Two others continuous-line are desire position after using input shaper and the response of the system applying LQR controller with this new reference
Figure 4.4 Positions Response of 3-D overhead crane on Y-Axis
In both cases, by LQR controller to track the positions, after 6-8 second system reach to the desire inputs As considering with the simulation result of nonlinear optimal con troll in Nguyen and Nguyen [25], the position results of these method are similar
Chapter 4: Input shaping method in 3-D overhead crane
Noted that, with the method of using LQR controller in Figure 4.1 and Figure 4.2, only position errors and derivatives of the position errors are computed, integrals of errors are ignored Hence, the overshoots of this controller can not be eliminated
Again, the purpose of this thesis is minimizing the sway angles, overshoots of LQR controller are also ignored If PID controller is used, this overshoot can be solved easily
The main point has to be cared is response of the sway angles, which showed in Figure 4.5 and Figure 4.6 Because ZV input shaper using two impulses to turning the reference inputs, amplitude of the first oscillation cycle in both sway angles reduce to near 50,6% This amplitude can be reduced smaller of ZVD or ZVDD input shaper are used However, of course the time of reaching to the desired point in X- axis, Y-axis increases
Figure 4.5 Angle θ response with and without input shaping method
Turn back to the figure 4.5 and figure 4.6, after the second impulse applied, the amplitudes of the oscillations are almost eliminated For example: θ at 9.78s with input shaper is equal only 10.34% as it without input shaper, φ is smaller These
Chapter 4: Input shaping method in 3-D overhead crane results are similar to the results of using a nonlinear optimal control of Nguyen and Nguyen [25]
Figure 4.6 Angle φ response with and without input
In first 20 seconds, the maximum amplitudes of, θ, φ are 0.0003574 radians and 0.001804 radians or 0.0205 o and 0.0134 o correspondently They are very almost eliminated; these angles are ignored evidently in reality When changing the desire inputs of X-axis and Y-axis as 80 seconds after, this controller also works well, it can eliminate the oscillation of the overhead crane after two impulses
The desired inputs of X, Y are change in each 20 seconds In the initial of each period, the amplitude of the first oscillating cycle is biggest The reason is the method of using input shaper ZV controller only uses two shaping impulses which cause the payload to oscillate in opposite phases This makes the payload oscillate 2 haft-cycle before reaching to the stable period If ZVD, ZVDD are used, the response of the sway angles will have three or four half-cycle correspondently
In conclusion, the effectiveness of ZV input command shaper are proved by
Chapter 4: Input shaping method in 3-D overhead crane simulation, and also input command shaping method in controlling 3-D overhead crane However, all of the simulation results presented in Figure 4.3, Figure 4.4, Figure 4.5, 4.6 are present for the case that the the payload height does not change
Figure 4.7 shows the length of cable which present for the height of payload in both cases As said before, this variable is kept unchanging by the force u l applied on the trolley
Figure 4.7 Length of the cable/height of the payload
A new problem appears when length of the cable changing through the process
ADAPTIVE INPUT SHAPING IN 3-D OVERHEAD CRANE
ZV input shaper with variable cable length of 3-D overhead crane
Return to the equation (3.22), it is clear that the natural frequencies of 3-D overhead crane depend on length of cable and mass of the components In these element, mass of girder and the trolley obviously unchanged When the control process starts, mass of the payload also rarely changes in reality Hence, mass of the payload is considered as a constant in a process
Figure 5.1 Desired length of cable in changing follow the process
So that, length of the cable is the remaining variable which need to be consider when applied input shaping method In chapter 4, this variable is a constant, so that the natural frequencies of 3-D overhead crane system also unchanged Because of
Chapter 5: Adaptive input shaping in 3-D overhead crane that, input shaper block in figure 4.3 which depends on natural frequency of the system also unchanged So that, the response of the crane with this ZV input shaping method is satisfied the desired references
Now, the reference input of l is replaced as figure 5.1 Supposing that the desired input of l changes each 20 seconds by some particular demands Follow the diagram control in figure 4.2, l is controlled by LQR tracking controller
Run simulation with ZV input shaper with and without changing in cable length to check the responses of these condition It is easy to see in Figure 5.2, Figure 5.3, Figure 5.4 that the response of X-axis, Y-Axis and l are still reach the desired point in each trip Noted that the weight matrixes Q, R is hold as equal as (4.6) and (4.7)
Values of the coefficients in related with the sway angles parameters are zeros
Figure 5.2 Positions Response on X-Axis (l is a variable)
Thus, a conclusion can be exported is LQR controller still working appropriately With the weight matrixes decided in (4.6), (4.7) and given desired positions of x, y, l after 7-9 seconds, the crane reaches to the desired positions in X-Axis, Y-axis, and 8-10 seconds in length of cable
Chapter 5: Adaptive input shaping in 3-D overhead crane
Figure 5.3 Positions Response on Y-Axis (l is a variable)
Chapter 5: Adaptive input shaping in 3-D overhead crane
Figure 5.5 Response of θ with input shaping method (l is a variable)
Figure 5.6 Response of with input shaping method (l is a variable)
Chapter 5: Adaptive input shaping in 3-D overhead crane
However, the responses of sway angles are not stable as the case in Chapter 4, where the cable length is a constant Because of the variation of the cable length makes the natural frequency of 3-D overhead crane changes Consequently, the initial input command shaping is inaccurate for the next steps Simulation results from Figure 5.5 and Figure 5.6 show that:
+ At the first period: the input command shaper is calculated by initial length of cable, which is hold unchanged in this period So that the sway angle is minimized and eliminated
+ At second period, the amplitude of sway angles increased 4-5 times as first period It can be explained that because the length of cable changes, lead to the natural frequencies of the system change So that the input shaper is not appropriated to new specification of this system
+ The more changing times of cable length, the bigger amplitudes of oscillation increase In the fourth and fifth periods, although the length of cable is back to the beginning values, but the oscillation still exists
In conclusion, the input command shaping method is not effective when the length of cable changes through the process In reality, the cable can not be always kept unchanged, because it need to be lift up and down to move the payload, or to avoid the obstacle on the way of its movement
So that, this thesis proposes a method to improve this method by adding an auto turning method for input command shaping to adapt with the change of the cable It will be presented in the next section.
Adaptive input shaping method (AIS) applied in 3-D overhead crane
Dealing with the problem presented in 5.1, an auto turning method is added to the controlling diagram to dealing with the diversified cable length It is called adaptive input command shaping method
The main part of input command shaping controller which minimizes the sway angles is also input shaping LQR controller then uses the output values from input command shaping to control the position of the 3-D overhead crane So that, to solve
Chapter 5: Adaptive input shaping in 3-D overhead crane the problem when the cable length changes, input command shaper is replaced by adaptive input command shaper in figure 5.7
In each sampling time of adaptive input command shaping controller, the cable length signal has to be feedback to the adaptive input command shaper This signal is applied for turning the specifications of the command shaping impulses, adapts with the change of cable length Initial desired input becomes new input signals which are called adaptive shaping inputs They are generated by the adaptive input command shaper and updated in each sampling time LQR controller will use them to control the 3-D overhead crane positions in respective axis Noted that LQR controller parameters will be kept as Chapter 4 This controller is also used to control length of cable l following the desired length as input command shaping method in section 5.1
Figure 5.7 Adaptive Input Shaping Method
Perform simulations of the adaptive command shaping method in Figure 5.7
The positions of trolley x, y, cable length l are showed in Figure 5.8, Figure 5.9 and Figure 5.10 It is really easy to see that they do not have significant change as the results of input command shaping method That because the LQR parameters are unchanged This controller remains as effective as the original input command shaping method In addition, it can be inferred that the difference of input shaper and adaptive input shaper affects to the LQR controller insignificantly
Chapter 5: Adaptive input shaping in 3-D overhead crane
Figure 5.8 X-axis response in IS method and AIS method
Figure 5.9 Y-axis response in IS method and AIS method
Chapter 5: Adaptive input shaping in 3-D overhead crane
Figure 5.10 Cable length responses in IS method and AIS method
However, the responses of the sway angles are improved markedly The results listed in Figure 5.11, Figure 5.12 proved that adaptive input shaping method can solve the new demand
In this example, the sway angles at the steady state vary from 0.003÷0.008 rad (0.17 ÷ 0.46 degree) The amplitudes of sway angles decrease 6,7÷9 times as input command shaping method
The difference of the input shaping method and the adaptive in put shaping method is the turning impulses specifications With input shaper, the turning impulses is set at the beginning of the process When the natural frequencies of the 3-D overhead crane change in the process, the input shaper is equal as initial Although the natural frequencies can be chosen by many other methods for optimizing the sway angle errors, they only can divide the sway angles errors to the mean value The sum of sway angle errors can not be decreased
Chapter 5: Adaptive input shaping in 3-D overhead crane
Figure 5.11 θ response in IS method and AIS method
Figure 5.12 φ response in IS method and AIS method
Chapter 5: Adaptive input shaping in 3-D overhead crane
But in the adaptive input command shaper, the turning impulses are always updated in each sampling time by computing new natural frequencies with feedback value l So that the reference inputs are also change to satisfy with new specifications of model over the controlling process To check the difference of the desired signal after using input command shaping method and adaptive command shaping method, equations (5.1), (5.2) are considered
Figure 5.13, figure 5.14 show that, the difference of input signals in both method is really small But actually, the effectiveness of this difference is important To understand it easier, please check the Figure 5.15, Figure 5.16 They show the differences of the inputs in (5.1), (5.2)
1 ref _ x afterinput shaper ref _ x afteradaptiveinput shaper
2 ref _ y afterinput shaper ref _ y after adaptiveinput shaper
Figure 5.13 X inputs after using Input Shaper and Adaptive Input Shaper
Chapter 5: Adaptive input shaping in 3-D overhead crane
Figure 5.14 Y inputs after using Input Shaper and Adaptive Input Shaper
Chapter 5: Adaptive input shaping in 3-D overhead crane
Figures 5.15, Figure 5.16 proved that the difference of the reference input signals after turning in both method only exits in a shot time at the initial point which the desired cable length changes But as showed in figure 5.11, 5.12, the sway angles are significantly decreased Hence, in input shaping method, the values of the turning impulses are very important A little change in length of cable can make the natural frequency of system change, lead to unsatisfied responses And these results prove that the adaptive input shaping method can operate better that original input shaping
It can adapt with the cable length change in reality process which make the natural specifications of 3-D overhead crane system change
Moreover, the sway angles in Figure 5.11 and Figure 5.12 still exist with tiny values and can not be eliminated completely Comparing with [25], Nguyen used nonlinear optimal control method, the sway angles after rising time are almost zeroes
In optimal control, the errors of sway angles have to be monitored and feedback to
Chapter 5: Adaptive input shaping in 3-D overhead crane calculate the control signals But this adaptive input shaping control is an open-loop controller, errors of sway angles are not measured, they have just been still controlled by input shaping method The sway angles after rising time can not be removed completely by LQR controller So that in general if there are disturbance, the close- loop controller is more effective than open-loop controllers But the open-loop controller is simpler because it needn’t the feedback of sway angles.
CONCLUSIONS
In this thesis, a linearized model of 3-D overhead crane was inferred from nonlinear model of 3-D overhead crane system This nonlinear is developed by using Euler-Lagrange equations, and generalized coordinates theory Based on the linearized model, state space function of this system is deduced with 10 variables
State space function is used to design the LQR controller, with the combination with adaptive input shaping method to reach two objectives:
(1) Driving the trolley to the desired positions
(2) Minimizing the sway angles even the natural frequency of system change
The main stream of this thesis is introduced the open-loop input shaping method to minimizing the sway angles LQR controller is kept unchanged and used to control the trolley positions for investigating the effectiveness of input command shaping method The adaptive input command shaping method in proposed base on original shaping method It was verified by numerical simulation in Matlab ® and experiment results
This controller is simple to design, applied because it does not use the sway angles in controlling process But if the disturbance of the working environment is significant, this controller can not be work effectively
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