Điều khiển cầu trục hoạt động trong không gian 3 chiều sử dụng adaptive command shaping
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ĐẠI HỌC QUỐC GIA TP. HCM
TRƯỜNG ĐẠI HỌC BÁCH KHOA
----------------------------------------
LÊ GIÁP
ĐIỀU KHIỂN CẦU TRỤC HOẠT ĐỘNG TRONG KHÔNG
GIAN 3 CHIỀU SỬ DỤNG ADAPTIVE COMMAND SHAPING
ADAPTIVE COMMAND SHAPING CONTROL OF
A 3D OVERHEAD CRANE
Chuyên ngành: Kỹ Thuật Cơ Điện Tử
Mã số: 60520114
LUẬN VĂN THẠC SĨ
TP. HỒ CHÍ MINH, tháng 07 năm 2018
CÔNG TRÌNH ĐƯỢC HOÀN THÀNH TẠI
TRƯỜNG ĐẠI HỌC BÁCH KHOA –ĐHQG -HCM
Cán bộ hướng dẫn khoa học : PGS.TS Nguyễn Quốc Chí ........................
Cán bộ chấm nhận xét 1 :PGS.TS Nguyễn Duy Anh ................................
Cán bộ chấm nhận xét 2 :PGS.TS Võ Hoàng Duy ....................................
Luận văn thạc sĩ được bảo vệ tại Trường Đại học Bách Khoa, ĐHQG Tp.
HCM ngày 04 tháng 07 năm 2018.
Thành phần Hội đồng đánh giá luận văn thạc sĩ gồm:
1.Chủ tịch hồi đồng: PGS.TS Nguyễn Tấn Tiến.....................................
2.Thư ký hội đồng: TS. Ngô Hà Quang Thịnh .......................................
3.Ủy viên phản biện 1: PGS.TS Nguyễn Duy Anh. ................................
4.Ủy viên phản biện 2: TS. Võ Hoàng Duy. ...........................................
5.Ủy viên hội đồng: TS. Phùng Trí Công. ..............................................
Xác nhận của Chủ tịch Hội đồng đánh giá LV và Trưởng Khoa quản lý
chuyên ngành sau khi luận văn đã được sửa chữa (nếu có).
CHỦ TỊCH HỘI ĐỒNG TRƯỞNG KHOA
PGS.TS NGUYỄN TẤN TIẾN PGS.TS NGUYỄN HỮU LỘC
ĐẠI HỌC QUỐC GIA TP.HCM CỘNG HÒA XÃ HỘI CHỦ NGHĨA VIỆT
TRƯỜNG ĐẠI HỌC BÁCH KHOA
NAM Độc lập - Tự do - Hạnh phúc
NHIỆM VỤ LUẬN VĂN THẠC SĨ
Họ tên học viên: LÊ GIÁP MSHV :1670311
Ngày, tháng, năm sinh : 31/03/1989 Nơi sinh : Thừa Thiên Huế
Chuyên ngành : Kỹ thuật cơ điện tử Mã số : 60520114
I. 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
II. NHIỆM VỤ VÀ NỘI DUNG:
- Xây dựng phương trình động lực học của cầu trục hoạt động trông không gian 3
chiều
- Xây dựng thuật toán điều khiển adaptive input shaping cho cầu trục hoạt động trong
không gian 3 chiều
III. NGÀY GIAO NHIỆM VỤ (10/7/2017):
IV. NGÀY HOÀN THÀNH NHIỆM VỤ (17/06/2018):
V. CÁN BỘ HƯỚNG DẪN : PGS.TS. NGUYỄN QUỐC CHÍ
Tp. HCM, ngày tháng năm 2018
CÁN BỘ HƯỚNG DẪN CHỦ NHIỆM BỘ MÔN ĐÀO TẠO
PGS.TS. NGUYỄN QUỐC CHÍ TS. PHẠM CÔNG BẰNG
TRƯỞNG KHOA
PGS.TS NGUYỄN HỮU LỘC
Acknowledgement
ACKNOWLEDGEMENT
After one year, eventually the thesis “Using adaptive command shaping to
control 3-d overhead crane” was accepted. This thesis obtained positive objectives
about theories and simulation experiment. To achieve these results, I was received
the significant support from my teachers, parents, and friends.
First, I would like to express my deep gratitude to my supervisor Prof. Nguyen,
Quoc Chi. His enthusiastic support and continuous pieces of advice contributed to the
completion of my thesis. His encouragement and constructive comments had led me
to the right direction. They enriched and improved my work and my knowledge.
Without his motivation and instruction, the thesis would have been impossible to be
done effectively and timely.
I would like to show my special thanks to my parents for their endless love,
care. Without their helping in real life, I can not have enough time to work with this
thesis. assistances and motivation that they have saved me. I also want to say thank
you to my younger brother, my wife for their support and care all the time.
As last, my deeply thanks to all my friends, especial in the Control and
Automation Laboratory during the time I study and work in Ho Chi Minh City
University of Technology. Their kindly help, care, motivation gave me the motivation
to overcome many troubles in researching, during the process to complete the thesis.
I declare that the thesis has been done without any plagiarism violations and
does not conflict with any issue in ethics.
Ho Chi Minh City, 17/6/2018
Giap Le
1
Abstract
ABSTRACT
In this thesis, an adaptive command shaping control of a 3-D overhead crane
is proposed. The proposed control scheme is to suppress the sway angles of the
payload and to drive the trolley of the overhead crane to the desired position. The
Euler-Lagrange method is employed to derive a dynamic model of the overhead
crane, which is a nonlinear model. For convenience to design the input shaper, the
nonlinear dynamic model is linearized. To track the command shaping generated
from the input shaper, LQR control scheme is employed. The LQR controller
guarantees the positions of the trolley and the length of cable, together with the
adaptive command shaping method is used to minimizing the sway angles of the
payload. Numerical simulations are carried out to verify the effectiveness of the
proposed control method.
2
Table of contents
TABLE OF CONTENTS
ACKNOWLEDGEMENT ..........................................................................................1
ABSTRACT ................................................................................................................2
TABLE OF CONTENTS ............................................................................................3
LIST OF FIGURES.....................................................................................................4
CHAPTER 1. INTRODUCTION ...............................................................................6
1.1. General review of cranes and some factual applications ........................................................ 6
1.2. Operation of overhead crane ................................................................................................... 8
1.3. Modeling of an overhead crane ............................................................................................. 10
1.4. Control of 3-D overhead cranes ............................................................................................ 12
1.5. Objectives of the thesis: ........................................................................................................ 14
1.6. Organization of the thesis: .................................................................................................... 15
CHAPTER 2. FUNDAMENTAL THEORY ............................................................16
2.1. Generalized coordinate [25]: ................................................................................................. 16
2.2. Euler-Lagrange equations [27]:............................................................................................. 17
2.3. Input shaping (IS) method:.................................................................................................... 20
CHAPTER 3. OVERHEAD CRANE DYNAMICS MODEL .................................25
3.1. Overhead crane modeling: .................................................................................................... 25
3.2. System linearizing and oscillation specifications: ................................................................ 31
CHAPTER 4. INPUT SHAPING METHOD ...........................................................34
4.1. ZV Input shaping with 3-D overhead crane: ......................................................................... 34
4.2. Experiment in Matlab® simulation: ....................................................................................... 37
CHAPTER 5. ADAPTIVE INPUT SHAPING IN 3-D OVERHEAD CRANE ......43
5.1. ZV input shaper with variable cable length of 3-D overhead crane ...................................... 43
5.2. Adaptive input shaping method (AIS) applied in 3-D overhead crane ................................. 47
CHAPTER 6. CONCLUSIONS................................................................................56
REFERENCES ..........................................................................................................57
3
List of figures
LIST OF FIGURES
Figure 1.1. Mobile cranes............................................................................................6
Figure 1.2. Tower cranes. ............................................................................................7
Figure 1.3. Floating crane. ..........................................................................................7
Figure 1.4. Overhead crane. ........................................................................................8
Figure 1.5. 3-D Overhead crane model. ......................................................................9
Figure 1.6. ODEs model............................................................................................10
Figure 1.7. PDEs model. ...........................................................................................12
Figure 2.1. Diagram of input shaping concept. .........................................................21
Figure 3.1. 3-D Overhead crank principle diagram. .................................................25
Figure 3.2. Sway angles of the payload. ...................................................................26
Figure 4.1. LQR Position Tracking Control Applied on 3-D Crane. ........................37
Figure 4.2. Input Shaping applied on 3-D crane control. ..........................................37
Figure 4.3. Positions Response of 3-D overhead crane on X-Axis. .........................38
Figure 4.4. Positions Response of 3-D overhead crane on Y-Axis. .........................39
Figure 4.5. Angle response with and without input. ..............................................40
Figure 4.6. Angle response with and without input. ............................................41
Figure 4.7. Length of the cable/height of the payload. .............................................42
Figure 5.1. Desired length of cable in changing follow the process. ........................43
Figure 5.2. Positions Response on X-Axis ( l is a variable). ....................................44
Figure 5.3. Positions Response on Y-Axis ( l is a variable). ....................................45
Figure 5.4. Cable Length Response. .........................................................................45
Figure 5.5. Response of with input shaping method ( l is a variable). .................46
Figure 5.6. Response of with input shaping method ( l is a variable). ...............46
Figure 5.7. Adaptive Input Shaping Method.............................................................48
Figure 5.8. X-axis response in IS method and AIS method......................................49
Figure 5.9. Y-axis response in IS method and AIS method......................................49
Figure 5.10. Cable length responses in IS method and AIS method. .......................50
Figure 5.11. response in IS method and AIS method. ...........................................51
Figure 5.12. response in IS method and AIS method. .........................................51
Figure 5.13. X inputs after using Input Shaper and Adaptive Input Shaper. ............52
4
List of figures
Figure 5.14. Y inputs after using Input Shaper and Adaptive Input Shaper. ............53
Figure 5.15. 1 . .........................................................................................................53
Figure 5.16. 2 . .........................................................................................................54
5
Chapter 1: Introduction
CHAPTER 1. INTRODUCTION
In this chapter, the principles and the applications of overhead crane system will
be introduced. Secondly, different methods about control overhead crane are
considered to clarify some techniques for control overhead crane. Therefore, the
advantages and disadvantages of some existing control method are recorded. Finally,
a brief control method for this thesis will be suggested to increase the effectiveness
of an old method in control overhead crane.
1.1. 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.
Figure 1.1. Mobile cranes.
(Images from http://www.palfinger-sany.com/en/psv/products/truck-crane/stc750 )
6
Chapter 1: Introduction
Figure 1.2. Tower cranes.
(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).
Figure 1.3. Floating crane.
(Images from https://www.breakbulk.com/events/breakbulk-china-
2018/preview/hkzm-bridge/
7
Chapter 1: Introduction
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.
Figure 1.4. Overhead crane.
(Images from http://www.crane-tec.com)
1.2. Operation of overhead crane
Overhead cranes are very adaptive with working condition of small space like
8
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
(1): Hook (2): Pulleys (3): Cable
(4): Lifting-lowering (6): Based frame
(5): Rails/bridge/Girder
system/Trolley structure
6
5
4
3
2
1
Figure 1.5. 3-D Overhead crane model.
(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
9
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.
1.3. 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].
1.3.1. ODE model:
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.
z
F
Trolley
d x
friction
x(t)
θ(t)
Payload
m
Figure 1.6. ODEs model [1].
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
10
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.
1.3.2. PDEs model:
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.
11
Chapter 1: Introduction
z
x(t)
F Trolley
M
d
friction
w(t)
Payload m
Figure 1.7. PDEs model [1].
1.4. 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,
12
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
13
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.
1.5. 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.
1.5.1. Modeling 3-D overhead crane:
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
14
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.
1.6. 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.
Chapter 6 summarizes the results.
15
Chapter 2: Fundamental theories
CHAPTER 2. FUNDAMENTAL THEORIES
This chapter will present basic theory about the generalized coordinate and
Euler-Lagrange equation. It will be used in Chapter 3 to construct the dynamic
equations of 3-D overhead crane. In addition, the input command shaping method is
also introduced before using in chapter 4.
2.1. Generalized coordinate [26]:
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:
ri ( xi , yi , zi ), i 1: N (2.1)
If this system has C holonomic constraint:
f k (r1 ,r2 ...,rN ,t)= 0 ,k = 1 : C (2.2)
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:
q = (q1 ,q2 ,...,qn ) (2.3)
The position vector ri of particle ith is a function of all the ri generalized
coordinates and time:
ri = (q(t),t) (2.4)
The corresponding time derivatives of qk are called the generalized velocities:
dq
q (q1, q2 ,..., qn ) (2.5)
dt
The velocity vector vk is the total derivative of rk with respect to time:
n
drk r r
vk rk k q j k , k 1: N (2.6)
dt j 1 q j t
Because of that, with holonomic constraints, vk is generally depends on the
generalized velocities and generalized coordinates.
Generalized forces can be obtained from the computation of the virtual work,
16
Chapter 2: Fundamental theories
δW, of the applied forces [27]. Let Fi, i=1: N are the applied forces, the virtual work
of Fi, acting on the particles Pi, i=1: N, is given by:
N
W Fi . ri (2.7)
i 1
Where δri is the virtual displacement of the particle Pi. Inferring from (2.4):
n
r
ri i q j , i 1: N (2.8)
j 1 q j
Then the virtual work for the system of particles (2.7) becomes:
n n
r r
W F1 1 qi ... FN N qi (2.9)
i 1 q j i 1 q j
Collect the coefficients of q j :
N N
ri r
W Fi q1 ... Fi i qn (2.10)
i 1 q1 i 1 qn
So that, the virtual work of a system of particles can be written in the form:
W Q1. q1 ... Qn . qn (2.11)
Where:
N
ri
Q j Fi , j 1: n (2.12)
i 1 q j
And Qj, 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:
F V
ri r V (2.13)
F V i
qi qi qi
Or:
V
Qi (i 1: n ) (2.14)
qi
In closed-loop systems, generalized force is opposite in sign with partial
derivative of potential energy of the systems follow as generalized coordinate
2.2. Euler-Lagrange equations [28]:
Now consider a system of particles: j=1: Np. Let r(k) denote the position of k(th)
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TRƯỜNG ĐẠI HỌC BÁCH KHOA
----------------------------------------
LÊ GIÁP
ĐIỀU KHIỂN CẦU TRỤC HOẠT ĐỘNG TRONG KHÔNG
GIAN 3 CHIỀU SỬ DỤNG ADAPTIVE COMMAND SHAPING
ADAPTIVE COMMAND SHAPING CONTROL OF
A 3D OVERHEAD CRANE
Chuyên ngành: Kỹ Thuật Cơ Điện Tử
Mã số: 60520114
LUẬN VĂN THẠC SĨ
TP. HỒ CHÍ MINH, tháng 07 năm 2018
CÔNG TRÌNH ĐƯỢC HOÀN THÀNH TẠI
TRƯỜNG ĐẠI HỌC BÁCH KHOA –ĐHQG -HCM
Cán bộ hướng dẫn khoa học : PGS.TS Nguyễn Quốc Chí ........................
Cán bộ chấm nhận xét 1 :PGS.TS Nguyễn Duy Anh ................................
Cán bộ chấm nhận xét 2 :PGS.TS Võ Hoàng Duy ....................................
Luận văn thạc sĩ được bảo vệ tại Trường Đại học Bách Khoa, ĐHQG Tp.
HCM ngày 04 tháng 07 năm 2018.
Thành phần Hội đồng đánh giá luận văn thạc sĩ gồm:
1.Chủ tịch hồi đồng: PGS.TS Nguyễn Tấn Tiến.....................................
2.Thư ký hội đồng: TS. Ngô Hà Quang Thịnh .......................................
3.Ủy viên phản biện 1: PGS.TS Nguyễn Duy Anh. ................................
4.Ủy viên phản biện 2: TS. Võ Hoàng Duy. ...........................................
5.Ủy viên hội đồng: TS. Phùng Trí Công. ..............................................
Xác nhận của Chủ tịch Hội đồng đánh giá LV và Trưởng Khoa quản lý
chuyên ngành sau khi luận văn đã được sửa chữa (nếu có).
CHỦ TỊCH HỘI ĐỒNG TRƯỞNG KHOA
PGS.TS NGUYỄN TẤN TIẾN PGS.TS NGUYỄN HỮU LỘC
ĐẠI HỌC QUỐC GIA TP.HCM CỘNG HÒA XÃ HỘI CHỦ NGHĨA VIỆT
TRƯỜNG ĐẠI HỌC BÁCH KHOA
NAM Độc lập - Tự do - Hạnh phúc
NHIỆM VỤ LUẬN VĂN THẠC SĨ
Họ tên học viên: LÊ GIÁP MSHV :1670311
Ngày, tháng, năm sinh : 31/03/1989 Nơi sinh : Thừa Thiên Huế
Chuyên ngành : Kỹ thuật cơ điện tử Mã số : 60520114
I. 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
II. NHIỆM VỤ VÀ NỘI DUNG:
- Xây dựng phương trình động lực học của cầu trục hoạt động trông không gian 3
chiều
- Xây dựng thuật toán điều khiển adaptive input shaping cho cầu trục hoạt động trong
không gian 3 chiều
III. NGÀY GIAO NHIỆM VỤ (10/7/2017):
IV. NGÀY HOÀN THÀNH NHIỆM VỤ (17/06/2018):
V. CÁN BỘ HƯỚNG DẪN : PGS.TS. NGUYỄN QUỐC CHÍ
Tp. HCM, ngày tháng năm 2018
CÁN BỘ HƯỚNG DẪN CHỦ NHIỆM BỘ MÔN ĐÀO TẠO
PGS.TS. NGUYỄN QUỐC CHÍ TS. PHẠM CÔNG BẰNG
TRƯỞNG KHOA
PGS.TS NGUYỄN HỮU LỘC
Acknowledgement
ACKNOWLEDGEMENT
After one year, eventually the thesis “Using adaptive command shaping to
control 3-d overhead crane” was accepted. This thesis obtained positive objectives
about theories and simulation experiment. To achieve these results, I was received
the significant support from my teachers, parents, and friends.
First, I would like to express my deep gratitude to my supervisor Prof. Nguyen,
Quoc Chi. His enthusiastic support and continuous pieces of advice contributed to the
completion of my thesis. His encouragement and constructive comments had led me
to the right direction. They enriched and improved my work and my knowledge.
Without his motivation and instruction, the thesis would have been impossible to be
done effectively and timely.
I would like to show my special thanks to my parents for their endless love,
care. Without their helping in real life, I can not have enough time to work with this
thesis. assistances and motivation that they have saved me. I also want to say thank
you to my younger brother, my wife for their support and care all the time.
As last, my deeply thanks to all my friends, especial in the Control and
Automation Laboratory during the time I study and work in Ho Chi Minh City
University of Technology. Their kindly help, care, motivation gave me the motivation
to overcome many troubles in researching, during the process to complete the thesis.
I declare that the thesis has been done without any plagiarism violations and
does not conflict with any issue in ethics.
Ho Chi Minh City, 17/6/2018
Giap Le
1
Abstract
ABSTRACT
In this thesis, an adaptive command shaping control of a 3-D overhead crane
is proposed. The proposed control scheme is to suppress the sway angles of the
payload and to drive the trolley of the overhead crane to the desired position. The
Euler-Lagrange method is employed to derive a dynamic model of the overhead
crane, which is a nonlinear model. For convenience to design the input shaper, the
nonlinear dynamic model is linearized. To track the command shaping generated
from the input shaper, LQR control scheme is employed. The LQR controller
guarantees the positions of the trolley and the length of cable, together with the
adaptive command shaping method is used to minimizing the sway angles of the
payload. Numerical simulations are carried out to verify the effectiveness of the
proposed control method.
2
Table of contents
TABLE OF CONTENTS
ACKNOWLEDGEMENT ..........................................................................................1
ABSTRACT ................................................................................................................2
TABLE OF CONTENTS ............................................................................................3
LIST OF FIGURES.....................................................................................................4
CHAPTER 1. INTRODUCTION ...............................................................................6
1.1. General review of cranes and some factual applications ........................................................ 6
1.2. Operation of overhead crane ................................................................................................... 8
1.3. Modeling of an overhead crane ............................................................................................. 10
1.4. Control of 3-D overhead cranes ............................................................................................ 12
1.5. Objectives of the thesis: ........................................................................................................ 14
1.6. Organization of the thesis: .................................................................................................... 15
CHAPTER 2. FUNDAMENTAL THEORY ............................................................16
2.1. Generalized coordinate [25]: ................................................................................................. 16
2.2. Euler-Lagrange equations [27]:............................................................................................. 17
2.3. Input shaping (IS) method:.................................................................................................... 20
CHAPTER 3. OVERHEAD CRANE DYNAMICS MODEL .................................25
3.1. Overhead crane modeling: .................................................................................................... 25
3.2. System linearizing and oscillation specifications: ................................................................ 31
CHAPTER 4. INPUT SHAPING METHOD ...........................................................34
4.1. ZV Input shaping with 3-D overhead crane: ......................................................................... 34
4.2. Experiment in Matlab® simulation: ....................................................................................... 37
CHAPTER 5. ADAPTIVE INPUT SHAPING IN 3-D OVERHEAD CRANE ......43
5.1. ZV input shaper with variable cable length of 3-D overhead crane ...................................... 43
5.2. Adaptive input shaping method (AIS) applied in 3-D overhead crane ................................. 47
CHAPTER 6. CONCLUSIONS................................................................................56
REFERENCES ..........................................................................................................57
3
List of figures
LIST OF FIGURES
Figure 1.1. Mobile cranes............................................................................................6
Figure 1.2. Tower cranes. ............................................................................................7
Figure 1.3. Floating crane. ..........................................................................................7
Figure 1.4. Overhead crane. ........................................................................................8
Figure 1.5. 3-D Overhead crane model. ......................................................................9
Figure 1.6. ODEs model............................................................................................10
Figure 1.7. PDEs model. ...........................................................................................12
Figure 2.1. Diagram of input shaping concept. .........................................................21
Figure 3.1. 3-D Overhead crank principle diagram. .................................................25
Figure 3.2. Sway angles of the payload. ...................................................................26
Figure 4.1. LQR Position Tracking Control Applied on 3-D Crane. ........................37
Figure 4.2. Input Shaping applied on 3-D crane control. ..........................................37
Figure 4.3. Positions Response of 3-D overhead crane on X-Axis. .........................38
Figure 4.4. Positions Response of 3-D overhead crane on Y-Axis. .........................39
Figure 4.5. Angle response with and without input. ..............................................40
Figure 4.6. Angle response with and without input. ............................................41
Figure 4.7. Length of the cable/height of the payload. .............................................42
Figure 5.1. Desired length of cable in changing follow the process. ........................43
Figure 5.2. Positions Response on X-Axis ( l is a variable). ....................................44
Figure 5.3. Positions Response on Y-Axis ( l is a variable). ....................................45
Figure 5.4. Cable Length Response. .........................................................................45
Figure 5.5. Response of with input shaping method ( l is a variable). .................46
Figure 5.6. Response of with input shaping method ( l is a variable). ...............46
Figure 5.7. Adaptive Input Shaping Method.............................................................48
Figure 5.8. X-axis response in IS method and AIS method......................................49
Figure 5.9. Y-axis response in IS method and AIS method......................................49
Figure 5.10. Cable length responses in IS method and AIS method. .......................50
Figure 5.11. response in IS method and AIS method. ...........................................51
Figure 5.12. response in IS method and AIS method. .........................................51
Figure 5.13. X inputs after using Input Shaper and Adaptive Input Shaper. ............52
4
List of figures
Figure 5.14. Y inputs after using Input Shaper and Adaptive Input Shaper. ............53
Figure 5.15. 1 . .........................................................................................................53
Figure 5.16. 2 . .........................................................................................................54
5
Chapter 1: Introduction
CHAPTER 1. INTRODUCTION
In this chapter, the principles and the applications of overhead crane system will
be introduced. Secondly, different methods about control overhead crane are
considered to clarify some techniques for control overhead crane. Therefore, the
advantages and disadvantages of some existing control method are recorded. Finally,
a brief control method for this thesis will be suggested to increase the effectiveness
of an old method in control overhead crane.
1.1. 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.
Figure 1.1. Mobile cranes.
(Images from http://www.palfinger-sany.com/en/psv/products/truck-crane/stc750 )
6
Chapter 1: Introduction
Figure 1.2. Tower cranes.
(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).
Figure 1.3. Floating crane.
(Images from https://www.breakbulk.com/events/breakbulk-china-
2018/preview/hkzm-bridge/
7
Chapter 1: Introduction
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.
Figure 1.4. Overhead crane.
(Images from http://www.crane-tec.com)
1.2. Operation of overhead crane
Overhead cranes are very adaptive with working condition of small space like
8
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
(1): Hook (2): Pulleys (3): Cable
(4): Lifting-lowering (6): Based frame
(5): Rails/bridge/Girder
system/Trolley structure
6
5
4
3
2
1
Figure 1.5. 3-D Overhead crane model.
(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
9
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.
1.3. 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].
1.3.1. ODE model:
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.
z
F
Trolley
d x
friction
x(t)
θ(t)
Payload
m
Figure 1.6. ODEs model [1].
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
10
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.
1.3.2. PDEs model:
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.
11
Chapter 1: Introduction
z
x(t)
F Trolley
M
d
friction
w(t)
Payload m
Figure 1.7. PDEs model [1].
1.4. 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,
12
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
13
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.
1.5. 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.
1.5.1. Modeling 3-D overhead crane:
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
14
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.
1.6. 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.
Chapter 6 summarizes the results.
15
Chapter 2: Fundamental theories
CHAPTER 2. FUNDAMENTAL THEORIES
This chapter will present basic theory about the generalized coordinate and
Euler-Lagrange equation. It will be used in Chapter 3 to construct the dynamic
equations of 3-D overhead crane. In addition, the input command shaping method is
also introduced before using in chapter 4.
2.1. Generalized coordinate [26]:
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:
ri ( xi , yi , zi ), i 1: N (2.1)
If this system has C holonomic constraint:
f k (r1 ,r2 ...,rN ,t)= 0 ,k = 1 : C (2.2)
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:
q = (q1 ,q2 ,...,qn ) (2.3)
The position vector ri of particle ith is a function of all the ri generalized
coordinates and time:
ri = (q(t),t) (2.4)
The corresponding time derivatives of qk are called the generalized velocities:
dq
q (q1, q2 ,..., qn ) (2.5)
dt
The velocity vector vk is the total derivative of rk with respect to time:
n
drk r r
vk rk k q j k , k 1: N (2.6)
dt j 1 q j t
Because of that, with holonomic constraints, vk is generally depends on the
generalized velocities and generalized coordinates.
Generalized forces can be obtained from the computation of the virtual work,
16
Chapter 2: Fundamental theories
δW, of the applied forces [27]. Let Fi, i=1: N are the applied forces, the virtual work
of Fi, acting on the particles Pi, i=1: N, is given by:
N
W Fi . ri (2.7)
i 1
Where δri is the virtual displacement of the particle Pi. Inferring from (2.4):
n
r
ri i q j , i 1: N (2.8)
j 1 q j
Then the virtual work for the system of particles (2.7) becomes:
n n
r r
W F1 1 qi ... FN N qi (2.9)
i 1 q j i 1 q j
Collect the coefficients of q j :
N N
ri r
W Fi q1 ... Fi i qn (2.10)
i 1 q1 i 1 qn
So that, the virtual work of a system of particles can be written in the form:
W Q1. q1 ... Qn . qn (2.11)
Where:
N
ri
Q j Fi , j 1: n (2.12)
i 1 q j
And Qj, 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:
F V
ri r V (2.13)
F V i
qi qi qi
Or:
V
Qi (i 1: n ) (2.14)
qi
In closed-loop systems, generalized force is opposite in sign with partial
derivative of potential energy of the systems follow as generalized coordinate
2.2. Euler-Lagrange equations [28]:
Now consider a system of particles: j=1: Np. Let r(k) denote the position of k(th)
17