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12
CHAPTER 2
LITERATURE SURVEY
2.1 INTRODUCTION
Robots have been finding applications in many fields in the recent years,
especially in the field of industrial automation. So different kinds of robots find
application in the industrial automation process. One such robot is called manipulator,
the main purpose of which is to pick and place an object in a specific position. In order
to pick and place an object accurately, it is needed to implement a control system
algorithm on the manipulator. This chapter deals with a brief survey of various control
systems presently used in the manipulator control. The introduction to the control
systems is followed by a review of the problems in control system. The later part of the
chapter explains the proposed Haar Wavelet approach and the relevant presents the
conclusions
2.2 A BRIEF SURVEY OF VARIOUS CONTROL SYSTEMS
In the past few decades, the motion control of the industrial manipulator has
been remaining as a central issue in the robotic research area. The purpose of robot
control is to maintain the dynamic response of the manipulator in accordance with
some pre-specified performance criterion. The motion control can be classified into
three major categories for the purpose of discussion.
13
Joint Motion Controls
Joint servomechanism
Computed torque control technique
Minimum time control
Variable structure control
Non linear decoupled control
Resolved Motion Controls (CARTESIAN SPACE CONTROL)
Resolved motion rate control
Resolved motion acceleration control
Resolved motion force control
Adaptive Controls
Model referenced adaptive control
Self- tuning adaptive control
Adaptive perturbation control with feed forward compensation
Resolved motion adaptive control
Each of the control methods will be described in the subsequent topics
2.3 JOINT MOTION CONTROL
2.3.1. Joint Servomechanism
Joint servomechanism is basically a proportional plus integral plus derivative
control method (PID).
14
With its three-term functionality covering treatment to both transient and steady-
state responses, proportional-integral-derivative (PID) control offers the simplest and
yet the most efficient solution to many real-world control problems. Since the
invention of PID control in 1910, and the Ziegler and Nichols (Z-N) (1942) addressed
tuning methods in 1942, the popularity of PID control has increased tremendously.
With the advances in digital technology, the science of automatic control now offers a
wide spectrum of choices for control schemes. However, more than 90% of industrial
controllers are still implemented based on the PID algorithms (Astrom and Hagglund
1995), particularly at low levels. This is because no other controller matches the
simplicity, clear functionality, applicability, and ease of use offered by the PID
controller. It’s wide application has stimulated and sustained the development of
various PID tuning techniques, sophisticated software packages, and hardware
modules. In order to increase the effectiveness of the controller, the tuning of the PID
parameter has to be identified. The selection of these parameters has been addressed by
many authors (Toscano 1996; Cominos and Munro 2002).
Takegaki and Arimoto (1981) stated that PD control with gravity compensation
has been proved to achieve global asymptotic stability whatever be the values of the
control gains, there by exploiting the properties of natural systems, namely
unconstrained time invariant systems lying in a conservative force field, a class to
which the robot arms belongs to. Arimoto and Miyazaki (1984) proposed a proof for
the stability of PID control of robot arms which is an extension of the well-known
proof of global asymptotic stability of PD control plus gravity compensation. Qu and
Dorsey (1991) proposed a similar proof for the uniform ultimate boundedness of the
error in the trajectory tracking problem.
Wen and Bayard (1988) claimed that PID control is inadequate to cope with
highly nonlinear systems, since the design of the control law is based solely on local
arguments worked out on the linearized system. The main drawback of this control
15
method is that the feedback gains are constant and pre-specified. It does not have the
capability of updating the feedback gains under varying payloads. Since an industrial
robot belongs to a highly nonlinear system, the inertial loading, the coupling between
the joints and the gravity effects are all either position dependent or position and
velocity dependent terms. The stability and robustness are obtained by tuning PID
parameters. The tuning is the basic requirement for obtaining good stability and
robustness of this control system.
Literatures related to finding tuning are presented in the subsequent paragraphs.
Siljak (1978).Mills and Goldenberg (1988), Wen and Bayard (1988); Astrom
and Hagglund (2001); Paolo Rocco (1996); Katebi and Moradi (2001); Ala Eldin
Abdallah Awouda et al (2003); M. H. Moradi (2003); Jing-Chug Shen et al (2004),
Dingyu Xue (2007) et al; Su et al (2009). Thiang et al (2009). From the above
mentioned papers, it has been observed that the effectiveness of the tuning and
robustness are still evolving and new technologies will emerge in future. Furthermore,
at a high speed the inertial term may vary drastically. Thus the above control method
that uses constant feedback gains to control a nonlinear system does not perform well
under varying speeds and pay loads.
2.3.2 Computed Torque Control Technique
The research on force control of rigid manipulators began as early as 1960’s but,
the algorithm was systemized between 1970’s to 1980’s. Raibert et al (1981) and
Mason (1981) developed and divided the force control mainly into hybrid
position/force control schemes and impedance control schemes. So, till date, force
control of rigid manipulators has been one of the hottest research topic. However, the
research on force control of flexible manipulators just began in 1985 by Fukuda. Chiou
16
and Shahinpoor (1988) pointed out that the link flexibility is the main cause of
dynamic instability. They have extended their research from planar one-link flexible
manipulator to two-link manipulator, analyzing the stability by applying ahybrid
position/force control schemes. Matsuno and Yamamoto (1993) have addressed a
quasi-static hybrid position/force control scheme and dynamic hybrid position/force
control scheme for a planer two DOF manipulator with flexible second link. For force
control of flexible manipulators, the inverse kinematic task is an essential, and thus has
been proposed by Svinin and Uchiyama (1994).
In end point force control strategies, the force signals are measured by a wrist
sensor. The sensor output is fed back to the controller to alter the system performance.
Many such closed - loop systems have been built using various force control algorithm
and more problems on stability have been experimented. A theoretical treatment of
environmentally imposed constraints is provided by Matson (1981) who also suggests a
control methodology to augment these natural constraints with an appropriate set of
artificial constraints. Raibert et al (1981) developed hybrid control architecture capable
of implementing Mason’s theory. Jin-Soo Kim et al (1996) discussed about the
dynamic properties of robot system namely instability and limited performance. Fu et
al (1987) also discussed about computed torque control. A Control scheme makes
direct use of the complete dynamic model of the manipulator to cancel the effect of
gravity, coriolis force, friction, and manipulator inertia tensor. It is similar to PID
control laws which have two portions – a model based portion and a servo portion.
From the above literature, it can be observed that the computer torque controller
employs nonlinear feedback to linearize error dynamics and this provides better
trajectory tracking performance than the linear controllers since it is based on a more
accurate dynamic model. But the biggest limitation of this approach is that, it is
computationally more intensive and hence, highly expensive as well. Inaccuracy in the
17
parameters of the manipulator in the dynamic model is the other factor, which limits
the manipulator’s performance.
2.3.3 Minimum Time Control
For manufacturing tasks, it is desirable to move a manipulator of its highest
speed to minimize the task cycle time. The problem of minimum time control of a
robot manipulator concerns with the determination of control effort and the
corresponding trajectories that will drive the end effectors of a manipulator from the
given initial positions to the final positions within a minimum span of time.
Kahn and Roth (1971) were the first to study this problem and investigate the
problem of optimal time control for manipulators. They proposed an approximate
solution to this problem which gave the better results in near minimum time control.
These problems can be categorized in terms of different constraints that affect
manipulator motion as follows:
There are constraints on the intermediate configurations of the robot
arms due to which the manipulator either follows a specified path or
performs the movement in order to avoide collisions with the
obstacles in the work space.
The motion path space is obstacle-free and unconstrained between
two end points.
For the first category, certain algorithms have been proposed for minimum time
control of robot manipulator along a specified geometric path (Shin and Mckay 1985;
Bobrow et al 1985; Chen and Desrochers 1989; Shiller and Dubowsky 1991).
18
Solving the minimum time control problem for the second category represents a
difficult and elusive point to point minimum time control problem. The approach to
this problem can be categorized under the following two groups.
Dynamic programming or space search methods.
Calculation of variation based methods.
The first group includes certain approximation methods such as artificial
intelligence (AI) approaches (Sahar and Hollerbach 1985; Rajan 1985; Shiller and
Dubowsky 1991) and non linear parameters optimization methods. The basic is the use
of an initial feasible trajectory in conjunction with an exhaustive search of the
unconstrained motion between two end points.
The calculation of variation is based on methods that use the techniques of
variational calculus and apply Pontryagin’s maximum principle to reduce the time
optimization into a two point boundary value problem. There arises a difficulty in
finding a solution for the two point boundary value problem. Since the state and the
constant variables at the initial values and final values are only partially known. Hence,
initial values must be guessed to match the correct set of final values. Most of the robot
manipulators being nonlinear systems, the numerical methods are used to solve the
nonlinear two point boundary value problem. It was found that, it is highly sensitive to
the initial constant values that were guessed. However, it is extremely difficult to have
any simple interpolation or physical meaning. In general, the values of time optimal
control solutions are very difficult to obtain.
Meier and Bryson (1990) suggested a fast algorithm for a planer two-link
manipulator to travel a specified distance in minimum time with unspecified initial and
final position. This method reduces the computation time by forcing the control inputs
to be ‘bang-bang’ on both regular and singular intervals, and thus makes it feasible to
19
find minimum time solution for various initial and final positions and various dynamic
properties of a robot.
Lin (1992) proposed an efficient two-level parallel algorithm for solving the
general time optimal problem in discrete form. Chen et al (1993) developed an
algorithm for computing the time optimal trajectory for robotic point to point motion
based on perturbation and continuation methods. Both non-singular and singular cases
can be handled by converting the original time optimal problem into a perturbed time
optimal problem.
Yang (1993) proposed a two step Lyapunov approach for generating near
minimal time point to point motion of a robot manipulator. This method is based on the
observation of the kinetic energy mechanisms in minimum time operations of robot
manipulators.
It is extremely difficult, if not impossible, to obtain an exact closed –form
solution to the problem of point to point minimum time control motion of robot
manipulator, due to the nonlinearity and the coupled nature of the robot manipulator
dynamics. The available numerical solutions are computational intensive, required
good initial boundary values guesses, or sensitive to the time step or some other
variables in the numerical procedures. In particular, the question of existence of time
optimal control containing singular arcs is open. As a result, the practical applicability
of available methods is very limited. (Tao Fan 2003).
This control method is usually too complex to be used for manipulator with 4 or
more degrees of freedom and neglects the effects of the unknown external loads.
2.3.4. Variable Structure Control
20
Variable structure control (VSC) of robotic manipulators has been receiving
more and more attention recently. The variable structure system (VSS) has several
interesting and important properties that cannot be easily obtained by other approaches.
Increasing demand for high performance robots has lead to the development of various
advance control techniques. Two general controller approaches may be considered for
robot manipulators, the model-based approach and the non-model-based approach. The
model based controllers consider some of the system structures in their designs. In
contrast non-model-based controllers do not take account of the system properties. The
later group of controllers has been largely used in industry because of the ease of
implementation and relatively good performances. However the increasing demand for
high speed response and accuracy leads to more and more interest in the model-based
controllers. The idea behind model-based controllers are uses the dynamic equations of
the system as feed forward terms of the control algorithm, transforming the non-linear
system into a new linear system incorporating the robot and its controller. This
linearization is only effective if the exact dynamic equations are available, which in
practice not the case is as uncertainties on the parameters as well as on the model
equations exist. To deal with these imperfections, a robust algorithm can be used.
Russian authors initially proposed the VS theory in the 60's using the
mathematical work of Filippov (Filippov 1964). Several researchers have been
studying it, both in continuous time (DeCarlo et al 1988; Fu and Liao 1990; Utkin
1992) and in discrete time (Gao Wang and Homaifa 1995).VS controllers provide an
effective and robust way for controlling nonlinear plants and its roots are the bang-
bang and relay control theory (DeCarlo Zak and Matthews 1988). The term VS control
arose because the controller's structure is intentionally changed according with some
rule to obtain the desired plant behaviour. Due to this structure change, the resulting
control law is nonlinear.
21
Young (1978) proposed the hierarchy approach to the control of robotic
manipulators. Slotine and Sastry (1983) developed a methodology of feedback control
to achieve accurate tracking of manipulators. Yeung and Chen (1988) and Chen, Mita
and Wakui (1990) proposed some control algorithms where the inverse of inertia
matrix was not required. Young (1988) presented a variable structure model following
control (VSMFC) design for robotic applications. Leung, Zhou and Su (1991) provided
an adaptive VSMFC design for robot manipulators which did not require knowledge of
nonlinear robotic systems.
Accurate trajectory following in robotics is an important problem, and many
control algorithms using VSC. and sliding mode have been proposed by (Slotine et al
1983; Bailey et al 1987; Wijesoma et al 1990; Yi-Feng Chen et al 1990; Gao et al
1993; and Yu Tang 1998). The limitation with the use of sliding mode is the high
frequency switching, commonly known as chattering. Chattering is unacceptable in
robotics as it may excite unmodeled high frequency modes, which could damage the
robot manipulator and high frequency plant dynamics (Kwatny and Siu 1987). The
general approach to overcome the problem is to replace the non-linear switching
function by a smooth one as in (Slotine et al 1991 and Ambrosino et al 1984). This
method however seriously alters the performance of the controller, because of the high
degree of smoothness needed to completely overcome chattering.
There exist several techniques to eliminate chattering. The widely-adopted
approach to chattering-free VSS is the so-called boundary layer one, where the
discontinuous VSS control law’s sign function is approximated using saturation
nonlinearity within small vicinity around the sliding surface (Slotine and Li 1991;
Hung et al 1993). Unfortunately, boundary layer controllers do not guarantee
asymptotic stability but rather uniform ultimate boundedness (Corless and Leitmann
1981; Esfandiari and Khalil 1991). As a consequence, there exists a trade-off between
22
the smoothness of control signals and the control accuracy. Some boundary layer width
modification techniques to improve tracking precision are discussed in (Slotine and Li
1991; Arashima et al 1986; Yu et al 1994; Chen et al 2002). Nevertheless, these
proposed methods in practice lead to a computational burden when implemented, or are
applicable only to linear systems. The proposed research uses these control system to
position the robot arm accurately.
2.3.5 Non Linear Decoupled Control
There is a substantial body of nonlinear control theory which allows one to
design a near – optimal control strategy for mechanical manipulators. Most of the
existing robot control algorithms emphasize compensation of the interactions among
the links, e.g., the computed torque technique. Hemami and Camana (1976) applied the
non linear feedback control technique to a simple locomotion system which has a
particular class of nonlinearity (Sine, cosine and polynomial ) and obtained decoupled
subsystems, postural stability, and desired periodic trajectories. Their approach is
different from the method of linear system decoupling where the system to be
decoupled must be linear. Saridis and Lee (1979) proposed an iterative algorithm for
sequential improvement of a non linear suboptimal control law. To achieve such a high
quality of control, this method also required a consider amount of computational time.
Falb and Wolovich (1967), Freund (1982) applied this scheme to compute nonlinear
decoupled controller for robot manipulator. If the manipulator is considered as six
independent, decouple, second order, time invariant systems, it is easy to compute and
estimation of parameters are easy. But practically possible under some extend only.
Modi and Karray (1995) implemented non linear decoupled control system in to space
platform in the presence of induced disturbances and the results suggest an unstable
behaviour of the uncontrolled system in the presence of the manipulator maneuvers.
Yuan et al (2002) uses these control systems in addition to the fuzzy controller to avoid
23
disturbances in cart- pole and Ball beam systems. Hung et al (2007) proposed a neural
adaptive based algorithm with decoupled controller to achieve asymptotic stability in
cart- pole and Ball beam systems. Newton-Euler equation of motion is used to
estimate parameters of the control system.
In general, these control systems exhibits unavoidable disturbances and
instability.
2.4. RESOLVED MOTION CONTROLS
In many applications, resolved motion control, which commands the
manipulator hands to move in a desired Cartesian direction in a coordinate position and
rate control. Resolved motion means that the motions of various joint motors are
combined and resolved in to separately controlled hand motion along the world
coordinate axes. This implies several joint motors must run simultaneously at different
time varying rates in order to achieve desired coordinated hand motion along any world
coordinate axes. This enables the user to specify the direction and speed along any
arbitrarily oriented path for the manipulator to follow. This motion control greatly
simplifies the specification of the sequence of motions for completing a task because
users are usually more adapted to the Cartesian coordinate system. The problem of
finding the location of the hand is reduced to finding the position and orientation of the
hand coordinate frame with respect to the inertial frame of the manipulator. Under this
system, there are three kinds control system namely resolved motion rate control,
resolved motion acceleration control and resolved motion force control.
2.4.1. Resolved Motion Rate Control
It means that the motions of the various joint motors are combined and run
simultaneously at different varying time rates in order to achieve steady hand motion
24
along any world coordinate axis. This is required to compute the inverse Jacobian
matrix. This control system was initiated by Whitney (1969). Luh et al (1980) extends
the concept of resolved motion rate control to include acceleration control. Klein et al
(1983) and Das et al (1988) discussed about resolved motion rate control to solve the
inverse kinematics problem. The computation of Jacobian is additional computation
which leads to increase the computation time and also complexity and still the research
is being carried out to compute Jacobian matrix.
2.4.2. Resolved Motion Acceleration Control
It presents an alternative position control which deals directly with the position
and orientation of the hand of a manipulator. Luh et al (1980) extends the concept of
resolved motion rate control to include acceleration control and stated as, In order to
cope with non linear and coupled nature of the manipulator dynamic model, inverse
dynamic control can be pursued with consists of designing a model based
compensating action which globally linearizes and decouples the mechanical system in
terms of resolved task space acceleration. The computation of rotation matrix and end
effectors position vectors very complex task. The drawback of this approach is the
occurrence of representation of singularities. Caccavale et al (1998) reviewed about the
resolved acceleration control technique for the tracking control problem of robot
manipulator in the task space. The stability analysis, computational burden and
tracking performance for various acceleration controls are in order concerning the pros
and cons of each scheme. They stated that, there is no guarantee to avoid the
occurrence of representation singularities even when good –end effectors orientation
tracking is achieved. Jinding Tan et al (2004) proposed a singularity-free tracking
algorithm for robot manipulator using hybrid approach. Musa Mailah et al (2005)
proposed a new algorithm called resolved acceleration control and active force control.
In this they have stated that resolved acceleration control was designed for handling the
25
kinematics problem and active force control was serial designed to facilitate the
dynamic aspects. Kim et al (2009) discussed path tracking control for underactuated
AUVs and found that the initial stage error is maximum. This control system uses
Newton – Euler equations of motion to find out the errors in position and velocities.
These control methods are characterized by extensive computational requirements;
singularities associated with the Jacobian matrix and need to plan a manipulator hand
trajectory with acceleration information.
2.4.3. Resolved Motion Force Control
The basic concept of this type of control scheme is to determine the applied
torque to the joint actuators in order to perform the Cartesian position control of the
robot arm. Wu and Paul (1982) developed these control system. The advantages of this
controller is that the control is not based on the complicated dynamics equation of
motion of the manipulator and still has the ability to compensate for changing arm
configurations, gravity loading forces on the links and internal friction.
In general, the resolved motion control lies in the fact that the inverse Jacobian
matrix requires intensive computation.
2.5. ADAPTIVE CONTROL
Most of the schemes discussed in the previous sections control the arm at the
hand or joint level and emphasize nonlinear compensations of the interaction forces
between the various joints. These control algorithms sometimes are inadequate because
they require accurate modeling of the robot arm dynamics and neglect the changes of
the load in a task cycle. These changes in the pay load of the controlled system often
26
are significantly enough to render the above feedback control strategies ineffective.
The result is reduced servo response speed and damping, which limits the precision and
speed of the end effectors. The use of wide range of manipulator and varying pay loads
requires the consideration of adaptive control techniques. The basic idea of adaptive
control is to change values of the gains or parameters in the control law according to
some on line algorithm. In this way the controller can learn an appropriate set of
parameters during the course of its operation. (Spong and Vidyasagar 2008).
2.5.1. Model Referenced Adaptive Control
Among various control methods, the model referenced adaptive control is the
mostly widely used and it is also relatively easy to implement. The concept of model
referenced adaptive control is based on selecting an appropriate reference model and
adaptation which modifies the feedback gain to the actuators of actual system.
Dubowsky and DesForges (1979) proposed a simple model reference adaptive
control for the control of mechanical manipulators. In their analysis, the payload is
taken into consideration by combining it to the final link, and the end effector
dimension is assumed to be small compared with the length of other links. A linear
second order time invariant differential equation is selected as the reference model for
each degree of freedom of the robot arm.
The fact that this control approach is not dependent on a complex mathematical
model is one of its major advantages, but stability consideration of the closed loop
adaptive system is critical. A stability analysis is difficult, and Dubowsky and
DesForges (1979) carried out an investigation of this adaptive system using a
linearized model. However, the adaptability of the controller can become questionable
if the interaction forces among the various joints are severe.
27
Meng Joo Er et al (2003) stated that the manipulators are multivariable
nonlinear coupling systems and frequently subjected to structured and /or unstructured
uncertainties even in a well – structured setting for industrial use. Structured
uncertainties are mainly caused by imprecision in the manipulator link properties,
unknown loads, and so on. The unstructured uncertainties are caused by unmodeled
dynamics, disturbances and high-frequency models of the dynamics. As result it is
difficult to obtain an accurate mathematical model. The control often suffers from
heavy computational burden and this hinders their real-time applications. Furthermore,
control performance is often degraded due to the existence of modeling errors and
external disturbances. Hence, there is a need for model-free adaptive control strategies
(Slotine and Li 1991; Astrom and Wittenmark 1995; Lewis et al 1993 and Ge, Lee, and
Harris 1998).
2.5.2. Adaptive Control Using an Autoregressive Model
This method is basically different from the reference model method. The
problem induced by the setting up of the control rule splits up here in to two steps. First
one is on line identification of the robot dynamic parameters. The second one is the
synthesis of an optimized control rule for these parameters (Leiniger 1981; Koivo Guo
1983). Koivo and Guo (1983) proposed an adaptive, self – tuning controller using an
auto regressive model to fit the input- output data from the manipulator. The control
algorithm assumes that the interaction forces among the joints are negligible. Bercu
and Portier (2002) investigated the stability and the optimality of the adaptive
tracking for a wide class of parametric nonlinear autoregressive models, via a new
martingale approach. Stylianos et al (2006) used the multi-model partitioning theory
for simultaneous order and parameter estimation of multivariate autoregressive models
to select the correct model. So the exact model needed for control the manipulator
28
model. In the above two control methods, it is assumed that the interaction forces
among the joints are negligible (Lee et al 1985)
2.5.3. Adaptive Perturbation Control
Based on perturbation theory, Lee and Chung (1985) proposed an adaptive
control strategy which tracks a desired time- based manipulator trajectory as closely as
possible for all times over a wide range of manipulator motion and payloads. Adaptive
perturbation control differs from the above adaptive schemes in the sense that it takes
all the interactions among the various joint into consideration. The adaptive control is
based on linearized perturbation equations about the referenced trajectory. There is a
need to drive appropriate linearized perturbation equations suitable for developing the
feedback controller which computes perturbation joint torque to reduce position and
velocity errors along the nominal trajectory.
2.5.4. Resolved Motion Adaptive Control
The resolved motion adaptive control is performed at the hand level and is based
on the linearized perturbation system along a desired time based hand trajectory.
The resolved motion adaptive control differs from the resolved motion
acceleration control by minimizing the position / orientation and angular and linear
velocities of the manipulator hand along the hand coordinate axes instead of position
and orientation errors. A low level microprocessor is used to control joint variable
because they still do not have the required speed to compute the controllers for
standard frequency.
Likewise the number of other types of adaptive control systems exists and it is
in still research (Abdallah et al 1991; Berghuis and Nijmeijer 1993; Burg et al 1996;
29
Yazarel et al 2002; Ngo 2005; Keng Peng Tee and Haizhou Li 2009). Because of the
instability, the calculations of inverse Jacobian matrix are very difficult. The main
drawback of the above control schemes consists in assuming the torque applied directly
to the robot link
2.6. A BRIEF SURVEY OF ROBOT ARM CONTROL PROBLEM
Murugasan et al (2004) have studied the parameters governing the arm model of
a robot control problem through RK–Butcher algorithm. The exact solution of the
system of equations representing the arm model of a robot have been compared with
the corresponding discrete solutions (approximate solutions) at different time intervals
using RK–Butcher algorithm and also the absolute error between the exact and discrete
solutions have been determined.
Devarajan Gopal et al (2006) attempted to solve the robot arm control problem
by considering the system of equations in terms of second order linear time invariant
case and obtained solution by one of the RK algorithm, But still there was error and
this simulation was not validated.
Senthilkumar (2009) focused on providing numerical solutions for system of
second order robot arm problem using RK-eight stage seventh order limiting formulae.
But still there was error and this simulation was not validated.
Ponalagusamy and Senthilkumar (2010) adapted RK-sixth order algorithm to
solve robot arm problem. Results and comparison showed that the efficiency of the
RK-sixth order algorithm based on the absolute error between the exact and
approximate solutions at different time intervals. The corrective measure has been
taken to obtain the stability polynomial, the ranges for the real part of λh and graphical
30
representation of the stability regions for the case of RK-Butcher algorithm, obtained
by Sekar et al (2004), Murugesan et al (2004, 2005). It is found that the error involved
in the numerical solution obtained by RK-sixth order algorithm is less in comparison
with that obtained by the RK-fifth order and RK-fourth order algorithms respectively.
They didn’t validate their work.
31
S.NoPublication
YearAuthors Method
Control
law
Reference
Robot arm
model
DOFNumerical
Values
ε
OrderTracking Remarks
1 2004
Murugasan.K,
Sekar.S.,
Murugash.V.
Park.J.Y.
RK–Butcher
AlgorithmVSC
Approx. L-E,
Time
invariant
2 DOF
revoluteNormal 10-6
NA
In correct
Stability range
2 2006
DevarajanGopal
V. Murugesh K.
Murugesan
RK–Butcher
Algorithm
Arithmetic mean
VSC
Approx. L-E,
Time
invariant
2 DOF
revoluteNormal 10-6 NA
In correct
Stability range
3 2008 S. SenthilkumarRK-eight stage
seventh orderVSC
Approx. L-E,
Time
invariant
2 DOF
revoluteNormal 10-9 NA
Controversial
Statement by the
author with the
previous method.
4 2009R. Ponalagusamy
S. Senthilkumar
Runge-Kutta
Sixth order
algorithm
VSC
Approx. L-E,
Time
invariant
2 DOF
revoluteNormal 10-9 NA
Sixth order yields
little error.
Table 2.1 Survey of robot arm control problem
Abbreviations: Normal: Simulated with normalized values, SimReal: Simulated on the model of a real robot, Real:
Implemented on a real robot. NA- Not Applicable (Tracking was not analyzed); ε- error values
32
But the proposed research proved that the measured error less as compared to RK-sixth
order algorithm.
The table 2.1 shows a brief survey about various authors’ contribution and their results.
The authors used approximate L- E model two DOF robot, with VSC control law to
justify their mathematical tool. But practically the proposed tools are not validated and
they mathematically viewed the problem. The proposed research uses the same model
and control system and proved that the proposed tool measures error exactly and
validated by using real time robot.
2.7 HAAR WAVELET SERIES
Haar functions have been used from 1910 and introduced by the Hungarian
mathematician Alfred Haar. Haar wavelets are the simplest wavelets among various
types of wavelets. They are step functions (piecewise constant functions) on the real
line that can take only three values. Haar wavelets, like the well-known Walsh
function, form an orthogonal and complete set of functions representing discretized
functions and piecewise constant functions. A function is said to be piecewise constant
if it is locally constant in connected regions.
Haar wavelets have been applied extensively for signal processing in
communications and physics research, and have proved to be a wonderful
mathematical tool. The pioneer work in system analysis via Haar wavelets was done by
Chen and Hsiao (1997) who first derived a Haar operational matrix for the integrals of
the Haar function vector and paved the way for the Haar analysis of dynamic systems.
The fundamental work in state analysis and parameters estimation of bilinear systems
via Haar wavelets was done by Hsiao et al (1997) who first derived a Haar product
matrix and a coefficient matrix. Hsiao and Wang (1998) proposed a key idea to
transform the time-varying function and its product with the state into a Haar product
33
matrix. Since then, all related algorithms can be implemented easily. All severe
mathematical conditions can be met perfectly. The main characteristic of this technique
is to convert a differential equation into an algebraic one; hence, the solution, the
identification, and the optimization procedures are either reduced or simplified
accordingly.
Chen and Hsiao (1997) analyzed lumped and distributed parameter dynamic
systems and established an operational matrix of integration based on Haar wavelets.
Hsiao (1997) analyzed state analysis of time delayed systems via Haar wavelets.
Based upon some useful properties of Haar functions, a special product matrix and a
related coefficient matrix are applied to solve the time-delayed systems. The unknown
Haar coefficient matrix is solved via the Kronecker product method. The high accuracy
and the wide applicability of Haar approach was proved by demonstrated with
numerical examples.
Chen and Hsiao (1999) examined to compare the fast capabilities of operational
matrices of integration formed by various functions. They found that Haar wavelet
operational matrix is the fastest. They formulated a simple and complete procedure for
optimizing a dynamic system based on the established matrix. They considered
different constraints and various conditions of the time-invariant system optimization
problems and solved with the new wavelet approach.
Hsiao and Wang (1999) attempted to solve the time-varying singular nonlinear
systems via Haar wavelets. Based upon some useful properties of Haar wavelets, a
special product matrix and a related coefficient matrix is applied to the time-varying
systems such that the state of time-varying singular nonlinear systems can be solved
easily.
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Hsiao and Wang (2000) made state analysis of time-varying singular bilinear
systems via Haar wavelets. Based upon some useful properties of Haar wavelets, a
special product matrix and a related coefficient matrix are applied to the time-varying
systems such that the state of time-varying singular bilinear systems has been solved
easily. The local property of Haar wavelets is advantageous to shorten the calculation
process in the task.
A simple and effective algorithm based on Haar wavelet was proposed to the
solution of nonlinear stiff problems by Hsiao and Wang (2001). The simulation result
showed that the whole computation time can be reduced to one tenth of the well-known
Runge–Kutta–Fehlberg approach, while the accuracy is nearly the same.
Hsiao (2004) established a clear procedure for the variational problem solution
via Haar wavelet technique. The variational problems are solved by means of the direct
method using the Haar wavelets and reduced to the solution of algebraic equations. It
has been proved that the approach is more powerful either than Ritz’s or Euler’s direct
methods for solving variational problems.
Hsiao (2004) proposed a simple and effective algorithm called Single Term
Haar Wavelet series method based on the Haar wavelets to obtain the solution for
linear stiff problems. The simulation result showed that the single-term Haar wavelet
method (STHWS) is better than the classical Runge–Kutta fourth-order method (CRK)
Kalpana and Raja Balachandar (2007) made an analysis for observer design in
the generalized state space or singular system of transistor circuits. It can be easily
implemented with a digital computer and it is found to be fast, flexible, convenient and
computationally attractive.
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Bujurke et al (2007) applied Single Term Haar Wavelet Series (STHWS)
approach to solve of nonlinear stiff differential equations arising in nonlinear
dynamics. Form the results it has been proved that the STHWS turns out to be more
effective in its ability to solve systems ranging from mildly to highly stiff equations
and is free from stability constraints.
Bujurke et al (2009) applied a novel Single-Term Haar Wavelet Series
(STHWS) method to solve the Duffing equations and Painleve’s transcendent (PI and
PII). The results, the STHWS method has shows the remarkable features as compared
to the other wavelet based techniques.
Yuanlu Li (2010) derived Chebyshev wavelet operational matrix of the
fractional integration and used to solve a nonlinear fractional differential equations.
Yuanlu Li and Weiwei Zhao (2010) derived the Haar wavelet operational matrix of the
fractional order integration, and used it to solve the fractional order differential
equations including the Bagley–Torvik, Ricatti and composite fractional oscillation
equations. The results obtained are in good agreement with the existing ones in open
literatures.
2.8. CONCLUSIONS
The number of research works has been reviewed about various robot
manipulator control methods. They are varying from a simple servomechanism to
advanced control schemes such as control with an identification algorithm. The control
techniques are discussed in joint motion control, resolved motion control and adaptive
control. Each control system has a unique features and also demerits. Apart from this
variable structure control have the unique features. The features are
The exact knowledge of the manipulator’s model is not needed
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Under certain conditions, the system performance is insensitive to
bounded external disturbances.
These properties are considered for the proposed research and this control
system is used in the two DOF, LR model, pick and place robot which is used for
validation purpose in this research.
The proposed research uses the Single Term Haar Wavelet series method
(STHWS) to solve the dynamic equation of motion by considering nonlinear,
decoupled second order differential equation into singular and non singular system
time varying and time invariant cases. The computer and real time simulation has been
made to obtain desired trajectory. STHWS method is derived from the haar wavelet by
taking P1x1= ½. The STHWS turns out to be more effective in its ability to solve any
kind of differential equations and is free from stability constraints.