# Intelligent Hybrid Control Strategy for Trajectory Tracking of Robot Manipulators

Authors: Yi Zuo; Yaonan Wang; Lihong Huang; Chunsheng Li
Journal: Journal of Control Science and Engineering
 (2008)

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## Abstract

We address the problem of robust tracking control using a PD-plus-feedforward
controller and an intelligent adaptive robust compensator for a rigid robotic manipulator with uncertain dynamics and external disturbances. A key feature of this scheme is that soft computer
methods are used to learn the upper bound of system uncertainties and adjust the width of the boundary layer base. In this way, the prior knowledge of the upper bound of the system uncertainties
does need not to be required. Moreover, chattering can be effectively eliminated, and asymptotic error convergence can be guaranteed. Numerical simulations and experiments of two-DOF rigid robots are presented to show effectiveness of the proposed scheme.


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## Body

## 1. Introduction

Robot control has been an active research field during the past decade.
A main concern of the control task is to find an effective controller to
achieve accurate tracking of desired motions. However, robotic manipulators
have to face many uncertainties in their dynamics, such as payload parameter,
friction, and disturbance. Yet, the work using PD-plus-feedforward on robust
control of robot [1–6] shows great attraction to
researches. In [1], a
PD-plus-feedforward controller is firstly proposed for robot manipulators which
guarantees exponential convergence. The adaptive version of this controller is
addressed later [2]
considering the robustness of velocity measurement. However, the main drawback
of them is that they require high controller gains in order to overcome the
uncertainty in the initial parameter errors. Combining the ideas of [1, 3], Berghuis [4] put forward a modified plan ensuring global
exponential convergence. Wen [5] treats the analysis of a robot controller based on
energy-like Lyapunov function, which leads to a new class of global stability
control design, but is still a high-gain method.
Kelly [6] designs a
class of PD-plus-feedforward controller on the assumption that parameters
are precisely known, nevertheless obtaining local but not global stability
results. Unfortunately, all above works do not consider the external
disturbances and unmodeled dynamics.Recently, considerable research efforts have been
directed towards applying intelligent control methods to robot control [7–14]. In [9–11], intelligent controllers are proposed for the
compensation of effects yielded by nonlinearities and uncertainties. Some
theses [8, 12, 13] use soft computer methods
to approximate the whole robot dynamics rather than its nonlinear components as
done by static intelligent means. Man [7] and Qi [15] make use of intelligent systems to replace the
constant switching control gain and the width of the boundary layer. However, [7] has the
shortcoming that the inverse of inertia matrix needs to be calculated.In this note, motivated by the PD-plus-feedforward
control idea and Man's work [7], a new intelligent adaptive robust control scheme is
proposed for robot manipulators with model uncertainties and external
disturbance. The novel control scheme proposed in this paper contains a
PD-plus-feedforward part and an intelligent robust compensational part. As
compared with afore-mentioned results, not only does we suppose
that the mathematical model of robot system is
totally unknown, but we do not require calculating the inverse of inertia
matrix. Moreover, the parameter of the proposed dynamic compensator is
adaptively updated by using the output of intelligent system that is used to
learn the unknown upper bound of system uncertainties, and a fuzzy logic is
used to adjust the width of the boundary based on the state. In this way, the
vibrancy of the controller will be effectively reduced. Application to a
two-link robotic manipulator is presented. Numerical simulations and
experiments are proposed to show the excellent tracking performance of our
controller.


## 2. Preliminaries

In this section, we firstly give some descriptions of
intelligent systems, and after that, present the robot manipulator model and
some dynamic properties which are crucial to our work.


### 2.1. Description of Fuzzy-Logic Systems

The basic configuration of the fuzzy system is
constructed from the fuzzy if-then rules using some specific inference,
fuzzification, and defuzzification strategies [16]. Let x=(x1,…,xN)T∈U⊂RN and y∈V⊂R be the input and output of the fuzzy system.
The fuzzifier maps a crisp point in U into a fuzzy set in U.
The fuzzy rule base consists of a collection of fuzzy if-then rules.R(l):
If x1 is F1l and ⋯ and xN is FNl,
then y is Gl, in which Fil and Gl are fuzzy sets, and l=1,…,M,
where M denotes the number of fuzzy if-then rules. The
fuzzy inference engine performs a mapping from fuzzy sets in U to fuzzy sets in V.
The defuzzifier maps a fuzzy set in V to a crisp point in V.
A more detailed description of these systems can be found in [16].The fuzzy systems with center-average defuzzifier,
product inference, and singleton fuzzifier can be expressed as
in the following form:(1)y(x)=∑l=1Mθl(∏i=1NμFil(xi))∑l=1M(∏i=1NμFil(xi)), where μFil(⋅) is the numbership function of the fuzzy set Fil and θl is the point at which μGl achieves its maximum value (it is assumed here
that μGl(θl)=1). By the universal approximation
theorem [16], for any
given real continuous function f(x) on a compact subset U⊂RN and arbitrary ϵ>0,
there exists a fuzzy system y(x) in the form of (1) such that maxx∈U∥f(x)−y(x)∥<ϵ.


### 2.2. Description of Neural-Network Systems

The basic configuration of the neural network system
is implemented by using massive connections among processing units. Letx∈U⊂RN be the input of the neural network system. A
two-layer network has a network output given by [17, 18] (2)y(x):=∑l=1MθlHl(∑j=1Mωjlxj+ml), where M is the number of hidden neurons, Hl(⋅) denotes the activation function which must be
a nonconstant, bounded, and monotonically increasing continuous function (e.g.,
hyperbolic tangent function), ωji is the first-to-second layer interconnection
weights, ml is the biases, and θi is the second-to-output interconnection
weights.The neural network system is also a popular class of
function approximators that can be used in a similar setting as the fuzzy logic
system described above. That is, for any given real continuous functionf(x) and arbitrary ϵ>0,
there exists a neural network system y(x) in the form of (2) such that maxx∈U∥f(x)−y(x)∥<ϵ.


### 2.3. Description of Robot Manipulator Model

The basics of robot dynamics and control are
sufficiently well known by now that we will be brief in our derivation of the
control algorithm. Thus, given the Euler-Lagrange dynamic equations for ann-link robot [19] (3)M(q)q¨+N(q,q˙)=τ,N(q,q˙)=C(q,q˙)q˙+G(q)+Fd(q˙)+Fs(q˙)+τd(t,q,q˙), where q,q˙,q¨∈Rn are the joint position, velocity, and
acceleration vectors, respectively; M(q) denotes the inertia matrix; C(q,q˙) expresses the coriolis, centripetal matrix; G(q) is the gravity vector; Fd,Fs(q˙)∈Rn×n represent the dynamic friction coefficient
matrix and static friction vector, respectively; τd(t,q,q˙) is the vector of disturbances and
unmodeled dynamics; τ is the control vector representing the torque
exerting on joints. For our purposes, we assume (3) to have the following
properties.Property 1.
M(q) is a positive symmetric matrix defined by Mm≤∥M(q)∥≤MM with Mm,MM>0 being known constants.Property 2.
C(q,q˙) defined by using the Christoffel symbols
satisfies that
(a)
M˙(q)−2C(q,q˙) is Skew symmetry;(b)
∥C(q,q˙)∥≤CM∥q˙∥ and C(q,x)y=C(q,y)x,x,y∈Rn,CM>0.Property 3.
Linear parameterization of the local
dynamics of each subsystem:(4)M(q)q¨+C(q,q˙)q˙+g(q)=Y(q,q˙,q¨)θ, where θ is a constant p-dimensional vector of inertia parameters and Y is an n×p matrix called regressor, which contains known
functions and is uniformly continuous in its arguments.Moreover, without losing generality, the following
assumptions on model (3) are made.Assumption.
The bounding
function of the external disturbances and unmodeled dynamicsτd(t,q,q˙) is assumed to be (5)∥τd(t,q,q˙)∥≤c0+c1∥q∥+c2∥q∥2+c3∥q˙∥+c4∥q˙∥2, where ci(i=0,…,4) are positive constants.Assumption.
The desired
trajectoryqd,q˙d, and q¨d are uniformly bounded.


## 2.1. Description of Fuzzy-Logic Systems

The basic configuration of the fuzzy system is
constructed from the fuzzy if-then rules using some specific inference,
fuzzification, and defuzzification strategies [16]. Let x=(x1,…,xN)T∈U⊂RN and y∈V⊂R be the input and output of the fuzzy system.
The fuzzifier maps a crisp point in U into a fuzzy set in U.
The fuzzy rule base consists of a collection of fuzzy if-then rules.R(l):
If x1 is F1l and ⋯ and xN is FNl,
then y is Gl, in which Fil and Gl are fuzzy sets, and l=1,…,M,
where M denotes the number of fuzzy if-then rules. The
fuzzy inference engine performs a mapping from fuzzy sets in U to fuzzy sets in V.
The defuzzifier maps a fuzzy set in V to a crisp point in V.
A more detailed description of these systems can be found in [16].The fuzzy systems with center-average defuzzifier,
product inference, and singleton fuzzifier can be expressed as
in the following form:(1)y(x)=∑l=1Mθl(∏i=1NμFil(xi))∑l=1M(∏i=1NμFil(xi)), where μFil(⋅) is the numbership function of the fuzzy set Fil and θl is the point at which μGl achieves its maximum value (it is assumed here
that μGl(θl)=1). By the universal approximation
theorem [16], for any
given real continuous function f(x) on a compact subset U⊂RN and arbitrary ϵ>0,
there exists a fuzzy system y(x) in the form of (1) such that maxx∈U∥f(x)−y(x)∥<ϵ.


## 2.2. Description of Neural-Network Systems

The basic configuration of the neural network system
is implemented by using massive connections among processing units. Letx∈U⊂RN be the input of the neural network system. A
two-layer network has a network output given by [17, 18] (2)y(x):=∑l=1MθlHl(∑j=1Mωjlxj+ml), where M is the number of hidden neurons, Hl(⋅) denotes the activation function which must be
a nonconstant, bounded, and monotonically increasing continuous function (e.g.,
hyperbolic tangent function), ωji is the first-to-second layer interconnection
weights, ml is the biases, and θi is the second-to-output interconnection
weights.The neural network system is also a popular class of
function approximators that can be used in a similar setting as the fuzzy logic
system described above. That is, for any given real continuous functionf(x) and arbitrary ϵ>0,
there exists a neural network system y(x) in the form of (2) such that maxx∈U∥f(x)−y(x)∥<ϵ.


## 2.3. Description of Robot Manipulator Model

The basics of robot dynamics and control are
sufficiently well known by now that we will be brief in our derivation of the
control algorithm. Thus, given the Euler-Lagrange dynamic equations for ann-link robot [19] (3)M(q)q¨+N(q,q˙)=τ,N(q,q˙)=C(q,q˙)q˙+G(q)+Fd(q˙)+Fs(q˙)+τd(t,q,q˙), where q,q˙,q¨∈Rn are the joint position, velocity, and
acceleration vectors, respectively; M(q) denotes the inertia matrix; C(q,q˙) expresses the coriolis, centripetal matrix; G(q) is the gravity vector; Fd,Fs(q˙)∈Rn×n represent the dynamic friction coefficient
matrix and static friction vector, respectively; τd(t,q,q˙) is the vector of disturbances and
unmodeled dynamics; τ is the control vector representing the torque
exerting on joints. For our purposes, we assume (3) to have the following
properties.Property 1.
M(q) is a positive symmetric matrix defined by Mm≤∥M(q)∥≤MM with Mm,MM>0 being known constants.Property 2.
C(q,q˙) defined by using the Christoffel symbols
satisfies that
(a)
M˙(q)−2C(q,q˙) is Skew symmetry;(b)
∥C(q,q˙)∥≤CM∥q˙∥ and C(q,x)y=C(q,y)x,x,y∈Rn,CM>0.Property 3.
Linear parameterization of the local
dynamics of each subsystem:(4)M(q)q¨+C(q,q˙)q˙+g(q)=Y(q,q˙,q¨)θ, where θ is a constant p-dimensional vector of inertia parameters and Y is an n×p matrix called regressor, which contains known
functions and is uniformly continuous in its arguments.Moreover, without losing generality, the following
assumptions on model (3) are made.Assumption.
The bounding
function of the external disturbances and unmodeled dynamicsτd(t,q,q˙) is assumed to be (5)∥τd(t,q,q˙)∥≤c0+c1∥q∥+c2∥q∥2+c3∥q˙∥+c4∥q˙∥2, where ci(i=0,…,4) are positive constants.Assumption.
The desired
trajectoryqd,q˙d, and q¨d are uniformly bounded.


## 3. Ntelligent Adaptive Robust Controller Designs

We now come to the
question of trajectory tracking of robot dynamic system (2). Setting q˜=q^−q,q˜˙=q^˙−q˙,s=q˜˙+Λq˜,q˙r=q˙d−Λq˜,q˙0=q˙−Λq˜,
with Λ a positive definite matrix. Let us consider
the following robust controller: (6)τ=Y(f1,…,f5)θ^−Kνq˜˙−Kpq˜+u(t),Y(f1,…,f5)θ^=M^(f1)q¨+C^(f2,f3)f4+f5,θ^˙=Proj[−Γ−1YT(f1,…,f5)s], where (⋅^) denotes the estimate of (⋅); f1,f2,f5∈{q,qd},f3,f4∈{q˙,q˙d,q˙r,q0}; Γ is a positive definite matrix; Proj[•] expresses a discontinuous projection
mapping [20], and particularly,
in our paper, it is defined as (7); u(t) represents the nonlinear control component
used to stabilize the closed-loop system and enhance the system robustness
against the external disturbances and unmodeled dynamics. The schematic diagram
of overall control system is shown in Figure 1 (set f4=defq˙d for ease of illustration). In the following
part, we consider how to design the robust compensator u(t) so that the controller (6) can ensure the
asymptotic convergence of tracking error: (7)Proj[ωi]={ωi,{θ¯i<θ^i<θ¯i,θ^i≥θ¯i,ωi≤0,θ^i≤θ¯i,ωi≥0,0,{θ^i≥θ¯i,ωi>0,θ^i≤θ¯i,ωi<0, where ωi=−Γi−1∑j=1rYjisj,θ¯i<θ^i(0)<θ¯i,i=1,2,…,p.Figure 1
Schematic diagram of proposed PD plus
feedforward robust control.Substituting (6) into (3) yields the error dynamic
system (8)E˙=AE+B(u(t)+Y(f1,…,f5)θ˜+N(t,f1,…,f5)), where E=[q˜,q˜˙]T,B=[0,M−1(q)]T,θ˜=θ^−θ,
and (9)A=(0I−M−1(q)(Kp+λC(q,q˙))−M−1(q)(Kv+C(q,q˙)+λM(q))),N(t,f1,…,f5)=Y(f1,…,f5)θ−[M(q)(q¨d−λq˜˙)+C(q,q˙(q˙d−λq˜))+G(q)+Fs(q˙)+τd(t,q,q˙)].Due to the model uncertainty and external disturbance,N(t,f1,…,f5) in (8) is an uncertain unknown nonlinear
function. In order to obtain the upper bound of N(t,f1,…,f5),
inspired by the idea of [21], we give the following lemma.Lemma 1.
Using
Properties1–3 and Assumptions 1-2, one obtains the
upper bound of N(t,f1,…,f5) as follows: (10)N(t,f1,…,f5)≤N¯(t)=d1+d2∥E∥+d3∥E∥2+d4∥E∥3 for some di>0,i=1,2,3,4.Proof.
Using Properties1, 2 and Assumption
1, we have (11)N(t,f1,…,f5)≤N(t)=ξ1+ξ2∥f1−q∥+ξ3∥q˙−f3∥⋅∥f4∥+ξ4∥q−f2∥⋅∥q˙∥⋅∥f4∥+ξ5∥f4∥⋅∥q˙∥+ξ6∥q˙∥⋅∥f4−q˙d∥+ξ7∥f5−q∥+ξ8∥q˜˙∥+ξ9∥q˙∥⋅∥q˜∥+∥τd(t,q,q˙)∥, where ξi(i=1,…,9) are known constants.
Due to the definition offi(i=1,…,5) and Assumption 2, the following inequalities
can be obtained immediately: (12)∥fi−q∥≤∥q˜∥≤∥E∥,i=1,2,5,∥fi−q∥≤∥q˜˙∥+∥Λq˜∥≤(1+Λ¯)∥E∥,Λ¯=λmax(Λ),i=3,4,∥f4∥≤b1+b2∥E∥,∥q˙∥≤∥q˜˙∥+∥q˙d∥≤b3+∥E∥,∥q∥≤∥q˜∥+∥qd∥≤b4+∥E∥, where bi(i=1,2,3,4) are known constants, λmax denotes the maximum eigenvalue. Substituting
(12) into (11) yields (10).Remark 2.
There are a lot of
literature [22, 23], even when a small
perturbation exists in system, pointing out that parameter drift occurs and
devastates the total control scheme. Therefore, a nonlinear projection mapping
Proj[•] is led into our controller for purpose of
guaranteeing globally asymptotical converges and distinct transient
performance.Remark 3.
According to the differential
choice offi,i=1,…,5,
as many as 23×42=128 kinds of global robust stability controllers
could be obtained. It can be seen that our control scheme encompasses a wide
range of circumstances.Remark 4.
From error (2), it is known that
the key to synthesis controller (3) is how to design a suitable adaptive
control compensator u(t) so that the effect of uncertain function N(t,f1,…,f5) can be effectively eliminated.


### 3.1. Routine Robust Compensator

When knowing the upper bound functionN¯(t) of concentrated uncertainty, the following
continued dynamic compensator can be designed as follows: (13)u(t)=−N¯(t)s∥s∥N¯(t)+α∥E∥2, where constant α>0 satisfies β:=λmin(Q)−α>0,λmin(⋅) denotes the minimum eigenvalue of matrix,
respectively, Q is a symmetrical positive matrix chosen
as (14)Q=(λKp00Kν). Apparently, λmin(Q)=min(λmin(λKp),λmin(Kν)).The following theorem gives the dynamic performance of
closed loop system based on controller (6) with compensator (13).Theorem 1.
Consider
the error dynamic model (8) of the robot system (3) with Assumptions 1-2. If
the dynamic compensator of controller (6) is given by expression (13), then,
the close-loop system is exponentially stable.Proof.
Choose the Lyapunov function
as(15)V=12ETPE+12q˜TPq˜+12θ˜TΓθ˜, with P the unique symmetrical positive matrix
solution to Lyapunov function (16)ATP+PA=−Q, where Q is defined as (14).
DifferentiatingV with respect to time along solution trajectory
of (8), and using Property 3, we get (17)V˙=ETPE˙+12ETP˙E+q˜TPq˜˙+θ˜TΓθ˜˙=ETP(AE+B(u(t)+Y(f1,…,f5)θ˜+N(t,f1,…,f5)))+12ETP˙E+q˜TPq˜˙+θ˜TΓθ˜˙=−ETQE+ET(00−Kp0)E+q˜TPq˜˙+θ˜TΓθ˜˙+ET(λII)(u(t)+Y(f1,…,f5)θ˜+N(t,f1,…,f5))=−ETQE+sT(u(t)+N(t,f1,…,f5))≤−λmin(Q)∥E∥2+sTu(t)+∥s∥N¯(t)=−λmin(Q)∥E∥2−N¯(t)2∥s∥2∥s∥N¯(t)+α∥E∥2+∥s∥N¯(t)≤−(λmin(Q)−α)∥E∥2, that is, (18)V˙≤−β∥E∥2. From expression (15), the
following inequality can be obtained: (19)(λmin(P)+λmin(Kp))∥E∥2≤2V≤(λmax(P)+λmax(Kp))∥E∥2. Setting (20)λ1=λmin(P)+λmin(Kp),λ2=λmax(P)+λmax(Kp), and then from expressions (18)
and (19), we get (21)∥E∥≤(λ2λ1)1/2∥E(t0)∥exp(−βλ2t). Hence, the close loop system is
globally exponentially stable.Remark 5.
In dynamic compensator (13), the
prior knowledge on the upper bound N¯(t) of system uncertainty is required. However, in
general, precise bound on system uncertainty is difficult to compute; and
moreover, the conventional estimating method may lead to a conservative design.
Therefore, we introduce a new dynamic compensator based on intelligent systems
due to the excellent nonlinear mapping.


### 3.2. Intelligent Adaptive Robust Compensator

Consider the upper boundN¯(t) in (10) that cannot be directly available in
the control design. An adaptive universal approximation system N¯^(xE,Θ^)∈Rm with input vector xE∈Ω for some compact set Ω∈Rκ,κ>0,
is proposed here to approximate the behavior of N¯(t) where xE is a function of E and t,
and Θ is a vector containing the tunable
approximation parameters. Let N¯^i(xE,Θ^i) denote the i th component of N¯^(xE,Θ^).
As in many previous studies [16–18], the linearly parameterized
fuzzy model and neural network model are adopted in the approximation
procedure, and so N¯^i(xE,Θ^i) can be expressed as (22)N¯^i(xE,Θ^i)=hiT(xE)Θ^i,i=1,…,m, where Θ^i=[Θ^i1,…,Θ^imi]T for some mi>0 is a parameter vector and hi(xE)=[hi1(xE),…,himi(xE)]T is a regressive vector with the regressor
defined for the adaptive fuzzy approximator as
follows: (23)hil(xE)=∏j=1κμN¯jl(xEj)∑l=1M(∏j=1MμN¯jl(xEj)),l=1,…,mi, and for the adaptive neural
network approximator (24)hil(xE)=eσil(xE)−e−σil(xE)eσil(xE)−e−σil(xE),l=1,…,mi. The membership functions μHjl(xEj) in the fuzzy system and the activation
functions σil(xE) in the neural network system are specified
beforehand.Consequently, the totally adaptive approximation
systemN¯^(xE,Θ^) can be expressed as (25)N¯^(xE,Θ^)=W(xE)Θ^, where W(xE)=diag[h1T(xE),…,hmT(xE)] denotes a basis matrix and Θ^=[Θ^1T,…,Θ^mT].
According to the universal approximation theorem [16–18], there is an optimal
approximation parameter Θ such that N¯^(xE,Θ^) can approximate N¯(t) as best as possible. Therefore, the upper
bound N¯(t) can be expressed as (26)N¯(t)=W(xE)Θ+ΔN¯(t), where ΔN¯(t) denotes the optimal approximation
error.Remark 6.
Since both
adaptive fuzzy system and adaptive neural network system possess the property
of universal approximation, the structure of upper boundN¯(t) can be allowed to be entirely unknown. On the
other hand, suppose that the structure of N¯(t) is known, then according to the design
philosophy of classical adaptive control, N¯(t) can be expressed as N¯(t)=W(E,t)Θ+ΔN¯(t),
that is, as in the form of (10), in which N¯(t) is a regressor matrix of known functions, Θ is a constant vector with components depending
on the plant parameters, and ΔN¯(t) denotes the residual part that cannot be
linearly parameterized.For further analysis, the following assumptions are
needed.Assumption 3.
Given an
arbitrary small positive constantω,
there exists an optimal weight vector Θ* so that the approximation error ΔN¯(t) of intelligent systems
satisfies (27)|ΔN¯(t)|=|Θ*TW(xE)−N¯(t)|<ω.Assumption 4.
Forω in expression (27), the upper bound N¯(t) of system uncertainty N(t,f1,…,f5) satisfies (28)N¯(t)−∥N(t,f1,…,f5)∥>ω.Now, we use intelligent systems (25) to design the
adaptive robust compensator u(t),
and get the following theorem.Theorem 2.
Consider the error equation (8)
for the robot manipulator system (3) with Assumptions 1–4. If the compensator u(t) in controller (6) is designed
as (29)u(t)=−(W(xE)Θ^)2s∥s∥(W(xE)Θ^)+α∥E∥2, and the weight vector Θ^ is updated by the following adaptive
mechanism: (30)Θ^˙=ηW(xE)∥s∥, with η>0 the adaptive rate, then
controllers (6), (29) can ensure that the tracking
error of robot system asymptotically converges to zero.Proof.
Define the following Lyapunov
function(31)V=12ETPE+12q˜TPq˜+12θ˜TΓθ˜+12Θ˜Tη−1Θ˜, where Θ˜=Θ*−Θ^,Θ˜˙=−Θ^˙, matrix P is defined by (15).
DifferentiatingV with respect to time along trajectory of error
equation (8), similar to the proof of Theorem 1, we
get (32)V˙=−ETQE+sT(u(t)+N(t,f1,…,f5))+Θ˜Tη−1Θ˜˙≤−ETQE+sTu(t)+∥s∥⋅∥N(t,f1,…,f5)∥+Θ˜Tη−1Θ˜˙=−ETQE+sTu(t)+∥s∥⋅(∥N(t,f1,…,f5)∥−N¯(t))+∥s∥⋅N¯(t)+Θ˜Tη−1Θ˜˙. Using both expressions (27) and (30),
and considering Assumption 4, we have (33)V˙≤−ETQE+sTu(t)+∥s∥⋅(∥N(t,f1,…,f5)∥−N¯(t))+∥s∥⋅(W(xE)Θ*T+ΔN¯(t))+(Θ*−Θ^)Tη−1Θ˜˙≤−ETQE+sTu(t)+∥s∥⋅(−ω+ΔN¯(t))+Θ^TW(xE)∥s∥≤−λmin(Q)∥E∥2−(W(xE)Θ^)2∥s∥2∥s∥(W(xE)Θ^)+α∥E∥2+Θ^TW(xE)∥s∥≤−(λmin(Q)−α)∥E∥2=−β∥E∥2. Hence, it is easily concluded
that E∈L2∩L∞,Θ˜∈L∞, and then s∈L∞.
Furthermore, from expression (8), we obtain that E˙∈L∞.
Using Barbarlat lemma, it follows that limt→∞q˜˙=0.In the dynamic compensator (29), α∥E∥2 represents the width of the boundary layer
which is chosen to be a constant. The constant-width boundary layer design can
reduce chattering of the control signals, but it decreases the control
accuracy. The compromise between the smoothness of control signals and the
accuracy of the control results is dictated by the choice of the boundary layer
width. Through simulation we know that, for the control with a small boundary
layer width, chattering occurs only during the transient stage when the system
state is far from the origin. When the state gets closer and closer to the
origin, the chattering phenomenon gradually disappears even though the boundary
layer width is enough small. Based on the observations, this note proposes a
method, which makes use of fuzzy boundary layer control [24].We selected∥E∥ and ∥dE/dt∥ as the control input, and the boundary layer width h as the control output. The fuzzy control rules
are set as follows.If∥E∥ is PB AND ∥dE/dt∥ is PB, then h is PM.If∥E∥ is PB AND ∥dE/dt∥ is PS, then h is PB.If∥E∥ is PS AND ∥dE/dt∥ is PB, then h is PM.If∥E∥ is PS AND ∥dE/dt∥ is PS, then h is PS.Where “PB, PM, PS" stand for positive big,
positive middle, and positive small. “AND” is defined by “μA AND μB=μA×μB” for any two membership values μA and μB over the fuzzy subsets A and B,
respectively. Product-sum-gravity method of the inference technique is
introduced for fuzzy logic control system.Now the control law is proposed to be(34)u(t)=−(W(xE)Θ^)2s∥s∥(W(xE)Θ^)+h.Theorem 3.
Consider
the error equation (8) for the robot manipulator system (3) with
Assumptions 1–4. If the compensator u(t) in controller (6) is designed
as (35)u(t)=−(W(xE)Θ^)2s∥s∥(W(xE)Θ^)+h, where constant h denotes the fuzzy boundary layer width, and
the weight vector Θ^ is updated by the following adaptive
mechanism: (36)Θ^˙=ηW(xE)∥s∥, with η>0 the adaptive rate, then
controllers (6), (34) can ensure that the tracking
error of robot system asymptotically converges to zero.Proof.
The proof is similar to that of
Theorem1 above, thus we choose to omit the details.Remark 7.
In compensator (13), α is a small positive constant. Moreover, it can
be seen from the above proof that feedback gain matrix Kp and Kν are only required to satisfy λmin(Q)−α>0.
Hence, the proposed control scheme does not require the higher feedback gain
matrix, whose robustness is guaranteed by intelligent adaptive dynamic
compensator that effectively eliminated the effects of system uncertainties.Remark 8.
It has been shown that the proposed
control scheme does need to calculate the inverse
of inertia matrix as it was in [7].


## 3.1. Routine Robust Compensator

When knowing the upper bound functionN¯(t) of concentrated uncertainty, the following
continued dynamic compensator can be designed as follows: (13)u(t)=−N¯(t)s∥s∥N¯(t)+α∥E∥2, where constant α>0 satisfies β:=λmin(Q)−α>0,λmin(⋅) denotes the minimum eigenvalue of matrix,
respectively, Q is a symmetrical positive matrix chosen
as (14)Q=(λKp00Kν). Apparently, λmin(Q)=min(λmin(λKp),λmin(Kν)).The following theorem gives the dynamic performance of
closed loop system based on controller (6) with compensator (13).Theorem 1.
Consider
the error dynamic model (8) of the robot system (3) with Assumptions 1-2. If
the dynamic compensator of controller (6) is given by expression (13), then,
the close-loop system is exponentially stable.Proof.
Choose the Lyapunov function
as(15)V=12ETPE+12q˜TPq˜+12θ˜TΓθ˜, with P the unique symmetrical positive matrix
solution to Lyapunov function (16)ATP+PA=−Q, where Q is defined as (14).
DifferentiatingV with respect to time along solution trajectory
of (8), and using Property 3, we get (17)V˙=ETPE˙+12ETP˙E+q˜TPq˜˙+θ˜TΓθ˜˙=ETP(AE+B(u(t)+Y(f1,…,f5)θ˜+N(t,f1,…,f5)))+12ETP˙E+q˜TPq˜˙+θ˜TΓθ˜˙=−ETQE+ET(00−Kp0)E+q˜TPq˜˙+θ˜TΓθ˜˙+ET(λII)(u(t)+Y(f1,…,f5)θ˜+N(t,f1,…,f5))=−ETQE+sT(u(t)+N(t,f1,…,f5))≤−λmin(Q)∥E∥2+sTu(t)+∥s∥N¯(t)=−λmin(Q)∥E∥2−N¯(t)2∥s∥2∥s∥N¯(t)+α∥E∥2+∥s∥N¯(t)≤−(λmin(Q)−α)∥E∥2, that is, (18)V˙≤−β∥E∥2. From expression (15), the
following inequality can be obtained: (19)(λmin(P)+λmin(Kp))∥E∥2≤2V≤(λmax(P)+λmax(Kp))∥E∥2. Setting (20)λ1=λmin(P)+λmin(Kp),λ2=λmax(P)+λmax(Kp), and then from expressions (18)
and (19), we get (21)∥E∥≤(λ2λ1)1/2∥E(t0)∥exp(−βλ2t). Hence, the close loop system is
globally exponentially stable.Remark 5.
In dynamic compensator (13), the
prior knowledge on the upper bound N¯(t) of system uncertainty is required. However, in
general, precise bound on system uncertainty is difficult to compute; and
moreover, the conventional estimating method may lead to a conservative design.
Therefore, we introduce a new dynamic compensator based on intelligent systems
due to the excellent nonlinear mapping.


## 3.2. Intelligent Adaptive Robust Compensator

Consider the upper boundN¯(t) in (10) that cannot be directly available in
the control design. An adaptive universal approximation system N¯^(xE,Θ^)∈Rm with input vector xE∈Ω for some compact set Ω∈Rκ,κ>0,
is proposed here to approximate the behavior of N¯(t) where xE is a function of E and t,
and Θ is a vector containing the tunable
approximation parameters. Let N¯^i(xE,Θ^i) denote the i th component of N¯^(xE,Θ^).
As in many previous studies [16–18], the linearly parameterized
fuzzy model and neural network model are adopted in the approximation
procedure, and so N¯^i(xE,Θ^i) can be expressed as (22)N¯^i(xE,Θ^i)=hiT(xE)Θ^i,i=1,…,m, where Θ^i=[Θ^i1,…,Θ^imi]T for some mi>0 is a parameter vector and hi(xE)=[hi1(xE),…,himi(xE)]T is a regressive vector with the regressor
defined for the adaptive fuzzy approximator as
follows: (23)hil(xE)=∏j=1κμN¯jl(xEj)∑l=1M(∏j=1MμN¯jl(xEj)),l=1,…,mi, and for the adaptive neural
network approximator (24)hil(xE)=eσil(xE)−e−σil(xE)eσil(xE)−e−σil(xE),l=1,…,mi. The membership functions μHjl(xEj) in the fuzzy system and the activation
functions σil(xE) in the neural network system are specified
beforehand.Consequently, the totally adaptive approximation
systemN¯^(xE,Θ^) can be expressed as (25)N¯^(xE,Θ^)=W(xE)Θ^, where W(xE)=diag[h1T(xE),…,hmT(xE)] denotes a basis matrix and Θ^=[Θ^1T,…,Θ^mT].
According to the universal approximation theorem [16–18], there is an optimal
approximation parameter Θ such that N¯^(xE,Θ^) can approximate N¯(t) as best as possible. Therefore, the upper
bound N¯(t) can be expressed as (26)N¯(t)=W(xE)Θ+ΔN¯(t), where ΔN¯(t) denotes the optimal approximation
error.Remark 6.
Since both
adaptive fuzzy system and adaptive neural network system possess the property
of universal approximation, the structure of upper boundN¯(t) can be allowed to be entirely unknown. On the
other hand, suppose that the structure of N¯(t) is known, then according to the design
philosophy of classical adaptive control, N¯(t) can be expressed as N¯(t)=W(E,t)Θ+ΔN¯(t),
that is, as in the form of (10), in which N¯(t) is a regressor matrix of known functions, Θ is a constant vector with components depending
on the plant parameters, and ΔN¯(t) denotes the residual part that cannot be
linearly parameterized.For further analysis, the following assumptions are
needed.Assumption 3.
Given an
arbitrary small positive constantω,
there exists an optimal weight vector Θ* so that the approximation error ΔN¯(t) of intelligent systems
satisfies (27)|ΔN¯(t)|=|Θ*TW(xE)−N¯(t)|<ω.Assumption 4.
Forω in expression (27), the upper bound N¯(t) of system uncertainty N(t,f1,…,f5) satisfies (28)N¯(t)−∥N(t,f1,…,f5)∥>ω.Now, we use intelligent systems (25) to design the
adaptive robust compensator u(t),
and get the following theorem.Theorem 2.
Consider the error equation (8)
for the robot manipulator system (3) with Assumptions 1–4. If the compensator u(t) in controller (6) is designed
as (29)u(t)=−(W(xE)Θ^)2s∥s∥(W(xE)Θ^)+α∥E∥2, and the weight vector Θ^ is updated by the following adaptive
mechanism: (30)Θ^˙=ηW(xE)∥s∥, with η>0 the adaptive rate, then
controllers (6), (29) can ensure that the tracking
error of robot system asymptotically converges to zero.Proof.
Define the following Lyapunov
function(31)V=12ETPE+12q˜TPq˜+12θ˜TΓθ˜+12Θ˜Tη−1Θ˜, where Θ˜=Θ*−Θ^,Θ˜˙=−Θ^˙, matrix P is defined by (15).
DifferentiatingV with respect to time along trajectory of error
equation (8), similar to the proof of Theorem 1, we
get (32)V˙=−ETQE+sT(u(t)+N(t,f1,…,f5))+Θ˜Tη−1Θ˜˙≤−ETQE+sTu(t)+∥s∥⋅∥N(t,f1,…,f5)∥+Θ˜Tη−1Θ˜˙=−ETQE+sTu(t)+∥s∥⋅(∥N(t,f1,…,f5)∥−N¯(t))+∥s∥⋅N¯(t)+Θ˜Tη−1Θ˜˙. Using both expressions (27) and (30),
and considering Assumption 4, we have (33)V˙≤−ETQE+sTu(t)+∥s∥⋅(∥N(t,f1,…,f5)∥−N¯(t))+∥s∥⋅(W(xE)Θ*T+ΔN¯(t))+(Θ*−Θ^)Tη−1Θ˜˙≤−ETQE+sTu(t)+∥s∥⋅(−ω+ΔN¯(t))+Θ^TW(xE)∥s∥≤−λmin(Q)∥E∥2−(W(xE)Θ^)2∥s∥2∥s∥(W(xE)Θ^)+α∥E∥2+Θ^TW(xE)∥s∥≤−(λmin(Q)−α)∥E∥2=−β∥E∥2. Hence, it is easily concluded
that E∈L2∩L∞,Θ˜∈L∞, and then s∈L∞.
Furthermore, from expression (8), we obtain that E˙∈L∞.
Using Barbarlat lemma, it follows that limt→∞q˜˙=0.In the dynamic compensator (29), α∥E∥2 represents the width of the boundary layer
which is chosen to be a constant. The constant-width boundary layer design can
reduce chattering of the control signals, but it decreases the control
accuracy. The compromise between the smoothness of control signals and the
accuracy of the control results is dictated by the choice of the boundary layer
width. Through simulation we know that, for the control with a small boundary
layer width, chattering occurs only during the transient stage when the system
state is far from the origin. When the state gets closer and closer to the
origin, the chattering phenomenon gradually disappears even though the boundary
layer width is enough small. Based on the observations, this note proposes a
method, which makes use of fuzzy boundary layer control [24].We selected∥E∥ and ∥dE/dt∥ as the control input, and the boundary layer width h as the control output. The fuzzy control rules
are set as follows.If∥E∥ is PB AND ∥dE/dt∥ is PB, then h is PM.If∥E∥ is PB AND ∥dE/dt∥ is PS, then h is PB.If∥E∥ is PS AND ∥dE/dt∥ is PB, then h is PM.If∥E∥ is PS AND ∥dE/dt∥ is PS, then h is PS.Where “PB, PM, PS" stand for positive big,
positive middle, and positive small. “AND” is defined by “μA AND μB=μA×μB” for any two membership values μA and μB over the fuzzy subsets A and B,
respectively. Product-sum-gravity method of the inference technique is
introduced for fuzzy logic control system.Now the control law is proposed to be(34)u(t)=−(W(xE)Θ^)2s∥s∥(W(xE)Θ^)+h.Theorem 3.
Consider
the error equation (8) for the robot manipulator system (3) with
Assumptions 1–4. If the compensator u(t) in controller (6) is designed
as (35)u(t)=−(W(xE)Θ^)2s∥s∥(W(xE)Θ^)+h, where constant h denotes the fuzzy boundary layer width, and
the weight vector Θ^ is updated by the following adaptive
mechanism: (36)Θ^˙=ηW(xE)∥s∥, with η>0 the adaptive rate, then
controllers (6), (34) can ensure that the tracking
error of robot system asymptotically converges to zero.Proof.
The proof is similar to that of
Theorem1 above, thus we choose to omit the details.Remark 7.
In compensator (13), α is a small positive constant. Moreover, it can
be seen from the above proof that feedback gain matrix Kp and Kν are only required to satisfy λmin(Q)−α>0.
Hence, the proposed control scheme does not require the higher feedback gain
matrix, whose robustness is guaranteed by intelligent adaptive dynamic
compensator that effectively eliminated the effects of system uncertainties.Remark 8.
It has been shown that the proposed
control scheme does need to calculate the inverse
of inertia matrix as it was in [7].


## 4. Computer Simulation

Consider (3) of a two-DOF planar robot arm, in which G(q)=0 (no gravity torque) for simplicity. The
entries of M(q),C(q,q˙),Fd,Fs(q˙), and τd(t,q,q˙) are (37)M(q)=[θ1+θ2+2θ3cos(q2)θ2+θ3cos(q2)θ2+θ3cos(q2)θ2],C(q,q˙)=[−θ3sin(q2)q˙2−θ3sin(q2)(q˙1+q˙2)θ3sin(q2)q˙10],Fd=diag[2,2],Fs(q˙)=sgn(q˙),τd(t,q,q˙)=[q˙1sint,q˙2cost]T, and one parameterization of this robot is given by (38)Y(q,q˙,q¨)θ=τ,θ∈R3,θ1=m1lc12+m2l12+I1,θ2=m2lc22+I2,θ3=m2l1lc2,y11=q¨1,y21=0,y12=q¨1+q¨2,y22=q¨1+q¨2,y13=cos(q2)(2q¨1)−sin(q2)(q˙22+2q˙1q˙2),y23=cos(q2)q¨1+sin(q2)q˙12.In the equations above, physical parameters and their
numerical values are shown in Table1. The reference trajectories specified
are (39)qd1=sint+0.1sin3t−0.2sin4t,qd2=−0.1sin2t−0.2sin3t+0.1sin4t, with selection of parameters T=1ms,Kν=diag[20,20],Kp=diag[100,100],Λ=diag[5,5],Γ=diag[50,10,10].
In simulation, the nominal robot parameter is θ=[2.9,1.0,0.9]T,
with choice of its bound 0≤θ1≤4,0≤θ2≤2,0≤θ3≤2.
For ease of illustration, let f1=f2=f5=qd,f3=f4=q˙d.Table 1m1m2l1l2Ic1Ic2I1I2105110.50.510/125/12For the case of adopting conventional robust
controller, namely, the upper bound on the parametric uncertainty is known, we
have the following simulation results as shown in Figure2(α=10) and Figure 3(α=100), by the choose of the upper bound function
as N¯(t)=20+7.5∥E∥+5.0∥E∥2.Response using proposed conventional
robust control law(α=10).
(a)(b)(c)(d)Response using proposed conventional robust control law(α=100).
(a)(b)(c)(d)The results of the intelligent hybrid robust control
algorithm, while the upper bound is assumed to be unknown a priori, are also
presented here in Figure4(α=10) and Figure 5(α=100) as a comparison, with selection of the
adaptive rate η=0.5.Response using proposed intelligent
robust control law(α=10,η=0.5).
(a)(b)(c)(d)Response using proposed intelligent
robust control law(α=100,η=0.5).
(a)(b)(c)(d)It can be observed from the above simulation results
that both the conventional robust control method and the proposed intelligent
hybrid robust control algorithm have satisfying robust results, and the latter
achieves more excellent tracking performance than the former, for the reason
that chattering of the control signals is obviously
weakened by the fuzzy boundary layer, and the control precision
is improved significantly. In addition, for the conventional robust control method,
simulation results indicate that largerα (e.g., α=100) may yield better performance in attenuating
the chattering of control torque than the case of smaller α (e.g., α=10); to the proposed intelligent hybrid robust
control algorithm, yet the system behavior in the case of larger α (e.g., α=100) is almost the same as the case of smaller α (e.g., α=10). This implies that our proposed intelligent
hybrid robust control algorithm has better robust
performance in its insensitivity to parameter variation than the conventional
robust control method.


## 5. Experiments

In this section, the experimental setup and system
realization are described. Then experimental data is discussed and analyzed.


### 5.1. Hardware

A 2-DOF direct-drive ShinMaywaR3C robot arm is used as testbed, see Figure 6. It
is a rigid-link, low-friction, planar robot with two joints.Figure 6
High-performance robot used
as testbed.The dimensions of the robot are given in Table2,
where subindexes 1 and 2 stand for the first and second
links, respectively, and Lc stands for the center of mass, Par means
parameters, I stands for inertia, and Δ defines joint limits. Joint angles q are measured with optical encoders at a high
resolution of 120 000 pulse/rad, and joint-velocity
signal q˙ is estimated from position signals using a
first-order filter with a constant τ=10.0.Table 2ParMassLengthLeIΔLink121.200.250.150.21±160Link115.220.450.190.18±145UnitsKgmmKgm2deg


### 5.2. Firmware

A Digital Signal Processor Loughborough Sound Images
Ltd.DPC/C40B board control system was integrated on a
16-bit expansion bus slot of a DX—486 personal computer. TMS320 floating point DSP C v.1.0.1 compiler provided the programming
environment. The control input is transmitted to the servomotors through the
Shinmaywa servosystem, which powers the pulse-wide modulators motor drives at
each joint.The performance of the proposed controller is shown in
comparison with the neurofuzzy adaptive controller [12] and fuzzy supervisory
sliding-mode controller [14].
Experiments are carried out at high velocities in order to show the performance
of the system at inertial dominated dynamics.Tuning the parameters is not obvious, and special
attention is paid to avoid misleading conclusions. Since we present a
comparison among similar but structurally different controllers, we set common
gains to the same value. All feedback gains, including those of [12, 14], are given
as (40)r=(120012),K=(1000080),I=(1001),Λ=(1005),C=(r,I),r0=(0.12000.12).Comparative experiments are difficult to obtain
because any comparative result can be dangerously biased. Besides, it is
difficult to qualitatively compare controllers that are structurally different.
Therefore, we show experimental results obtained “at first run,” that is,
when reasonable performance, as dictated by the theory, is obtained without
perhaps obtaining the best plots.


### 5.3. Experimental Results

For better visualization of the plots, some figures
are shown in two windows. Window 1 shows the time
interval[0,1) s,
while Window 2 shows the time interval [1,5] s.Figure7 shows a comparative plot of position tracking
errors at high velocities of the neurofuzzy adaptive controller [12] and our intelligent hybrid
control. Desired paths at each joint are qd1=0.5sin(6πt) and qd2=0.2sin(πt+π/2).
Initial conditions are set to −5.0∘ with zero initial velocity. After a transient
of 1 second,
our controller yields tracking errors threefold and fourteenfold smaller at
each joint, respectively. In this figure, we can observe that our controller
yields better tracking accuracy with smooth control input, even when [3] is
started with smaller position tracking errors, see controllers in Figure 8.Comparative plots for tracking errors of both joints between the
controller [12] (.–) and our proposed controller (–).
(a)(b)(c)(d)Comparative plots for control inputs of
both joints between the controller [12] (.–) and our proposed controller (–).
(a)(b)Figure9 shows a comparative plot of position tracking
errors at high velocities of the the fuzzy supervisory sliding-mode controller [14] and our
Intelligent hybrid control. Desired paths at each joint are qd1=1.25−(7/5)e−t+(7/20)e−4t and qd2=1.25+e−t−(1/4)e−4t.
Initial conditions are set to 1.0 with zero initial velocity. In this figure, we
can observe that our controller yields better tracking accuracy with smooth
control input, even when [14] is started with smaller position tracking errors, see
controllers in Figure 10.Comparative plots for tracking errors
of both joints between the controller [14] (.–) and our proposed controller (–).
(a)(b)(c)(d)Comparative plots for control inputs of
both joints between the controller [14] (.–) and our proposed controller (–).
(a)(b)


## 5.1. Hardware

A 2-DOF direct-drive ShinMaywaR3C robot arm is used as testbed, see Figure 6. It
is a rigid-link, low-friction, planar robot with two joints.Figure 6
High-performance robot used
as testbed.The dimensions of the robot are given in Table2,
where subindexes 1 and 2 stand for the first and second
links, respectively, and Lc stands for the center of mass, Par means
parameters, I stands for inertia, and Δ defines joint limits. Joint angles q are measured with optical encoders at a high
resolution of 120 000 pulse/rad, and joint-velocity
signal q˙ is estimated from position signals using a
first-order filter with a constant τ=10.0.Table 2ParMassLengthLeIΔLink121.200.250.150.21±160Link115.220.450.190.18±145UnitsKgmmKgm2deg


## 5.2. Firmware

A Digital Signal Processor Loughborough Sound Images
Ltd.DPC/C40B board control system was integrated on a
16-bit expansion bus slot of a DX—486 personal computer. TMS320 floating point DSP C v.1.0.1 compiler provided the programming
environment. The control input is transmitted to the servomotors through the
Shinmaywa servosystem, which powers the pulse-wide modulators motor drives at
each joint.The performance of the proposed controller is shown in
comparison with the neurofuzzy adaptive controller [12] and fuzzy supervisory
sliding-mode controller [14].
Experiments are carried out at high velocities in order to show the performance
of the system at inertial dominated dynamics.Tuning the parameters is not obvious, and special
attention is paid to avoid misleading conclusions. Since we present a
comparison among similar but structurally different controllers, we set common
gains to the same value. All feedback gains, including those of [12, 14], are given
as (40)r=(120012),K=(1000080),I=(1001),Λ=(1005),C=(r,I),r0=(0.12000.12).Comparative experiments are difficult to obtain
because any comparative result can be dangerously biased. Besides, it is
difficult to qualitatively compare controllers that are structurally different.
Therefore, we show experimental results obtained “at first run,” that is,
when reasonable performance, as dictated by the theory, is obtained without
perhaps obtaining the best plots.


## 5.3. Experimental Results

For better visualization of the plots, some figures
are shown in two windows. Window 1 shows the time
interval[0,1) s,
while Window 2 shows the time interval [1,5] s.Figure7 shows a comparative plot of position tracking
errors at high velocities of the neurofuzzy adaptive controller [12] and our intelligent hybrid
control. Desired paths at each joint are qd1=0.5sin(6πt) and qd2=0.2sin(πt+π/2).
Initial conditions are set to −5.0∘ with zero initial velocity. After a transient
of 1 second,
our controller yields tracking errors threefold and fourteenfold smaller at
each joint, respectively. In this figure, we can observe that our controller
yields better tracking accuracy with smooth control input, even when [3] is
started with smaller position tracking errors, see controllers in Figure 8.Comparative plots for tracking errors of both joints between the
controller [12] (.–) and our proposed controller (–).
(a)(b)(c)(d)Comparative plots for control inputs of
both joints between the controller [12] (.–) and our proposed controller (–).
(a)(b)Figure9 shows a comparative plot of position tracking
errors at high velocities of the the fuzzy supervisory sliding-mode controller [14] and our
Intelligent hybrid control. Desired paths at each joint are qd1=1.25−(7/5)e−t+(7/20)e−4t and qd2=1.25+e−t−(1/4)e−4t.
Initial conditions are set to 1.0 with zero initial velocity. In this figure, we
can observe that our controller yields better tracking accuracy with smooth
control input, even when [14] is started with smaller position tracking errors, see
controllers in Figure 10.Comparative plots for tracking errors
of both joints between the controller [14] (.–) and our proposed controller (–).
(a)(b)(c)(d)Comparative plots for control inputs of
both joints between the controller [14] (.–) and our proposed controller (–).
(a)(b)


## 6. Conclusion

A new adaptive intelligent hybrid robust controller
based on sliding mode is proposed in this paper for the trajectory tracking of
robot manipulators with unknown nonlinear dynamics. We use intelligent systems
to approximate the upper bound on the parametric uncertainty, and propose the
idea of fuzzy bound later used to weaken the chattering
effect. The system global stability and the tracking error convergence are
proved by Lyapunov techniques. Simulation results and experimental data show
that the intelligent hybrid controller make a significant improvement in the
tracking performance.



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*Source: 520591-2008-06-19.xml*