2000 Santibañez - Fuzzy PD+ Control for Robot Manipulators.pdf

Proceedings of the 2000 IEEE
International Conference on Robotics & Automation
San Francisco, CA April 2000
Fuzzy PD+ Control for Robot Manipulators *
Victor Santibaiiez
t Instituto Tecnol6gico de la Laguna
Apdo. Postal 49, Adm. 1
Torrebn, Coahuila, 27001 MEXICO
e-mail: vsantiba@itZaguna. edu. mx
Rafael Kelly and Miguel A. Llamat
Divisi6n de Fisica Aplicada
CICESE
Ensenada , B.C., 22800 MEXICO
e-mail: rkeZZy@cicese.mx
Abstract
This paper deals with applications of fuzzy logic systems
to motion control of robot manipulators. In the
proposed application, the fuzzy logic system plays the
role of a tuner of the robot controller gains. This idea
is evoked for the gains of the so-called PD+ control
scheme and we demonstrate, by taking into account the
full non-linear and multivariable nature of the robot
dynamics, that the overall closed loop system is globally
asymptotically stable. Experimental results on a
two degrees of freedom direct-drive arm show the usefulness
of the proposed control approach.
1 Introduction
In recent years fuzzy logic has been widely used as a
successful practical approach in robotics for designing
and implementing control systems [3, 15, 13, 7, 6, 41.
In this paper we use the potential of fuzzy self-tuning
schemes to motion control of robot manipulators in
order to design a methodology for on-line parameter
selection of a robot motion controller. Particular attention
is paid to provide a rigorous stability analysis
including the robot nonlinear dynamics.
A basic problem in controlling robots is the so-called
motion control where a manipulator is requested to
track a desired position trajectory. A number of
such robot motion controllers having rigorous stability
proofs have been reported in the literature
and robotics textbooks [2, 12, 111. Among these
motion controllers, the PD+ control introduced by
Koditschek [5] is simple and attractive. The structure
of PD+ control consists of a linear PD feedback plus a
specific compensation of the robot dynamics. F’urthermore,
this control strategy has the nice feature that it
reduces to the PD control with gravity compensation
in the particular case of set-point control.
The first complete stability analysis of the PD+ control
strategy was provided by Paden and Panja [8] who
coined the name “PD+ control”. The stability study
was possible thanks to the use of the Matrosov’s theorem.
Some time later, Whitcomb et a1 [14] presented
an alternative stability analysis by introducing a strict
Lyapunov function in an adaptive context. Both analysis
assume that the controller parameters, that is,
the proportional and derivative gain matrices are constants.
The main contribution of this paper is the extension
of the PD+ control to the case where the gains are allowed
to vary according to a fuzzy logic system which
depends on the robot state. This is an important
ingredient to deal with practical specifications such
as keeping asymptotically the tracking error within
prescribed bounds without saturating the actuators.
A stability analysis incorporating the nonlinear robot
dynamics guarantees global asymptotic stability. The
feasibility of the proposed control approach is illustrated
through experiments on a robotic system composed
by a two degrees-of-freedom direct drive vertical
arm.
Throughout this paper, we use the notation X,{A}
and XM{A} to indicate the smallest and largest eigen-
values, respectively, of a symmetric positive definite
bounded matrix A(x), for any x E Etn. The norm of
vector x is defined as IIxlI = &%.
*Work partially supported by CONACyT and COSNET.
0-7803-5886-4/00/$10.00~ 2000 IEEE 2112

2 Robot dynamics and
control formulation
motion
The dynamics of a serial n-link rigid robot can be
written as [12]:
M(qk + C(q, 414 + S(Q) = (1)
where q is the n x 1 vector of joint displacements, q
is the n x 1 vector of joint velocities, r is the n x 1
vector of actuators applied torques, M(q) is the n x n
symmetric positive definite manipulator inertia matrix,
C(q,q)q is the n x l vector of centripetal and
Coriolis torques and g(q) is the n x 1 vector of gravitational
torques obtained as the gradient of the potential
energy U(q) due to gravity. We assume the robot
joints are joined together with revolute joints.
Assume that joint position q and joint velocity q are
available for measurement. Let the desired joint position
qd be a twice differentiable vector function. We
define a motion controller as a controller to determine
the actuator torques r in such a way that the following
control aim be achieved
The control system is said to be globally asymptotically
stable if the control aim is guaranteedl irrespective
of the robot initial configuration q(0) and q(0).
3 fizzy PD+ Control
The structure of the PD+ control strategy introduced
by Koditschek [5] can be written as
= Kpq + K ~ + 6 M(q)qd + c(q, 4)qd + g(q) (2)
where qd,qd and qd are the n x 1 vectors of desired
position, desired velocity and desired acceleration respectively,
q = q d - q is the n x 1 vector of position
errors, q = qd - q is the n x 1 vector of velocity errors,
and Kp and K, are the n x n proportional and
derivative gain matrices respectively.
In this paper we generalize the PD+ control by keeping
intact its structure but permitting the gains be
state dependent. Specifically, we propose the control
law
. .
r=Kp(q)q+K,(q,q)q+M(g)iid+C(q,il)qdfg(4)
(3)
lStrictly speaking, in addition to global attraction also Lyapunov
stability of the control system must be established.
where Kp(q) and K,(q, G) are n x n diagonal positive
definite matrices whose diagonal entries are denoted
by kpi (&) and k,i (&, Qi) respectively.
Freedom to select the Proportional and Derivative
gain matrices in a nonlinear manner may be of worth
in real applications where manipulators are under effects
of disturbances and constraints. Two important
real constraints on robot manipulators are the friction
in the manipulators joints and the technological
limitation of torque (or force) capability in robot actuators.
Friction produces bias in positioning while
torque capability reduces the class of desired position
trajectories. The entries Icpi(@) and k,,i(&, &) of matrices
Kp(G) and K,,(q,q) may be tuned on-line to
get small gains for big tracking error and thus avoid
torque saturation; and high gains for small tracking
errors to obtain good accuracy in presence of friction.
This is the rational behind the role of the fuzzy logic
system in our approach.
Fuzzy logic is suitable as a mechanism to determine
the nonlinear Proportional and Derivative gains of the
PD+ control law according to previous practical specifications.
In order to tune the proportional gains
kpi(&) and the derivative gains Ic,,i(&, &) according to
the input liil, in this paper we define one conceptual
Fuzzy Logic Tuner (FLT) . In summary, 2n elementary
FLT will be involved in computation of n proportional
gains and n derivative gains.
3.1 Basic fuzzy logic tuner
Let the conceptual FLT have one input 1x1 and the
corresponding output y. The FLT can be seen as a
H: Et+ --+ IR
static mapping H defined by
I4 Y.
The universes of discourse of 1x1, and g are partitioned
into 3 fuzzy sets: B (Big), M (Medium), and S (Small)
with each attribute being described by a membership
function. We shall employ trapezoidal membership
functions for input variables and singleton for output
variables. In order to simplify notation, let us use
the following convention. With reference to figure 1,
the corresponding Small, Medium and Big membership
functions for the input variable x are denoted
respectively by: Ps(lzl ;Pl,PZ), PM(lzl ;Pl,P2,P3,P4),
and PB(l”l ;P3,P4).
For convenience we define the following vector
1
PS(14 ;PlrP2)
PM(b? ;Pi,Pz,P3,P4) where P =
PB(14 ;Ps,P4)
[Pi, Pz, P3, P41.
21 13

Since we attempt real-time implementation of the
fuzzy self-tuning algorithm, this should have as few
rules as possible in order to reduce the computational
effort. The selected rules into the Mamdani’s rule base
are the following
IF PS(lXl ;Pl,P2) THEN Pfkk3)
IF PM(lzl ;Pl,P2,P3,p4) THEN vi‘(.;k2)
IF PB(bl ;P3,P4) THEN P:(+w
Evaluation of the rules under the ‘scale’ criterion leads
to a real-valued vector function h(l.1 , .) : IR+ x IR -i
IR3 given by
Figure 1: Input membership functions
P(l4 ;PI
-1 I [
4 cw )
0 fi. 22
7j 1
0
*.
22
0 ! 22
kJ
Figure 2: Output membership functions
With reference to the output variable y (see Figure
2), the singleton membership functions corresponding
to Small, Medium and Big are denoted by:
p: (.; ki), py( *; k2) , and ,U: (.; k3), respectively.
It is also convenient for the to define where
the following vector: py(.) =
2#
21 14
Owing to singleton membership functions, for a given
s E IR we have
h)~~(bl
1
[ 1
P(I4 ;P3,P4P(S- kl)
h(l.1 ,S> =
;Pi,P2)
Pf(S; k2)PM(bl ;Pi7P2,P3,P4)
P;b; k1)PB(14 ;P3,P4)
PS(14 ;Pl,mMS- k3)
= PuM(lzl ;Pl,B,P3,P4)6(s- k2) (4)
where 6(.) denotes the ‘Dirac function’ [l]. The following
feature of Dirac functions will be evoked later.
For a real valued function + : IR -+ IR and any SO E IR,
the Dirac function has the following property
+(s)6(s - s ow = +(so). (5)
The defuzzification strategy chosen in this paper is the
center of area method, also called “centroid method”.
Therefore, the output y can be computed as y =
E:=, J-: hi(l4Pb ds
. Invoking this formula and (4)-
Eh, J-; hi(l4,s)ds
(5), the defuzzification procedure reduces to
Y=
kTP(14 ;P)
IlP(14 ;p)II1
(6)
= ICz k31T, 11.111 for the norm
and (.)T denotes transpose. Therefore, for a given input
1x1 the output y can be computed straightforward
from (6).

This FLT has the feature that under weak conditions
its output y is bounded away from a strictly positive
constant. This is stated in the following
Lemma 1. Consider the described FLT. Assume that
IIp( 1x1 ; p) )I > 0 for all 2 E IR, and k3 > > kl > 0.
Then
y(I2:l) 2 kl > 0
v 2 E R.
vvv
Proof. Since by definition all entries of ~(1x1 ;p) are
nonnegative and by assumption k3 > k:! > kl > 0,
then [kl k2 k3lP(14 ;P) 2 kl IlP(l4 ;P)lll.
Incorporating this expression in (6) we get the desired
conclusion.
0
Assumption Ilp(lzl ;p)lll > 0 means that the intersection
of the input membership functions is nonempty
for all input 1x1. This is easy to check by testing the
following obvious inequalities
3.2 Tuning the gains
Pl < P2, (7)
P3 P4. (8)
As previously described, the basic FLT is invoked
to determine the proportional and derivative gains.
Thus, a set of 2n FLTs are defined, that is
R+ -+
IGil '+ kpi
and
R
I4il I-+ kvi
HkVi: R+ -+
for i = 1, . . . , n. Notice that for the.sake of simplicity,
the derivative gains are allowed to depend only on
the position error \&I instead of both (position and
velocity errors). This was done mainly to reduce the
computational cost for real-time implementation.
A block diagram of the proposed self-tuning variable
gains PD+ control scheme is depicted in Figure 3.
4 Stability analysis
The closed-loop system is obtained by combining the
robot dynamic model (1) with control law (3). The
resulting equation is given by
Figure 3: Block diagram
4
4
. .
C(s7C)Q]
(9)
which is a nonautonomous equation and the origin of
the state space is the unique equilibrium.
We assume that each FLT is designed according to
(7)-(8) and the fuzzy set supports for the output variables
are strictly positive (i.e. the corresponding kl ,
kg and k3 are strictly positive). These assumptions together
with lemma 1 implies that the FTL guarantee
the existence of E > 0 such that
[ M(4)-' [-Kp(Q)G -Am, -
0 kpi(ii) 2 E for all Gi E R, and i = l,...,n
0 k,i(&,&) ~~forall~~,~~~R,andi=l,...,n
and therefore, matrices Kp(q) and K,(ij,q) are uniformly
positive definite.
The stability analysis of the closed-loop system (9)
is greatly simplified because under above assumptions
the control law (3) belongs to the family of potential
energy shaping motion controllers studied in [lo]. This
family of motion controllers is defined by
. .
= v;yu,(q)+Kv(~,q)q+lM(4)ii,+c(q,q)C, (10)
where matrix K,(q,q) is a n x n diagonal positive
definite matrix for all q, 6 E R" and Ua(q) is a differentiable
function called artificial potential energy defined
as: &(q) = I .r~(q) -U(q), with U~(ij) being any
continuously differentiable radially unbounded positive
definite function with an unique minimum point
at q = O E Et".
2115

Straightforward calculations show that the proposed
fuzzy PD+ control (3) can be derived from (10) considering
the particular function
The global asymptotic stability of potential energy
shaping motion controllers, such as the proposed controller,
can be proven by using the following strict Lyapunov
function [IO]:
where s(q) = A and y is a constant that satisfies
1fllQll
with k1 and kcl suitable positive constants.
Finally, as a consequence of asymptotic stability, the
position tracking aim limt,, q(t) = qd(t) is achieved.
5 Experimental evaluation
Experiments on a direct-drive robot arm, whose dynamics
entries (1) are given in [9], have been carried
out in order to evaluate the performance of the proposed
fuzzy PD+ controller. The structure of the desired
position trajectory qd (see equation (13)) has
been chosen with the end of exploiting the arm in its
fastest motion but without invading the actuators saturating
zone, and to demand an initial big torque
[ l:::]
qd = t
r0.78[1 - e-’.’ t3] + 0.17[1 - ew2.O t3] sin (wit) 1
Fuzzy partitions of the universes of discourse of the
tracking errors I and I&/ are characterized respectively
by the sets: pql = {0.5,1,10,15} [deg] and
P, = {2,4,10,15} [degl, where = {P~,PZ,...,P~
denotes the supports of the membership function of &
according to our convention (see figure 1).
On the other hand, the universes of discourse of the
proportional gains kpi were determined according to
the following criterion. A high priority was paid to
avoid actuators saturation, thus the smaller values of
the proportional gains were computed for the worst
case in our experimental set-up where the larger position
error was defined equal to 27r [rad].
The partition of the universes of discourses for the proportional
gains were: kkpl = {0.3,4.0,40.0} [Nm/deg]
and kkpZ = {0.03,0.4,5.0} [Nm/deg], where kkp, =
{kl, tip, kg} denotes the supports of the membership
function of output k,i according to our convention (see
figure 2).
The partition of the universe of discourse for
the derivative gains kvi were chosen as follows:
kkul = {0.087,0.5,1.57} [Nm- sec/deg] and kkvZ =
{0.0087,0.05,0.157} [Nm- sec/deg].
Fuzzy partitions chosen ensure that the fuzzy logic
systems deliver proportional and derivative gains in
agreement with conditions of Lemma 1.
Implementation of the full control system composed
by the PD+ control law and the fuzzy logic algorithm
was executed in 0.3 msec.
Position tracking errors q for the fixed and fuzzy gains
PD+ controller are shown in Figures 4 and 5 respectively.
It can be seen in base on the experimental results,
that the fixed gains PD+ controller is not able
to suitably follow the desired position trajectory (13).
This, because it is not possible increase the gains without
saturating the actuators to reduce the position
tracking errors. This type of trajectories are too severe
to be followed by the classical tracking control
schemes of robot manipulators. The main source of
severity in the desired trajectory is the addition of
a step reference to a smooth tracking reference. In
contrast, the fuzzy self-tuning approach has shown to
have the ability to face up to the tracking of such type
of trajectories without exceeding the torque limits of
the actuators.
[1.04[1- e-1.8 t3] + 2.18[1 - t3] sin (wpt)
(13)
where w1 and wp represent the frequency of desired
trajectory for the shoulder and elbow joints respect
ively.
2116

degrees
IOU
0 2 4 6 sec
Figure 4: Position errors of the fixed gains scheme
degrees
t 100
40
0 2 4 6 sec
Figure 5: Position errors of the fuzzy gains scheme
6 Conclusions
A fuzzy logic approach for on-line tuning of the Proportional
and Derivative gains of a PD+ controller for
robot manipulators has been presented. A stability
analysis incorporating the nonlinear robot dynamics
guarantees global asymptotic stability.
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