2001 Parra - Nonlinear PID control with sliding modes for tracking of robot manipulators.pdf

Proceedings of the 2001 IEEE International
Conference on Control Applications
September 5-7,2001 Mexico City, Mexico
Nonlinear PID Control with Sliding
Modes for Tkacking of Robot Manipulators
V. Parra-Vega5 and S. Arimotot
SMechatronics Division, CINVESTAV, AP 14-740, M6xic0, DF, Mdxico, vparra@mail.cinvestav.mx
$Department of Robotics, Ritsumeikan Univ., 1-1-1 Noji-Higashi, Kusatsu, Shiga 525-8577, Japan.
Abstmct-It is well-known that PI) controller plus gravity
compensation yields the global asymptotic stabllitjr for regdatiun
tasks for robot manipulators [l], [a]. The impressive
succew of these controllers in real-time tasks lie in the fact
that they do neither compensate inertial nor coriolis forces,
so neither inertial nor coriolis matrices are needed to implement
the controller. However this linear state feedback controllers
with gravity compeneation mnnot render asymptotic
stability for tmcking tab. In this paper, a simple decentralized
continuous nonlinear PID controller that yields local
exponential stability for tracking tasks is proposed. The
controller does neither need the inertial nor the Coriolis matrix.
Comparative experimental data versus [l] and [SI for
a rigid mbot arm validates our design.
Kegwd- PID, Sliding Mode Control, Robot Manipulators
I. INTRODUCTION
Studies on the control theory of serial mechanical SYStems
have been subject of intensive and profitable research
over the last two decades. In particular, industrial robot
prototypes have been controlled poinbhpoint using the
linear decentralized PID controllers without gravity compensation.
Most of these controllers have been designed using
linear models, or linearized ones, and some interesting
nonlinear PID structures have been proposed to overcome
the limitations of traditional linear PID controllers for regulation
tasks of nonlinear mechanical plants [4]-[ll]. The
tracking capability of these control systems over the entire
domain of the nonlinear dynamics is still a challenging
research topic.
A. Regulation
The breakthrough PD controller proposed by Takegaki
and Arimoto in 1981 [l] paved the way to establiih theoretical
foundations to controlling robotic systems. The
successful application of this controller for regulation tasks
in many physical systems becomes apparent because there
are available explicit tuning procedures, as well as online
compensation techniques of the gradient of the potential
energy [6],[ll],[9],[7]. Recently PID-like controllers have
been proposed which do not require La’SaUe’s arguments
[4], [lo], however any of these controllers are not useful to
obtain tracking. An outstanding mechatronic design together
with a PID-like controller may be good enough for
many robot based industrial applications. The success of
these kind of studies are driven by the simplicity and useful-
ness of non-model based robot controllers. Therefore, it is
interesting the class of model-free decentralised controllers
that can overcome some of the typical deficiencies of linear
regulators but at the same time can render at least similar
real time performance compared to some model-based
tracking controllers [3].
B. lhckang
On the other hand, when model-based nonlinear control
law is designed, a complex nonlinear controller is obtained,
and typically no easy and no clear tuning procedures are
proposed [13] except for some well-studied robotic prote
types in reaearch laboratories. In the laboratory, these controllers
certainly yields high performance over the whole
domain of the dynamical system but in the industrial floor
the lack of tuning procedures and the high implement&
tion and computational cost deprive us to fully obtain the
advantages of model-based nonlinear controllers over the
decentralized model-free linear controllers [13], [18], [19].
C. Contribution
In this paper, motivated by a mechatronics design, and
considering the real-time performance of some nonlinear
controllers, a new structure of a continuous nonlinear PID
control law for continuous mechanical plants with tracking
capability is proposed. The nonlinear I-tame control
input compensates for inertial, coriolis and gravitational
forces, while the PD action comprises for an stabilizer of
closed loop system trajectories. Experimental data d i -
dates our design where, surprisingly enough, in some cases
this model-free controller yields better performance in comparison
with the model-based adaptive controller [3].
In Section 11, it is presented the class of nonlinear systems
considered in this article. Section I11 shows the control
law and presents its stability analysis, and Section lV
discusses some aspects of the control structure. Section
V shows the experimental set up and discusses the closedloop
real-time performance. Finally, in Section VI some
conclusions axe presented.
11. NONLINEARDBOT DYNAMICS
The dynamic model of a rigid n-link serial non-redundant
robot manipulator with all actuated revolute joints de-
Owe do not intend to offer an extensive review on the subject of
nonlinear PID controller, we review briefly the literature of a class of
passivity-based controllers for robot arms.
0-7803-6733-2/01/$10.00 0 2001 IEEE 351

scribed in joint coordinates is given as follows
Wda + a 1414 + Bo4 + G(q) = 71 (1)
where H(q) E Rnxn denotes a symmetric positive definite
inertial matrix, BO E Rnxn stands for a diagonal positive
definite matrix composed of damping friction coefficients
€or each joint, C(q,q) E RnXn stands for the coriolis and
centrifugal forces, G(q) E 92" models the gravity forces,
and r E 8" stands for the torque input.
A. Open Loop Emr Dynamics
Since equation (1) is linearly parametrizable [3] by the
product of a regressor Y = Y (q,cj,d,q) E RnxP, composed
of known nonlinear functions, and a vector 8 E RP
which represents unknown but constant parameters, then
the parametrization Y8 can be written in terms of a nominal
reference &, to be defined yet, and its derivative & as
follows
H(q)ir + (BO + C(qi 4)) 4r + G(Q) = Kei (2)
where the regressor Yr = Yr (q, Q, Qrl Q) E Rnxp. Equ*
tion (2) into (1) yields the open loop error dynamics in
error coordinates S, as follows
H(q)Sr + {BO + C(qi 4)) sr = * - Yre, (3)
where Sr is defined by
s, =q-q,. (4)
Now, consider the following nominal reference qr and its
derivative defined as follows 1131
qr = qd - CUAq + Sd "/U, (5)
b = wn(AS),
where a, 7 are diagonal positive definite n x n matrices,
function sgn(*) stands for the signum function of (*), and
for IC
AS = s-sd (6)
S = AQ+aAq (7)
sd = S(to)mp-'S(t-to) (8)
> 0 and S(to) stands for S(t) at t = to. Notice
that & = - aAq + & - ysgn(AS) k discontinuous, and
AS(b) = 0 V for any initial condition. Equation (5) into
(4) gives rise to the dynamic error coordinates
Sr = AS + TU. (9)
We now introduce some useful properties for the stability
analysis.
Properties: There exists positive scalars Pi, where i =
0, ..., 5 such that
2 Po > 0, &(A) 5 PI < 00 stand for the
where X,(A)
minimum and maximum eigenvalues of an A E Rnxn matrix,
respectively. Norms IlAll = d-,
and llbll of
vector b E Rn stand for the induced Frobenius and vector
Euclidean norms, respectively. These constants can be
computed from the state of the system, desired trajectories,
feedback gains, and a conservative upper bounds of the dynamic
model of the robot arm, besides that it is assumed
that qd E c2.
111. NONLINEAR PID CONTROLLER
A decentralized model-free nonlinear PID controller is
stated in a theorem.
Theorem 1: Consider the robot dynamics (1) in closed
loop with the controller given by:
T = -Kdsr (11)
t
= -KpAq - K,Aq + KdSd - Ki 1 sgn(AS(F))&
where sr is given in (Q), and Kd is a n X n diagonal symmetric
positive definite matrix, and Kp = &a, K, = Kd,
and Ki = &"/. Then, if error on initial condition are small
enough, then local exponential tracking is assured provided
that 7 in (9) is tuned accordingly to the inequality (21)
given in the proof.
A. Stability Analysis
Equation (11) into (3) renders the following closed-loop
error dynamics
H(q)Sr = - {K + C(q, 4)) sr - Yre (12)
where K = Kd + Bo. A passivity-motivated analysis yields
the following Lyapunov function
whose total derivative along its solution (12) is given by
V = -S,TKSr - S,'Yr8. (14)
The norm of the function Yre in (12) is upper bounded
according to the following derivations
Yre 5 IIH(S)IIIISll+ II (BO + C(qi4))dr + llG(q)Il
5 XM(H(q)) {allAdl +P6) $. {XM(BO)+
+hlQI) (allAcrll+ P4 3- 7lloll) + P3
5 Pl4lAQll + XMM(B0) + PzIlOll(~llAqI1+ P4 +
+7Il4) +A
5 +q, Ad1 a, Pi) (15)
where ps = PIPS+& and v(Aq, A& a, A) is a scalar. Then
according to (15), equation (14) becomes
Q 5 -llK~sr112 + Ilsrllv(&,Aq,o,Pi) (16)
where K = G Kl. Since Sr = v'(Aq,Aq,Sd,a), then if
initial conditions me such that to) belongs to a compact
tD
352

set ne= the equilibrium S, = 0, we have that by invoking
Lyapunav arguments, there exist 0 < K1 < 00 large enough
such that S, converges into a neighborhood e > 0 with
radius r > 0 centered in the equilibrium S, = 0. Thus, the
boundedness of S, can be concluded, namely
Sr+Eo 89 t+W, (17)
for a bounded constant EO, and €1 > 0 stands for the
upper bound of S,. This result stands for local stability
of Sr provided that the state is neax the desired trajectories
for any initial conditions. Thus, boundedness of
S, = q'(Aq, Aq, sd, .) together with boundedness of feedback
gains and desired trajectories imply the boundedness
of Deltaq, Aq, sd, 6, and therefore the boundedness of Y,Q.
In virtue that H(q) %,positive dehite, we can also conclude
the boundedness of Sr as follows
s, = -H(q)-l {(K + C(q, q))sr - &e}
I h(H(Q)-l) {(hm + P211~11)~1+ 9)
I C(Q9 Q, dA4, Ad, 4) (18)
where the bounded function

which yields
t
Sr=AS+TLAS(C)e& (25)
where AS and s d are given in (7)-(9), and a, dd = 2j + 1,
j are non-negative integer. Then, a continuous terminal
attractor is induced in finite time provided that 7 in (9) is
tuned according to (21). A
In equation (25), the coefficients dn and dd give ad&-
tiond degrees of freedom to shape the dynamic surface S,.
Surface (25) together with control (11) yields fast asymp
totic convergence with a certain degree of robustness (see
Figure 5).
On the other hand, several interesting results have been
obtained by using saturated filtered errors ([ll],[4]. These
schemes basically introduce timevarying feedback gains,
among other technicd advantages.
Pmpsition 3: Consider the robot dynamics (1) in closed
loop with the controller (ll), provided that y in (9) is tuned
according to (21) the we have the following:
Hyperbolac Tangent.
Consider the following dynamic change of coordinates using
qr = qd-atanh(A&) +sdl-Tlsql(r)e&
AS = & + a tanh(XAq) - S ~(to)~-"(~-~)
I
-' *
s1
sd I
Then, a wntinuous saturated terminal attractor is induced
in finite time.
Satunrted Sine.
Consider the following dynamic change of coordinates
t
Qr = qd - crSin(AAq) f Sd2 - 7 lo AS(c)%d<
AS = Aq + aSin(XAq) - Sz(t~)ezp-"(~-~~)
-'
t
Sa
SdZ
t
sr = AS + 7 J, AS(~)%~C (27)
A. Hardware
A two degrees of freedom direct drive ShinMaywa R3C
robot arm is used as testbed. It is a rigid-link, low-friction,
planar robot with two joints. Dimensions of the robot
are given in Table I, where subindex 1 and 2 stand for
first and second link, respectively, and L, stands for the
center of mass, Pur means parameters, I stands for inertia,
and A defines joint limits. Joint position is measured
with optical encoders with resolution of 120,000 pulselrad
and joint velocity signal is estimated from position signals
using a first order filter with a constant T = 10.0.
Viscous friction damping coefficients were estimated using
an adaptive controller with persistent exciting trajectories
[3] running during 5min to obtain nominal coefficients are
Bo = (B11 = 1.674, B22 = 1.472)Nms.
B. Firmware
A Digital Signal Processor Loughborough Sound Images
Ltd DPCIC40B board control system was integrated on
a 16bit expansion bus slot of a DX - 486 personal computer.
TMS320 floating point DSP C v.1.0.1 compiler provided
the programing environment. The control input is
transmitted to the servomotors through the Shmmaywa
servosystem which powers the pulse wide modulators motor
drives at each joint.
C. Experimental Data
The performance of the proposed controller is shown in
comparison to the standard PD [l] and the baseline modelbased
adaptive controller [3]. Experiments are carried out
at high velocities in order to show the performance of the
system at inertial dominated dynamics.
C.1 Initial conditions, desired trajectories, and gains
Desired trajectories at each joint are Qd(t) =
AX:=, sin (7) where A = $! deg/rad is the amplitude
and fm = 5 s and fn = 2.5 S, for middle and high
velocities respectively, stands for the period. The irequencies
w, are w1 = 1.0 Ha, w2 = 2.0 Hz and w3 = 4.0 Hz.
The peak desired velocity is 87.96 deg/s and the peak desired
acceleration is 304.64 deg/s2. Initial conditions are
set about -5.0 deg with zero initial velocity. Feedback
gains are given in the "able 11.
Then, a continuous saturated terminal attractor is induced.
A
Surface (26) and (27) yield fast asymptotic convergence
with a certain degree of robustness. Since the system is
very sensitive to the feedback gain a, the tuning of X can
improve signiscantly the response of the closed-loop system,
(see Figure 7).
V. EXPERIMENTAL RESULTS
In this section the experimental setup and then the experimental
data is discussed and analyzed.
C.2 Tuning control gains
Wing the parameters is nothing obvious and special attention
is paid to avoid misleading conclusions. Since we
Par I( Mass I Lank I L, ( I ( A
DOFl II 21.20 I 0.25 I 0.15 I 0.21 I f160
DOF;
Units
15.22 0.45 0.19 0.18 A145
Kg n m Kgm2 deg
354

TABLE 11
FEEDBACK GAINS
present a comparison among similar but structurally Merent
controllers, we set common gains to the same value, see
Table 11, where values stand for the diagonal entry times
an identity matrix of proper dimensions, and w,, stands for
the desired natural frequency and T) the damping ration of
the PD controller [l].
c.3 Results
Comparative experiments are difficult to obtain because
any comparative result can be biased dangerously, besides
that it is difficult to compare qualitatively controllers that
are structurally Werent. To this end, we have limited our
study to pasivity based controllers [3], [l], and [2] that have
some common gains. We show experimental results obtained
”at kst run”, that is when reasonable performance,
as dictated by the theory, is obtained without perhaps obtaining
the best plots. Surprisingly enough, in all cases our
controller outperforms 131. To have a better view on the
results some plots are shown in two windows.
Figure 1 shows a comparative plot of positions tracking
errors at high velocities of the controllers PID 111, and our
Sliding-PD control. After a transient of 1 s, our controller
respect to [Z] yields tracking errors at each joint of threefold
and fourteenfold smaller respectively,
Figure 3 depicts the establishment of a terminal attractor
at AS = 0 of control (11) and (26). When % = 1.0 we
have in fact a PI controller in the error space AS. When
& < 1.0 a terminal attractor or slider is induced at AS =
0. Figure 4 shows control and position tracking errors when
we use Sliding-PD control for Werent dues of K in sd.
We can see that by simply tuning K we can slow down or
speed up the response of the system. Figure 5 shows the
effects of setting different values of saturation, via X in (26).
The higher the X the faster the response and the smaller
tracking errors.
Figure 6 shows comparative results between the controller
Sliding-PD and [3] where we can o k e that our
controller yields better tracking accuracy with smooth control
input, even when (31 started with smaller position
trackiig errors. It has been set the maximum feedback
control gains that [3] can afford without going into stability
or saturated problems.
VI. CONCLUSIONS
An approach to control a class of robot manipulators using
a very simple decentralized model-free PID controller
has been proposed. Experimental data shows the performance
of the proposed controller where it is concluded the
vaIidity of the proposed scheme. In this particular experimental
set-up, our model-free decentrdi controller af-
fords at least similar performance compared to the modelbased
adaptive controller [3] for tracking tasks.
ACKNOWLEDGMENTS
The authors acknowledge the anonymous reviewers for
their helpful comments. This work was carried out under
the Japan Society for Promotion of Science and the
Institute for Transfer of Industrial Technology Fellowships
of Japan, and written during an Alexander von Humboldt
Fellowship in the Institute for Roboticas and Mechantronics,
Germany, all held by the first author.
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Fig. 1. Gomparative wormance between [2] and our Controller (12)
in terms of position tracking errors. Dash line stands for PID 111
and solid line stands for our controller.
4 6
nma [sj
............ i ............ j .............
-200 2 4
nms [si
Fig. 4. Performance of Sliding-PD control for different values of n
(n = 1.0(.), = 5.0(.-), = io.o~), = 20.0(-)).
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different values ofratio 2 (E(-.), E(:), %(-I, &(.I). "k.acking errom in two windows.
356