2012

International Symposium on Power Electronics,

Electrical Drives, Automation and Motion

One Step Ahead Nonlinear Predictive Control of

Two Links Robot Manipulators
K. Bdirina*, R. Hajer**, M. Boucherit*, D. Djoudi*** and D. Rabehi***
* National polytechnic school (ENP), Hacen Badi avenue El Harrach, (Algeria)
** Department of Computer Engineering, King Saud University P.O. Box 51178 Riyadh, (Kingdom of Saudi Arabia)
*** Electronics Departements, Djelfa University, P.O. Box 3117 Djelfa 17000, (Algeria)

Abstract--Rigid link manipulators have attracted more II. NONLINEAR PREDICTIVE CONTROL
and more attention from robot control theorists and robot
users because of its various potential advantages. However,
Consider the nonlinear system
their nonlinear dynamics present a challenging control
problem, since traditional linear control approaches do not ‫ݔ‬ሶ ൌ ݂ሺ‫ݔ‬ሻ ൅ σ௠
௜ୀଵ ݃௜ ሺ‫ݔ‬ሻ‫ݑ‬௜ ሺ‫ݐ‬ሻ
൜ ሺͳሻ
easily apply. For a while, the difficulty was mitigated by the ‫ݕ‬ሺ‫ݐ‬ሻ ൌ ݄ሺ‫ݔ‬ሻ
fact that manipulators were highly geared, thereby strongly
reducing the interactive dynamic effects between links. In Where x (t) is the vector of state variables, u (t) is the
this work a fixed one step ahead nonlinear predictive control vector and y (t) is the output vector, the functions
control has been applied to compute time optimal solutions
f, g, h are assumed to be real and have continuous partial
for a two link manipulator operating in the horizontal plane
subject to control angle positions, where the solution is
derivatives.
obtained by minimizing a cost function. Tracking The classical goal in control is to impose the output of
performances of the controller are investigated via some the controlled system to achieve a setpoint as quickly as
simulations. possible [6]. In the predictive context, the predicted
tracking error is minimized over a finite horizon. The
Index Terms-- One step ahead predictive control, cost model prediction of a nonlinear system is a continuous
function, robot manipulator, nonlinear system, function that allows us to calculate the system output at
optimization future time (t + h), where h> 0 is the prediction horizon.
The predictive model output based on the Taylor series
expansion is given by,
I. INTRODUCTION
The concept of predictive control is the creation of an ‫ݕ‬ሺ‫ ݐ‬൅ ݄ሻ ൌ ‫ݕ‬ሺ‫ݐ‬ሻ ൅ ܸ௬ ሺ‫ݔ‬ǡ ݄ሻ ൅ ሺ݄ሻܹሺ‫ݔ‬ሻ‫ݑ‬ሺʹሻ
anticipatory effect, this control structure, developed for
linear systems, has experienced a real boom as advanced Where
control technology since the 80s [1]. This growth is due
to its robustness vis-à-vis the structured or unstructured ܸ௬ ሺ‫ݔ‬ǡ ݄ሻ ൌ ሺ‫ݒ‬ଵ ሺ‫ݔ‬ǡ ݄ሻ ‫ݒ‬ଶ ሺ‫ݔ‬ǡ ݄ሻ ǥ Ǥ ‫ݒ‬௠ ሺ‫ݔ‬ǡ ݄ሻሻ் ;
uncertainties. In general, the dynamic model of physical
processes is nonlinear and the establishment of predictive With
control laws for these processes requires minimizing the
cost function online, which is an operation very complex ݄ଶ ଶ ݄ ௥೔ ௥೔
‫ݒ‬௜ ሺ‫ݔ‬ǡ ݄ሻ ൌ ݄‫ܮ‬௙ ݄௜ ሺ‫ݔ‬ሻ ൅ ‫ܮ‬௙ ݄௜ ሺ‫ݔ‬ሻ ൅ ‫ ڮ‬൅ ‫ ݄ ܮ‬ሺ‫ݔ‬ሻ
[2]. To avoid this problem of online optimization, ʹǨ ‫ݎ‬௜ Ǩ ௙ ௜
nonlinear predictive control several off-line have been
proposed [3]–[4]–[5]. The prediction of tracking discard ݄௥భ ݄௥మ ݄௥೘
at one step is obtained using Taylor expansion of order ri ሺ݄ሻ ൌ ݀݅ܽ݃ ൬ ǡ ǡǥǡ ൰
‫ݎ‬ଵ Ǩ ‫ݎ‬ଶ Ǩ ‫ݎ‬௠ Ǩ
of the output signal and reference, where ri is the relative
degree of the ith system output, the solution of the ܹሺ‫ݔ‬ሻ ൌ ሺ‫ݓ‬ଵ ‫ݓ‬ଶ ǥ ‫ݓ‬௠ ሻ் ;
minimization a quadratic criterion at one step establishes
the control law. With
௥ ିଵ ௥ ିଵ
In this paper a fixed one step ahead nonlinear ‫ݓ‬௜ ሺ‫ݔ‬ሻ ൌ ൫‫ܮ‬௚ଵ ‫ܮ‬௙೔ ݄௜ ሺ‫ݔ‬ሻ ǥ ‫ܮ‬௚௠ ‫ܮ‬௙೔ ݄௜ ሺ‫ݔ‬ሻ൯
predictive control with input constraints is applied to two
link manipulator system. After presenting the principle of A. Reference Trajectory
a fixed one step ahead predictive control, a mathematical
For the output y (t) of nonlinear system (1) can follow
model is developed for our nonlinear system to validate
the reference trajectory yref (t), it must be r differentiable,
and test the proposed control, simulations were conducted
where r is the relative degree of the output y (t). This
through which the control performance is evaluated.
condition ensures the controllability of the output along
the setpoint yref (t) [7].

978-1-4673-1301-8/12/$31.00 ©2012 IEEE

1219
 Therefore we can apply the Taylor expansion of order III. ONE STEP AHEAD PREDICTIVE CONTROL OF RIGID
r to the reference signal: MANIPULATOR
The robot system is described by nonlinear model
‫ݕ‬௥௘௙ ሺ‫ ݐ‬൅ ݄ሻ ൌ ‫ݕ‬௥௘௙ ሺ‫ݐ‬ሻ ൅ ݀ሺ‫ݐ‬ǡ ݄ሻሺ͵ሻ [8]–[9], with q being the joint angles,  being the joint
inputs, rigid robot manipulator can be generally
expressed as
Where
݀ሺ‫ݐ‬ǡ ݄ሻ ൌ ሺ݀ଵ ሺ‫ݐ‬ǡ ݄ሻ ݀ଶ ሺ‫ݐ‬ǡ ݄ሻ ǥ ݀௠ ሺ‫ݐ‬ǡ ݄ሻሻ் ‫ݍ‬ሷ ሺ‫ݐ‬ሻ ൌ െ‫ܯ‬ሺ‫ݍ‬ሻିଵ ൫‫ܥ‬ሺ‫ݍ‬ǡ ‫ݍ‬ሶ ሻ‫ݍ‬ሶ ൅ ‫ܩ‬ሺ‫ݍ‬ሻ൯ ൅ ‫ܯ‬ሺ‫ݍ‬ሻିଵ ߬ሺ‫ݐ‬ሻ (8)
௛మ ௛ ೝ೔ ሺ௥ ሻ
With: ݀௜ ሺ‫ݐ‬ǡ ݄ሻ ൌ ݄‫ݕ‬ሶ ௥௘௙௜ ൅ ‫ݕ‬ሷ ௥௘௙௜ ൅ ‫ ڮ‬൅ ‫ݕ‬௥௘௙௜
೔
Optimal control is applied to the robot manipulator to
ଶǨ ௥೔
resolve the problem of tracking a reference trajectory for
the joint positions, the reference trajectory is a specified
In case this is not checked, a trajectory model of
by continuous function qref(t) for t [t0, tf]. The
exponential type is used to generate the reference
problem consists to find a control law of Torque (t)
trajectory yref (t) from the setpoint yd(t) [5]. The reference
such that the tracking error is minimized at the instant
trajectory yref (t) is in this case the solution of differential
(t+h), where h is an increment of time, in a way that the
equation:
݀‫ݕ‬௥௘௙ ݀‫ݕ‬௥௘௙ ݀‫ݕ‬௥௘௙ predicted output q (t + h) coincides with the set-point
‫ݕ‬௥௘௙ ሺ‫ݐ‬ሻ ൅ ߛଵ ൅ ߛଶ ǥ ൅ ߛ௥ ൌ ‫ݕ‬ௗ ሺͶሻ qref (t + h). Thus, the tracking error at time (t + h) will be
݀‫ݐ‬ ݀‫ݐ‬ ݀‫ݐ‬ defined by

B. One Step Ahead Predictive Control ݁ሺ‫ ݐ‬൅ ݄ሻ ൌ ‫ݍ‬ሺ‫ ݐ‬൅ ݄ሻ െ ‫ݍ‬௥௘௙ ሺ‫ ݐ‬൅ ݄ሻ (9)
The objective of one step ahead predictive control is to
find a control law u (t) which coincides the output y (t) The easiest way to predict the influence of the torque
with the reference trajectory yref (t) at time (t + h) [3]–[8]. (t) on the output q (t + h) is to use the Taylor expansion
So the criterion is to minimize the following functional: [10] –[11] to second order (r = [2.2]), we get:

ଵ
‫ܬ‬ଵ ൫‫ݕ‬ǡ ‫ݕ‬௥௘௙ ǡ ܴǡ ܳǡ ‫ݑ‬൯  ൌ  ฮ‫ݕ‬ሺ‫ ݐ‬൅ ݄ሻ െ ‫ݕ‬௥௘௙ ሺ‫ ݐ‬൅ ܶሻฮ  ൅
ଶ ݄ଶ
ଶ ொ ‫ݍ‬ሺ‫ ݐ‬൅ ݄ሻ ൌ ‫ݍ‬ሺ‫ݐ‬ሻ ൅ ݄‫ݍ‬ሶ ሺ‫ݐ‬ሻ ൅ ‫ݍ‬ሷ ሺ‫ݐ‬ሻሺͳͲሻ
ଵ ʹ
 ԡ‫ݑ‬ሺ‫ݐ‬ሻԡଶோ  (5)
ଶ
From equation (10), we deduce the model predictive
By replacing equations (3) and (2) in (5) the cost output at time (t + h):
function is then written: ݄ଶ
‫ݍ‬ሺ‫ ݐ‬൅ ݄ሻ ൌ ‫ݍ‬ሺ‫ݐ‬ሻ ൅ ܸሺ‫ݍ‬ǡ ‫ݍ‬ሶ ǡ ݄ሻ ൅ ܹሺ‫ݍ‬ሻ߬ሺ‫ݐ‬ሻሺͳͳሻ
ʹ
ଵ
‫ܬ‬ଵ ൫‫ݕ‬ǡ ‫ݕ‬௥௘௙ ǡ ܴǡ ܳǡ ‫ݑ‬൯ ൌൌ  ฮ൫‫ݕ‬ሺ‫ݐ‬ሻ ൅ ܸ௬ ሺ‫ݔ‬ǡ ݄ሻ ൅
ଶ
ଶ ଵ Where
ሺ݄ሻܹሺ‫ݔ‬ሻ‫ݑ‬ሻ െ ሺ‫ݕ‬௥௘௙ ሺ‫ݐ‬ሻ ൅ ݀ሺ‫ݐ‬ǡ ݄ሻሻฮ ൅ ԡ‫ݑ‬ሺ‫ݐ‬ሻԡଶோ (6) ݄ଶ ିଵ
ொ ଶ
ܸሺ‫ݍ‬ǡ ݄‫ݍ‬ሶ ǡ ݄ሻ ൌ ݄‫ݍ‬ሶ ሺ‫ݐ‬ሻ െ ‫ ܯ‬ሺ‫ݍ‬ሻሺ‫ܥ‬ሺ‫ݍ‬ǡ ‫ݍ‬ሶ ሻ‫ݍ‬ሶ ൅ ‫ܩ‬ሺ‫ݍ‬ሻሻ
ʹ
ܹ ൌ ‫ܯ‬ሺ‫ݍ‬ሻିଵ
m×m
Where Q  R is a definite positive matrix and R 
Rm×m is a positive semi definite matrix. The optimal Similarly, we apply the Taylor expansion of order 2 to
solution is then obtained by minimizing the criterion (6) the reference signal qref (t + h), we have:
for the nonlinear system (1) compared to control vector
u (t) ݄ଶ
‫ݍ‬௥௘௙ ሺ‫ ݐ‬൅ ݄ሻ ൌ ‫ݍ‬௥௘௙ ሺ‫ݐ‬ሻ ൅ ݄‫ݍ‬ሶ ௥௘௙ ሺ‫ݐ‬ሻ ൅ ‫ݍ‬ሷ ሺ‫ݐ‬ሻ
ʹ ௥௘௙
‫ݑ‬ሺ‫ݐ‬ሻ ൌ െሾሺܹሻ் ܹܳ ൅ ܴሿିଵ ሺܹሻ் ܳൣ݁ሺ‫ݐ‬ሻ ൅ ൌ ‫ݍ‬௥௘௙ ሺ‫ݐ‬ሻ ൅ ݀ሺ‫ݐ‬ǡ ݄ሻ (12)
ܸ௬ ሺ‫ݔ‬ǡ ݄ሻ െ ݀ሺ‫ݐ‬ǡ ‫ݔ‬ሻሿ (7)

The dynamic tracking at time (t + h) depending on the

e (t) is the tracking error, input torque can be then written:

݁ሺ‫ ݐ‬൅ ݄ሻ ൌ ‫ݕ‬ሺ‫ ݐ‬൅ ݄ሻ െ ‫ݕ‬௥௘௙ ሺ‫ ݐ‬൅ ݄ሻ ݁ሺ‫ ݐ‬൅ ݄ሻ ൌ ݁ሺ‫ݐ‬ሻ ൅ ܸሺ‫ݍ‬ǡ ‫ݍ‬ሶ ǡ ݄ሻ െ ݀ሺ‫ݐ‬ǡ ݄ሻ
݄ଶ
൅ ܹሺ‫ݍ‬ሻ߬ሺ‫ݐ‬ሻሺͳ͵ሻ
C. Stability ʹ
Let h be a positive real number and Q a positive The aim of control in the one step ahead predictive
definite matrix. Predictive optimal control described in control is to find the torque  (t) which forwards to the
(7) linearizes the dynamics of the tracking error and output q (t) the reference trajectory qref (t) at time (t + h )
allows a continuation of the asymptotic trajectory while seeking the least possible actuators. This then leads
reference if and only if r  4 for i = 1, 2,..... m. [7] – [10]. to an optimization criterion of the form:

1220
 ଵ ଵ
‫ܬ‬௟ ሺ݁ǡ ߬ǡ ݄ሻ ൌ ݁ሺ‫ ݐ‬൅ ݄ሻ் ܳ݁ሺ‫ ݐ‬൅ ݄ሻ ൅ ߬ሺ‫ݐ‬ሻ் ܴ߬ሺ‫ݐ‬ሻ g1 = (m1 (lcl + l1 ) + m2l1 + mLl1) g cos q1 + (m2 (lc 2 + l2 ) + mL (l2 + lcl )) g cos(q1 + q2 )
ଶ ଶ
(14) g 2 = (m2 (lc 2 + l2 ) + mL (l2 + lcl )) g cos(q1 + q2 )

From the first derivative of the cost function J, we find Where

the optimal solution for the control signal as:
IL the inertia of the load organe Link
߬ሺ‫ݐ‬ሻ ൌ Icl is the center of gravity of the load .
்
ିଵ
௛ర ௛మ J1 and J2 represent the inertia of the motors.
െ ቂ ܹሺ‫ݍ‬ሻ் ܹܳሺ‫ݍ‬ሻ ൅ ܴቃ ቆ ܹሺ‫ݍ‬ሻቇ ܳ൫݁ሺ‫ݐ‬ሻ ൅
ଶ ଶ

ܸ‫ݍ‬ǡ‫ݍ‬ǡ݄݀‫ݐ‬ǡ݄ (15) The numerical values of the parameters of the

represented manipulator are given in the table below

Consider the manipulator similar to the Fig. 1. The

TABLE I. PHYSICAL PARAMETERS OF TWO LINKS MANIPULATOR
manipulator consists of two rigid bodies of masses m1
and m2, lengths l1and l2. The angles of the joints are
l 1 = 0.45 m1 = l C1 = I1 = 6.25 J1 = 4.77
respectively q1 and q2 and a load mass mL of diameter Link 1
m 100 Kg 0.15 m Kg m2 Kg m2
lL.
m2 = l C2 = I2 = 0.61 J2 = 3.58
Link 2 l 2 = 0.20 m
25 Kg 0.15 m Kg m2 Kg m2

݉௅ ǡ ‫ܫ‬௅ mL = l CL= IL= 7.68

40 Kg 0.15 m Kg m2

݉ଶ ǡ ‫ܫ‬ଶ ݈ଶ ൅ ݈௖௅ The reference models chosen in continuous time are:

‫ݍ‬ଶ ‫ݍ‬௥௘௙ଵ
݈௖ଶ ‫ݍ‬௥௘௙ ൌ ቚ‫ݍ‬ ቚ
௥௘௙ଶ

Where:
‫ܬ‬ଶ
݉ଵ ǡ ‫ܫ‬ଵ ఠమ
‫ݍ‬௥௘௙௜ ሺ‫ݏ‬ሻ ൌ ‫ݎ‬௜ ሺ‫ݏ‬ሻ ǥ ǥ ݅ ൌ ͳǡʹ (16)
௦ ାଶకఠ௦ାఠమ
మ
݈௖ଵ
݈ଵ
‫ݍ‬ଵ
The nonlinear predictive controller is used to track
‫ܬ‬ଵ these desired trajectories
Fig. 1. Schematic of Two axis robot ߨ
‫ Ͳ݅ݏ‬൑ ‫ ݐ‬൑ ʹǤͷ‫ݏ‬
‫ݎ‬ଵ ሺ‫ݐ‬ሻ ൌ ൝ ʹ
ͲǤͻሺͳ െ ‘•ሺͳǤʹ͸‫ݐ‬ሻሻ‫ ݐ݅ݏ‬൐ ʹǤͷ‫ݏ‬

െߨ‫ Ͳ݅ݏ‬൑ ‫ ݐ‬൑ ͵‫ݏ‬

‫ݎ‬ଶ ሺ‫ݐ‬ሻ ൌ ቊ ߨ
‫ ݐ݅ݏ‬൐ ͵‫ݏ‬
The dynamic model is described by equation (10) with Ͷ
the following components [7] – [10]:
The initial displacements and velocities are chosen as:
A11 = m2l12 + ml l12 + I1 + I 2 + Il + J1 + 2 (m2l1(lc2 + l2 ) + ml l1(l2 + lcl )) cos q2
A12 = A21 = I 2 + I l + (m2l1(lc2 + l2 ) + ml l1(l2 + lcl )) cos q2
q1(0)=q2(0)=0°, ‫ݍ‬ሶ ଵ ሺͲሻ ൌ ‫ݍ‬ሶ ଶ ሺͲሻ ൌ 0
A22 = I 2 + Il + J 2
The choice of the damping coefficient and natural
c11 = − q 2 ( m 2 l1 (l c 2 + l 2 ) + m L l1 ( l 2 + l cL )) sin q 2 ; angular frequency of the filter is as follows:
c12 = − ( q1 + q 2 )( m 2 l1 (l c 2 + l 2 ) + m L l1 (l 2 + l cL )) sin q 2 ;
c 21 = q1 ( m 2 l1 (l c 2 + l 2 ) + m L l1 (l 2 + l cL )) sin q 2 ; =1, w1=w2=5rad/s.
c 22 = 0 .

1221
 position: q1, qref1 position: q2, qref2 Tracking:q1-qref1 Tracking: q2-qref2
1.8 1 0.06 0.04
position q1 position q2
1.6 poistion qref1 0.5 poistion qref2 0.05
0.02
1.4 0
0.04

1.2 -0.5 0
0.03
1 -1

(rad)

(rad)
(rad)

(rad)
0.02 -0.02
0.8 -1.5
0.01
0.6 -2 -0.04

0
0.4 -2.5
-0.06
0.2 -3 -0.01

0 -3.5 -0.02 -0.08

0 500 1000 0 500 1000 0 500 1000 0 500 1000
time(s) time(s) time(s) time(s)

(a) (b)

Torque r1 Torque r2
600 800

500
600

400

400
300
Nm

Nm

200 200

100
0

-200
-100

-200 -400
0 500 1000 0 500 1000
time(s) time(s)

(c)
Fig. 2. One step ahead predictive control of two link robot: (a) Angle positions and references, (b) tracking errors and (c) applied torques

The nonlinear controller has been tested by obtained by minimizing the quadratic error between
simulation where the control parameters Q=105 In, them.
R=10-12 In and h is set to 0.005. Simulation results are This approach has been applied to a nonlinear
show in Fig. 2. These Figures give the angular position system, that of two links manipulator to control the
(q1(t),q2(t)), angle references (qref1(t),qref2(t)) and the angle positions. Simulation results clearly show the
position tracking errors also the applied torques. The effectiveness of this approach in terms of references
simulation results show clearly the effectiveness of the tracking (angle positions) where the control objective is
one step ahead controller in terms of references achieved with good accuracy.
tracking (angle positions). Note here the presence of
undulation in the torque signal; this problem can be REFERENCES
solved by imposing of incremental variation of control [1] Koo, T. J.: stable model reference adaptive fuzzy control
signal constraints in control objective. Finally the of a class of non linear systems. IEEE transactions on
major drawback of one step ahead predictive control is Fuzzy systems, vol. 09, no4, pp. 624-636, August 2001.
the need to perform several simulations to obtain [2] Feng, G.:An approach to adaptive control of dynamics
numerical values of optimal parameters settings. systems. IEEE, vol. 10, no 2, pp. 268-275, Apil 2002.
[3] D, W.; Mohtadi, C.: Generalized predictive control,
IV. CONCLUSION Part1” The basic algortithm. Part II “Extension and
interpretation”, Automatica, Vol.32,no.2,1987,pp.137-
This work has focused on the contribution to the 160.
development of original control structure: The one step [4] Henson, E. M. A.; Seborg, D. E.: Non linear process
ahead predictive control, a strategy based on nonlinear control. New Jersey, Prentice Hall, 1997 . H. Poor,
predictive model where optimization mechanism of the An Introduction to Signal Detection and Estimation.
quadratic criterion is used to extract the control law. New York: Springer-Verlag, 1985, ch. 4.
The principle of the one step ahead predictive control, [5] Ping, L.: Non linear predictive control of continuous
nonlinear systems. Journal of Guidance, control and
is based on the Taylor series expansion of the predicted dynamics, vol.17, no.3,May-June 1994,pp.553-560.
output and the reference where the control law is [6] Singh, S.N.; Steinberg, M.; Digirolamo, R.D.: Nonlinear
predictive control of feedback linearizable systems and

1222
 flight control system design. Journal of Guidance ,
control and dynamics, vol. 18, no5, 1995, pp.1023-1028.
[7] Hadjer, R.: Contribution à l’analyse et à la synthèse de
commandes adaptatives et predictive. Application à un
processus physique. Doctoral thesis, USTHB university,
Algeria, 2000.
[8] Slotine, J. E.: Applied nonlinear control. Prentice Hall,
1991, chap.9.
[9] Hedjar, R.; Boucher, P.: Nonlinear receding horizon
control of rigid link robot manipulators. International
journal of advanced robotic systems, Vol.2, No.1, pp.
15-24, March 2005.
[10] Hedjar, R. ; Toumi, R. ; Boucher, P. ; Dumur, D. :
Commande prédictive non linéaire d’un robot
manipulateur rigide. Journal Européen des systèmes
Automatisés (JESA) Vol.36, No.6, pp.845-863, 2002.
[11] Chu, Y. S.; Stepanenko, Y.: On the robust control of
robot manipulators including actuator dynamics. Journal
of robotic Systems, vol.13, no1, 1996, pp.1-10

1223