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1
eyðsÞ
k0 s2
2
2 q
2
2
0
0
2
2 p
2
0
ðs
2
x_ 1
ðs
u2
x 2
2
2
x_ 1
2
2
x_ 1
u2
eyðsÞ k2eyðsÞ k1
0
2
x_ 1
0
2 þ x_
2 þ x_
2
2
x_ 1
0
0
0
2
2
2 þ x_
2
x_ 1
2 p
2
x_ 1
k1; 1 ðt e1ðsÞds k1; 0 ðt
k2; 1 ðt e2ðsÞds k2; 0 ðt
3 Problem Formulation
Fig. 1 The single axis car
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5 Numerical Results
6 Experimental Results
1
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7
7
7
7
7
x 2
e
2
x_1
5
x 1ðtÞ ¼ R sinð3xt þ gÞsinð2xt þ gÞ x 2ðtÞ ¼ R
sinð3xt þ gÞ cosð2xt þ gÞ
4 ðx_ 1ðtÞÞ2 þ ðx_ 2ðtÞÞ2
þ x_
x_1
6
6
6
6
6
2 q ðx_ 1ðtÞÞ2 þ ðx_ 2ðtÞÞ2
with x ¼ 0.009 (s The
desired characteristic equations, Eqs.
(43)
and
(44)
were set to exhibit the following parameters,
f1 ¼ 1.9, f2 ¼ 2, xn1 ¼ 1.2, xn2 ¼ 1.25. The initial conditions of the system were chosen to be close
to the reference trajectory, x1 (0) ¼ 0.63, x2 (0) ¼ 0.02.
x_1
þ k2; 1s þ k2; 0 ¼ 0
converges toward the identity matrix. By virtue of the feedforward linearization control law design
(see Subsection
2.2),
a linear con trol law is required, which helps in keeping the flat outputs, x1, x2,
on the desired reference trajectories. In this case, the linearization scheme becomes equivalent to
controlling two independent chains of integrators. The linear parts of the control tasks are performed
by the discussed GPI controllers, given by Eqs.
(32)
and
(33).
The following ultimate description of
the closed-loop tracking errors is obtained:
þ 2f2xn2 þ x2 n2Þ
MAY 2015, Vol. 137 / 051001-5
Proof. Applying Eq.
(31)
in Eq.
(30):
þ 2f1xn1 þ x2 n1Þðs
4 ðx_ 1ðtÞÞ2 þ ðx_ 2ðtÞÞ2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiff
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiff
x € 2
q ðx_ 1ðtÞÞ2 þ ðx_ 2ðtÞÞ2
þ k1; 2e € 1 þ k1; 1e_1 þ k1; 0e1 ¼ 0
(42)
x_ 1ðtÞ
Figure
6
shows a block diagram of the experimental framework.
x_
1
x_2
2 2 p
þ x_
ðtÞ
x_2
x_
1ðtÞ 5 ðtÞþðx_
2ðtÞÞ2
Pd1ðsÞ¼ðs
2 2 p x_2
6
6
6
with xni, fi > 0, i ¼ 1, 2, 3, 4, and where xni, and, fi, respectively, represent desired natural frequency
and desired damping factor values.
(40)
pe2ðsÞ ¼ s
x_ 2ðtÞ
6
6
6
6
6
The initial orientation was set to be h (0) ¼ p / 2, and u1ð0Þ
5
By means of numerical simulations, it is verified that the pro posed controllers achieve accurate
tracking results. To illustrate the methodology, let us assume that it is desired to track a “six petals
flower” drawn in the X1 X2 plane. The following desired trajectory, expressed in parametric form,
was proposed:
þ 2f3xn3 þ x2 n3Þðs
(45)
5 ðx_ 1ðtÞÞ2 þ ðx_ 2ðtÞÞ2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffifffi
The pole placement has to be such that both characteristic equa tions assure asymptotic
stability of the underlying linear systems.
ffiffiffiffiffiffiffiffiffiffiffiffiffiffifffi
The experimental implementation results of the control law based
3
e
1
If the initial conditions are set sufficiently close to the given nominal trajectory, their corresponding
time derivatives x_1ðtÞ; x_2ðtÞ asymptotically evolving toward x_ 1ðtÞ; x_ 2ðtÞ. Using the exact
linearization principle (Lemmas 4 and 5), the matrix product
þ x_
x 2
þ k1; 1s þ k1; 0 ¼ 0
6
6
6
(43)
ðx_ 1ðtÞÞ2 þ ðx_ 2ðtÞÞ2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiff
3
x_ 2ðtÞ
þ 2f4xn4 þ x2 n4Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiff
þ k2; 2e € 2 þ k2; 1e_2 þ k2; 0e2 ¼ 0
7
7
7
2 q
x_ 1ðtÞ
), R ¼ 0.6 (m), g ¼ p.
The characteristic equations of the tracking errors, in closed-loop, are given by
x_2
4
(46)
(38)
ffiffiffiffiffiffiffiffiffiffiffiffiffiffifffi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffifffi
7
7
7
3
ðtÞ
þ k2; 2s
This can be achieved by forcing the desired tracking error dynam ics to coincide with prescribed
Hurwitz polynomials, pd1 (s), pd2 (s). For the sake of simplicity, the proposed desired Hurwitz
polynomials are represented by the product of two second-order Hurwitz polynomials of the form
(41)
ðx 1ð0ÞÞ2 þ ðx 2ð0ÞÞ2 . Figures
2
and
3
depict the simulation results for the feedforward
linearization control scheme, where the proposed control scheme works well with appropriate control
values in order to carry out experimental tests. Figure
3
also shows a detailed view of the tracking
results in which the tracking errors are confined to a small bounded zone of origin.
The experimental prototype used in the experiments is a paral lax Boe – Bot mobile robot (see
Fig. 4). The robot parameters are the following: The wheel's radius is R ¼ 0.7 (m); the axis length is
given by L ¼ 0.125 (m). Each wheel radius includes a rubber band to reduce slippage. The car
motion is provided by two DC motors, with a 6 (V) power supply. The acquisition system for the
planar position information is a 352 288 pixels resolution color web camera (Fig. 5) fixed on the
ceiling of the laboratory. The corresponding image processing is carried out by the MATLAB image
acquisition toolbox, and the computed control signal is sent to the internal robot microcontroller by
means of a wireless communication arrangement. The main function of the robot microcontroller is
to pulse width modulate (PWM) the control sig nals for the motor inputs. The microcontroller is a
BASIC STAMP 2 with a bluetooth communication card. The proposed output trajec tory tracking task
was the same as that defined in the simulations example. The controller parameters were defined
exactly as in Sec.
5
and the required sampling time was set to be 0.015 (s).
Journal of Dynamic Systems, Measurement, and Control
ðx_ 1Þ
pe1ðsÞ ¼ s
þ x_
ðx_ 1ðtÞÞ2 þ ðx_ 2ðtÞÞ2
7
7
7
7
7
q
x € 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiff
x_1
q
(39)
¼
3
Pd2ðsÞ¼ðs
4
(44)
2
2
2
2
2 2 x_ þ x_
1 2
2 p
2
x_ 1
2
x_ 1
2
x_ 1
4
ð3Þ þ k2; 3e
4
2
2
2
2 þ x_
2
2
ð3Þ þ k1; 3e
ð4Þ
þ k1; 2s
1
ð4Þ
1
2 2 x_ þ x_
1 2
2
2
x_ 1
2
2
2 þ x_
2
3
3
2
2 p
2
x_ 1
1
1
2
þ k1; 3s
þ k2; 3s
2
x_ 1
Fig. 2 Simulation control inputs
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Fig. 3 Simulation trajectory tracking
Fig. 4 Mobile robot prototype
Fig. 5 Image acquisition device
Fig. 6 Experimental control schematics
Fig. 7 Experimental control inputs
2o
u2
1o
Transactions of the ASME
5
3
u_1
ki2 ð ðx2ðsÞ x 2ðsÞÞds
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiff
2oðtÞ ¼ x € 2ðtÞ kd2ð_ x ^ 2ðtÞ
4
_
In order to analyze the controller's behavior, an experimental
test was provided using the same feedback control law, but,
instead of using the feedforward linearization, PI controllers
were implemented through a gain matrix using observer based
esti mates of the velocities. The following control law was used:
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiff
_
The same detailed view in relation to the numerical simulation
tracking result is provided to show the differences between the
nu merical and experimental performances.
ki1 ð ðx1ðsÞ x 1ðsÞÞds
x ^ 1ðtÞ
7
7
7
7
1oðtÞ ¼ x € 1ðtÞ kd1ð_ x ^ 1ðtÞ
051001-6 / Vol. 137, MAY 2015
_
x_ 2ðtÞÞ kp2ðx2ðtÞ x 2ðtÞÞ
6
6
6
6
with
2 q ð_ x ^ 1ðtÞÞ2 þ ð_ x ^ 2ðtÞÞ2 x ^
2ðtÞ ð_ x ^ 1ðtÞÞ2
þ ð_ x ^ 2ðtÞÞ2
q ð_ x ^ 1ðtÞÞ2 þ ð_ x ^ 2ðtÞÞ2 x
^ 1ðtÞ ð_ x ^ 1ðtÞÞ2
þ ð_ x ^ 2ðtÞÞ2
The experimental results were compared to the ones obtained
in the numerical simulations. Notice that the control inputs
(experimental and numerical) may be different, and the error
_
compensation in practice (see Fig. 9) needs additional control
efforts due to the unmodeled dynamical effects, the wireless
com munication process, the noisy measurement of the position,
among other unmodeled effects and unexpected disturbances.
x ^ 2ðtÞ
¼
are depicted in Figs.
7
and
8
where a good tracking is depicted.
x_ 1ðtÞÞ kp1ðx1ðtÞ x 1ðtÞÞ