![](bg4.png)
Three experiments had been made, for V , V
n
and
W separately, see Fig. 6. The accelerations(
dV
dt
,
dV
n
dt
and
dW
dt
) were calculated from the velocities(V ,
V
n
and W ), using the Euler’s method(Franklin et al.,
1997)). We used the range of data, where the forces
and torque were null to estimate the parameters, as in
Fig.7.
0 10 20 30 40 50 60 70 80
0
0.5
1
1.5
V[m/s]
Samples
(a) V (t)(m/s)
0 10 20 30 40 50 60 70 80
0
0.2
0.4
0.6
0.8
1
Vn[m/s]
Samples
(b) V
n
(t)(m/s)
0 10 20 30 40 50 60 70 80
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
W[rad/s]
Samples
(c) W (t)(rad/s)
Figure 6: Velocities behaviour.
0 5 10 15 20 25 30 35
−1.5
−1
−0.5
0
0.5
1
1.5
Samples
Velocity [m/s]
Acceleration[m/s.s]
(a) V (t),
dV (t)
dt
0 5 10 15 20 25
−1.5
−1
−0.5
0
0.5
1
Samples
Velocity[m/s]
Acceleration[m/s.s]
(b) V
n
(t),
dV
n
(t)
dt
0 2 4 6 8 10 12
−6
−5
−4
−3
−2
−1
0
1
2
3
Samples
Velocity[rad/s]
Aceleration[rad/s.s]
(c) W (t),
dW (t)
dt
Figure 7: Velocities and accelerations.
In order to estimate the viscous and the coulomb
frictions, the mass M and the inertia moment J
values are necessary. As there is no inertia mo-
ment J value, an estimation of J was firstly ob-
tained. We used the equation 18, the velocity curve
W (t) and the acceleration curve
dW (t)
dt
shown in Fig.
7(c). The viscous(B
w
) and the coulomb(C
w
) fric-
tions, estimated with method 1, were used on equa-
tion 18. The estimated robot inertia moment was
J = 1.358[kg.m
2
].
Since we have the values of robot mass, velo-
city and acceleration, we can apply on equations 16
and 17, to obtain viscous frictions(B
v
and B
v n
) and
coulomb frictions (C
v
and C
v n
). The table 3 shows
the equations obtained through least-squares method
and the friction values.
Table 3: Resulting equations and frictions - method 2.
Equations B
v
, C
v
,
B
vn
C
vn
V
dV (t)
dt
= −0.11V (t) − 0.86
4.01 30.35
V
n
dV n(t)
dt
= −0.10V
n
(t) − 0.87
3.77 30.76
3.3 Method 3 - Inertia Moment
Identification
A practical method for estimating the moment of iner-
tia J of the robot, to compare with the estimation
of the methods 1+2, is presented. The robot was
hanged from the ceiling by wire, see figure 8, to eli-
minate frictions between the robot and the floor. The
mass(m
o
) was hanged at the disc attached to the ro-
bot, by a wire.
Figure 8: Schematic of the experience.
By applying a known torque(Γ(t)) to robot body,
and measuring the resultant angular position(θ(t)), by
a external odometry system based on vision, we can
compute the moment of inertia of the robot. Based
in rotational equation of motion, see equation 3, and
considering null friction values (B
w
e B
cw
), we get:
Γ(t) = J
dW (t)
dt
(19)
The torque Γ(t) and the applied force f
m
(t) are
Γ(t) = f
m
(t)r
m
(20)
f
m
(t) = m
o
g (21)
where g is the acceleration of gravity (≈
9.8[m/s
2
]), m
o
is the mass and r
m
is the radius
disc(0.25[m]).
Four experiments had been made to identify the
value of J. The experiments 1, 2 and 3 were per-
formed with an object with mass equal 0.510 Kg, and
the experiment 4 with a object with mass equal 1 Kg.
Our objective was to verify the repeatability of the ex-
periments.
The figure 9(a) shows the angular position (θ(t))
curve for the first experiment. The angular velocities
(W (t)) were calculated using the derivative of the an-
gular position. The figure 9(b) shows the angular ve-
locity and the best fitting line to the angular velocity,
calculated by least-squares line method, that give us
the value of the angular accelerations.
The table 4 shows the torque values(Γ), angu-
lar accelerations(
dW (t)
dt
) and inertia moments(J) ob-
tained by experiments. The inertia moments were
similarly in all experiments.
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