J Braz. Soc. Mech. Sci. Eng. (2017) 39:543–551
DOI 10.1007/s40430-016-0693-5
TECHNICAL PAPER
Experimental evaluation of acceleration‑enhanced velocity
estimation algorithms using a linear motion stage
Yu‑Sheng Lu1
· Chung‑Heng Lee1
Received: 27 June 2016 / Accepted: 24 November 2016 / Published online: 3 December 2016
© The Brazilian Society of Mechanical Sciences and Engineering 2016
Abstract In this paper, several velocity estimation algorithms are redesigned by incorporating an acceleration signal into conventional schemes. These algorithms include a
state-space velocity observer (SSVO), a dynamically compensated velocity observer (DCVO), a tracking differentiator (TD), and a differentiator that uses the super-twisting
algorithm (STA). These approaches are practically realized
and experimentally compared to evaluate their utility for
velocity estimation. This paper also shows that an accelerometer-enhanced velocity observer can be used to improve
tracking performance for a feedback system. In contrast to
conventional velocity observers, which merely use position
information, an accelerometer-enhanced velocity observer
combines a position sensor and an accelerometer to produce an improved velocity estimation. Experimental results
are presented to show that an accelerometer-enhanced
velocity estimator gives a better tracking performance for a
linear motion stage. More specifically, a sliding-mode controller (SMC) is used to control the position of the payload
on a linear motion stage, which allows accurate positioning
within the limits of the resolution of the sensor, using an
acceleration-enhanced velocity estimation.
Keywords Accelerometer · Feedback control system ·
Linear motion stage · Velocity estimator · Velocity observer
Technical Editor: Sadek C. Absi Alfaro.
* Yu‑Sheng Lu
luys@ntnu.edu.tw
1
Department of Mechatronic Engineering, National
Taiwan Normal University, 162, He‑ping East Rd., Sec. 1,
Taipei 106, Taiwan
List of symbols
aAcceleration, m/s2
ciParameter of the DCVO, i = 1, 2, 3
CiPositive constant for tuning the STA, i = 1, 2
dUncertain input disturbance to the plant
DDynamic compensator of the DCVO
ePositional tracking error, m
gInput gain of the plant
hViscous coefficient of the plant
kiParameter related to a switching function in the SMC,
i = 1, 2, 3
KiSwitching gain in the SMC, i = 1, 2
liParameter of the SSVO, i = 1, 2
rPositional reference, m
uPlant’s torque-producing input, V
vVelocity, m/s
xPosition, m
Greek symbols
αParameter of the STA
εPositional estimation error
γ Parameter of the TD
ξCorrection term in the DCVO
Parameter of the STA
ωdcParameter for tuning the DCVO
ωnParameter for tuning the SMC
ωssParameter for tuning the SSVO
σ Switching function in the SMC
Subscripts
0Relative to initial condition
mRelative to measurement
Superscripts
∧Relative to estimation
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1 Introduction
Velocity feedback plays an important role in motion control
systems, because the velocity feedback can increase system
damping, which prevents the motion control systems from
becoming unstable. Effective velocity estimation is also
required for friction compensation [5]. However, it is not a
straightforward task to estimate the velocity. Motion control
systems are generally equipped with position encoders that
produce A- and B-phase quadrature signals. Conventionally, these quadrature signals are used to produce positional
information and to estimate the velocity. The common
approaches that use these quadrature signals to estimate
the velocity are the finite-difference method (FDM) and the
inverse time method (ITM). Both the FDM and the ITM
only use positional information to estimate the velocity.
The FDM is also called the fixed-time method, because it
calculates the backward difference in the encoder position
counts at fixed-time periods. The ITM evaluates the velocity by measuring the time between two consecutive encoder
pulses. This is also called the fixed-position method. However, these methods have limitations for either high-speed
or low-speed regimes. To estimate a velocity using encoder
pulses, Brown et al. [2] provided a review of existing techniques, such as backward difference expansions, Taylor
series expansions, and least-squares fits. Non-linear algorithms that only use positional information for velocity
estimation have been reported, including enhanced differentiators [12], tracking differentiators (TD) [4], and differentiators with the super-twisting algorithm (STA) [10].
Advances in MEMS technology mean that accelerometers have become cheap and ubiquitous. Accelerometers
measure acceleration, which is the derivative of velocity with respect to time. Theoretically, the velocity can
be obtained by integrating the acceleration signal with
respect to time. Silva et al. [11] calculated the longitudinal velocity of an all-wheel-drive mobile robot using the
initial velocity and the integral of the acceleration in the
longitudinal direction. In some cases, this approach is not
feasible, because there is an uncertain initial velocity. The
acceleration signal that is produced by an accelerometer
also contains a dc offset, which causes a significant drift in
the velocity estimate when the acceleration signal is integrated with respect to time [1]. This problem is alleviated
by integrating the acceleration signal using the positional
information. In combination with an appropriate velocity estimation algorithm, the use of an accelerometer and
a position sensor gives a promising approach to obtaining
a high-quality velocity signal. This paper focuses on several velocity estimation algorithms that use the acceleration
signal. These algorithms include the state-space velocity
observer (SSVO) [6, 13], the dynamically compensated
velocity observer (DCVO) [9], the tracking differentiator
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J Braz. Soc. Mech. Sci. Eng. (2017) 39:543–551
(TD), and a differentiator that uses the super-twisting algorithm (STA). The SSVO and the DCVO are revisited in
this paper. While the TD [4] and the STA [10] have been
used to estimate the velocity, using the positional information only, they are enhanced in this paper by incorporating
the acceleration signal. An experimental evaluation of the
FDM, ITM, SSVO, DCVO, TD, and STA is also presented.
These schemes are practically realized and experimentally
compared.
2 Acceleration‑enhanced velocity estimation
algorithms
2.1 Revisiting of the state‑space velocity observer
(SSVO)
Let the measured position and acceleration be denoted as
xm and am, respectively. The SSVO can be described as
x̂˙ = v̂ + l1 (xm − x̂),
(1)
v̂˙ = am + l2 (xm − x̂),
(2)
in which x̂ and v̂, respectively, denote the estimated position and velocity, and l1 and l2 are constant observer gains.
The characteristic equation for this second-order SSVO is
s2 + l1 s + l2 = 0 [9]. Therefore, the observer gains, l1 and
l2, are designed using the pole-assignment technique.
2.2 Revisiting of the dynamically compensated velocity
observer (DCVO)
The DCVO [9] is given by
x̂˙ = v̂,
(3)
v̂˙ = am + ξ ,
(4)
in which ξ = D(s)ε and ε = xm − x̂. D(s) is a dynamic
compensator. Let the dynamic compensator be of the form:
D(s) = (c2 s + c1 )/(s + c3 ),
(5)
in which c1 , c2, and c3 are constant design parameters. It
is shown that the characteristic equation for this DCVO
is s3 + c3 s2 + c2 s + c1 = 0 [9]. As with the SSVO, the
observer gains, c1 , c2, and c3, are designed using the poleassignment technique.
2.3 Redesign of the tracking differentiator (TD)
With the addition of an acceleration signal, the TD [4] is
redesigned as
x̂˙ = v̂,
(6)
J Braz. Soc. Mech. Sci. Eng. (2017) 39:543–551
v̂
v̂
+ am ,
v̂˙ = −γ sign x̂ − xm +
2γ
545
(7)
in which γ is a constant design parameter. A conventional
TD is obtained by removing am from the redesigned TD.
The use of acceleration for a conventional TD is beneficial,
because it augments the feedback correction with feed-forward compensation.
2.4 Redesign of the differentiator that uses the
super‑twisting algorithm (STA)
Using an acceleration signal, the differentiator that uses the
STA [10] can be redesigned as
˙ = v̂(t) − |ε(t)|1/2 sign(ε(t)),
x̂(t)
(8)
˙ = −αsign(ε(t)) + am ,
v̂(t)
(9)
in which ε = xm − x̂, and and α are constant design
parameters. In one study [8], two methods for tuning and
α were presented and these are investigated in the experimental study. A conventional differentiator that uses the
STA is also obtained by omitting am from the redesigned
differentiator. Using acceleration for a conventional differentiator is advantageous, because measuring the acceleration allows feed-forward compensation.
3 Experimental system
3.1 Structure of the experimental system
A linear motion stage is the experimental system, whose
photo and schematic are, respectively, shown in Figs. 1 and
2. A permanent-magnet synchronous ac motor is driven by
a regulator current converter that receives a torque-producing command in the form of analog voltage from a DAC
interface with a controller core. The controller core is a
DSP/FPGA-based system with DIO, ADC, and DAC interfaces. A field-programmable gate array (FPGA) is configured to interface with an optical linear encoder for position
counting and velocity detection that implements the ITM.
The ADC interface in the FPGA acquires acceleration from
an accelerometer. A digital signal processor (DSP) reads
feedback information on acceleration, position, and velocity (when the ITM is used) through the DSP interface in
the FPGA, calculates velocity estimation and control algorithms, and sends the control effort to the regulator current
converter through the DAC interface. The motor’s output
shaft is connected to a ball screw that translates rotational
motion of the rotor to linear motion of the payload. Both
velocity estimation and positional control of the payload
are considered in this paper.
Fig. 1 Photo of the experimental system
Accelerometer
Optical Encoder
Motor
FPGA
ADC +
Analog Processing
ADC
Interface
DAC +
Regulated Current
Converter
DAC
Interface
Payload
Linear Stage System
Position
Counting
DSP
Interface
Velocity
Detection
Digital Signal Processor
TI TMS320C6713
Fig. 2 Schematic of the experimental system
As shown in Figs. 1 and 2, the plant is a screw-driven
linear motion stage equipped with a linear optical encoder
and an accelerometer, where the accelerometer is mounted
to the slider of the linear scale, and both are attached to the
payload to measure the payload’s acceleration and position.
The accelerometer is of model ADXL320 from Analog
Device, Inc., having a full-scale range of ±5 g. The optical linear encoder is of model WBT5-0600MM from Carmar Technology Co. with a grating pitch of 20 μm, which
gives a resolution of 5 μm after the signal processing by
the FPGA. The FPGA also acquires acceleration with an
ADC. The observer core is a TMS320C6713 DSP, which
obtains the position and acceleration data from the FPGA,
and calculates the observer algorithms. The sampling rate
was chosen to be 12.2 kHz.
Although the resolution of the used optical encoder is
5 μm, its resolution is artificially degraded to 20 μm in
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J Braz. Soc. Mech. Sci. Eng. (2017) 39:543–551
3.2 Experiments with the ITM and FDM
200
velocity (mm/s)
most of the following experiments. In this way, the ITM
with a positional resolution of 5 μm is regarded as a reference, denoted as Reference ITM in figure legends.
100
0
-100
-200
3.3 Experiments with acceleration‑enhanced velocity
estimation algorithms
3.3.1 Experiments with the TD
3.3.2 Experiments with the STA
0.1
Method 2 :
1/2
= 0.5C1 ,
1/2
= C2 ,
α = 4C1 ,
α = 1.1C2 ,
(11)
in which C1 and C2 are positive constants to be chosen, which
reduce the original two-dimensional parametric search problem to a one-dimensional search problem. Using Method 1,
the value of C1 was found experimentally. The experimental
results show that C1 = 150 is appropriate. The experimental
13
0.25
0.3
0.15
0.2
0.25
0.3
0.15
0.2
0.25
0.3
0.15
0.2
0.25
0.3
velocity (mm/s)
-100
-200
ITM w. 20 m-resolution
0
0.05
0.1
time (s)
Fig. 3 ITM with various position resolutions
200
100
0
-100
FDM w. 5 m-resolution
0
0.05
0.1
time (s)
velocity (mm/s)
200
100
0
-100
-200
Reference ITM
0
0.05
0.1
time (s)
Fig. 4 FDM with a 5-μm resolution
Accelerometer
(10)
0.2
0
In designing the STA, there are two parameters, and α.
In one study [8], two methods for tuning and α were presented, which are
Method 1 :
0.15
time (s)
100
-200
Figure 5 shows the block diagram for a system that is used
to evaluate acceleration-aided velocity estimation algorithms. In terms of the design of the TD, only one parameter, γ , needs to be determined. Figures 6, 7, respectively,
show the results for γ = 35, 000 and γ = 70, 000, in
which the positional resolution is 20 μm. It is seen that
whereas the TD for γ = 70, 000 produces noisy velocity
estimation, the TD for γ = 35, 000 produces an offset in
velocity estimation. The TD for γ = 70, 000 is chosen for
comparison with other schemes.
0.05
200
velocity (mm/s)
Most of the following experiments employ a sensor’s resolution of 20 μm, whereas the ITM with a positional resolution of 5 μm is considered a reference. The linear motion
stage was driven by a sinusoidal input, and Fig. 3 shows the
ITM with various position resolutions. It can be seen that
because of the averaging property, the ITM with the 20-μm
resolution gives less estimation noise when compared with
the ITM with the 5-μm resolution. However, during zerocrossing of the velocity trajectory that occurs at a relatively
low-speed condition, the ITM with the 20-μm resolution
shows a significant phase lag in velocity estimation. Figure 4 shows the FDM with a positional resolution of 5 μm,
demonstrating that the FDM has much poorer performance
than the ITM with the 5-μm resolution.
ITM w. 5 m-resolution (Reference ITM)
0
Acceleration
Signal
ADS8361
Position
Counting
Optical Encoder
Position Signal
ADC
Interface
DSP
Interface
FPGA
Digital Signal Processor
Fig. 5 System for evaluating acceleration-aided velocity estimation
algorithms
20
TD w. =35000
Reference ITM
0
0.005
0.01
0.015
time (s)
0.02
0.025
120
110
100
TD w. =35000
Reference ITM
90
80
0.05
0.055
time (s)
0.06
10
STA w. C1=150
0
Reference ITM
0
0.005
0.01
0.015
time (s)
0.02
0.025
0.03
120
110
100
STA w. C1=150
Reference ITM
0.055
time (s)
0.06
0.065
20
0
TD w. =70000
Reference ITM
0
0.005
0.01
0.015
time (s)
0.02
0.025
110
100
TD w. =70000
Reference ITM
80
0.05
0.055
time (s)
0.06
results also show that C2 = 900 is appropriate when Method
2 is used. Figures 8, 9, respectively, show the velocity estimates using STAs with C1 = 150 and with C2 = 900, in
which the positional resolution is 20 μm. These two STAs
are then chosen for comparison with other schemes.
30
20
10
Reference ITM
0
0.005
0.01
0.015
time (s)
0.02
0.025
0.03
120
110
100
STA w. C2=900
Reference ITM
0.055
time (s)
0.06
0.065
Fig. 9 STA with C2 = 900
60
Reference ITM
40
20
TD w. =70000
STA w. C1=150
0
STA w. C2=900
-20
-40
3.4 Experiments that are subjected to an accelerometer
offset
0
velocity (mm/s)
The accelerometer’s output usually has an uncertain dc offset. In this study, a voltage of 0.2 V is artificially added to
the accelerometer’s output, for the purposes of the experiment. Figure 10 shows the responses for the TD and the
STAs due to this dc accelerometer offset, in which the positional resolution is 20 μm. It is seen that the STAs are sensitive to this dc offset. Although the TD is more robust to
the dc offset, there is also an offset in its estimate.
STA w. C2=900
0
90
0.05
0.065
Fig. 7 TD with γ = 70,000
40
-10
0.03
120
90
velocity (mm/s)
50
velocity (mm/s)
velocity (mm/s)
velocity (mm/s)
20
Fig. 8 STA with C1 = 150
40
-20
30
90
0.05
0.065
Fig. 6 TD with γ = 35,000
40
-10
0.03
velocity (mm/s)
0
velocity (mm/s)
50
40
-20
velocity (mm/s)
547
velocity (mm/s)
velocity (mm/s)
J Braz. Soc. Mech. Sci. Eng. (2017) 39:543–551
0.005
0.01
0.015
time (s)
0.02
0.025
0.03
120
110
100
Reference ITM
90
80
0.05
TD w. =70000
0.055
time (s)
0.06
0.065
Fig. 10 TD and STA subject to an acceleration offset
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J Braz. Soc. Mech. Sci. Eng. (2017) 39:543–551
25
Reference ITM (ITM w. 5 m-resolution)
40
ITM w. 20 m-resolution
20
Reference ITM
SSVO w/o an acc offest
SSVO w. an acc offset
-20
-40
-60
0
0.005
0.01
0.015
time (s)
0.02
0.025
0.03
140
velocity (mm/s)
DCVO w. 20 m-resolution
20
0
Reference ITM
SSVO w/o an acc offest
SSVO w. an acc offset
120
velocity (mm/s)
velocity (mm/s)
60
15
10
5
100
0
80
0.05
0.055
time (s)
0.06
0.065
Fig. 11 SSVO with and without an accelerometer offset
0
0.002 0.004 0.006 0.008
0.01
time (s)
0.012 0.014 0.016 0.018
0.02
Fig. 13 Magnified view with the 20-μm resolution
40
Reference ITM
DCVO w/o an acc offest
DCVO w. an acc offset
-40
0
0.005
0.01
0.015
time (s)
0.02
0.025
0.03
velocity (mm/s)
0
-20
-60
velocity (mm/s)
200
20
120
0
-100
Reference ITM (ITM w. 5 m-resolution)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.15
0.2
0.25
0.3
time (s)
200
110
100
Reference ITM
DCVO w/o an acc offest
DCVO w. an acc offset
90
80
0.05
100
-200
0.055
time (s)
0.06
0.065
velocity (mm/s)
velocity (mm/s)
60
100
0
-100
-200
DCVO w. 5 m-resolution
0
0.05
0.1
time (s)
Fig. 12 DCVO with and without an accelerometer offset
Fig. 14 DCVO with a 5-μm resolution
In terms of the DCVO, the observer gains, c1 , c2, and
c3, are designed using the pole-assignment technique. Let
2 ), in
s3 + c3 s2 + c2 s + c1 = (s + 5ωdc )(s2 + 2ωdc s + ωdc
which ωdc = 200. In terms of the SSVO, the
observer gains, l1 and l2, are designed by letting
2 , in which ω is a params2 + l1 s + l2 = s2 + 2ωss s + ωss
ss
eter. To compare the performance with that of the DCVO,
let ωss = 663, so that the high-frequency response for the
SSVO to accelerometer noise is the same as that for the
DCVO. Figures 11, 12, respectively, show the responses for
the SSVO and the DCVO, in which the positional resolution is 20 μm. It is seen that the DCVO is insensitive to this
dc accelerometer offset, but the SSVO produces an offset in
its velocity estimate.
13
3.5 Experiments with the DCVO using various
positional resolutions
Figure 13 shows that for the same positional resolution of
20 μm, the DCVO outperforms the ITM. Figures 14, 15
show the responses for the DCVO and the ITM for a positional resolution of 5 μm. Figure 14 shows that the velocity estimates that are produced by both the DCVO and the
ITM are similar, but the ITM’s velocity estimate is noisier
than that of the DCVO. Figure 15 shows that the DCVO
can detect extremely slow motion, but the ITM cannot
because of the limitation in the position sensor’s resolution.
J Braz. Soc. Mech. Sci. Eng. (2017) 39:543–551
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position (mm)
25
Reference ITM (ITM w. 5 m-resolution)
DCVO w. 5 m-resolution
error (mm)
15
10
control (V)
5
0
3.6 Positional control using an accelerometer‑enhanced
velocity observer
3.6.1 A design for a sliding‑mode controller for a linear
motion stage
A linear motion stage is used to evaluate the performance
of two velocity observers: one of which is an accelerometer-enhanced velocity estimator, the DCVO, and the other
of which is a conventional estimator that uses the ITM. The
plant is modeled as a second-order system that is described
by
(12)
in which h = 4.857, g = 11, 432, x and u, respectively,
denote the plant’s positional output and torque-producing
input, and d denotes an uncertain input disturbance. Let the
tracking error be defined as e = x − r, in which r denotes
the positional reference. The feedback controller uses the
sliding-mode control (SMC) law with integral compensation [3, 7]. A switching function is defined as
t
σ = ė + k2 e + k1
edt − (ė0 + k2 e0 ),
(13)
0
in which k2 = 2ωn, k1 = ωn2, and ωn = 30. Let the SMC
law be
in which k3 = ωn, K2 = 0.4g−1, and K1 = 0.1.
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.5
0.6
0.7
0.8
0.9
1
0.5
0.6
0.7
0.8
0.9
1
time (s)
-0.1
1.5
1
0.5
0
-0.5
time (s)
time (s)
Fig. 16 Step response by the ITM with a 20-μm resolution
In terms of phase, it is also seen that the DCVO’s velocity
estimate leads that for the ITM.
u = g−1 [hẋ − (−r̈ + k2 ė + k1 e + k3 σ )]
− [K2 |−r̈ + k2 ė + k1 e| + K1 ]sgn(σ ),
0
0.02
Fig. 15 Magnified view with the 5-μm resolution
ẍ + hẋ = g(u + d),
5
0
(14)
position (mm)
0.012 0.014 0.016 0.018
error (mm)
0.01
time (s)
control (V)
0.002 0.004 0.006 0.008
15
10
5
0
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.5
0.6
0.7
0.8
0.9
1
0.5
0.6
0.7
0.8
0.9
1
time (s)
0.1
0
-0.1
1.5
1
0.5
0
-0.5
time (s)
time (s)
Fig. 17 Step response by the DCVO with a 20-μm resolution
0.04
error (mm)
0
10
0.1
0.02
0
-0.02
-0.04
velocty (mm/s)
velocity (mm/s)
20
15
DCVO
Reference w. 5 m-resolution
0
0.1
0.2
0.3
0.4
0.5
time (s)
0.6
0.7
0.8
0.9
1
DCVO
100
Reference ITM (ITM w. 5 m-resolution)
50
0
0
0.1
0.2
0.3
0.4
0.5
time (s)
0.6
0.7
0.8
0.9
1
Fig. 18 5-μm reference and the DCVO with a 20-μm resolution
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J Braz. Soc. Mech. Sci. Eng. (2017) 39:543–551
position (mm)
0
-0.05
DCVO w. 20 m-resolution
10
5
0
0.2
0.3
0.4
0.5
time (s)
0.6
0.7
0.8
0.9
1
0.05
-0.1
1.5
1
0.5
0
-0.5
0
ITM w. 20 m-resolution
DCVO w. 20 m-resolution
1
1.5
2
time (s)
2.5
3
3.5
4
Fig. 19 Output error with a resolution of 20 μm
3.6.2 Positioning results and discussion
Two references are used in the following experiments:
a step reference of 10 mm and a sinusoidal reference
of 0.25 Hz. Figures 16, 17 show the respective step
responses for the ITM and DCVO. It is seen that the
switching frequency for the control input that is associated with the DCVO is much faster than that for the ITM,
so the DCVO allows faster correction of output errors
than the ITM. Figure 18 shows a magnified view of the
response that is shown in Fig. 17, in which signals with a
5-μm resolution are also used for reference. It is worthy
of note that this response by the DCVO is obtained using
a sensor’s resolution of 20 μm, but the reference signals
for a 5-μm resolution are not used for real-time feedback control. As seen from the reference signals that are
shown in Fig. 18, it is suggested that a positional accuracy
within the sensor’s resolution of 20 μm is achieved using
the acceleration-enhanced velocity estimation scheme.
The upper sub-plot in Fig. 19 shows the error responses
to the step reference, and the lower sub-plot shows the
error responses to the sinusoidal reference. It is seen that
compared with the ITM, the DCVO has a smaller output
error, because it uses an accelerometer. In terms of regulation control, output precision is limited by the sensor’s
resolution.
In the subsequent experiments, the resolution of the
positional sensor was reduced to 5 μm. Figures 20, 21,
respectively, show the step responses for the ITM and
DCVO. It is seen that the DCVO allows a control input
with a faster switching frequency than the ITM, which
gives a prompter reaction to system perturbation and a
more accurate positional response. To compare performance, the upper sub-plot in Fig. 22 shows the error
responses to the step reference and the lower sub-plot
13
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
time (s)
Fig. 20 Step response by the ITM with a 5-μm resolution
position (mm)
0.5
error (mm)
0
control (V)
-0.1
15
10
5
0
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.5
0.6
0.7
0.8
0.9
1
0.5
0.6
0.7
0.8
0.9
1
time (s)
0.1
0
-0.1
1.5
1
0.5
0
-0.5
time (s)
time (s)
Fig. 21 Step response by the DCVO with a 5-μm resolution
0.1
error (mm)
-0.05
0
0.1
error (mm)
0.1
control (V)
0
error (mm)
15
ITM w. 20 m-resolution
-0.1
0.05
0
-0.05
ITM w. 5 m-resolution
-0.1
DCVO w. 5 m-resolution
0
0.1
0.2
0.3
0.4
0.5
time (s)
0.6
0.7
0.8
0.9
1
0.05
error (mm)
error (mm)
0.1
0.05
0
-0.05
ITM w. 5 m-resolution
-0.1
DCVO w. 5 m-resolution
0
0.5
1
1.5
2
time (s)
2.5
Fig. 22 Output error with a resolution of 5 μm
3
3.5
4
J Braz. Soc. Mech. Sci. Eng. (2017) 39:543–551
shows the error responses to the sinusoidal reference. It
is obvious that the DCVO outperforms the ITM. Because
of advances in MEMS technology, accelerometers have
become cheap and ubiquitous, so an accelerometerenhanced velocity estimator is a cost-effective scheme that
improves performance.
4 Conclusions
This paper revisits the SSVO and DCVO and redesigns the
TD and STA using an acceleration signal in the original
designs. Although the experimental results show that the
redesigned TD and STA are not very robust to an accelerometer offset, the redesigned TD and STA are more effective than the traditional TD and STA. The experimental
results show that the SSVO is sensitive to an accelerometer
offset, but the DCVO is nearly not affected by this offset.
Compared with the ITM, the DCVO can detect very lowspeed motion and produces a less noisy velocity estimate.
In this paper, two velocity estimation schemes are experimentally evaluated using the same SMC law. The experimental results show that the DCVO enables higher speed
error correction and more precise tracking than a conventional estimator that uses the ITM. One significant aspect
of the experimental results is that positional accuracy for a
specific sensor’s resolution is increased for an accelerationaugmented velocity observer.
Acknowledgements The authors would like to thank the Ministry
of Science and Technology for their support of this research under
Grants No. MOST 104-2221-E-003-011-MY2.
551
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