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1 HIGH RESOLUTION ANALOGICAL MEASUREMENT OF THE ANGULAR VELOCITY OF A MOTOR USING A LOW RESOLUTION OPTICAL ENCODER José G. N. de Carvalho Filho 1 , Elyson A. N. Carvalho 1,2 , Lucas Molina 1,3 , Eduardo O. Freire 1 and Benedito A. Luciano 2 1 Electrical Engineering Nucleus of Federal University of Sergipe NEL/UFS Av. Marechal Rondon, S/N, São Cristóvão-SE, Brazil, 49100-000 [email protected]; [email protected]; [email protected]; [email protected] 2 Electrical Engineering Departament of the Federal University of Campina Grande DEE/UFCG Av. Aprígio Veloso, 882, Bodocongó, Campina Grande-PB, Brazil, 58109-900 [email protected] 3 COPPE of the Federal University of Rio de Janeiro COPPE/UFRJ Cidade Universitária, Ilha do Fundão, Rio de Janeiro-RJ, Brazil, 21945-970 Abstract: Dead-reckoning is the most widely used method to determine the instantaneous pose of a mobile robot. As a way to improve dead-reckoning, an analogical system to measure the angular velocity of a motor based on the use of an optical encoder was proposed and implemented. The encoder’s output signal is fed into a PLL (phase-locked loop) and its VCO (voltage-controlled oscillator) output is proportional to the angular velocity of the motor. Using an A/D converter the information is available to be used in digital control by a microcontroller. The obtained exactness and response time depend only on the used A/D converter. The paper presents a briefly overview about dead-reckoning and a detailed description of the proposed method. Simulation results are then presented to illustrate the proposed system performance. Keywords: angular velocity measurement, optical encoders, PLL, dead-reckoning. 1. INTRODUCTION Nowadays, the robots are no longer confined to the well structured and known industrial environments, where the ability to perform just repetitive tasks may be just enough. They are already being used to perform several tasks that demand a certain degree of “intelligence”. The necessity of safe and effective interaction with non-trained personal and increasingly mobility are some of the greatest challenges that are currently motivating the research in the field of robotics. As a consequence, mobile robots are already being designed and used for industrial applications, where the accomplishment of tasks with a high level of precision, exactness and speed, at a low cost, is required. Mobile robots are also used to replace the man in dangerous tasks or hostile environments, like to disarm bombs or space exploration. Now, they are being used even in the search of comfort, playing some domestic tasks, like cleanness and organization. Such applications often demand a high degree of precision and exactness of the robots during the accomplishment of the attributed tasks. These characteristics are obtained, generally, from the use of feedback controllers. Optimal estimators may also be applied, due to the fact that robot’s pose, and maybe other state variables, must to be known during the execution of the task. Dead-reckoning is the most widely used method to determine the instantaneous pose of a mobile robot [1]. Two basic approaches are commonly used: the absolute and the relative ones, and their detailed description are presented in [1]. The absolute position measurement methods are usually based on the use of active or passive landmark detection, map searching, or satellite data, etc. On the other hand, relative position measurement methods infer the position of the robot in the scene by the integration of velocity measurements and the knowledge of the its initial pose. The integrative nature of the relative approaches results in incremental localization errors. Despite of this, they are more used than the absolute systems due to their easier implementation, and for this reason this paper is focused on such relative methods. In [1] the dead-reckoning errors are classified as systematic or non-systematic errors. In structured and semi- structured environments the systematic errors are dominant and are the main cause of imprecision [1]. As a way to significantly reduce the systematic errors due to encoder limitations and to improve dead-reckoning performance, an analogical system to measure the angular velocity of a motor based on the use of an optical encoder was proposed and implemented. This paper is organized as follows: Section 2 is about dead-reckoning; in Section 3, the proposed approach is described; several simulation results are presented and discussed in Section 4; finally, in Section 5, some conclusions are presented and possible future works are indicated.

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Page 1: HIGH RESOLUTION ANALOGICAL MEASUREMENT OF THE … · High Resolution Analogical Measurement of the Angular Velocity of a Motor Using a Low Resolution Optical Encoder José G. N. de

1

HIGH RESOLUTION ANALOGICAL MEASUREMENT OF THE ANGULAR VELOCITY OF A

MOTOR USING A LOW RESOLUTION OPTICAL ENCODER

José G. N. de Carvalho Filho

1, Elyson A. N. Carvalho

1,2, Lucas Molina

1,3, Eduardo O. Freire

1 and Benedito A. Luciano

2

1 Electrical Engineering Nucleus of Federal University of Sergipe – NEL/UFS

Av. Marechal Rondon, S/N, São Cristóvão-SE, Brazil, 49100-000

[email protected]; [email protected]; [email protected]; [email protected]

2 Electrical Engineering Departament of the Federal University of Campina Grande – DEE/UFCG

Av. Aprígio Veloso, 882, Bodocongó, Campina Grande-PB, Brazil, 58109-900

[email protected]

3 COPPE of the Federal University of Rio de Janeiro – COPPE/UFRJ

Cidade Universitária, Ilha do Fundão, Rio de Janeiro-RJ, Brazil, 21945-970

Abstract: Dead-reckoning is the most widely used method

to determine the instantaneous pose of a mobile robot. As a

way to improve dead-reckoning, an analogical system to

measure the angular velocity of a motor based on the use of

an optical encoder was proposed and implemented. The

encoder’s output signal is fed into a PLL (phase-locked

loop) and its VCO (voltage-controlled oscillator) output is

proportional to the angular velocity of the motor. Using an

A/D converter the information is available to be used in

digital control by a microcontroller. The obtained exactness

and response time depend only on the used A/D converter.

The paper presents a briefly overview about dead-reckoning

and a detailed description of the proposed method.

Simulation results are then presented to illustrate the

proposed system performance.

Keywords: angular velocity measurement, optical encoders,

PLL, dead-reckoning.

1. INTRODUCTION

Nowadays, the robots are no longer confined to the well

structured and known industrial environments, where the

ability to perform just repetitive tasks may be just enough.

They are already being used to perform several tasks that

demand a certain degree of “intelligence”. The necessity of

safe and effective interaction with non-trained personal and

increasingly mobility are some of the greatest challenges

that are currently motivating the research in the field of

robotics.

As a consequence, mobile robots are already being

designed and used for industrial applications, where the

accomplishment of tasks with a high level of precision,

exactness and speed, at a low cost, is required. Mobile

robots are also used to replace the man in dangerous tasks or

hostile environments, like to disarm bombs or space

exploration. Now, they are being used even in the search of

comfort, playing some domestic tasks, like cleanness and

organization.

Such applications often demand a high degree of

precision and exactness of the robots during the

accomplishment of the attributed tasks. These characteristics

are obtained, generally, from the use of feedback controllers.

Optimal estimators may also be applied, due to the fact that

robot’s pose, and maybe other state variables, must to be

known during the execution of the task.

Dead-reckoning is the most widely used method to

determine the instantaneous pose of a mobile robot [1]. Two

basic approaches are commonly used: the absolute and the

relative ones, and their detailed description are presented in

[1]. The absolute position measurement methods are usually

based on the use of active or passive landmark detection,

map searching, or satellite data, etc. On the other hand,

relative position measurement methods infer the position of

the robot in the scene by the integration of velocity

measurements and the knowledge of the its initial pose. The

integrative nature of the relative approaches results in

incremental localization errors. Despite of this, they are

more used than the absolute systems due to their easier

implementation, and for this reason this paper is focused on

such relative methods.

In [1] the dead-reckoning errors are classified as

systematic or non-systematic errors. In structured and semi-

structured environments the systematic errors are dominant

and are the main cause of imprecision [1]. As a way to

significantly reduce the systematic errors due to encoder

limitations and to improve dead-reckoning performance, an

analogical system to measure the angular velocity of a motor

based on the use of an optical encoder was proposed and

implemented.

This paper is organized as follows: Section 2 is about

dead-reckoning; in Section 3, the proposed approach is

described; several simulation results are presented and

discussed in Section 4; finally, in Section 5, some

conclusions are presented and possible future works are

indicated.

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2. DEAD-RECKONING

To be able to autonomously navigate in its operating

environment, a mobile robot needs some information about

such environment. This problem is known as Robotic

Mapping Problem [2]. To acquire the necessary information

regarding the environment, a wide range of sensors, such as

cameras, sonar, laser, infrared sensors, contact sensors,

radars, GPS, etc., are used [2].

Due to its easy and low cost implementation, dead-

reckoning is often used combined with other kind of sensors

to build a map of the robot operating environment. When a

map of the environment is already known, the robot can be

endowed with the capability of autonomous navigation just

through the determination of the robot pose that can be more

easily done using dead-reckoning.

Dead-reckoning is commonly implemented using optical

encoders, which consist in a disc coupled to the motor axis

and an infrared Tx/Rx pair. The disc contains holes that

allow the infrared light to pass. As the motor axis turns, the

infrared light is sequentially blocked and non-blocked,

resulting in a square-wave with a frequency proportional to

the rotor’s speed [3-6].

Knowing the angular velocity of the motors and the

robot kinematics model, it is possible to calculate the linear

and angular velocities of the robot and its pose. However, as

previously mentioned, since the position is the integral of

the velocity, the uncertainty about the real pose of the robot

tends to accumulate during the trajectory, what, many times,

makes impracticable the use of dead-reckoning.

In [1] the dead-reckoning errors are classified as

systematic or non-systematic errors. In structured and semi-

structured environments the systematic errors are dominant

and are the major cause of imprecision [1]. Some examples

of systematic errors are [1]:

Unequal wheel diameters;

Average of both wheel diameters differs from

nominal diameter;

Misalignment of wheels;

Uncertainty about the effective wheelbase (due to

non-point wheel contact with the floor);

Limited encoder resolution;

Limited encoder sampling rate.

In [1] a calibration method to reduce uncertainty

accumulation when applying dead-reckoning is presented.

Errors due to the unequal wheel diameter, misalignment of

wheels and uncertainty about the effective wheelbase where

taking into account. The systematic errors associated with

limited encoder resolution and limited encoder sampling rate

were not considered.

The new method proposed in this paper to measure the

angular velocity of a motor aims to significantly reduce the

systematic errors due to encoder limitations.

Several methods are used to determine the angular

velocity of a motor from the signal output of the encoder.

The most important ones are: M method [6], T method [6],

M/T method [6], and S method [5-6].

Among the above mentioned methods, the most used one

in the M method. In this method the number of pulses (me)

during a fixed interval of time (TS), is counted. From me and

TS it is possible to calculate the angular velocity of the rotor

[6]. This method is easy to implement and the motor model

is not necessary, but the measure exactness and the response

time are directly dependent on TS. So, in low speeds the

measure exactness deteriorates. As data acquisition is

discrete, a high resolution optical encoder is required.

In T method, the time between two pulses is measured

(Te) and the rotor velocity is calculated from Te and the

angular displacement of the rotor during Te [6]. The

advantages of this method are the easy implementation and

the fact that the model of the motor is not necessary. On the

other hand, the measure exactness and the response time are

inversely dependent on the motor angular velocity. In this

case a high resolution optical encoder is also required.

M/T method results from the combination of methods M

and T. At low speeds, the T method is used, and, at high

speeds, the M method is used [6]. The major advantages and

disadvantages of both methods, in a softly way, are

encountered in the M/T method.

The S method is obtained from the T method. To

increase the velocity range of operation, the velocity curve

is segmented and so, the velocity to each segment is

calculated as if it had started from zero. The real velocity is

given by the sum of the velocity relative to each segment

with the maximum velocity of the respective anterior

segment. Thus, the exactness obtained to lower velocities is

extended to the higher ones [5-6]. However, dependence

between exactness, response time and the angular velocity

of the motor still exists.

As a way to avoid the disadvantages of the above

mentioned methods it is possible to use the resistance Rsense

[7-8]. This method is based on the use of a resistance in

series with the motor electrical circuit, making possible to

infer the motor velocity through the measurement of the

electrical current used to feed it. Even obtaining shorter

Fig. 1. Optical encoder. a) perspective view b) lateral view.

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response time, easy implementation and with lower size and

cost than the other methods, the S method is not widely used

due to the fact that, in this case, the model of the motor is

required, and besides, the electrical current varies as a

function of the motor load, resulting in non-reliable

measurements.

In this paper a high precision analogical method to

determine the angular velocity of a motor using optical

encoders is proposed, implemented and tested. To do so, a

PLL (phase-locked loop) is used to measure the frequency

of the encoder’s output signal, and a linear transformation is

then used to determine the angular velocity of the motor. As

the resultant measurement is in analogical form, an A/D

converter must to be used to convert it to a digital signal.

The proposed system may reach high resolutions, and

short response times, both depending on the used A/D

converter. The model of the motor is not necessary and the

costs are low, since the use of high resolution encoders is no

longer required.

3. THE PROPOSED APPROACH

3.1. The Robot Kinematics Model

The determination of the linear and angular velocities of

the robot is made based on its kinematics model. The

definition of such model is very important, since modeling

errors will produce velocity measurement errors, which will

result in cumulative positioning errors.

The unicycle model is assumed and in Fig. 2, xc and yc

are, respectively, the x and y position of the robot center of

mass. The robot linear and angular velocities are represented

by u and ω, respectively, whereas φ is the heading angle.

Using this kinematics model, the linear and angular

velocities of the robot can be obtained from the velocities of

the right and left wheels, respectively, ur and ul, according to

(1) and (2) [1, 9]:

(1)

(2)

where D is the distance between the wheels.

The kinematics model of the robot is given by [9]:

(3)

3.2. The Proposed System

This paper proposes to measure the angular velocity of a

motor (ωm) through the use of an optical encoder like shown

in Fig. 1. The angular velocity of the motor is related to the

linear velocity of the wheel (uw) by

(4)

where r is the radius of the wheel.

By its turn, the angular velocity of the motor is related to

its angular frequency (fm) by

(5)

Considering an optical encoder with n holes, the angular

velocity of the motor may be obtained from the

measurement of the encoder’s output signal frequency (fe),

by

(6)

From (4) and (6):

(7)

Thus, the encoder’s output signal frequency is a linear

function of the wheel linear velocity (uw), and one can be

easily obtained from the measurement of the other.

One way to perform frequency measurement consists in

converting frequency in voltage through the use of a PLL. A

PLL is a circuit that works through a feedback structure in

which the input signal is compared with a signal generated

by a VCO (voltage-controlled oscillator). The VCO

frequency is adjusted according to the feedback voltage

which is the result of the phase difference between those

two signals. Thus, the circuit makes use of feedback to make

the VCO frequency equal to the input signal frequency. As

the VCO makes a linear transformation from voltage to

frequency, the voltage value at the VCO input corresponds

to a measurement of its oscillation frequency, and as the

VCO frequency is equal to the encoder’s output signal

frequency, the input voltage of the VCO is an indirect

measure of the angular frequency of the encoder. By the use

of an A/D converter, this analogical measurement may be

converted into a digital form. The proposed system is

depicted in Fig. 3.

The voltage comparator and the signal conditioner

circuits shown in Fig. 3 were just used to transform the

pulsed signal provided by the optical encoder into a square-

wave with a 50% duty cycle.

Fig. 2. The unicycle model.

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The fact that the measurement is in the analogical form

is valuable, since in this kind of measurement the major

disadvantages of digital measurement are attenuated or even

eliminated, such as low measurement speed; inverse

relationship between response speed and accuracy;

dependence between the encoder’s frequency and the

measurement accuracy; and the necessity to know the motor

model; are attenuated or even eliminated when using the

approach here proposed. Besides, the PLL is a very common

and costless device.

The proposed system is not hard to mount and

reproduce. For example, to the PLL circuit, in this case the

CD4046 was used, the parameters to be adjusted are just the

values of two resistors and a capacitor in the VCO circuit,

and the low-pass filter. Thus, the same circuit may be

adapted to be used under different circumstances.

These two resistors and the capacitor in the VCO must

be chosen in way to allow that the frequency in the VCO

can vary over the entire range of values that the encoder’s

angular frequency can assume. Such choice is made using

the data extracted from the graphics found in the datasheet

of the component.

The A/D converter is the part of the circuit that will

determine the data acquisition rate and the exactness of the

motor angular measurement. A/D converters with 12 and 16

bits resolution are commonly found, and the time needed to

perform a conversion is usually under 50 µs, what means a

resolution and an acquisition rate very superior to the

systems currently used in dead-reckoning. A/D converters

like these are commonly included in many commonly used

microcontrollers.

The design of the low-pass filter is determinant to a

proper operation of the proposed system, and will be

detailed in the next sub-section.

3.3. The Low-Pass Filter

Any low-pass filter, passive or active, may be used in

this system, since it has a unit gain. For simplicity, a basic

RC filter was used, whose 3dB frequency is

(8)

In order to allow a strong attenuation of the modulation

generated by the comparison between the output signal of

the encoder and the signal generated by the VCO, the 3dB

frequency of the filter (fc) must be conveniently chosen

considering the smallest possible frequency of the encoder’s

output signal (femin). Such relation is presented in (9).

(9)

However, the filter 3dB frequency should be also chosen in

order to do not attenuate the signal frequency which is

correspondent to the angular velocity of the motor. So, the

maximum possible frequency (fvmax) should also be

determined, which represents the highest frequency

component of the signal amplitude spectrum corresponding

to the encoder frequency and it is calculated using the

parameters of the robot kinematics model and the robot

control system. So, the 3dB frequency should also obey the

following relation:

(10)

The 3dB frequency limits presented in equations (9) and

(10) are shown in Fig. 4.

From equation (9), the proposed system is not able to

measure zero velocities. As this is an isolated case, such

condition may be detected just observing the encoder output

signal.

The filter order should be as high as possible in order to

avoid ripples around the measured signal. So, a digital filter

may be used in order to obtain an even more exact measure.

As the maximum variation of frequency, corresponding

to the highest variation of motor velocity, is limited by the

smaller frequency of the encoder’s output signal, to increase

the value of fvmax it is necessary to use an encoder with a

higher number of holes. Thus, for slow-varying systems, it

is possible to use encoders with a very small resolution.

It is important to notice that the encoder resolution do

not affects the measurement exactness, just affecting the

superior limit of the motor velocity.

4. RESULTS

Two systems, one based on the proposed approach and

the other based on the M method were modeled using

MATLAB/SIMULINK. The proposed approach was

Fig. 4.a) Motor velocity limits b) Encoder’s output frequency

limits c) encoder frequency variation limits.

Fig. 3. The proposed system.

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compared with the M method, due to the fact that it is the

most used in dead-reckoning.

The answers of the two implemented systems to three

different kinds of input were simulated. The inputs were: a

square wave, a triangular wave, and a sine wave with an

increasing frequency. Each input represents the output

velocity signal (reference) sent to the motor by some

hypothetic control system.

The motor’s response to the control signal was modeled

by the following second order transfer function:

(11)

In Fig. 5, each used input and the motor’s response to it

is shown.

In Table 1 the absolute maximum values of the relevant

parameters are presented. They were taken from the Pioneer

2-DX mobile robot datasheet. Table 1. Pioneer 2 DX parameters.

Parameters Value

Maximum linear velocity

Maximum acceleration

Maximum deceleration

Wheel diameter 18.5 cm

A VCO was used to simulate the encoder’s response

when the motor is turning; transforming the reference

velocity into a signal whose frequency is proportional to

such velocity.

To simulate the system based on the M method, several

encoders with different resolutions were considered.

Encoders with 38, 128, 512, 1024, 2048 and 4096 holes

were tested. On the other hand, to illustrate the high

resolution attained by the proposed approach to perform

velocity measurement while using low resolution encoders,

the proposed system was always simulated considering the

lowest resolution encoder here used (38 holes).

Low resolution encoders are easily found, due to its low

cost and easy manufacturing. Some of the main applications

of low resolution encoders are in mouse’s circuitry. In

robotics, the use of such encoders is dramatically restricted

due to the fact that the most used methods to perform

velocity measurement are extremely dependent of the

encoder’s resolution.

The motor’s angular velocity and its measurement to

each of the different input signals using the proposed

approach are presented in Fig. 6.

The oscillations verified at the beginning of each

simulation are due to model limitations of the proposed

system. However, the system output response may oscillate

according with the design of the filter.

Fig. 6. Motor’s angular velocity (blue) and the measured angular

velocity (red) using the proposed approach with a low resolution

encoder (38 holes).

Fig. 5. Reference (blue) and output (red) angular velocities of the

motor.

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High Resolution Analogical Measurement of the Angular Velocity of a Motor Using a Low Resolution Optical Encoder José G. N. de Carvalho Filho, Elyson A. N. Carvalho, Lucas Molina, Eduardo O. Freire and Benedito A. Luciano

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The low-pass filter was designed to a cut frequency of

0.72 Hz. When the sine signal presented in Fig. 6 reaches a

frequency equal to 0.71 Hz (at time 17.5 s), which

corresponds to a motor acceleration 12.665 rad/s2, the

system starts to work beyond its operational range and the

measured velocity is no longer reliable.

The motor’s angular velocity and its measurement to

each of the different input signals using the M method are

presented in figures 7, 8 and 9. In each figure an encoder

with a different resolution is used. Such figures are provided

for the sake of comparison with the results obtained using

the proposed approach, presented in Fig. 6.

Two criteria were used to evaluate the obtained results

using the proposed approach and the M method using

encoders with different resolutions. The first one is the

correlation between the motor’s angular velocity and its

measurement. The second criterion is the relative mean error

of the measurement with respect to the motor’s angular

velocity.

The values obtained for such parameters using the

proposed approach and the M method using encoders with

different resolutions, considering the motor response for a

square signal, a triangular signal, and a sine signal with an

increasing frequency are presented in Tables 2, 3 and 4,

respectively.

As can be noticed by inspection of Tables 2, 3 and 4,

considering the motor’s response to the several reference

velocity signals, the correlation value for the output signal

of the proposed system using a low resolution encoder (38

holes) is very close to “1”, and the mean relative error is

close to zero. Such performance is comparable with the

performance obtained using the M method (a classical one)

based on an encoder with a resolution of 2048 holes. When

considering the motor’s response for a square and triangular

signal, the performance of the proposed approach was better

than the performance of the M method using an encoder

with 2048 holes, while the former method attained a better

result than the proposed approach when considering the

motor’s response for a sine signal with an increasing

frequency.

Table 2. Correlation and absolute mean error considering the systems

under evaluation and the motor’s response for a square signal.

Fig. 8. Measured angular velocity based on the M method (red)

using encoders with different resolutions, considering the motor

response for a triangular wave (blue).

Fig. 7. Measured angular velocity based on the M method (red)

using encoders with different resolutions, considering the motor

response for a square wave (blue).

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Table 3. Correlation and absolute mean error considering the systems

under evaluation and the motor’s response for a triangular signal.

Table 4. Correlation and absolute mean error considering the systems

under evaluation and the motor’s response for a sine signal with an

increasing frequency.

When compared with the M method using an encoder

with 4096 holes (a very good resolution) the proposed

system reached a slightly inferior result. Despite of this, the

proposed approach result should be considered satisfactory,

since it was obtained using an encoder with a resolution

more than 100 times lower.

5. CONCLUDING REMARKS

The paper focused on the proposition of a high precision

analogical system to measure the angular velocity of a motor

using low resolution encoders. It was implemented and

tested by simulation with good and promising results. The

system is based on the use of a PLL and represents a good

alternative to improve the performance of dead-reckoning

based localization systems using low resolution encoders.

The proposed system has a low implementation cost and

it can be easily adapted to measure the angular velocity of

any motor, by just adjusting the value of some resistors and

capacitors, since no knowledge about the motor model is

necessary.

The resolution of the proposed system is limited by the

resolution of the A/D converter chosen to be used.

Considering the A/D converters commonly available, the

resolution that may be obtained is very high when compared

with other methods used to perform the same task. So, the

encoder resolution, a common drawback to the currently

using systems, does not limit the resolution of the proposed

approach. The encoder resolution only affects the maximum

velocity variation value that the system is able to support.

Thus, high resolution measurements may be obtained even

using encoders with a very low resolution, since this will

just impose an upper limit to the velocity variation.

Another limiting factor to the use of dead-reckoning is

the encoder data acquisition rate. For the proposed

approach, this parameter is also limited by the A/D

converter. So, the presented approach is also superior to

other methods since A/D converters with high data

acquisition rates are easily found in the market with

accessible costs.

Besides, the fact that the resolution and the acquisition

rate are limited just due to the A/D converter makes the

system’s acquisition rate and resolution independent of the

angular velocity of the motor. This contributes to enlarge the

application range of the proposed approach.

An important drawback is that the proposed system is

not suitable to measure the zero velocity and the motor’s

rotation orientation. Such isolated cases should be detected

and treated using another approach.

Future works consists in the realization of experiments

using a prototype of the proposed approach and to mount it

onboard a mobile robot.

REFERENCES

[1] J. Borenstein and L. Feng, “Measurement and Correction

of Systematic Odometry Errors in Mobile Robots,” IEEE

Transactions on Robotics and Automation, Vol. 12, No.

Fig. 9. Measured angular velocity based on the M method (red)

using encoders with different resolutions, considering the motor

response for a sine wave with an increasing frequency (blue).

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