HIGH RESOLUTION ANALOGICAL MEASUREMENT OF THE ANGULAR VELOCITY OF A MOTOR USING A LOW RESOLUTION OPTICAL ENCODER

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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, j.gilmar@click21.com.br; ecarvalho@ufs.br; lmolina@ufs.br; efreire@ufs.br 2 Electrical Engineering Departament of the Federal University of Campina Grande DEE/UFCG Av. Aprígio Veloso, 882, Bodocongó, Campina Grande-PB, Brazil, benedito@ufcg.edu.br 3 COPPE of the Federal University of Rio de Janeiro COPPE/UFRJ Cidade Universitária, Ilha do Fundão, Rio de Janeiro-RJ, Brazil, 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 semistructured 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. 1

2 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 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, deadreckoning 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 semistructured environments the systematic errors are dominant and are the major cause of imprecision [1]. Some examples of systematic errors are [1]: Fig. 1. Optical encoder. a) perspective view b) lateral view. 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 (m e ) during a fixed interval of time (T S ), is counted. From m e and T S 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 T S. 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 (T e ) and the rotor velocity is calculated from T e and the angular displacement of the rotor during T e [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 R sense [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 2

3 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, x c and y c 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, u r and u l, according to (1) and (2) [1, 9]: where D is the distance between the wheels. (1) (2) The kinematics model of the robot is given by [9]: 3.2. The Proposed System This paper proposes to measure the angular velocity of a motor (ω m ) 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 (u w ) by where r is the radius of the wheel. By its turn, the angular velocity of the motor is related to its angular frequency (f m ) by (3) (4) (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 (f e ), by From (4) and (6): (6) (7) Thus, the encoder s output signal frequency is a linear function of the wheel linear velocity (u w ), 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 squarewave with a 50% duty cycle. Fig. 2. The unicycle model. 3

4 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 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 The Low-Pass Filter Fig. 3. The proposed system. 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 In order to allow a strong attenuation of the modulation generated by the comparison between the output signal of (8) the encoder and the signal generated by the VCO, the 3dB frequency of the filter (f c ) must be conveniently chosen considering the smallest possible frequency of the encoder s output signal (f emin ). 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 (f vmax ) 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 f vmax 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. Fig. 4.a) Motor velocity limits b) Encoder s output frequency limits c) encoder frequency variation limits. 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 4

5 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. 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. Fig. 5. Reference (blue) and output (red) angular velocities of the motor. 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 Maximum linear velocity Maximum acceleration Maximum deceleration Value Wheel diameter 18.5 cm A VCO was used to simulate the encoder s response when the motor is turning; transforming the reference Fig. 6. Motor s angular velocity (blue) and the measured angular velocity (red) using the proposed approach with a low resolution encoder (38 holes). 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. 5

6 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 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 rad/s 2, 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. 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). 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. 6

7 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. 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). 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. 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. 7

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