A Novel High-Resolution Optical Encoder With Axially ... - IEEE Xplore

IEEE SENSORS JOURNAL, VOL. 18, NO. 14, JULY 15, 2018

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A Novel High-Resolution Optical Encoder With Axially Stacked Coded Disk for Modular Joints: Physical Modeling and Experimental Validation Sarbajit Paul , Student Member, IEEE, Junghwan Chang , Member, IEEE, John Edward Fletcher , Senior Member, IEEE, and Subhas Mukhopadhyay , Fellow, IEEE

Abstract— A novel approach to realize a high-resolution absolute encoder with compact disk size for the application in a light weight robotic arm modular joint is proposed in the present research. First, the high resolution coded disk of the absolute encoder is designed using graph theory-based Hamiltonian cycle. Unlike the traditional 1-D binary coded tracks, the proposed generated code is a 2-D n × n matrix code with 0s and 1s as the matrix elements and n numbers of uniquely coded tracks. The 2-D matrix code improves the code density of the encoder system 2 by 2n compared with that of the traditional 1-D codes (2n ). Second, the coded tracks are arranged axially along Z-axis to stack the tracks within a constant disk diameter and to avoid the radially divergent track patterns. These combined operations result in a high-resolution absolute encoder with a compact constant disk diameter which is essential to comply with the size constraint of the modular joint. With the proposed designed framework, a prototype with n = 2 is manufactured using 3-D printing technology. The designed encoder with two tracks is tested on a rotary system and the absolute angle values are obtained using the unique codes generated by the photosensors installed in the prototype. Index Terms— Absolute encoder, axially stacked disk, directed graph, high resolution, modular joint.

I. I NTRODUCTION

A

BSOLUTE optical encoders are non-contacted electromechanical sensors which employ unique coded patterns to find real-time absolute positions. They are considered as an integral part of any automation and precision controlling system. Due to their high accuracy, high resolution, long lifetime, high reliability and wide measurement range, they can be used in high-performance servo applications, precise

Manuscript received April 19, 2018; accepted May 26, 2018. Date of publication June 1, 2018; date of current version June 26, 2018. This work was supported by Dong-A University Research Fund. The work of S. Paul was supported by the Australia Award Endeavour Fellowship, Department of Education and Training, Australian Government. The associate editor coordinating the review of this paper and approving it for publication was Prof. Kazuaki Sawada. (Corresponding author: Junghwan Chang.) S. Paul and J. Chang are with the Mechatronics System Research Laboratory, Department of Electrical Engineering, Dong-A University, Busan 49125, South Korea (e-mail: [email protected]; [email protected]). J. E. Fletcher is with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia (e-mail: [email protected]). S. Mukhopadhyay is with the Department of Engineering, Macquarie University, Sydney, NSW 2109, Australia (e-mail: subhas.mukhopadhyay@ mq.edu.au). Digital Object Identifier 10.1109/JSEN.2018.2841982

position controlling of robotic arms, radar systems and various astronomical and avionic apparatus etc [1]–[7]. Traditional absolute optical encoder consists of encoder disk, light source, and a decoding unit, respectively. The encoder disks are generally patterned with binary gray coded tracks [8]–[11]. Even though the generation of the gray coded disk is relatively simple, they suffer from the inherent problem of quantization noise on account of the limited resolution they possess. The resolution of the gray coded absolute optical encoder increases at the cost of increasing the number of tracks. With the increase of the number of tracks of the coded disk, the size and the weight of the encoder system also increase. However, with the rapid improvement in technology for microsensors and the increase of the application field of the absolute encoders, high precision and miniaturization of the optical encoder with advanced intelligence have become an important topic of research. For coded disk, being the core component of an optical encoder, a number of studies have been performed since last two decades to achieve miniaturization of the disk without compromising with the resolution of the encoder [12]–[18]. [12], [13] proposed the inclusion of Vernier method with the gray coded track to improve the resolution of the encoder system. However, Vernier encoders require a large number of sensor heads for the detection circuit. In articles [14]–[16], to reduce the number of coded tracks of the encoders, single track disks were proposed. Some researchers have used pseudorandom coded patterns [14] or De Burjin sequence [15] to design the coded track. Also, M-code was used to design the absolute encoder disk pattern [16]. However, all these single track disk has the track at its outermost periphery. Therefore, in terms of mechanical design, the disk size increases with the increase of the number of bits because the patterns can only be accommodated at the outermost track. In [17], the encoder design shows a promising improvement of resolution with a limited number of coding patterns. However, they face trouble with high-speed operation due to the error in synchronization rate and divergence characteristic of the coded disk. The synchronization error was addressed by Das et al. [19], where the authors employed sequential logic circuit to achieve the synchronization with the object speed. However, the divergence character of the coded tracks is not yet studied in literature before. To Summarize, all the above

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mentioned coding pattern shares one common aspect, that the tracks are arranged along the radial direction of the coded disk. However, for the applications such as the light weight robotic arm modular joint, where the system design is constrained with the compactness of its different components, a radially stacked track disk is not the ideal choice. Motivated by this challenge to achieve a high resolution encoder within the constraint of compactness for the light weight modular joints [23], this paper proposes a novel absolute optical encoder design where the encoder coded disk is designed based on the graph theory approach which ensures 2 an ultra-high resolution (2n ). Instead of using the radially stacked coded disk axially stacking is considered to design the coded disk. The idea of axial stacking can frequently be found in electric motor design, in which motor parts namely, stator and rotor are stacked axially over a common axial shaft [20]–[22]. Previously, some researches on axially stacked encoder were documented in literature which employs individual disks stacked axially over a common shaft. However, to the best of our knowledge, there are no earlier reports in the literature that specifically address the design of an absolute optical encoder with axially stacked two dimensional tracked disk using the Hamiltonian cycle of the graph theory. Furthermore, axially stacked disk reduceds the overall size of the conventional modular joint by incorporating the use of the preaxisting outer covering of the modular joint of the robotic arm which uses high power density permanent magnet motors with hollow shaft. In agreement with the aforementioed requirements, the design fulfills the need in the light weight robotic where the space and resolution are considered as the design constraints. Moreover, the whole modular joint system weight is considered including the high resolution axial encoder while the design process is performed. The paper is organized as follows. Section II introduces the operating principle of an absolute encoder. The framework of a 2-dimensional coded disk design is presented in section III. While section IV presented the prototype of the proposed absolute encoder module and the experimental analysis. The paper concludes in section V with a discussion on the overall performance. II. O PERATING P RINCIPLE OF A BSOLUTE O PTICAL E NCODER The schematic of the commercial robototic arm with the modular joints is shown in Fig. 1. The enlarged sectional view of the modular joint in Fig. 1 shows that the modular joint comprises of the the harmonic drive, gearing mechanism, motor and the position and torque sensors. The sensors provide the required information for the two level control system of robotic arm. The two level control system consisting of low and the high-level controllers support the multicontrol modes including the position control, current control, velocity control, trajectory planning, dynamic sensor data tracking and error detection etc. The position sensor section of the modular joint which in general employs the rotory optical encoders is highlighted in the yellow box, as shown in the Fig. 2(a). A schematic of the conventional abosulute encoder

Fig. 1. Schematic of the robotic arm showing the modular joint with different components and the two level commercial control setup based on the position signal obtained from the encoder attached to the modular joint.

Fig. 2. Sectional view: (a) the modular joint highlighting the optical encoder attached at the end and (b) the traditional absolute encoder with radially stacked binary tracked disk.

with radially stacked tracked binary gray coded disk is shown in the Fig. 2(b). As illustrated in Fig. 2(b), the absolute encoder system is composed of three different sections, namely, the coded disk, the light source, and the photodetector assembly. In Fig. 2(b), the coded disk has three tracks. For three tracks of the coded disk, three light sources and three photodetectors are needed. In general, for encoders using the binary gray code, to achieve n bits of resolution, n numbers of coded tracks are needed which are decoded using n numbers of sensor heads facing each tracks. The unique numerical codes thus generated are converted into absolute angular position information. The resolution of the optical encoder system is determined by the coded disk track numbers. For an n track binary gray coded absolute encoder, the expression for resolution, R can be written as follows, R=

360 2n

(1)

From (1) it is evident that a high resolution radially stacked track disk absolute optical encoder needs a large number of tracks. Consequently, this increases the diameter of the disk and thus the size and weight of the overall encoder system. Thus for an lightweight robotic arm modular joint which has a restriction over the size and weight of its components, the traditional gray coded radially tracked disk encoder are not

PAUL et al.: NOVEL HIGH-RESOLUTION OPTICAL ENCODER WITH AXIALLY STACKED CODED DISK

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the best choice. From the sectional view of the modular joint in Fig. 2(a), it is noteworthy that size of the modular joint can significanlty be reduced if the a conventional gray coded encoder system is substituted with a more superior encoder, designed considering the constraint such as the space and weight of the modular joint. Motivated by this requirement in the forhcoming sections a novel high resolution optical encoder is proposed with an axial track stacking approach. III. F RAMEWORK FOR THE T OW-D IMENSIONAL C ODED D ISK As mentioned in the section II, the aim of this paper is to design a compact and high resolution absolute optical encoder that can easily be accomodated into the conventional modular joint without increasing the size and weight of the whole system extensively. To achieve this goal, in this section an axially stacked track Hamiltonian cycle based disk design is proposed. The objective of the proposed research can be broadly classified into: (a) Introduction of a two dimensional binary coding system that provides a ultra-high resolution compared with the resolution obained using the conventional one dimensional binary coding (e.g. gray code, M-code); (b) ultilization of the existing conventional structure of the modular joint, illustrated on Fig. 2(a) to accomodate the novel coded disk absolute encoder without increasing the size of the overall system. To start with, the following subsections formulate the first objective of designing an ultra-high resolution disk. A. Preliminaries To achieve an ultra-high resolution for the intended novel encoder system, it is essential to revisit and modify the resolution expression, (1) of conventional radially tracked encoder system. It must be kept in mind while designing a rotory optical encoder that the coded disk design, which is considered as the most critical component, must satisfy two very basic requriments such as the code should be a (a) bijective and (b) cyclic code. in other words each elements of a set of n elements must occure only once and the lee distance of the code should be equal to unity. Preexisited conventional codes such as gray code and M-code satify these two abovementioned requriements. To comply with the objective (a) mentioned above, the proposed research utilized the well estiblished mathematical concept called directed graph theory, to achieve the two-dimensional binary code [23]–[28]. In particular the concept of Hamiltonian cycle, first proposed by Sir William Rowan Hamilton in 1875 is used to find the two-dimensional cyclic pattern which fulfills the requriements (a) and (b) mentioned aboved. Definition 1: Given a (connected) graph G(V, E), a circuit can be found that starts at a vertex of G, passes through every vertex exactly once and returns to the starting vertex. This circuit is called Hamiltonian Cycle [24]. Fig. 3(a) shows the graph of a dodecahedron to illustrate the concept of Hamiltonian Cycle. The path highlighted in red shows the Hamiltonian Cycle of the dodecahedron with every vertex traveled once, which is in accordance with the requriement (a).

Fig. 3. Idea of the two dimensional coding based on: (a) Hamiltonian cycle; (b) arrangement of the coded patterns showing the movement of the photodetectors while fulfilling the left-right submatrix condition of the directed graph theory; and (c) arrangement over a cycle.

Under the idea of this Hamiltonian path, the aim is to find a Hamiltonian Cycle for a matrix sequence of n × n binary matrices (with matrix elements 0 or 1) which can be used to design a unique two-dimensional pattern disk for the absolute encoder. B. Initialization and Termination Conditions Using the Hamiltonian Cycle Based n × n Binary Matrix Sequence To find the Hamiltonian cycle for an n × n matrix sequence X, let us consider, any directed graph D = (V, E), where V(D) is the set of vertices of binary square matrix sequence X and E is the set of arcs between two vertices with mapping, init: (i) and ter: (i+1) with i being the number of vertices. Definition 2: Let a and b are two square matrices order n. The right submatrix of a is a matrix formed by selecting the subsets of all columns of a, except the first column. For b, the left submatrix of b is constructed using all the columns of b except the last column. Mathematically, (2) a, b ∈ R n×n , a = b ⊂ X (n+1)×n n×(n−1) = [a1 , a2 , . . . , an ]; L = [b1 , b2 , . . . , bn ] R (3) where, ai , bi ∈ Rn × n are the column matrices, made of the columns of a and b, R and L are the right and the left submatrices, respectively. Proposition 3: The arc E (ab) between two matrices a and b is possible when the right submatrix of a is equal to the left submatrix of b. Mathematically, using (3), R (n+1)×n = L n×(n−1) ⇒ E(ab) ∈ E(D)

(4)

if (4) satisfies, D is connected and is a balanced directed graph. It can be expressed as, deg− (v i ) = deg+ (v i )

(5)

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where, {v i }2i=0−1 ⊆ V is a vertex of D. deg− (v) and deg+ (v) are the indegree and the outdegree of v, respectively. if both (4) and (5) are true for D, D has an Eulerian cycle. Moreover, for a diagraph D which is Eulerian, it has a Hamiltonian cycle. C. Design of the Two-Dimensional Coded Disk and Track Arrangement With the proposal of Hamiltonian cycle mentioned in subsection B, a coded disk is designed with a binary square matrix sequence of order 2. The matrix sequence can be listed as follows, 0 0 1 0 0 1 v1 = v2 = v0 = 0 0 0 0 0 0 1 1 0 0 1 0 v3 = v4 = v5 = 0 0 1 0 1 0 0 1 1 1 0 0 v6 = v = v = 1 0 7 1 0 8 0 1 1 0 0 1 1 1 v9 = v 10 = v 11 = 0 1 0 1 0 1 0 0 0 1 1 0 v 12 = v 13 = v 14 = 1 1 1 1 1 1 1 1 v 16 = 1 1 n2

where, {v i }2i=0−1 ∈ V (D). Using (5), a Hamiltionian cycle of D that can fulfill the condition in (4), can be framed as, v 0 → v 2 → v 1 → v 8 → v 4 → v 10 → v 7 → v 3 → v 9 → v 12 → v 14 → v 13 → v 6 → v 11 → v 15 → v 5 → v 0 The abovementioned Hamiltonian cycle will form the two-dimensional code of the disk for the proposed encoder. The binary square matrix associated with each vertex in the above generated Hamiltonian cycle is further divided into upper and lower row vectors and arranged in two tracks as shown in Fig. 3(b). To detect the patterns on both the tracks in matrix form, 4 photodetectors (in red circle) are arranged as shown in Fig. 3(b). For a movement from time t to t + t, the upper and the lower track sensors are moved for one pattern section, which makes the right submatrix of vi equal to the left submatrix of v i + 1. If the two tracks, thus generate are arranged radially over a circular disk as shown in Fig. 3(c). The resolution of the encoder designed with the coded disk in Fig. 3(c) can be expressed as, R=

360 2 2n

(6)

From (1) and (6), using the proposed coded disk, the 2 resolution of the absolute encoder can be improved by 2n /2n . Thus from (6), it is evident that the coded disk designed using the Hamiltonian cycle gives a very high resolution compared with the traditional gray coded optical encoders. However, as illustrated in Fig. 3(c), when the trackes are assembled radially, they are stacked in a radially outward fashion. In other words, from the illustration Fig. 3(c), all the

Fig. 4. Proposed designs: (a) axially stacked two-dimensional Hamiltonian cycle based coded disk illustrating data decoding and (b) arrangement of the coded disk and the auxillary electronic unit of the encoder system into the modular joint.

2

tracks contain 2n numbers of pattern sections to generate an 2 n × n matrix code. As a result, to accommodate 2n number pattern sections of the first track in a circular fashion, a definite inner diameter has to be fixed while considering the mechanical tolerance and the remaining tracks are placed over the first track. This ultimately increases the overall diameter of the disk with the increase of track numbers. Mathematically, if s is the arc length of the pattern section, the inner radius can be written as, 360 s (7) r1 = ; θ = 2 θ 2n Now with inner radius, r1 and the thickness of a single track t, for n × n matrix code, the outer radius will be, r2 = r1 + n × t

(8)

Here, the thickness, t depends on the size of the photo detector module. For an instance, high-resolution CCD sensors have dimension within 10 mm × 10 mm × 10 mm. Thus, t can be arranged accordingly. Similarly, an assembly of n2 , light dependent resistors (LDR) in n × n matrix will increase the thickness more than 10mm. Thus, the overall diameter of the coded disk increases. This radially outward stacking of the tracks restricts the use of this ultra-high resolution encoder system in the desired modular joint application. Therefore, summarizing the discussion so far, it is evident that an alternative approch is needed to arrange the two-dimensional coded pattern so that the overall disk diameter can be reduced by eliminating the divergence characteristic shown in the radial arrangement. This brings to the second objective (b) mentioned at the starting of section III, which is explained in the following section. IV. P ROTOTYPING AND R ESULTS A. Coded Disk Design for the Application in Modular Joint As mentioned in section III, the two-dimensional coding pattern based on Hamiltonian cycle possesses divergence character in track distribution when the tracks are arranged along radial direction, which eventually increases the overall coded disk size. Therefore, to reduce the code disk diameter and make it feasible to install the proposed coded pattern with the conventional modular joint, an alternative approach shown in Fig. 4 which involves an axial stacking of the


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TABLE I A NGULAR P OSITION I NFORMATION BASED ON THE T WO -D IMENSIONAL M ATRIX

two dimensional tracks is proposed in the proposed research. The two individual trackes of Fig. 2(b) are placed over the circumference of a hollow cylinder as shown in Fig. 4(a), stacked along Z direction. The hollow cylindrical disk provides two distinct adventages for the intended application of the modular joint. Firstly, a constant diameter of the tracks can be maintained, which is same as that of the hollow cylinder. This eliminates the increase in diameter with the mounting of extra tracks on the outer periphery of the preexisting tracks in case of the radial disk, shown in Fig. 3(c). Secondly, as illustrated in the sectional diagram in Fig. 4(b), the hollow disk can be mounted over the motor section along the outer covering and the electronic components including the light source, photo detectors and the data acquisition circuits can be placed over the outer covering of the moular joint. It is noteworthy that, even though by stacking the two dimensional tracks along axial direction to form the cylindrical coded disk restricts the the overall size of the encoder disk along radial direction to a fixed value, it may not alone fullfill the requriement of reducing the weight of the existing modular joint. Howoever, when compared with the commercially available radial disk encoders, the size and weight of the modular joint can be reduced if the proposed design is used. In other words, a comparison between Fig. 2(a) and Fig. 4(b) clearly shows the benfit of the proposed cylindrical disk based modular joint system, which helps in eliminating the extra encoder section of the external housing added next to the motor by housing the encoder over the motor. Furthermore, unlike the commercial encoders which uses speparate coxial disks for light sources and photodetectors with data acquisition circuits housed inside a metal housing, as mentioned above the proposed system uses the inner wall of outer housing of the modular joint to install all the electronic components. which further compensate any potential increase in weight beacuse of the axial stacking of the tracks and makes the overall system compact compared with the initial model of Fig. 2(a). Now, to understand further the data acquisition from the axially stacked tracked disk using the photo sensor, Fig. 4(a) shows the matrix code decoding method using the 2 × 2 binary coded matrix sensors. this binary coded 2 × 2 matrix changes with angular rotation of the cylindrical disk along with the motor. The transformation of the 2 × 2 matrix code, thus obtained into the absolute angular position is shown in Table. I.

Fig. 5. Prototype: (a) Sketch of the designed two-dimensional axially stacked track disk absolute encoder module using SOLIDWORKS and (b) manufactured 2 × 2 absolute encoder module using 3D printing technology.

Fig. 6. Experimental setup to test the designed two dimensional absolute encoder.

B. Designed Prototype and Experimental Analysis The manufactured two dimensional axially stacked tracked disk encoder prototype is shown in Fig. 5. The sketch of the encoder module assembly designed using the SOLIDWORKS software is shown in Fig. 5(a), showing the three-dimensional view of the cylindrical disk and the shaft. The manufactured encoder module using 3D printing technology is shown in Fig. 5(b). The manufactured module is designed in such a way that it can be coupled with motors and the laboratory testing can be performed. The experimental setup of the manufactured encoder prototype is shown in Fig. 6. The encoder shaft is coupled with a

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TABLE II S PECIFICATION OF THE E XPERIMENTAL S ETUP

Fig. 8. Deviation of the photosensors output from the fixed reference values. TABLE III C OMPARISON A MONG D IFFERENT E XISTING E NCODER S YSTEMS

information as shown in Table I. Also, the percentage deviation of the sensor data from the reference output value is shown in Fig. 8. The deviation arises due to the errors in the sensor installation and manufacturing tolerances present in the prototype. C. Discussion

Fig. 7. Output waveform of the photosensors with the movement of the coded disk.

rotary system (BLDC Motor) to provide an angular velocity. As shown in Fig. 6, the light source and the photosensor matrix are installed on either side of the coded disk. The specifications of different components of the experimental setup are listed in Table II. Fig. 7 shows the output waveforms of the photosensor matrix. It is clearly visible from Fig. 7 that with the movement of the two-dimensional disk, the four sensor modules trace four different information of the tracks. The output data of photosensor 1 and 2 are combined to form the upper track information and that of photosensor 3 and 4 provides the information of lower track, as mentioned before in Fig. 4(a). Combining them and transforming into the binary equivalent matrix can finally provide the angular position

From above-mentioned proposal and analysis, following inferences can be made: (a) The manufactured 2 × 2 encoder module with two tracks provide a resolution of 22.5◦. To achieve the same resolution using a gray coded binary encoder, four coded tracks are needed. Similarly, for a 3×3 coded disk with only 3 coded tracks, the resolution can be improved to approximately 0.703◦, whereas 9 tracks are needed to achieve the same resolution in case of a traditional gray coded encoder. Thus, in general, by using an n×n encoder module an improvement 2 of (2n )/ (2n ) can be achieved compared with the traditional gray coded encoder system. The proposed encoder system with n tracks is compared with other existing methods for the absolute optical encoder in terms of resolution, number of tracks and number of photodetectors and presented in Table III. Table III guarantees that the proposed method provides a high resolution with less track and sensors compared with the other traditional methods.


(b) The reduction of the track numbers without compromising with the resolution ultimately reduces the overall size of the encoder module which make it suitable for the light weight robotic arm modular joint application. Furthermore, the axial stacking provideds a superior assembling option when used in the lightweight modular joint. (c) The prototype used to validate the proposed encoder design method is considered only to validate the proposal. Further improvement in terms of commercial use can be achieved by introducing a black and white color printed track which can be detected by a CCD camera sensor and image processing technique. (d) Use of a CCD camera sensor will reduce the error due to the noise, incurred in the present prototype. V. C ONCLUSION A new high resolution two-dimensional absolute optical encoder design for the use in a light weight robot arm modular joint is proposed in this paper. The design of coded disk of the encoder module is achieved using the directed graph theory. The Hamiltonian cycle is used to generate the coded tracks of the encoder disk. Generated code improves the code density by compared with the traditional absolute encoders. To eliminate the divergence behavior of the generated code, codes are stacked along the axial direction, which restricts the diameter of the tracks to a constant value. Thus within a constant diameter high resolution of the optical encoder can be realized. A prototype using 2 × 2 matrix code is manufactured using 3D printing technology and tested over a rotary system. The result validates the capability of the encoder module to provide a precise absolute angular position without violating the overall size constraint of the modular joint. ACKNOWLEDGMENT This present work is part of the invention that is the subject of the patent with application number 10-2017-0103591. R EFERENCES [1] K.-M. Lee, J. Choi, and Y.-B. Bang, “Shaft position measurement using dual absolute encoders,” Sens. Actuators A, Phys., vol. 238, pp. 276–281, Feb. 2016. [2] T. S. Sarkar, S. Das, B. Chakraborty, and H. S. Dutta, “Fuel level measurement system based on absolute shaft encoder,” Sens. Actuators A, Phys., vol. 259, pp. 77–84, Jun. 2017. [3] J. Choi, C. H. Lee, and Y. B. Bang, “Initial positioning of a smart actuator using dual absolute encoders,” in Proc. 13th Int. Conf. IEEE Autom. Syst. (ICCAS), Oct. 2013, pp. 1588–1592. [4] F. Ariznavarreta-Fernández, C. González-Palacio, A. Menéndez-Díaz, and C. Ordoñez, “Measurement system with angular encoders for continuous monitoring of tunnel convergence,” Tunnelling Underground Space Technol., vol. 56, pp. 176–185, Jun. 2016. [5] G. Hao, “A multiaxis, large-output, sensing framework of integrating linear optical encoders for nanopositioning systems,” IEEE Sensors Lett., vol. 1, no. 3, Jun. 2017, Art. no. 5500304. [6] T. A. Tameh, R. Kashyap, and M. Sawan, “Self-referenced broadrange optical rotation sensor for flight control applications,” J. Lightw. Technol., vol. 36, no. 10, pp. 2000–2009, May 15, 2018. [7] T. A. Tameh, M. Sawan, and R. Kashyap, “Novel analog ratio-metric optical rotary encoder for avionic applications,” IEEE Sensors J., vol. 16, no. 17, pp. 6586–6595, Sep. 2016. [8] W. Qiu-Hua, W. Yuan-Yuan, S. Ying, and Y. Shou-Wang, “A novel miniature absolute metal rotary encoder based on single-track periodic Gray code,” in Proc. 2nd Int. Conf. Instrum., Meas., Comput., Commun. Control (IMCCC), Dec. 2012, pp. 399–402.

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[9] M. Schwartz and T. Etzion, “The structure of single-track Gray codes,” IEEE Trans. Inf. Theory, vol. 45, no. 7, pp. 2383–2396, Nov. 1999. [10] F. Zhang and H. Zhu, “Single-track Gray codes with non-k-spaced heads,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jul. 2013, pp. 311–315. [11] F. Zhang, H. Zhu, Y. Li, and C. Qiu, “Upper bound of single-track Gray codes and the combined coding method of period 2n ,” in Proc. 8th Int. Forum Strategic Technol. (IFOST), Jun. 2013, pp. 405–409. [12] S. Wekhande and V. Agarwal, “High-resolution absolute position Vernier shaft encoder suitable for high-performance PMSM servo drives,” IEEE Trans. Inst. Meas., vol. 55, no. 1, pp. 357–364, Feb. 2006. [13] H. Ai and L. Li, “The study of encoding principle and encoding disc design of a new-style Vernier absolute encoder,” in Proc. Int. Conf. Elect. Control Eng., Sep. 2011, pp. 3271–3273. [14] E. M. Petriu, “Absolute-type position transducers using a pseudorandom encoding,” IEEE Trans. Instrum. Meas., vol. TIM-36, no. 4, pp. 950–955, Dec. 1987. [15] G. H. Tomlinson, “Absolute-type shaft encoder using shift register sequences,” Electron. Lett., vol. 23, no. 8, pp. 398–400, Apr. 1987. [16] Y. Matsuzoe, N. Tsuji, T. Nakayama, K. Fujita, and T. Yoshizawa, “High performace absolute encoder using multitrack and M-code,” Opt. Eng., vol. 42, no. 1, pp. 124–131, 2003. [17] T. Dziwinski, “A novel approach of an absolute encoder coding pattern,” IEEE Sensors J., vol. 15, no. 1, pp. 397–401, Jan. 2015. [18] S. Paul and J. H. Chang, “Design of absolute encoder disk coding based on affine n digit N-ary Gray code,” in Proc. IEEE Int. Conf. Instrum. Meas., May 2015, pp. 1–6. [19] S. Das, T. S. Sarkar, B. Chakraborty, and H. S. Dutta, “A simple approach to design a binary coded absolute shaft encoder,” IEEE Sensor J., vol. 16, no. 8, pp. 2300–2305, Apr. 2016. [20] F. Marignetti, G. Volpe, S. M. Mirimani, and C. Cecati, “Electromagnetic design and modeling of a two-phase axial-flux printed circuit board motor,” IEEE Trans. Ind. Electron., vol. 65, no. 1, pp. 67–76, Jan. 2018. [21] A. Dwivedi, S. K. Singh, and R. K. Srivastava, “Analysis and performance evaluation of axial flux permanent magnet motors,” IEEE Trans. Ind. Appl., vol. 54, no. 2, pp. 1765–1772, Mar./Apr. 2016. [22] M. Andriollo, F. Graziottin, and A. Tortella, “Design of an axialtype magnetic gear for the contact-less recharging of a heavy-duty bus flywheel storage system,” IEEE Trans. Ind. Appl., vol. 53, no. 4, pp. 3476–3484, Jul./Aug. 2017. [23] H. Fang, L. Guo, and S. Bai, “A light weight arm designed with modular joints,” in Recent Advances in Mechanism Design for Robotics, vol. 33, S. Bai and M. Ceccarelli, Eds. Cham, Switzerland: Springer, 2015. [24] J. A. Bondy and U. S. R. Murty, Graph Theory With Applications. London, U.K.: Springer, Sep. 2011. [25] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory. New York, NY, USA: Springer-Verlag, 2012, doi: 10.1007/978-1-46144529-6. [26] J. B. Jensen and G. Gutin, Digraphs Theory, Algorithms and Applications. London, U.K.: Springer-Verlag, 2007, doi: 10.1007/978-1-84800998-1. [27] R. Diestel, Graph Theory. Berlin, Germany: Springer-Verlag, 2006, doi: 10.1007/978-3-662-53622-3. [28] V. Borkar, V. Ejov, J. Filar, and G. Nguyen, Hamiltonian Cycle Problem and Markov Chains. New York, NY, USA: Springer-Verlag, 2012, doi: 10.1007/978-1-4614-3232-6. Sarbajit Paul (S’15) received the M.Sc. degree in electrical engineering from the Mechatronics System Research Laboratory, Department of Electrical Engineering, Dong-A University, Busan, South Korea, in 2016, where he is currently pursuing the Ph.D. degree in device design and control. He was a Korean Government Research Scholar with the Mechatronics System Research Laboratory, DongA University, from 2014 to 2016, under the sponsorship of the National Institute of International Education, Ministry of Education, South Korea. He received the Australia Award Endeavor Research Fellowship from 2017 to 2018 to perform part of his research with the University of New South Wales, where he developed different winding designs for printed circuit board Micromachines. He was a recipient of the 2016 Travel Grant Award by the IEEE Instrumentation and Measurement Society and the 2017 Best Research Paper Award by the IEEE Region 10 (Asia-Pacific). His current research interests include motor design and drive, and instrumentation and measurement.

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Junghwan Chang (M’05) received the B.S. and M.S. degrees in electrical engineering and the Ph.D. degree in precision mechanical engineering from Hanyang University, Seoul, South Korea, in 1994, 1997, and 2001, respectively. From 2001 to 2002, he was with the Institute of Brain Korea 21, Hanyang University, where he developed micro drive and high-speed spindle motor. From 2002 to 2003, he was a Research Fellow with the University of California at Berkeley, Berkeley, CA, USA, where he analyzed and developed electrically controlled engine valve system. From 2003 to 2009, he was a Technical Leader with the Korea Electrotechnology Research Institute, South Korea, where he was involved in the developments of special purpose machines. Since 2009, he has been an Associate Professor with the Department of Electrical Engineering, Dong-A University, Busan, South Korea. His current research interests include the design and analysis of electromechanical systems, such as electrically driven machine tools and magnetic gear. Dr. Chang is a member of the Korea Institute of Electrical Engineers, South Korea. He was a Steering Committee Member and the Technical Program Chair in different conferences, such as the IEEE ITEC Asia-Pacific 2016 and ICEMS 2013. He is a Reviewer of the IEEE T RANSACTIONS ON M AGNETICS , the IEEE T RANSACTIONS ON I NDUSTRIAL A PPLICATION , the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS , and the IEEE T RANSACTIONS ON S UPERCONDUCTIVITY.

John Edward Fletcher (SM’11) received the B.Eng. (First Class Hons.) and Ph.D. degrees in electrical and electronic engineering from HeriotWatt University, Edinburgh, U.K., in 1991 and 1995, respectively. Until 2007, he was a Lecturer with HeriotWatt University. From 2007 to 2010, he was a Senior Lecturer with the University of Strathclyde, Glasgow, U.K. He is currently a Professor with the University of New South Wales, Sydney, Australia. His research interests include distributed and renewable integration, silicon carbide electronics, pulsed-power applications of power electronics, and the design and control of electrical machines. Dr. Fletcher is a Charted Engineer in U.K. and a Fellow of the Institution of Engineering and Technology.


Subhas Mukhopadhyay (M’97–SM’02–F’11) received the B.E.E. (Hons.) degree, the M.E.E. degree, the Ph.D. degree in India, and the Dr. Eng. degree in Japan. He has over 26 years of teaching, industrial and research experience. He was a Professor of Sensing Technology with Massey University, New Zealand. He is currently a Professor of Mechanical/Electronics Engineering, Macquarie University, Australia, and a Discipline Leader of the Mechatronics Engineering Degree Program. He has supervised over 40 postgraduate students and over 100 honors students. He has examined over 50 post-graduate theses. He has published over 400 papers in different international journals and conference proceedings, written six books, 30 book chapters, and edited 15 conference proceedings. He has also edited 28 books with Springer-Verlag and 17 journal special issues. He has delivered 298 presentations, including keynote, invited, tutorial, and special lectures. His research interests include smart sensors and sensing technology, instrumentation techniques, wireless sensors and network, numerical field calculation, and electromagnetics. He has organized over 20 international conferences as either general chairs/co-chairs or technical program chair. Dr. Mukhopadhyay is a Distinguished Lecturer of the IEEE Sensors Council from 2017 to 2019. He is a Fellow of IET, U.K., and IETE, India. He is a Topical Editor of the IEEE S ENSORS J OURNAL and an Associate Editor of the IEEE T RANSACTIONS ON I NSTRUMENTATION AND M EASUREMENTS . He is the Chair of the IEEE IMS Technical Committee 18 on Environmental Measurements.