A New Rotor Position Measurement Method for Permanent Magnet Spherical Motors
A New Rotor Position Measurement Method for Permanent Magnet Spherical Motors
Lu, Yin;Hu, Cungang;Wang, Qunjing;Hong, Yi;Shen, Weixiang;Zhou, Chengquan
2018-11-28 00:00:00
applied sciences Article A New Rotor Position Measurement Method for Permanent Magnet Spherical Motors 1 , 2 2 , 3 , 2 , 3 1 , 4 5 Yin Lu , Cungang Hu *, Qunjing Wang , Yi Hong , Weixiang Shen and Chengquan Zhou School of Electronics and Information Engineering, Anhui University, Hefei 230601, China; wwwluyinlove@163.com (Y.L.); hongyi@163.com (Y.H.); chengquanzhouahu@163.com (C.Z.) Power Quality Engineering Research Center of Ministry of Education, Hefei 230601, China; wqunjing@sina.com National Engineering Laboratory of Energy-Saving Motor and Control Technology, Hefei 230601, China China Electronics Technology Group No.38 Research Institute, Hefei 230601, China Faculty of Science, Engineering and Technology, Swinburne University of Technology, Melbourne 3122, Australia; wshen@swin.edu.au * Correspondence: hcg@ahu.edu.cn; Tel.: +86-158-5511-5115 Received: 24 October 2018; Accepted: 25 November 2018; Published: 28 November 2018 Abstract: This paper proposes a new high-precision rotor position measurement (RPM) method for permanent magnet spherical motors (PMSMs). In the proposed method, a LED light spot generation module (LSGM) was installed at the top of the rotor shaft. In the LSGM, three LEDs were arranged in a straight line with different distances between them, which were formed as three optical feature points (OFPs). The images of the three OFPs acquired by a high-speed camera were used to calculate the rotor position of PMSMs in the world coordinate frame. An experimental platform was built to verify the effectiveness of the proposed RPM method. Keywords: permanent magnet spherical motor; rotor position measurement; optical feature point; image processing 1. Introduction A spherical motor can make complex motions of three degree-of-freedom (DOF) with its simple structure, which can be applied to many applications, such as robotics, aerospace and military. It has advantages over traditional three DOF motors, which are composed of several single-DOF [1,2], such as low manufacturing cost and high efficiency. Many researchers have studied and developed different kinds of spherical motors. For example, a spherical induction motor was developed by Williams and Laithwaite as early as 1959 [3]. Lee et al. developed a spherical stepper wrist motor based on the principle of variable reluctance spherical motor [4]. Son et al. studied the control methods and working characteristics of a spherical wheel motor [5]. A permanent magnet spherical motor (PMSM), with variable pole pitch and 96 stator poles, was proposed by Kahlen et al. [6]. Chirikjian et al. studied the kinematic design and commutation of a spherical stepper motor [7]. A three-DOF cylindrical spherical ultrasonic motor was developed by Takefumi et al. [8]. The research topics that encompass the field of spherical motors include structural design, magnetic field analysis, rotor position measurement, control strategy, and drive circuit design. Rotor position measurement is a necessary precondition that must be taken into account when attempting to achieve precise control of spherical motors. Appl. Sci. 2018, 8, 2415; doi:10.3390/app8122415 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 2415 2 of 21 Rotor position measurement (RPM) in spherical motors is necessary when considering rotation angles in three directions. It is much more complicated than the RPM in traditional single-DOF motors when only one rotation angle needs to be calculated. Many multi-DOF RPM methods have been proposed, which are generally divided into contact type and non-contact type methods. The contact type method adds a mechanical detection mechanism to the rotor [9–14]. This type of the RPM can achieve high-precision results, but the heavy structure of the RPM system increases the moment of inertia to the rotor and brings huge friction resistance to the bearings. In order to avoid extra moment of inertia and friction resistance, the non-contact type method has been put forward to measure rotor position. A non-contact RPM method based on a photoelectric sensor has been proposed by Lee et al. [14–17]. Garner et al. have proposed a non-contact RPM method based on machine vision [18]. Both mothods provide high-precision results by increasing the density of the grid pattern, but it is difficult to ensure clear grid pattern on the spherical shell when the spherical motor is moving. A non-contact laser-based orientation RPM method has been proposed by Yan et al. [19]. This method is capable of achieving high-precision results, but the bulky structure makes it difficult to be installed in spherical motors. Hall-effect sensors have been used to measure the rotor position for spherical motors [20–28], however the magnetic field varies so slowly that the signal induced in the Hall-effect sensor cannot be used to differentiate different rotor positions with high precision. In addition, the terrestrial magnetic field may influence the Hall-effect sensor, leading to a large error in the RPM. Given the limitations and trade-offs observed from the existing techniques, a novel high-precision non-contact RPM method based on machine vision is proposed for PMSMs in this paper. A LED light spot generation module (LSGM) was installed at the top of the rotor shaft in a spherical motor to form three optical feature points (OFPs). A high-speed camera was used to obtain the images of these three OFPs to compute rotation angles in three directions through image processing, obtaining the rotor position of PMSMs. Compared with other non-contact RPM methods, the proposed method provided reliable and accurate RPM with a simple structure. It was not affected by the environmental field or the moving surface of spherical rotors and, as such, is suitable for other types of spherical motors. The remaining content of this paper is organized as follows. Section 2 presents the structure of a PMSM. Section 3 introduces the composition of the measurement device, and the principle of the proposed RPM method. Section 4 shows the experimental results for validation of the proposed RPM method. Conclusions are summarized in Section 5. 2. Structure of a PMSM The structure of a PMSM used in this paper [29] is shown in Figure 1. The PMSM consists of two parts: A spherical rotor and a spherical-shell stator. The radius of the rotor is 65 mm, and the length of the rotor shaft is 40 mm. There are 40 NdFeB permanent magnets on the spherical rotor, which are divided into four layers symmetrically distributed around the equatorial plane of a rotor. 24 air-core coils are assembled on the spherical-shell stator, which are divided into two layers and evenly distributed on both sides of the equator. Figure 2 shows three-DOF motion of a PMSM. Figure 2a shows the motion range of three-DOF PMSM’s rotor shaft. A stator coordinate frame (SCF) and a rotor coordinate frame (RCF) are used in a PMSM to describe the motion of a spherical rotor. The SCF is stationary relative to the earth, and the center of sphere is defined as the origin (O). The RCF also defines the center of sphere as the origin (o). The centers of the SCF and RCF are completely coincided at the initial position. Figure 2b,c shows the spinning motion and tilting motion of a PMSM, respectively. Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 21 Appl. Sci. 2018, 8, 2415 3 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 21 Figure 1. Structure of the PMSM Figure 2 shows three-DOF motion of a PMSM. Figure 2a shows the motion range of three-DOF PMSM’s rotor shaft. A stator coordinate frame (SCF) and a rotor coordinate frame (RCF) are used in a PMSM to describe the motion of a spherical rotor. The SCF is stationary relative to the earth, and the center of sphere is defined as the origin (O). The RCF also defines the center of sphere as the origin (o). The centers of the SCF and RCF are completely coincided at the initial position. Figure 1. Structure of the PMSM Figure 1. Structure of the PMSM Figure 2b,c shows the spinning motion and tilting motion of a PMSM, respectively. Figure 2 shows three-DOF motion of a PMSM. Figure 2a shows the motion range of three-DOF PMSM’s rotor shaft. A stator coordinate frame (SCF) and a rotor coordinate frame (RCF) are used in a PMSM to describe the motion of a spherical rotor. The SCF is stationary relative to the earth, and the center of sphere is defined as the origin (O). The RCF also defines the center of sphere as the origin (o). The centers of the SCF and RCF are completely coincided at the initial position. Figure 2b,c shows the spinning motion and tilting motion of a PMSM, respectively. (a) (b) (c) Figure Figure 2. 2. Three degree-of-free Three degree-of-freedom dom (DOF) motion of a (DOF) motion of a permanent magnet spherical motor ( permanent magnet spherical motorP (PMSM): MSM): (a) Rotor shaft motion and the coordinate frames; (b) spinning motion; (c) tilting motion. (a) Rotor shaft motion and the coordinate frames; (b) spinning motion; (c) tilting motion. 3. Rotor Position Measurement for a PMSM 3. Rotor Position Measurement for a PMSM 3.1. The Structure of the Measurement Device 3.1. The Structure of the Measurement Device Figure 3 shows the structure of the RPM device for the PMSM, which consists of two parts: Figure 3 shows the structure of the RPM device for the PMSM, which consists of two parts: A A high-speed camera and a LED LSGM. The LSGM was installed at the top of the rotor shaft. high-speed camera and a LED LSGM. The LSGM was installed at the top of the rotor shaft. The The distance between the bottom of the rotor shaft and the top surface of the LSGM is 90 mm, distance between the bottom of the rotor shaft and the top surface of the LSGM is 90 mm, and the and the distance between the lens of the high-speed camera and the top of the LSGM is l. (a) (b) (c) distance between the lens of the high-speed camera and the top of the LSGM is l . Figure 2. Three degree-of-freedom (DOF) motion of a permanent magnet spherical motor (PMSM): (a) Rotor shaft motion and the coordinate frames; (b) spinning motion; (c) tilting motion. 3. Rotor Position Measurement for a PMSM 3.1. The Structure of the Measurement Device Figure 3 shows the structure of the RPM device for the PMSM, which consists of two parts: A high-speed camera and a LED LSGM. The LSGM was installed at the top of the rotor shaft. The distance between the bottom of the rotor shaft and the top surface of the LSGM is 90 mm, and the distance between the lens of the high-speed camera and the top of the LSGM is l . Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 21 Appl. Sci. 2018, 8, 2415 4 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 4 of 21 Figure 3. Schematic diagram of rotor position measurement. Three LEDs were installed at the top surface of the LSGM, which were arranged in a straight line, as shown in Figure 4. The LED L is located at the center, the distance between L and L c c s Figure 3. Schematic diagram of rotor position measurement. Figure 3. Schematic diagram of rotor position measurement. was 7 mm and the distance between and was 10 mm. The three optical feature points (OFPs) L L c l Three LEDs were installed at the top surface of the LSGM, which were arranged in a straight line, Three LEDs were installed at the top surface of the LSGM, which were arranged in a straight were identified through the different distances between them. The three LEDs are represented by L , as shown in Figure 4. The LED L is located at the center, the distance between L and L was 7 mm c c s line, as shown in Figure 4. The LED L is located at the center, the distance between L and L c c s and the distance between L and L was 10 mm. The three optical feature points (OFPs) were identified c l * * was 7 mm and the distance between L and L was 10 mm. The three optical f eature poi nts (OFPs) c l through the different distances between them. The three LEDs are represented by L , L , and L in the c s L L , and in the image coordinate frame (ICF), respectively. s l image coordinate frame (ICF), respectively. were identified through the different distances between them. The three LEDs are represented by L , * * L , and L in the image coordinate frame (ICF), respectively. s l Figure 4. Arrangement of three LEDs on the top surface of light spot generation module (LSGM). The parameters of the high-speed camera are shown in Table 1. Figure 4. Arrangement of three LEDs on the top surface of light spot generation module (LSGM). Figure 4. Arrangement of three LEDs on the top surface of light spot generation module (LSGM). The parameters of the high-speed camera are shown in Table 1. The parameters of the high-speed camera are shown in Table 1. Appl. Sci. 2018, 8, 2415 5 of 21 Table 1. Parameters of the high-speed camera Maximum Resolution 2320 1720 Pixel Size 7 mm 7 mm Frame Rate Resolution 96 2320 1720 180 1920 1080 Frame rate 360 1024 1024 (USB3.0 Interfaces) 490 1080 720 1000 640 480 1400 512 512 Dynamic Range 60 dB Accuracy 8 bit Sensitivity 5200 DN/Lux.S, 550 nm Exposure time >2 S Size 82 mm 77 mm 57.5 mm Figure 5 shows the images of three OFPs photographed by the high-speed camera at different exposure times. It can be seen that, when the exposure time of the high-speed camera was 5000 s, only three OFPs in the image could be identified, which brought great convenience to the subsequent image processing and could be used to extract the coordinates from the three light spots in the ICF. Figure 5. Images taken by a high-speed camera at different exposure times. 3.2. The Principle of Camera Imaging Based on Pin-Hole Model A pin-hole model is often used to establish the mathematical model of images in machine vision, which is shown in Figure 6. Appl. Sci. 2018, 8, 2415 6 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 21 (a) (b) Figure 6. Pin-hole model in machine vision: (a) Camera pin-hole model; (b) Image coordinate system. Figure 6. Pin-hole model in machine vision: (a) Camera pin-hole model; (b) Image coordinate system. In Figure 6, the point o represents the optical center of the camera, and o , X , Y , and Z form c c C C C In Figure 6, the point o represents the optical center of the camera, and o , X , Y , and Z the camera coordinate system c (CCF). The ICF includes an imaging plane I, the horizontal or c dinates C CX , C and a, which are parallel and the vertical ordinates Y and b, which are parallel too, and the optical form the camera coordinate system (CCF). The ICF includes an imaging plane I , the horizontal axis of camera Z , which is vertical to the imaging plane. The intersection of the optical axis and the ordinates , and , which are parallel and the vertical ordinates and β , which are parallel X α Y C C imaging plane is the origin of the ICF, which is expressed as O . The distance between o and O is f , ab ab too, and the optical axis of camera Z , which is vertical to the imaging plane. The intersection of the which is the focal length of the camera. A point (u, v) in the imaging plane can be expressed as: optical axis and the imaging plane is the origin of the ICF, which is expressed as O . The distance 0 1 2 30 1 αβ 0 u u 0 a B C 6 7B C O 1 between o and is f , which is the focal length of the camera. A point (u , v ) in the imaging c αβ 0 v v = b (1) @ A 4 0 5@ A 1 1 plane can be expressed as: 0 0 1 where (u , v ) is the coordinate values of O , d and d is thepixel size of the camera. 0 0 a ab b 0 u We define a world coordinate frame (WCF) which constitutes X , Y , and Z to describe the W W W u α position of the camera in the CCF. The transformation relationship between the WCF and CCF can be expressed as: v = 0 v β (1) 0 1 0 1 0 1 X Xβ X C W W 1 1 B C B C B C 0 0 1 Y Y Y B C R t B C B C C W W = = M (2) B C B C B C @ Z A @ Z A @ Z A C 0 1 W W 1 1 1 O d Where (u , v ) is the coordinate values of , d and is the pixel size of the camera. 0 0 αβ α β where: R is a rotation matrix of 3 3; t is a translation vector of 3 1; M is a external parameter We define a world coordinate frame (WCF) which constitutes , , and to describe X Y Z W W W matrix of 4 4. As observed from triangulation in Figure 6, we can get the position of the camera in the CCF. The transformation relationship between the WCF and CCF can be expressed as: f X a = (3) f Y X X X b = C W W Y R t Y Y C W W = = M (2) T Z 0 1 Z Z C W W 1 1 1 Where: R is a rotation matrix of 3 × 3 ; t is a translation vector of 3 ×1 ; M is a external parameter matrix of 4 × 4 . As observed from triangulation in Figure 6, we can get fX α = Z (3) fY β = Z C equation (3) can be expressed in the form of matrix as: Appl. Sci. 2018, 8, x FOR PEER REVIEW 7 of 21 Appl. Sci. 2018, 8, 2415 7 of 21 X α f 0 0 0 Equation (3) can be expressed in the form of matrix as: Z β = 0 f 0 0 (4) 0 1 0 1 0 1 Z CX 1 0 0 1 0 a f 0 0 0 B C 1Y B C B CB C C Z b = 0 f 0 0 (4) @ A @ AB C @ Z A 1 0 0 1 0 Substituting equations (1) and (2) into equation (4) leads to X Substituting Equations (1) and (2) into Equation (4) leads to u a 0 u 0 α 0 R t Y 0 1 Z v = 0 a v 0 = M M p (5) C β 0 a b W T 0 1 0 1 0 1 Z W ! W u a 0 u 0 ! a 0 B C 1 0 0 1 0 Y B C B C R t B C Z @ v A = @ 0 a v 0 A B C = M M p (5) C b 0 b @ A 0 1 W 1 0 0 1 0 where: , a = ; is the intrinsic parameters matrix of a camera, which is determined by a = M α a f f where: a = , a = ; M is the intrinsic parameters matrix of a camera, which is determined a b a d d a b a 、 a 、 u and v ; M is the extrinsic parameters matrix of a camera, which is determined by the β 0 0 b by a , a , u and v ; M is the extrinsic parameters matrix of a camera, which is determined by the a b 0 0 b relationship between the CCF and WCF. Equation (5) can be used to calculate the position of objects, relationship between the CCF and WCF. Equation (5) can be used to calculate the position of objects, such as rotor position in this study. In the following analysis and experiments, the high-speed camera such as rotor position in this study. In the following analysis and experiments, the high-speed camera was fixed on a tripod and its distance to the top of LSGM was l ∈ M , which was the main parameter was fixed on a tripod and its distance to the top of LSGM was l 2 M , which was the main parameter in the experiments. in the experiments. 3.3. Analysis of Rotor Motion in a PMSM 3.3. Analysis of Rotor Motion in a PMSM When the PMSM was working in the maximum motion range of three-DOF with the maximum When the PMSM was working in the maximum motion range of three-DOF with the maximum tilting angle of 37.5 , the output shaft tip of the rotor produced a spherical trajectory (gray color) and tilting angle of 37.5°, the output shaft tip of the rotor produced a spherical trajectory (gray color ) and the midpoint LED in the LSGM also produced another spherical trajectory (red color), as shown in the midpoint LED in the LSGM also produced another spherical trajectory (red color), as shown in Figure 7. The sphere centers of the two trajectories were the same, the two tilting angles were also Figure 7. The sphere centers of the two trajectories were the same, the two tilting angles were also the the same, but the radii of them were different. Therefore, the spherical trajectory generated by the same, but the radii of them were different. Therefore, the spherical trajectory generated by the midpoint LED could be used to determine the position of the rotor shaft. midpoint LED could be used to determine the position of the rotor shaft. Figure 7. Rotor motion range of a PMSM. Figure 7. Rotor motion range of a PMSM. 3.3.1. Calculation of the Rotor Position in a PMSM 3.3.1. Calculation of the Rotor Position in a PMSM The rotor position in a PMSM can be represented by three rotation angles: The tilting angle_q, The rotor position in a PMSM can be represented by three rotation angles: The tilting angle_ θ , the yaw angle_j, and the spinning angle_w. Three rotation angels can be calculated by computing the yaw angle_ ϕ , and the spinning angle_ ω . Three rotation angels can be calculated by computing the coordinate values of three OFPs in the ICF, which are captured by a high-speed camera. Figure 8 the coordinate values of three OFPs in the ICF, which are captured by a high-speed camera. Figure 8 shows the procedure to calculate the three rotation angles. shows the procedure to calculate the three rotation angles. Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 21 Appl. Sci. 2018, 8, 2415 8 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 21 Figure 8. The procedure to calculate three rotation angles for rotor position in a PMSM. Figure 8. The procedure to calculate three rotation angles for rotor position in a PMSM. Figure 8. The procedure to calculate three rotation angles for rotor position in a PMSM. 3.3.2. Calculation of the Tilting angle_ 3.3.2. Calculation of the Tilting angle_q 3.3.2. Calculation of the Tilting angle_ θ When the tilting angle of the rotor shaft is θ , Figure 9 shows the relationship between the When the tilting angle of the rotor shaft is q, Figure 9 shows the relationship between the camera camera and the midpoint LED, where R is the distance between the sphere center and the midpoint When the tilting angle of the rotor shaft is θ , Figure 9 shows the relationship between the and the midpoint LED, where R is the distance between the sphere center and the midpoint LED. LED. camera and the midpoint LED, where R is the distance between the sphere center and the midpoint LED. Figure 9. Figure 9. Calcu Calculation lation of of the t theilt tilting ing ang angle_ le_ θq.. Figure 9. Calculation of the tilting angle_ θ . Appl. Sci. 2018, 8, x FOR PEER REVIEW 9 of 21 When θ = 0° , i.e., the rotor shaft is perpendicular to the horizontal plane, the distance between the lens of high-speed camera and the midpoint LED is l . When θ varies between 0° and 37.5°, the angle δ can be computed by d = R × sin θ (6) Δr = R(1 − cosθ ) (7) d Num tan δ = = (8) l + Δr f Appl. Sci. 2018, 8, 2415 9 of 21 (9) Num=−() u u × d +(v− v)× d [] 00xy When q = 0 , i.e., the rotor shaft is perpendicular to the horizontal plane, the distance between the lens of high-speed camera and the midpoint LED is l. When q varies between 0 and 37.5 , the angle d can be computed by Where d and Δr are the deflection distances in the horizontal and vertical directions, respectively, d = R sin q (6) Num is the pixel coordinate distance in the ICF. Dr = R(1 cos q) (7) Substituting equations (6) and (7) into equation (8) yields d Num tan d = = (8) l + Dr f R × sin θ Num tan δ = = 2 (10) Num = [(u u ) d ] + (v v ) d (9) x y 0 0 l + R(1 − cosθ ) f where d and Dr are the deflection distances in the horizontal and vertical directions, respectively, Num is the pixel coordinate distance in the ICF. Then, the tilting angle is Substituting Equations (6) and (7) into Equation (8) yields Num(l + R) Num R sin q Num θ = arcsin − arctan tan d = = (10) (11) 2 2 l + R(1 cos q) f R f + Num Then, the tilting angle is Num(l + R) Num q = arcsin arctan (11) 3.3.3. Calculation of the Yaw angle_ ϕ 2 2 R f + Num 3.3.3. Calculation of the Yaw angle_j When the rotor moves within the maximum angle, the high-speed camera captures the image of When the rotor moves within the maximum angle, the high-speed camera captures the image of the three OFPs, which are located within a circle at the radius of VR (the value of VR is max max the three OFPs, which are located within a circle at the radius of VR (the value of VR is max max correlation with the experimental parameters), as indicated in Figure 10. correlation with the experimental parameters), as indicated in Figure 10. Figure 10. Calculation of yaw angle_j. Figure 10. Calculation of yaw angle_ ϕ . Similarly, the angle ϕ ' can be computed by ϕ'a = rctan (12) Appl. Sci. 2018, 8, x FOR PEER REVIEW 10 of 21 Where: and are the distances between the midpoint LED and the origin in the ICF. d d u v With the calculated ϕ ' , the tilting angle_ ϕ in the range of 0~2π can be expressed as: ϕϕ = ' The first quadrant ϕπ=−ϕ ' The second quadrant (13) ϕπϕ =+ ' The third quadrant ϕπ=−2'ϕ The fourth quadrant Appl. Sci. 2018, 8, 2415 10 of 21 Similarly, the angle j0 can be computed by Thus, the rotor position expressed by the tilting angle_ θ and yaw_ ϕ in a spherical coordinate j0 = arctan (12) frame can be converted to a position [ x , y , z ] in a rectangular coordinate frame as where: d and d are the distances between the midpoint LED and the origin in the ICF. u v With the calculated j0, the tilting angle_j in the range of 0~2 can be expressed as: x sin θ cosϕ > j = j0 The first quadrant j = p j0 The second quadrant y = R sin θ sin ϕ (14) (13) > j = p + j0 The third quadrant j = 2p j0 The fourth quadrant z cosθ Thus, the rotor position expressed by the tilting angle_q and yaw_j in a spherical coordinate frame can be converted to a position [x, y, z] in a rectangular coordinate frame as Where R is the distance between the tip of the rotor shaft and sphere center. 2 3 2 3 x sin q cos j 6 7 6 7 = R (14) 4 y 5 4 sin q sin j 5 z cos q 3.3.4. Calculation of the Spinning angle_ ω where R is the distance between the tip of the rotor shaft and sphere center. While the rotor shaft moved within the maximum angle, the rotor was spinning around z axis 3.3.4. Calculation of the Spinning angle_w While the rotor shaft moved within the maximum angle, the rotor was spinning around z axis in in the RCF as shown in Figure 11 and the spinning angle ω is needed to determine the rotor position. the RCF as shown in Figure 11 and the spinning angle w is needed to determine the rotor position. Figure 11. Spinning motion of the rotor. Figure 11. Spinning motion of the rotor. Figure 12 shows the images of the three OFPs taken by the high-speed camera when the rotor shaft was spinning. Figure 12 shows the images of the three OFPs taken by the high-speed camera when the rotor shaft was spinning. Appl. Sci. 2018, 8, x FOR PEER REVIEW 10 of 21 Where: d and d are the distances between the midpoint LED and the origin in the ICF. u v With the calculated ϕ ' , the tilting angle_ ϕ in the range of 0~2π can be expressed as: ϕϕ = ' The first quadrant ϕπ=−ϕ ' The second quadrant (13) ϕπϕ =+ ' The third quadrant ϕπ=−2'ϕ The fourth quadrant Thus, the rotor position expressed by the tilting angle_ θ and yaw_ ϕ in a spherical coordinate y z frame can be converted to a position [ x , , ] in a rectangular coordinate frame as x sin θ cosϕ y = R sin θ sin ϕ (14) z cosθ Where R is the distance between the tip of the rotor shaft and sphere center. 3.3.4. Calculation of the Spinning angle_ ω While the rotor shaft moved within the maximum angle, the rotor was spinning around z axis in the RCF as shown in Figure 11 and the spinning angle ω is needed to determine the rotor position. Figure 11. Spinning motion of the rotor. Figure 12 shows the images of the three OFPs taken by the high-speed camera when the rotor Appl. Sci. 2018, 8, 2415 11 of 21 shaft was spinning. Figure 12. Images of three optical feature points (OFPs) for calculation of spinning angle_w. Appl. Sci. 2018, 8, x FOR PEER REVIEW 11 of 21 The angle w0 can be computed by Figure 12. Images of three optical feature points (OFPs) for calculation of spinning angle_ ω . w0 = arctan (15) The angle ω ' can be computed by O where O is the distance between L and L inω the'a = dir rctaection n of v axis, and O is the distance between v u s (15) L and L in the direction of u axis. * * Where O is the distance between L and L in the direction of v axis, and O is the distance With the rotation angle w0, the spinning angle_w in the range of 0~2 can be expressed in the four v l s u * * between L and L in the direction of u axis. quadrants as: l s With the rotation angle ω ' , the spinning angle_ ω in the range of 0~2π can be expressed in the w = w0 The first quadrant four quadrants as: < w = p w0 The second quadrant (16) ωω = ' The first quadrant w = p + w0 The third quadrant ωπ = - ω ' The second quadrant w = 2p w0 The fourth quadrant (16) ωπ=+ω ' The third quadrant ωπ =2' -ω The fourth quadrant 4. Experimental Results According to the proposed method, the tilting angle_q, the yaw angle_j, and the spinning angle_w 4. Experimental Results can be calculated to determine a rotor position of the PMSMs. In order to verify the RPM method According to the proposed method, the tilting angle_θ , the yaw angle_ ϕ , and the spinning for PMSMs, an experimental platform, as shown in Figure 13a, was constructed, which consisted of angle_ ω can be calculated to determine a rotor position of the PMSMs. In order to verify the RPM a PMSM and its control circuit, a LSGM and its driver circuit, a high-speed camera (Revealer, Hefei, method for PMSMs, an experimental platform, as shown in Figure 13a, was constructed, which Anhui, China) and tripod, a power supply (Tradex, Beijing, China), and a computer (Lenovo, Beijing, consisted of a PMSM and its control circuit, a LSGM and its driver circuit, a high-speed camera (Revealer, Hefei, Anhui, China) and tripod, a power supply (Tradex, Beijing, China) , and a computer China). Figure 13b shows the block diagram of the control system. The block diagram of the rotor (Lenovo, Beijing, China). Figure 13b shows the block diagram of the control system. The block position measurement system is shown in Figure 13c. diagram of the rotor position measurement system is shown in Figure 13c. (a) Figure 13. Cont. Appl. Sci. 2018, 8, 2415 12 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 12 of 21 (b) (c) Figure 13. Experimental platform for measuring the position of the PMSM rotor: (a) Experimental Figure 13. Experimental platform for measuring the position of the PMSM rotor: (a) Experimental prototype; (b) Block diagram of the control system; (c) Block diagram of rotor position measurement prototype; (b) Block diagram of the control system; (c) Block diagram of rotor position measurement system, micro-electro-mechanical system(MEMS): MPU6050. system, micro-electro-mechanical system(MEMS): MPU6050. Table 2 shows the parameters of the spherical motor driver circuit. Table 2 shows the parameters of the spherical motor driver circuit. Table 2. Parameters of the spherical motor driver circuit. Table 2. Parameters of the spherical motor driver circuit. Working voltage 24 V Working voltage 24 V Maximum input voltage 55 V Maximum input voltage 55 V Total driver unit number 24 Total driver unit number 24 Maximum output current ±5 A Maximum output current (per every driver unit) 5 A (per every driver unit) Constant-current precision ±20 mA Constant-current precision 20 mA Maximum output frequency 200 Hz Maximum output frequency 200 Hz In the following experiment, we set the resolution, the sampling time, and the exposure time of the high-speed camera as 1100 × 1100, 20 ms and 5000 us, respectively. The distance between the In the following experiment, we set the resolution, the sampling time, and the exposure time of LSGM’s tip and high-speed camera lens l was 420mm, we can measure through equation (11) that the high-speed camera as 1100 1100, 20 ms and 5000 s, respectively. The distance between the the maximum value of θ is 42°, (the maximum tilting angle of PMSM is 37.5°). Consequently, the LSGM’s tip and high-speed camera lens l was 420 mm, we can measure through Equation (11) that the intrinsic parameter matrix of the camera generated by the calibration method [30] is maximum value of q is 42 , (the maximum tilting angle of PMSM is 37.5 ). Consequently, the intrinsic 2295.71 0 555.51 0 parameter matrix of the camera generated by the calibration method [30] is M = 0 2295.71 554.87 0 (16) 0 1 2295.71 0 555.51 0 0 0 1 0 B C M = 0 2295.71 554.87 0 (16) @ A From Table 1, d = d = 0.007mm , the focal length of the camera can be computed as: α β 0 0 1 0 f = a × d = 16 .07 mm (17) α α From Table 1, d = d = 0.007 mm, the focal length of the camera can be computed as: a b Measurement results obtained by the MEMS were set as the reference (analytical results), Table 3 shows the parameters of the MEMS. Although MEMS can get a precise position of PMSM in a f = a d = 16.07 mm (17) a a certain time (about 10 min) after calibration, it is known that the measurement error will gradually increase after a certain time. Every measurement time was 10 s in the following experiment. Measurement results obtained by the MEMS were set as the reference (analytical results), Table 3 shows the parameters of the MEMS. Although MEMS can get a precise position of PMSM in a certain time (about 10 min) after calibration, it is known that the measurement error will gradually increase after a certain time. Every measurement time was 10 s in the following experiment. Appl. Sci. 2018, 8, x FOR PEER REVIEW 13 of 21 Appl. Sci. 2018, 8, 2415 13 of 21 Table 3. Parameters of the MEMS. Module Type MPU6050 Table 3. Parameters of the MEMS. Acceleration 3-DOF Module Type MPU6050 Measurement dimension Angular velocity 3-DOF Acceleration 3-DOF Measurement dimension Attitude angle 3-DOF Angular velocity 3-DOF Acceleration ±16 g Attitude angle 3-DOF Measurement range Angular velocity ±2000°/s Acceleration 16 g Measurement range Acceleration 0.01 g Angular velocity 2000 /s Acceleration 0.01 g Resolution Angular velocity 0.05°/s Resolution Angular velocity 0.05 /s Attitued angle 0.01° Attitued angle 0.01 Data output frequency 100 Hz Data output frequency 100 Hz 4.1. Experimental Measurement on Tilting Motion of PMSM Rotor 4.1. Experimental Measurement on Tilting Motion of PMSM Rotor The rotor shaft can make a tilting motion with respect to the z axis in the SCF, the tilting angle The rotor shaft can make a tilting motion with respect to the z axis in the SCF, the tilting angle varied between 0° to 37.5°. The initial position was determined by drawing two lines on the stator varied between 0 to 37.5 . The initial position was determined by drawing two lines on the stator shell and spherical rotor, respectively, as shown in Figure 14. shell and spherical rotor, respectively, as shown in Figure 14. Figure 14. Tilting motion of PMSM rotor. Figure 14. Tilting motion of PMSM rotor. Figure 15a shows the images obtained by the high-speed camera when the rotor made a tilting Figure 15a shows the images obtained by the high-speed camera when the rotor made a tilting motion; the coordinate values of three OFPs in the ICF were calculated by image processing. Further, motion; the coordinate values of three OFPs in the ICF were calculated by image processing. Further, we get the position of rotor output shaft tip by the tilting angle_q through Equation (11) and the yaw angle_j through Equation (13), which is shown in Figure 15b. we get the position of rotor output shaft tip by the tilting angle_ θ through equation (11) and the yaw angle_ ϕ through equation (13), which is shown in Figure 15b. Appl. Sci. 2018, 8, 2415 14 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 14 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 14 of 21 (a) (b) (a) (b) Figure 15. Experimental measurement on tilting motion: (a) Images captured by high-speed camera Figure 15. Experimental measurement on tilting motion: (a) Images captured by high-speed camera and coordinate values of optical feature points; (b) Position of rotor shaft tip. and coordinate values of optical feature points; (b) Position of rotor shaft tip. Figure 15. Experimental measurement on tilting motion: (a) Images captured by high-speed camera and coordinate values of optical feature points; (b) Position of rotor shaft tip. Figure 16a shows the tilting angle_ θ when the PMSM made a tilting motion. The maximum Figure 16a shows the tilting angle_q when the PMSM made a tilting motion. The maximum tilting tilting angle was about 22°, and the rotor wiggled 11.5 times in 10 s. The maximum difference between angle was Figabout ure 16a 22 sh,ows the tiltin and the rotor g an wiggled gle_ θ wh 11.5 en the PM times in SM 10 made s. The a timaximum lting motion. The ma difference xim between um the experimental and analytical results of the tilting angle was about 0.32°, as shown in Figure 16b. tilting angle was about 22°, and the rotor wiggled 11.5 times in 10 s. The maximum difference between the experimental and analytical results of the tilting angle was about 0.32 , as shown in Figure 16b. Figure 16c shows the yaw angle_ ϕ . It moved between 90° and 270° in the first 5 s, which means the the experimental and analytical results of the tilting angle was about 0.32°, as shown in Figure 16b. Figure 16c shows the yaw angle_j. It moved between 90 and 270 in the first 5 s, which means rotor made a tilting motion about 5 times near the XZ plane. In the next 5 s, the rotor made a tilting Figure 16c shows the yaw angle_ ϕ . It moved between 90° and 270° in the first 5 s, which means the the rotor made a tilting motion about 5 times near the XZ plane. In the next 5 s, the rotor made a motion about 6.5 times near the YZ plane and the yaw angle moved between 180° and 360°. The rotor made a tilting motion about 5 times near the XZ plane. In the next 5 s, the rotor made a tilting tilting motion about 6.5 times near the YZ plane and the yaw angle moved between 180 and 360 . maximum difference between the experimental and analytical results of the yaw angle was about 0.3°, motion about 6.5 times near the YZ plane and the yaw angle moved between 180° and 360°. The The maximum difference between the experimental and analytical results of the yaw angle was about as shown in Figure 16d. Figure 16e shows the spinning angle_ ω . It was about 330° in the first 5 s and maximum difference between the experimental and analytical results of the yaw angle was about 0.3°, 0.3 , as shown in Figure 16d. Figure 16e shows the spinning angle_w. It was about 330 in the first 5 s was changed into about 255° in the next 5 s. The maximum difference between the experimental and as shown in Figure 16d. Figure 16e shows the spinning angle_ ω . It was about 330° in the first 5 s and and was changed into about 255 in the next 5 s. The maximum difference between the experimental analytical results of the spinning angle was about 0.31°, as shown in Figure 16f. was changed into about 255° in the next 5 s. The maximum difference between the experimental and and analytical results of the spinning angle was about 0.31 , as shown in Figure 16f. analytical results of the spinning angle was about 0.31°, as shown in Figure 16f. (a) (b) (a) (b) (c) (d) (c) (d) Figure 16. Cont. Appl. Sci. 2018, 8, x FOR PEER REVIEW 15 of 21 Appl. Sci. 2018, 8, 2415 15 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 15 of 21 (e) (f) Figure 16. Experimental results of rotor tilting motion: (a) Experimental and analytical results of ; (b) difference between experimental and analytical results of ; (c) experimental and analytical (e) (f) ϕ ϕ results of ; (d) difference between experimental and analytical of ; (e) experimental and Figure 16. Experimental results of rotor tilting motion: (a) Experimental and analytical results θ of q; Figure 16. Experimental results of rotor tilting motion: (a) Experimental and analytical results of ; ω ω analytical results of ;(f) difference between experimental and analytical of . (b) difference between experimental and analytical results of q; (c) experimental and analytical (b) difference between experimental and analytical results of ; (c) experimental and analytical results of j; (d) difference between experimental and analytical of j; (e) experimental and analytical ϕ ϕ results of ; (d) difference between experimental and analytical of ; (e) experimental and results of w; (f) difference between experimental and analytical of w. 4.2. Experimental Measurement on Spinning Motion of PMSM Rotor at the Center Point ω ω analytical results of ;(f) difference between experimental and analytical of . 4.2. Experimental Measurement on Spinning Motion of PMSM Rotor at the Center Point When the rotor was rotating around the z axis at the center point, i.e., the tilting angle_ was 4.2. Experimental Measurement on Spinning Motion of PMSM Rotor at the Center Point When the rotor was rotating around the z axis at the center point, i.e., the tilting angle_q was 0 , 0°, as shown in Figure 17. The spinning angle of the rotor shaft can be calculated from the images When the rotor was rotating around the z axis at the center point, i.e., the tilting angle_ θ was as shown in Figure 17. The spinning angle of the rotor shaft can be calculated from the images which 0°, as shown in Figure 17. The spinning angle of the rotor shaft can be calculated from the images which were taken by the high-speed camera. were taken by the high-speed camera. which were taken by the high-speed camera. Figure 17. Spinning motion of PMSM rotor at center point. Figure 18a shows the images taken by the high-speed camera and the coordinate values of the three OFPs in the ICF. After calculating the tilting angle_ θ and yaw angle_ ϕ , we can get the position of rotor output shaft tip, which is shown in Figure 18b. Figure 17. Spinning motion of PMSM rotor at center point. Figure 17. Spinning motion of PMSM rotor at center point. Figure 18a shows the images taken by the high-speed camera and the coordinate values of the three OFPs in the ICF. After calculating the tilting angle_q and yaw angle_j, we can get the position of Figure 18a shows the images taken by the high-speed camera and the coordinate values of the rotor output shaft tip, which is shown in Figure 18b. three OFPs in the ICF. After calculating the tilting angle_ θ and yaw angle_ ϕ , we can get the position of rotor output shaft tip, which is shown in Figure 18b. Appl. Sci. 2018, 8, 2415 16 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 16 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 16 of 21 (a) (b) Figure 18. Experimental measurement on the spinning motion of PMSM rotor at center point: (a) Figure 18. Experimental measurement on the spinning motion of PMSM rotor at center point: (a) (b) Images captured by high-speed camera and coordinate values of optical feature points; (b) Position (a) Images captured by high-speed camera and coordinate values of optical feature points; (b) Position of Figure 18. Experimental measurement on the spinning motion of PMSM rotor at center point: (a) of rotor shaft tip. rotor shaft tip. Images captured by high-speed camera and coordinate values of optical feature points; (b) Position of rotor shaft tip. Figure 19a shows the tilting angle_q when the PMSM rotor made a spinning motion at the center Figure 19a shows the tilting angle_ when the PMSM rotor made a spinning motion at the point, center poi however nt, how the maximum ever the matilting ximum til angle ting a was ngl about e was a 2.8 bout 2 . This .8°. T may hisbe may be caused by the p caused by the position ositi dir on ect Figure 19a shows the tilting angle_ when the PMSM rotor made a spinning motion at the (PD) direct (PD control algorithm ) control alg and oritthe hm and time t delay he time d of position elay of detection position det applied ection in applie this d PMSM. in thisThe PMS maximum M. The center point, however the maximum tilting angle was about 2.8°. This may be caused by the position maximum difference between the experimental and analytical results was about 0.25°, as shown in difference between the experimental and analytical results was about 0.25 , as shown in Figure 19b. direct (PD) control algorithm and the time delay of position detection applied in this PMSM. The Figure 19b. Figure 19c shows the yaw angle_ ϕ , it varied from 90° to 210°, the maximum difference maximum difference between the experimental and analytical results was about 0.25°, as shown in Figure 19c shows the yaw angle_j, it varied from 90 to 210 , the maximum difference between the Figure 19b. Figure 19c shows the yaw angle_ ϕ , it varied from 90° to 210°, the maximum difference between the experimental and analytical values was about 0.3°, as shown in Figure 19d. Figure 19e experimental and analytical values was about 0.3 , as shown in Figure 19d. Figure 19e shows the shows the spinning an between the experimgle ent_ alω and . It ca ann be seen tha alytical values t w the PMSM rotor rota as about 0.3°, as shown ted cl in F ockwi igure se 1a 9td. a Fi bout 5.5 gure 1 t 9u e rns spinning angle_w. It can be seen that the PMSM rotor rotated clockwise at about 5.5 turns in 10 s, shows the spinning angle_ ω . It can be seen that the PMSM rotor rotated clockwise at about 5.5 turns in 10 s, and the maximum difference between the experimental and analytical values was about 0.3°, and the maximum difference between the experimental and analytical values was about 0.3 , as shown in 10 s, and the maximum difference between the experimental and analytical values was about 0.3°, as shown in Figure 19f. in Figure 19f. as shown in Figure 19f. (a) (b) (a) (b) (c) (d) (c) (d) Figure 19. Cont. Appl. Sci. 2018, 8, x FOR PEER REVIEW 17 of 21 Appl. Sci. 2018, 8, 2415 17 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 17 of 21 (e) (f) Figure 19. Experimental results on spinning motion of PMSM rotor at center point: (a) Experimental θ θ and analytical results of ; (b) difference between experimental and analytical results of ; (c) ϕ ϕ experimental and analytical results of ; (d) difference between experimental and analytical of (e) (f) ; (e) experimental and analytical results of ; (f) difference between experimental and analytical of ω Figure 19. Experimental results on spinning motion of PMSM rotor at center point: (a) Experimental Figure 19. Experimental results on spinning motion of PMSM rotor at center point: (a) Experimental θ θ and analytical results of q; (b) difference between experimental and analytical results of q; and analytical results of ; (b) difference between experimental and analytical results of ; (c) (c) experimental and analytical results of j; (d) difference between experimental and analytical of j; ϕ ϕ experimental and analytical results of ; (d) difference between experimental and analytical of (e) experimental and analytical results of w; (f) difference between experimental and analytical of w. 4.3. Experimental Measurement on Edge Spinning Motion of PMSM Rotor ; (e) experimental and analytical results of ; (f) difference between experimental and analytical of 4.3. Experimental . Measurement on Edge Spinning Motion of PMSM Rotor Figure 20 shows that the PMSM rotor shaft spinning at a tilted angle. When the rotor shaft was Figure 20 shows that the PMSM rotor shaft spinning at a tilted angle. When the rotor shaft was in 4.3. Experimental Measurement on Edge Spinning Motion of PMSM Rotor in motion in three-DOF, the spherical rotor span around the tilted z axis at the same time. motion in three-DOF, the spherical rotor span around the tilted z axis at the same time. Figure 20 shows that the PMSM rotor shaft spinning at a tilted angle. When the rotor shaft was in motion in three-DOF, the spherical rotor span around the tilted z axis at the same time. Figure 20. Edge spinning motion of PMSM rotor. Figure 21 shows the experimental measurement on the edge spinning motion of PMSM rotor. Figure 20. Edge spinning motion of PMSM rotor. Figure 20. Edge spinning motion of PMSM rotor. Figure 21 shows the experimental measurement on the edge spinning motion of PMSM rotor. Figure 21 shows the experimental measurement on the edge spinning motion of PMSM rotor. Appl. Sci. 2018, 8, 2415 18 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 18 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 18 of 21 (a) (b) (a) (b) Figure 21. Experimental measurement on the edge spinning motion: (a) Image captured by high- Figure 21. Experimental measurement on the edge spinning motion: (a) Image captured by high-speed Figure 21. Experimental measurement on the edge spinning motion: (a) Image captured by high- speed camera and coordinate values of optical feature points; (b) Position of rotor shaft tip. camera and coordinate values of optical feature points; (b) Position of rotor shaft tip. speed camera and coordinate values of optical feature points; (b) Position of rotor shaft tip. Figur Fige ure 22 2 a2shows a shows t the he tilting tilting angle_ angle_q when when the the PMSM PMSM rotor ma rotor made de a an n edge spi edge spinning nning moti motion, on, itsits Figure 22a shows the tilting angle_ when the PMSM rotor made an edge spinning motion, its average value was about 24°. The variation of the tilting angle might be caused by the PD control average value was about 24 . The variation of the tilting angle might be caused by the PD control average value was about 24°. The variation of the tilting angle might be caused by the PD control algorithm and the time delay of position detection used in the PMSM. The maximum difference of algorithm and the time delay of position detection used in the PMSM. The maximum difference of algorithm and the time delay of position detection used in the PMSM. The maximum difference of the tilting angle between the experimental and analytical results was about 0.22°, which is shown in the tilting angle between the experimental and analytical results was about 0.22 , which is shown the tilting angle between the experimental and analytical results was about 0.22°, which is shown in Figure 22b. Figure 22c shows the yaw angle_ ϕ . It can be seen that the PMSM rotor is rotating anti- in Figure 22b. Figure 22c shows the yaw angle_j. It can be seen that the PMSM rotor is rotating Figure 22b. Figure 22c shows the yaw angle_ ϕ . It can be seen that the PMSM rotor is rotating anti- clockwise at the approximate 4.5 turns around the z axis in the SCF. The maximum difference of the anti-clockwise at the approximate 4.5 turns around the z axis in the SCF. The maximum difference of clockwise at the approximate 4.5 turns around the z axis in the SCF. The maximum difference of the yaw angle between the experimental and analytical results was about 0.3°, as shown in Figure 22d. the yaw angle between the experimental and analytical results was about 0.3 , as shown in Figure 22d. yaw angle between the experimental and analytical results was about 0.3°, as shown in Figure 22d. Figure 22e shows the spinning angle_ ω , we can see that the PMSM rotor was rotating clockwise at Figure 22e shows the spinning angle_w, we can see that the PMSM rotor was rotating clockwise at Figure 22e shows the spinning angle_ ω , we can see that the PMSM rotor was rotating clockwise at the approximate 4 turns around the z axis in the RCF. The maximum difference of the spinning angle the approximate 4 turns around the z axis in the RCF. The maximum difference of the spinning angle the approximate 4 turns around the z axis in the RCF. The maximum difference of the spinning angle between the experimental and analytical results was about 0.45°, as shown in Figure 22f, this between the experimental and analytical results was about 0.45 , as shown in Figure 22f, this difference between the experimental and analytical results was about 0.45°, as shown in Figure 22f, this difference was mainly caused by the three LEDs in the LSGM which were not at the same level when was mainly caused by the three LEDs in the LSGM which were not at the same level when the rotor difference was mainly caused by the three LEDs in the LSGM which were not at the same level when the rotor was tilted. was tilted. the rotor was tilted. (a) (b) (a) (b) Figure 22. Cont. Appl. Sci. 2018, 8, 2415 19 of 21 Appl. Sci. 2018, 8, x FOR PEER REVIEW 19 of 21 (c) (d) (e) (f) Figure Figure 22. 22. Experimental Experimentalr r esults esults o off e edge dge s spinning pinning m motion otion oof f PPMSM MSM ro rtotor: or: (a) Experimental and (a) Experimental and θ θ analytical results of q; (b) difference between experimental and analytical results of q; (c) experimental analytical results of ; (b) difference between experimental and analytical results of ; (c) and analytical results of j; (d) difference between experimental and analytical results of j; experimental and analytical results of ; (d) difference between experimental and analytical results (e) experimental and analytical results of w; (f) difference between experimental and analytical of ; (e) experimental and analytical results of ; (f) difference between experimental and results of w. analytical results of . 4.4. Comparison of Different Rotor Position Measurement Methods 4.4. Comparison of Different Rotor Position Measurement Methods To date, four types of sensors have been widely used for the RPM including MEMS, photoelectric To date, four types of sensors have been widely used for the RPM including MEMS, photoelectric sensor, Hall-effect sensor, and high-speed camera The experimental results obtained by the MEMS sensor, Hall-effect sensor, and high-speed camera The experimental results obtained by the MEMS were set as the reference values because the MEMS has been found to have the highest accuracy among were set as the reference values because the MEMS has been found to have the highest accuracy four of them. The experimental results obtained by the other three methods were compared with those among four of them. The experimental results obtained by the other three methods were compared by the MEMS. Table 4 shows their differences. It can be seen that the high-speed camera based RPM with those by the MEMS. Table 4 shows their differences. It can be seen that the high-speed camera has shown higher accuracy than the other two methods. based RPM has shown higher accuracy than the other two methods. Table 4. Comparison of different rotor position measurement methods. Table 4. Comparison of different rotor position measurement methods. Measurement Method Photoelectric Sensor Hall-Effect Sensor High-Speed Camera Measurement Method Photoelectric Sensor Hall-Effect Sensor High-Speed Camera ee ( (°) ) 0.65 1.55 0.32 0.65 1.55 0.32 Tilting motion e ( ) 0.58 1.62 0.3 Tilting motion (°) 0.58 1.62 0.3 e ( ) 0.49 1.47 0.31 e (°) e ( ) 0.590.49 1. 1.2947 0.250.31 ωq Spinning motion at e ( ) 0.51 1.36 0.3 Center point e (°) 0.59 1.29 0.25 e ( ) 0.47 1.28 0.3 Spinning motion at e (°) 0.51 1.36 0.3 e ϕ ( ) 0.69 1.87 0.22 Center point Edge spinning motion e ( ) 0.62 1.83 0.3 e (°) 0.47 1.28 0.3 e ( ) 0.51 1.31 0.45 e (°) 0.69 1.87 0.22 * e is the error of tilting θ angle_q; e is the error of yaw angle_j; e is the error of spinning angle_w. j w e (°) Edge spinning motion 0.62 1.83 0.3 5. Conclusions (°) e 0.51 1.31 0.45 Rotor position measurement (RPM) is a precondition for closed-loop operation of spherical motors. e ϕ * e is the error of tilting angle_θ ; is the error of yaw angle_ ; e is the error of spinning θ ϕ ω This paper presents a novel RPM method for a PMSM based on image processing. In the proposed RPM, angle_ ω . a LSGM was installed at the top of a PMSM rotor shaft, where three LEDs in the LSGM were arranged in a straight line with different distances to form three optical feature points (OFPs). A high-speed Appl. Sci. 2018, 8, 2415 20 of 21 camera was used to capture the images of these three OFPs. The coordinate values of the three OFPs in the images were extracted to compute the tilting angle_q, the yaw angle_j and the spinning angle_w of the PMSM rotor, and thus obtain the rotor position of a PMSM. As there was no physical contact between a high-speed camera and a PMSM, extra moment of inertia and friction resistance, which may compromise the working performances of spherical motors, were avoided. The experimental platform was set up to verify the effectiveness of the proposed RPM method with high detection precision. In the future, we will install a tiny camera in a PMSM to measure the rotor position, which has negligible influence on the motion of rotor and structure of a PMSM. Combining with the other sensors, we will use the multiple sensor fusion method to further improve the precision of the RPM. Author Contributions: Y.L. and C.H. designed the architecture of rotor position measurement device, Q.W., W.S. and Y.H. helped the experiments. All of the authors wrote and revised the paper. Funding: This work is supported by the key project of the China National Natural Science Foundation (51637001). Conflicts of Interest: The authors declare no conflict of interest. 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