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Magnetic levitation with unlimited omnidirectional rotation range
Mechatronics 24 (2014) 252–264
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Magnetic levitation with unlimited omnidirectional rotation range Muneaki Miyasaka a, Peter Berkelman b,⇑ a b
University of Washington, Department of Mechanical Engineering, Seattle WA, USA University of Hawaii, Department of Mechanical Engineering, Honolulu HI, USA
a r t i c l e
i n f o
Article history: Received 2 April 2013 Accepted 3 February 2014 Available online 26 February 2014 Keywords: Magnetic levitation Motion control Electromagnetic design and modeling
a b s t r a c t We have realized a magnetic levitation device in which the motion of a levitated body can be stably controlled in any orientation, with no limits on its spatial rotation range. The system consists of a planar array of cylindrical coils on a fixed base, a levitated frame containing disc magnets and LED position markers, and an optical motion tracking sensor for feedback control of levitation. This system combines the capabilities of fine positioning, vibration isolation, and a spherical motor, with potential applications in omnidirectional antenna and camera pointing, user interaction, manipulation, and simulated spaceflight dynamics and control. The device design is presented including the magnet and coil configuration, analysis and control methods, and position and rotation trajectory control results. The system development process consisted of numerical analysis of electromagnetic forces and torques between coils and magnets, to find the maximum required coil currents for levitation and the condition numbers of the transformation matrices between coil currents and forces and torques generated on the levitated body, for various configurations of coils and magnets, over their full 3D translation and rotation ranges. As a result, a magnetic levitation setup consisting of an array of 27 coils and a levitated object with six disk-shaped permanent magnets was selected. The setup achieved levitation in six degrees of freedom and unlimited rotation about any axis at a fixed height of 40 mm (a 4 mm minimum height above the coil array). The performance was verified with levitated trajectory following rotation command experiments in roll, pitch, yaw, and including 360° rotations about non-principal axes. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The benefits of magnetic levitation systems for motion control applications include isolation from external vibrations and the elimination of contact friction, backlash, and wear. Generally, the disadvantage of magnetic levitation motion control systems has been their small motion ranges, but we have developed control and analysis methods to use pre-computed electromagnetic force and torque data between each magnet and coil with real-time feedback of the position and orientation of a levitated object to enable levitated motion over large translation and rotation ranges in all directions. The position and orientation of the levitated object are sensed using an optical motion tracker and LED markers, and the translation and rotation motions of the levitated object are controlled by proportional-derivative feedback control of each degree of freedom. The coil currents are controlled independently by analog output signals sent from a computer via amplifiers to generate required forces and torques on the levitated object. The entire ⇑ Corresponding author. Tel.: +1 8083935049. E-mail addresses: [email protected] (M. Miyasaka), [email protected] (P. Berkelman). http://dx.doi.org/10.1016/j.mechatronics.2014.02.001 0957-4158/Ó 2014 Elsevier Ltd. All rights reserved.
picture of the magnetic levitation system setup is shown in Fig. 1. The motion tracking system can be operated either in wired or wireless modes to flash the LED markers in a set sequence i synchronization with the motion tracking sensor. Wireless operation is more advantageous as it eliminates the possibility of any wired connection restricting the free motion of the levitated body, but it is more difficult to implement since the levitated object must carry a wireless strober box and battery, increasing its mass and the coil currents required for levitation. The objective of this research was to achieve unlimited roll, pitch, and yaw rotation of a levitated body by analyzing various magnet configurations for the levitated body and the configuration of coils used to produce forces and torques for stable levitation. The novelty of this research is in the realization of this unlimited rotation range in all directions, as typical magnetic levitation systems in applications such as transportation or bearings have very restricted controlled rotation ranges. Unlimited controlled rotation ranges in magnetic levitation enables application in areas such as camera or antenna pointing and rigid-body manipulation. The approach taken in magnetic levitation method is to calculate detailed electromagnetic models of forces and torques in all directions generated between each magnet and coil in the system as
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Fig. 1. Magnetic levitation setup.
functions of their relative position and orientation, combine the actuation forces and torques from a large, redundant array of coils into a current to force and torque transformation, and use the pseudoinverse of the transformation matrix at each control update to calculate the least-squares set of coil currents necessary to generate the forces and torques required for stable levitation. The range of motion in the horizontal plane can easily be increased by employing more coils in the array. On the other hand, improving rotation in roll and pitch angles requires extended analysis of the coil and magnet setup. As the levitation height increases, it becomes more difficult to generate sufficient levitation forces and torques due to the rapid attenuation and complexity of the magnetic field. Thus, the analysis was done at a position close to the coils, allowing the translation in the z axis to be reduced to fulfill the goal. As mentioned earlier, the wireless setup requires extra mass to be levitated compared to the wired one. Since stable levitation is easier to achieve with a less massive body, this research focuses on the wired setup. Prior related work published by others and the authors is reviewed in the following sections. Section 2 describes our modeling and design methods in more detail and Section 3 presents the development of the magnet and coil design for levitation with unlimited controlled rotation. Sections 4 and 5 describe position sensing and feedback control for our implementation, and Sections 6, 7 and Section 8 present results, discussion, and conclusions. 1.1. Related work Magnetic levitation has been often applied to fine manipulation and positioning. A magnetic levitation system applied to a micro-robotic system for hazardous environments is described in [1]. This micro-robot can be remotely operated with three degrees of freedom (DOF) by using eight electromagnets and it is capable of manipulating objects up to 1.5 g within a 29 29 26 mm translation range with a precision of 0.05 mm. Another magnetic levitation micromanipulation system investigated in [2] consists of seven electromagnets and provides 3 DOF motion of a micro-robot. The system demonstrated repositioning of objects with 0.1 mm height and 1 mm diameter within a 4 5 4 mm motion range. A 6 DOF micro-positioning system that employs three horizontally and three vertically arranged coils successfully levitated a stage consisting of six permanent magnets with a 10 lm resolution [3]. In addition, [4] discusses various 6 DOF magnetically levitated micro-positioning stages with fine precisions developed in different institutions and analyzes methods for developments of those systems. A Lorentz force magnetic levitation system for haptic interactions described in [5] levitated a hemispherical shell containing six thin coils arranged in two layers and connected to a handle.
The system achieved 6 DOF motion with 50 mm translation and ±30° rotation ranges in all directions. The most closely related work that focuses on large air gap levitation and large rotation ranges was carried out in [6–8]. Levitation of a cylindrical permanent magnet was investigated using five, six, seven, and eight air core electromagnets in planar array configurations and extensive analysis of analytical model of the system and control methods were presented. They reported 0.1 m air gap levitation with 360° yaw rotation over five electromagnets in a circular arrangement with 5 DOF. Furthermore, the research was extended to develop analytical models for different orientations of a cylindrical permanent magnet [9]. Our work in unlimited rotation levitation has similarities also to research and development in spherical motors [10,11], however with the critical difference that in magnetic levitation systems, position must be stably controlled as well as orientation throughout the rotation range. 1.2. Previous work Our initial magnetic levitation research system performed levitation of a single 37.5 mm diameter and 12.5 mm thick disk permanent magnet with a mass of approximately 125 g, using a set of five 28 mm wide by 28 mm long cylindrical coils [12]. Due to rotational symmetry, the yaw rotation of a single levitated disk magnet cannot be controlled. This degree of freedom therefore rotates freely, and the remaining 5 DOF of the magnet motion can be controlled using five coils in an appropriate configuration. In the research, the actuation force and torque between a single coil and magnet were measured experimentally over a range of vertical and horizontal separation distances, performances of different coil configurations with five coils were investigated, and the basic feasibility of the magnetic levitation control methods was demonstrated. A setup that employed ten actuator coils was tested with two different levitation methods; selection of coil subsets and redundant actuation, as described in [13]. The subset selection method chooses the optimal set of five coils for levitation from the full set of 10 depending on the position and orientation of the magnets, and the redundant actuation method uses the full set of 10 coils together. As a result, a redundant actuation method using the Moore–Penrose pseudoinverse [14,15] to calculate the least square sum of currents to generate necessary force and torque for levitation control showed better performance. The setup was able to provide an 80 60 25 mm motion range with rotations about x and y axes using a single 37.5 mm diameter and 12.5 mm thick disk permanent magnet in horizontal orientation. Levitation of the same single disk permanent magnet in horizontal and vertical orientations was successfully demonstrated
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Fig. 2. Previous levitation coil dimensions and arrangement.
(0.5 in.). Two magnets were arranged horizontally with a separation of 48 mm as shown in Fig. 3. The total mass of the levitated object was 375 g including the mass of two magnets, the magnet frame, three LED markers, a wireless strober, and a battery. Since the setup had redundant actuation inputs, the leastsquares minimum coil current was determined based on a static electromagnetic actuation model obtained using Mathematica [18] and Radia [19] analysis software. In the paper, it was revealed that the calculated results agreed closely with the measured data and the details about the formulation of transform matrix, inverting current to force and torque transformation, and levitated body motion control are explained. 2. Electromagnetic modeling methods Fig. 3. Previous magnet dimensions and arrangement.
with a setup of 16 12.5 mm id 25 mm od 30 mm thick coils in [16]. Also, a two disk permanent magnet platform that provided a rotation about the vertical axis was levitated with both coil setups. The research confirmed that a system with redundant coil actuation was able to provide 6 DOF motion control of a levitated body, and adding more coils to the array reduced the overall maximum coil current for levitation and increased the vertical and horizontal ranges. 1.3. Most recent work Our most recent previous levitation system employs a two disk permanent magnet platform with a sixteen coil setup and is described in [17]. This system uses a Linux 2.6 operating system kernel running on a 1.8 GHz Intel Core 2 Duo processor PC using real time, preemption, multithreaded, multipriority programming for levitation control and its user interface. An optical motion tracker with a 0.01 mm resolution and a maximum 860 Hz update rate was used to provide real-time feedback of the position and orientation of the levitated object. Each coil has an outer diameter of 25.4 mm and an inner diameter of 12.5 mm, with a height of 27 mm. The coils are arranged in a hexagonal array with 35 mm and 30 mm separation in x and y directions. The dimensions and arrangement of the coils are shown in Fig. 2. Copper was used for the spool of the coils for avoiding magnetization and obtaining effective heat dissipation. Magnet wire with diameter of 0.402 mm (AWG 26) and thermal grade of 155 °C was wound 1000 times for each coil. The levitated object contained two axially magnetized NdFeB (Neodymium-Iron-Boron) permanent disk magnets with grade N50 (Brmax: 14,700 Gauss, BHmax: 50 MGOe). Diameter of each magnet was 38.1 mm (1.5 in.) with thickness of 12.7 mm
In order to evaluate the performance of a coil and magnet setup, forces and torques generated on the magnet setup from a single coil need to be calculated as a function of the horizontal and vertical distances from the coil to the magnet setup, the orientation of the levitated object, and the coil current. Based on the force and torque models, information that determines the performance of the system such as the required current in each coil to levitate a magnet setup and the controllability of a system is evaluated. 2.1. Force and torque calculation Forces and torques can be obtained from experimental measurement or finite element modeling (FEM) methods. The main problem with experimental measurement is that it requires a high precision device that can hold various sizes and shapes of magnet in place and capable of changing the position and orientation of a magnet with a small increment. Finite element methods eliminate the problem of having such a device, however FEM normally requires long calculation times and since we need to take a large number of position and orientation samples to find a desired setup, FEM is not appropriate. Hence, the best way for force and torque calculation is to employ Radia and Mathematica software. Radia uses a boundary integral method and analytical expressions to calculate fields. Due to its efficient computation approach, the CPU time for the computation is reduced significantly compared to FEM [19]. For this reason, Mathematica and Radia were used to do all modeling and other computations in this research. 2.2. Coil current calculation Magnetization within the setup is assumed to be zero because non-magnetic materials are employed near the setup. Also, self and mutual magnetic inductions in the coils are negligibly small. Hence, the forces and torques acting on a magnet due to the magnetic field generated by a coil are proportional to the amount of
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current applied. Also, the resultant forces and torques from all the coils are calculated by the sum of the forces and torques from each coil. As a result, when N coils are employed, the resultant forces and torques of the system from all the coils as a function of current in each coil (I) are represented by the transformation F ¼ AI, where
3 2 2 3 Fx aFx ðr 1 ; z;h;/; wÞ I1 6F 7 6 a ðr ; z;h;/; wÞ 6 7 Fy 1 6 y7 6 6 I2 7 6 7 6 6 7 6 Fz 7 6 a Fz ðr 1 ; z;h;/; wÞ 6 I3 7 7 F ¼6 7; A ¼ 6 6 s 7; I ¼ 6 6 a ðr ; z;h;/; wÞ 7 6 6 x7 6 sx 1 6 .. 7 6 7 6 4. 5 4 sy 5 4 asy ðr 1 ; z;h;/; wÞ I N a ðr ; z;h;/; wÞ s 2
z
sz
1
.. . aFx ðr N ; z; h; /;wÞ
3
.. . aFy ðr N ;z; h; /;wÞ 7 7 7 .. . aFz ðr N ; z; h; /;wÞ 7 7 .. . asx ðr N ; z; h; /;wÞ 7 7 7 .. . asy ðr N ; z; h; /;wÞ 5
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given position and orientation of the levitated object, the condition number is calculated by the highest singular value of A divided by the smallest singular value of A. In such a way that the magnitude of forces and torques approximately matches, torques are measured in [N-cm] while forces are in [N]. A system is wellconditioned if the condition number is close to 1 and it is illconditioned if the condition number is high. When the condition number is infinite, a system is singular and has loss of at least one degree of freedom and stable levitation cannot be performed. 3. Unlimited rotation design
.. . asz ðr N ; z; h; /;wÞ ð1Þ
so that A is a 6xN matrix. By knowing each coil current, the forces and torques on the levitated object can be calculated for any position and orientation of the levitated object. Also, by setting F z equal to the weight of the levitated object and the other forces and torques equal to zero and solving the equation above for I, the currents in each coil to levitate the magnet setup for any arbitrary position and orientation can be obtained. Then, the absolute value of each coil current is checked to verify that the system does not require too much current for levitation.
To achieve magnetic levitation of a magnet platform in any orientation, we evaluated the stability and current requirements various combinations of magnet and coil arrangements, calculated over a sampled set of positions and orientations. The general design criteria to be satisfied were that maximum currents required for levitation would be under 5 A to avoid coil overheating, and transformation condition numbers would be 10 or less so that feedback control in all directions would be stable and robust with regards to sensing, actuation, and model errors. The goal of this design is to find a coil and magnet arrangement which satisfies these criteria for the levitated body in any orientation.
2.3. Evaluation of controllability 3.1. Coil configuration Depending on the position and orientation of the levitated object with respect to the array of coils, there may be some situations, where the system loses one or more degrees of freedom. This happens because the system cannot generate required forces or torques to control one or more of translations in x, y and z axes and rotations about x, y and z axes independently. In order to evaluate whether the system is controllable or not, the condition number of the transform matrix is calculated. The condition number of a system measures the sensitivity of the solution to small perturbations in the data and thus, it indicates the relative magnitudes of currents in each coil that are required to cancel out disturbances in unnecessary translation and rotation. For a linear system, the condition number is given by the ratio of maximal singular value and minimal singular value. Our system is represented as Eq. 1 and at a
As an initial design candidate, levitation of an object containing six 25.4 mm (1 in.) square permanent magnets with a thickness of 9.53 mm (3/8 in.) in a cube configuration over the existing coil setup was investigated. The grade of the magnets was N52, with axial magnetization and magnetic fields with the same pole facing outward. The total mass of the levitated object was approximately 300 g including the magnets, frame, and three LED markers. To evaluate the performance of the setup, the maximum coil currents required to levitate the object and the current to force–torque transformation condition numbers were calculated for two different orientations of the levitated object. The orientations include no rotation (orientation 1) and 45° and 50° rotations in yaw and roll (orientation 2) as illustrated in Fig. 4.
Fig. 4. Sample orientations of first magnet design.
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Fig. 5 shows the values of the transformation condition numbers as a function of horizontal position, with a 10 mm vertical air gap, and Table 1 shows each coil current required for levitation measured at the x ¼ 52 mm and y ¼ 30 mm position (with the coordinate origin at the center of the bottom left coil) where they have excellent performances in the work plane. The smallest condition numbers for both orientations 1 and 2 are 10 which are sufficiently low for stable control. However, since the highest absolute current required for orientation 1 is 39.0 A and for orientation 2 is 14.6 A, this configuration of coils and magnets cannot be used for successful levitation. To make the magnetic field denser and reduce the levitation coil currents, the coils are arranged closer together with 27 mm spacing leaving 2 mm gap for the current supply wiring. The difference in the original and packed arrangements can be seen in Fig. 6. To make the result from the packed coil arrangement comparable to the results from the original coil setup, the data were taken at the same height and at right above the second coil from the left in the second row from the bottom (x ¼ 52 mm, y ¼ 30 mm for the original and x ¼ 39 mm, y ¼ 22 mm for the closely packed coil arrangement). The currents required for levitation with the packed coil arrangement are also shown in Table 1. As a result, for orientation 1, the required maximum absolute current was improved from 39.0 A to 5.2 A, and from 14.6 to 9.8 A for orientation
2. Therefore, all the following analysis was performed with the packed coil arrangement. 3.2. Magnet configuration When operated at a low levitation height, the fact that the magnetic fields are directed outward causes the bottom magnet to be repelled while the top magnet is attracted. Therefore, the effect of the separation distance between magnets to the levitation performance was examined. In addition, we explored the best magnet shape and size. Due to the limited commercial availability of N52 magnets, research was limited to rectangular and disk shaped magnets. To investigate six magnet designs with different separation distances, the maximum current and condition number were plotted at different combinations of yaw and pitch orientations. We tested the previous block magnet design and a design with 25.4 mm diameter and 9.53 mm thick disk magnets. The total mass of the six disk magnet setup is approximately 260 g. The position of the magnets was fixed at x ¼ 47 mm and y ¼ 40 mm with a 5 mm air gap, where the maximum current and transformation condition numbers were measured over the orientation angles. Figs. 7 and 8 show the results for square and disk shaped magnet designs with 15 mm separations, which produced the lowest numbers for both cases evaluated within a range of 0–30 mm of magnet separation.
Fig. 5. Transformation matrix condition numbers for initial magnet design across planar range in two orientations.
Table 1 Coil currents for 16-coil levitation with six-magnet design for two different orientations and coil arrangements. Coil number
Orientation 1 original coils current (A)
Orientation 2 original coils current (A)
Orientation 1 packed coils current (A)
Orientation 2 packed coils current (A)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
22.9 2.4 12.2 12.4 10.6 2.0 39.0 12.6 6.6 18.8 7.1 5.1 6.4 20.8 9.2 5.7
2.7 3.4 2.7 0.9 0.9 6.7 2.5 2.7 1.1 1.0 1.8 2.4 7.0 14.6 5.0 1.6
2.0 3.4 4.0 0.2 5.2 3.1 0.5 0.2 2.0 0.2 1.0 3.2 3.3 1.7 1.3 0.3
2.2 5.5 3.9 1.1 2.8 1.2 7.3 1.8 1.4 5.7 4.0 0.7 2.1 9.8 2.3 0.2
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Fig. 6. Original and packed coil arrangements.
Fig. 7. Square magnet levitation maximum currents and condition numbers at x, y, z = (47, 40, 5) mm with 16 coils.
Fig. 8. Disk magnet levitation maximum currents and condition numbers at x, y, z = (47, 40, 5) mm with 16 coils.
The square magnet design has better condition numbers, but the disk magnet design required less maximum currents. However, with both candidate designs, the system is not capable of producing sufficient force, torque, and stability for successful levitation at every orientation because the required current exceeds 5 A for the
square magnet design and the transformation condition number goes up to 37 for the disk magnet design at select yaw and pitch angles. We have tested various single magnet designs and designs with combinations of 2–12 magnets with different dimensions. As a
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Fig. 9. Nineteen coil and 27 coil arrangements.
result, we have confirmed that the designs tested above (the six 25.4 25.4 9.53 mm block magnet design and the six 25.4 mm diameter and 9.53 mm thick disk magnet designs) are the best among those for the magnet diameters and lengths of 1 ± 0.5 inch and thicknesses of 0.5 ± 0.25 inch. 3.3. Additional coils Due to the large size of the design with 15 mm magnet separation, the levitated object covers most of the area over the packed coil setup and some orientations result in magnets being at the edge of the workspace. Therefore to increase the motion workspace in translation, improve levitation stability and provide stronger maximum forces and torques, more coils were added. In particular, we tested levitation using the 19 and 27 coil setups as shown in Fig. 9. Figs. 10 and 11 show the maximum current and the condition number plots for rectangle and disk magnet designs over the 19 and 27 coil setups together with the original 16 coil setup at their optimal positions. On the 16, 19, and 27 coil setups, the maximum currents of the rectangular magnet design range between 2.4 and 6.8 A for 16 coils, 2.4 and 5.2 A for 19 coils, and 2.3 and 4.5 A for 27 coils. The condition numbers are between 14 and 36 for 16 coils, 13 and 34 for 19 coils, and 13 and 27 for 27 coils. Also, the maximum currents for the disk magnet design range between 2 and 4.9 A for 16 coils, 1.9 and 4.5 A for 19 coils, and 1.6 and 3.7 A for 27 coils, and the condition numbers are between 13 and 44 for 16 coils, 12 and 42 for 19 coils, and 13 and 28 for 27 coils respectively. The reduction in the maximum coil current from 4.5 A to 3.7 A reduces the maximum coil heating by 1/3, as coil heating is proportional to current squared. Thus, it was concluded that the system with more coils provided significantly better performance, and it is clear that the disk magnet design performs better than the block magnet design using the 27 coil setup. 4. Position and orientation sensing Position and orientation sensing for feedback levitation control is provided by an optical motion tracking system1 which senses the locations of infrared LED position markers fixed to a rigid body to 1
Northern digital OptoTrak certus.
calculate its spatial position and orientation. At least three LED markers must be visible to the tracking sensor to allow both the position and orientation of the rigid body to be calculated. Each LED position marker is visible to the sensor within an angle of approximately 70° to the normal of the surface on which it is mounted. Accordingly, three LED position markers are sufficient for motion tracking of a rigid body only up to a tilt angle of 70° from its initial orientation, and additional markers with different orientations are necessary in order to track a rigid body with arbitrary spatial orientation. There is however a disadvantage to adding more LED markers in that the update rate of the motion tracker decreases when the number of LEDs increases, as each LED is individually activated in sequence by the motion tracker. The motion tracker maximum update rate is given by the manufacturer as f ¼ 4500=ðn þ 2Þ, where f is the update rate in Hz and n the number of LEDs. The reduction of the sensing update rate negatively affects the stability and performance of levitated motion control. For optimal performance of the omnidirectional levitation system, the minimum number of LEDs should be used which ensures that at least three are visible to the motion tracker in all orientations of the rigid body. Symmetry considerations suggest that a regular distribution of LED positions and orientations on a spherical surface would be the most efficient arrangement to minimize the number of LEDs for omnidirectional motion tracking. For example, if 8 LEDs are arranged on the surface of a sphere so that each one is located at the center of one of the faces of the regular octahedron enclosing the sphere, then the normal vector of each LED forms a 70.5° angle with the normals of each of its three neighbors, as the dihedral angle between adjoining faces of the octahedron is 109.5°. This LED arrangement is not suitable for omnidirectional motion tracking, because if the normal vector of one LED points directly vertical, then none of its neighbors will be visible to the sensor. If 12 LEDs are arranged with three on each face of a tetrahedron, then at least one face of the tetrahedron and its 3 LEDs will be visible to the motion tracking sensors at all times. Yet if one of the vertices of the tetrahedron points directly upward, then the three visible faces will each be at a 70.5° angle to the horizontal plane, corresponding to the tetrahedron dihedral angle. As this angle is beyond the angular limit for visibility, this LED arrangement is not suitable for motion tracking. An LED arrangement on the 12 vertices of a regular icosahedron is equivalent to the arrangement on the centers of the 12 faces of a dodecahedron, as the dodecahedron is the dual polyhedron of the icosahedron. If 12 LEDs are used in a regular dodecahedron
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Fig. 10. Best square magnet design maximum currents and condition numbers with 19 and 27 coil arrangements.
arrangement, then all five neighboring LEDs will still be visible to the sensor when one LED points directly upwards, as the dihedral angle of the dodecahedron is 116.5°, producing an angle between neighboring LED normals of 63.5°. However the geometry of the dodecahedron LED configuration would be difficult to fabricate on a common frame with the six magnet configuration described in Section 3.2. An LED arrangement on the 12 faces of a rhombic dodecahedron (or vertices of a cuboctahedron) is also feasible for omnirotational motion tracking. With a dihedral angle of 120°, the four neighboring LEDs to an LED oriented directly upwards will be oriented at a 60° angle and visible to the motion tracking sensors. This LED arrangement is especially suitable for use with the six-magnet arrangement, as the LEDs can be placed on the beams of the frame supporting the magnets as shown in Fig. 12. 5. Feedback control for levitation At a sufficiently high control update rate, the motion of the levitated body can be controlled by independent proportionalderivative (PD) digital controllers for orientation and for each direction in translation. A constant feedforward term is added to the vertical position controller for gravity compensation. With 3
LED emitter position markers, position and orientation can be sensed and calculated by the motion tracker at a 860 Hz rate and these simple digital controllers work well for levitation and position control. With the addition of more emitters to extend the detectable rotation range of the motion tracker, the update rate of the feedback controller must be reduced, to 440 Hz for eight emitters, 375 Hz for 10 emitters, or 320 Hz for 12 emitters. At the slower update rates, these simple digital PD controllers cannot be made stable with any combination of gains. To realize stable levitation and motion control at these lower control update rates, discrete-time modeling and state-space methods [20] were used to implement feedback control while accounting for discrete-time sampling, sensor noise, and velocity estimation in the controller. Second-order discrete-time controllers for rotation and translation of the levitated rigid body were formulated in Matlab and its Control System Toolbox using the acker function for estimator and controller pole placement and the c2d (‘tustin’) function to convert continuous time models to discrete time. Passive damping may also be introduced into the magnetic levitation system to improve stability. When a conductive non-ferrous plate is placed above the coil array, then the motion of any levitated magnets above the plate generates viscous damping forces
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Fig. 11. Best disk magnet design maximum currents and condition numbers with 19 and 27 coil arrangements.
Fig. 12. Ball-shaped levitated platform with magnets and LED position markers.
on the levitated body due to the generation of eddy currents in the plate from the moving magnetic field. The degree of passive damping can be modulated by varying the thickness of the conductive plate. Due to the large rotation range of this levitation setup, it is unsuitable to represent orientation with combinations of Euler or fixed axis rotation angles such as ‘‘roll–pitch–yaw’’, as any of these representations contain distortions, discontinuities, and singularities at certain spatial orientations. Instead, 3 3 rotation matrices
are used in the motion tracking and control calculations to calculate the angular error and velocity vectors used in rotational control. 6. Experimental results The coil configuration and magnet assembly designs from Section 3 were fabricated and used with our existing magnetic levitation setup shown in Fig. 1. Large scale rotational motions with
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rotations of over 180° about each axis were recorded to demonstrate the rotational capabilities of the levitation platform. 6.1. Setup The dimensions of the coils were kept unchanged except for an inner diameter which is 5 mm less than earlier setups and the magnet wire for some coils which was changed to one having better thermal resistance with the same diameter of 0.402 mm. The previous setup had a 9.5 mm (3/8 in.) thick aluminum base plate and a 3.21 mm (1/8 in.) thick aluminum top plate, but to improve heat dissipation the thickness of the base plate was increased to 12.7 mm and the top plate was taken off to operate the levitated object as close to the coils as possible. The levitated object design contains six axially magnetized 25.4 mm diameter, 9.53 mm thick disk magnets with a grade of N52. The magnets are placed in a cube configuration with 15 mm separation with common poles facing out as shown in Fig. 12. The total mass of the six magnets is 217 g. The frame for the magnets was designed using CAD software and printed with a rapid prototyping 3D printer using ABS plastic as model material. The actual picture is shown in Fig. 13. In order to track the object motion at any orientations, 12 LED markers are employed and placed on each joint of the frame. The mass of the frame is 30 g and total mass of 12 markers and the wires are 45 g, resulting in a levitated object mass of 292 g. The control system was implemented with a Linux 2.6 operating system low latency kernel running on a quad core 3.4 GHz Intel i7-2600 processor PC using real time, preemption, multithreaded, and multipriority programming to realize stable levitation control. Real-time feedback of the position and orientation of the levitated object is provided by the same optical motion tracker with 0.01 mm resolution at a maximum 860 Hz sampling rate. 6.2. Experiments The actuation force and torque models were obtained using Mathematica and Radia analysis software with sampled resolutions of 1 mm in translation and 10° in rotation. Multidimensional linear interpolation of these sampled models is used to calculate the coil current to force and torque vector transformation at the sensed position and orientation at each control update of the levitation control. Steady levitation is more successful when the levitated object is close to the coils, however it is prudent to levitate the magnets somewhat higher to reduce the possibility of magnet-coil collisions during levitated motions. To demonstrate stable levitation over large rotation angles in all directions, two rotational motion command trajectories were generated to be executed by the levitation control system. During both
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trajectories, the center of the levitated object was fixed at x ¼ 56 mm, y ¼ 24 mm, and z ¼ 40 mm (with a 4 mm minimum gap between coils and magnets) during both trajectories. The origin of the coordinate system is at the center of top surface of the first coil in the second row from the bottom with the x and y axes on the plane of the array of the coils and the z axis in the axial direction of the coils. Coil currents were limited to ±5 A. The first trajectory consisted of ±100° rotations commanded about the x and y horizontal axes, followed by ±360° rotations about the z vertical axis, over a total time interval of 18 s. Horizontal axis rotational velocities were approximately 60°/s and vertical axis rotations were 180°/s, as vertical rotations are more stable with the levitation system. The rotations in this first commanded trajectory necessitated the use of eight position markers and a control update rate of 440 Hz. Fig. 14 shows commanded motion (dotted lines) and measured responses (solid lines) during the first motion trajectory. The nine elements of the rotation matrices are shown on top, the center position of the levitated object is shown in the middle, and the axis rotation angles calculated from the rotation matrices are shown below. This rotational motion sequence is also shown in the attached Video segment. The second trajectory consisted of sequences 360° rotations in both directions, first about the vertical z or ½0; 0; 1 axis, then about h pffiffiffi. pffiffiffi. pffiffiffi. i the 3 3; 3 3; 3 3 axis, and finally about the h pffiffiffi. pffiffiffi. i 2 2; 2 2; 0 axis, all executed an angular velocity of approximately 100°/s, over a time interval of approximately 20 s. This sequence of rotations required 10 position markers to be used, producing a control update rate of 370 Hz. The results of this rotation trajectory command are shown in Fig. 15. As before, the commanded (dotted lines) and measured (solid lines) elements of the rotation matrix are shown in the top plot, with the position results shown directly underneath. The bottom two plots show the rotational motion results in the axis-angle representation, in which the orientation of a rigid body can be described by a single rotation angle about a unit vector which defines the rotation axis. This representation shows the rotation results more clearly than any roll–pitch–yaw representations about fixed or body axes, as these representations have distortions, singularities, and discontinuities over 360° rotations. For the axis-angle representation, the rotation axis is undefined when the rotation angle is exactly zero; in the result plot the zero vector is displayed in this case. Vertical lines can be seen in the top two plots of Fig. 15 after 8 and 12 s, defining two brief periods with no position and orientation results. During these periods, the rotation of the levitated body caused one or more of the position markers to be occluded by the marker wiring, and the motion tracking data signal was lost. Both of these periods were sufficiently brief that stable levitation was recovered afterwards.
Fig. 13. Support frame for levitation magnets and LED position markers.
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6.3. Discussion Compared to previous magnetic levitation setups, higher derivative gain was required for the rotation control and the system became more sensitive to noise. Additionally, since more LED markers were employed and they needed to be flashed individually in sequence, the frequency for the motion tracking needed to be reduced from 860 Hz to 440 or 375 Hz resulting in reduced stability in the levitation control. In general, the rotational motion responses closely followed the command trajectory, however the rotational motions produced error motions in translation and in the other rotational axes commanded to remain constant. These errors were most prevalent during the more rapid axis rotations, in which position errors exceeded 2 mm and rotation errors exceeded 10° for brief periods. The possible sources of these errors in positions and orientations are variations in the coils, especially in the inner diameters, the windings of the magnet wire, variations in the magnetization of the magnets, the resolution and accuracy of the actuation models, and the non-uniform mass distribution of the levitated object. One or more of the coil amplifiers are also likely to have reached their
±5 A limits during this period, producing errors in control force and torque generation. Position errors and vibration are more pronounced in the Fig. 15 results. This is due to the slower sensing and control update rate during this trajectory, as 10 position markers were used instead of eight as was the case during the first trajectory. The slower 375 Hz update rate reduces the stability and motion accuracy of the system compared to the 440 Hz rate, as the response speed of the controller is reduced with the increased latency in the position sensing.
7. Conclusions and future work The setup demonstrates the feasibility of a magnetic levitation system with unlimited pitch, roll, and yaw rotation capabilities. However, the fluctuations of the sensed positions and orientations of the levitated object by the motion tracker are not negligible and the overall levitation current is higher than what is expected. The errors in the results could be reduced by having a more accurate actuation model with higher precision and selecting the optimal
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set of PD gains or implementing more sophisticated control methods. Also, incorporating a dynamic model of the levitated object as a rigid body could reduce overshoots and increase the stability. In order to reduce the levitation coil current further and make the levitation control easier, the coils can be designed to be longer. Furthermore, increasing the current density by winding more turns in the same space using thinner magnet wire could generate a stronger magnetic field, however it overheat the coils much faster as the coil wire resistance would be increased due to both the increased wire length, and the reduced thickness. If it is possible to perform intensive calculations of actuation models and to process the data in the controller with sufficiently high frequency, the most effective way to minimize the levitation coil current would to employ iron cores for all the coils. This approach would require recalculation of the magnetization of all the coil cores at each control update. The performance of this levitation system is limited by the update rate and position signal noise in the position sensing motion tracking system. The current optical motion tracker needs to flash all the LED markers in sequence although only three are necessary for tracking motion. As a result, the control rate of the levitation system must be reduced as more markers are added so that at least three are visible to the sensor across the rotation range of the levitated object. Thus, employing a motion tracker that runs with
higher frequency and better accuracy, or using a motion tracker that can be modified to activate only three visible markers based on a predicted orientation of the levitated object, would definitely improve the stability of the system. It may be possible to sense and calculate levitated magnet positions and orientations using an array of gaussmeters rather than the optical motion tracker, reducing the cost and complexity of the system and eliminating the visibility issues of the LED position markers. Acknowledgment The authors gratefully acknowledge support from NSF awards CNS05-51515 and IIS08-46172, and the University of Hawaii College of Engineering. Sebastian Bozlee contributed to the programming, calibration, and tuning of the magnetic levitation control system.
Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.mechatronics. 2014.02.001.
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References [1] Khamesee MB, Kato N, Nomura Y, Nakamura T. Design and control of a microrobotic system using magnetic levitation. IEEE/ASME Trans Mech 2002;7(1):1–14. [2] Elbuken C, Khamese MB, Yavuz M. Design and implementation of a micromanipulation system using a magnetically levitated MEMS robot. IEEE/ ASME Trans Mech 2009;14(4):434–45. [3] Guang L, Yu Z, Ming Z, Guanghong D, Tomizuka M. Six-DOF maglev nano precision microstage development. In: IEEE International conference on mechanic automation and control engineering, Wuhan; 2010. p. 938–41. [4] Chen M-Y, Lin T-B, Hung S-K, Fu L-C. Design and experiment of a macro–micro planar maglev positioning system. IEEE Trans Ind Electron. 2012;59(11):4128–39. [5] Berkelman P, Dzadovsky M. Actuation model for control of a long range lorentz force magnetic levitation device. In: IEEE/RSJ international conference on intelligent robots and systems, Taipei; 2010. p. 4668–73. [6] Groom NJ, Britcher CP. Open-loop characteristics of magnetic suspension systems using electromagnets mounted in a planar array. Tech. rep. NASA-TP 3229, Hampton (Virginia): NASA Langley Research Center; September 1992. [7] Groom NJ, Britcher CP. A description of a laboratory model magnetic suspension testfixture with large angular capability. In: IEEE conference on control applications, Dayton; 1992. p. 454–9. [8] Britcher CP, Ghofrani M. A magnetic suspension system with a large angular range. Rev Sci Instrum 1993;64(7):1910–7. [9] Groom NJ. Expanded equations for torque and force on a cylindrical permanent magnet core in a large gap magnetic suspension system. Tech. rep. NASA-TP 3638, Hampton (Virginia): NASA Langley Research Center; February 1997.
[10] Chirikjian GS, Stein D. Kinematic design and commutation of a spherical stepper motor. IEEE/ASME Trans Mech 1999;4(4):342–53. [11] Yan L, Chen I-M, Lim CK, Yang G, Lin W, Lee K-M. Torque modeling of spherical actuators with double-layer poles. In: IEEE/RSJ international conference on intelligent robots and systems, Beijing; 2006. p. 5447–52. [12] Berkelman P, Dzadovsky M. Single magnet levitation by repulsion using a planar coil array. In: IEEE international conference on control applications, San Antonio; 2008. p. 108–13. [13] Berkelman P, Dzadovsky M. Magnet levitation and trajectory following motion control using a planar array of cylindrical coils. In: ASME dynamic systems and control conference, Ann Arbor; 2008. p. 767–74. [14] Moore EH. On the reciprocal of the general algebraic matrix. Bull Am Math Soc 1920;26:394–5. [15] Penrose R. A generalized inverse for matrices. Proc Camb Philos Soc 1955;51:406–13. [16] Berkelman P, Dzadovsky M. Novel design, characterization, and control method for large motion range magnetic levitation. IEEE Magnetics Letters 2010;1:#0500104. [17] Berkelman P, Dzadovsky M. Magnetic levitation over large translation and rotation ranges in all directions. IEEE/ASME Trans Mech 2013;18(1):44–52. [18] Wolfram S. The mathematica book. 5th ed. Wolfram Media; 2003. [19] Chubar O, Elleaume P, Chavanne J. A three-dimensional magnetostatics computer code for insertion devices. J Synchrotron Radiat 1998;5:481–4. [20] Franklin GF, Powell JD, Workman ML. Digital control of dynamic systems. Addison Wesley; 1997.
Contents lists available at ScienceDirect
Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Magnetic levitation with unlimited omnidirectional rotation range Muneaki Miyasaka a, Peter Berkelman b,⇑ a b
University of Washington, Department of Mechanical Engineering, Seattle WA, USA University of Hawaii, Department of Mechanical Engineering, Honolulu HI, USA
a r t i c l e
i n f o
Article history: Received 2 April 2013 Accepted 3 February 2014 Available online 26 February 2014 Keywords: Magnetic levitation Motion control Electromagnetic design and modeling
a b s t r a c t We have realized a magnetic levitation device in which the motion of a levitated body can be stably controlled in any orientation, with no limits on its spatial rotation range. The system consists of a planar array of cylindrical coils on a fixed base, a levitated frame containing disc magnets and LED position markers, and an optical motion tracking sensor for feedback control of levitation. This system combines the capabilities of fine positioning, vibration isolation, and a spherical motor, with potential applications in omnidirectional antenna and camera pointing, user interaction, manipulation, and simulated spaceflight dynamics and control. The device design is presented including the magnet and coil configuration, analysis and control methods, and position and rotation trajectory control results. The system development process consisted of numerical analysis of electromagnetic forces and torques between coils and magnets, to find the maximum required coil currents for levitation and the condition numbers of the transformation matrices between coil currents and forces and torques generated on the levitated body, for various configurations of coils and magnets, over their full 3D translation and rotation ranges. As a result, a magnetic levitation setup consisting of an array of 27 coils and a levitated object with six disk-shaped permanent magnets was selected. The setup achieved levitation in six degrees of freedom and unlimited rotation about any axis at a fixed height of 40 mm (a 4 mm minimum height above the coil array). The performance was verified with levitated trajectory following rotation command experiments in roll, pitch, yaw, and including 360° rotations about non-principal axes. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The benefits of magnetic levitation systems for motion control applications include isolation from external vibrations and the elimination of contact friction, backlash, and wear. Generally, the disadvantage of magnetic levitation motion control systems has been their small motion ranges, but we have developed control and analysis methods to use pre-computed electromagnetic force and torque data between each magnet and coil with real-time feedback of the position and orientation of a levitated object to enable levitated motion over large translation and rotation ranges in all directions. The position and orientation of the levitated object are sensed using an optical motion tracker and LED markers, and the translation and rotation motions of the levitated object are controlled by proportional-derivative feedback control of each degree of freedom. The coil currents are controlled independently by analog output signals sent from a computer via amplifiers to generate required forces and torques on the levitated object. The entire ⇑ Corresponding author. Tel.: +1 8083935049. E-mail addresses: [email protected] (M. Miyasaka), [email protected] (P. Berkelman). http://dx.doi.org/10.1016/j.mechatronics.2014.02.001 0957-4158/Ó 2014 Elsevier Ltd. All rights reserved.
picture of the magnetic levitation system setup is shown in Fig. 1. The motion tracking system can be operated either in wired or wireless modes to flash the LED markers in a set sequence i synchronization with the motion tracking sensor. Wireless operation is more advantageous as it eliminates the possibility of any wired connection restricting the free motion of the levitated body, but it is more difficult to implement since the levitated object must carry a wireless strober box and battery, increasing its mass and the coil currents required for levitation. The objective of this research was to achieve unlimited roll, pitch, and yaw rotation of a levitated body by analyzing various magnet configurations for the levitated body and the configuration of coils used to produce forces and torques for stable levitation. The novelty of this research is in the realization of this unlimited rotation range in all directions, as typical magnetic levitation systems in applications such as transportation or bearings have very restricted controlled rotation ranges. Unlimited controlled rotation ranges in magnetic levitation enables application in areas such as camera or antenna pointing and rigid-body manipulation. The approach taken in magnetic levitation method is to calculate detailed electromagnetic models of forces and torques in all directions generated between each magnet and coil in the system as
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Fig. 1. Magnetic levitation setup.
functions of their relative position and orientation, combine the actuation forces and torques from a large, redundant array of coils into a current to force and torque transformation, and use the pseudoinverse of the transformation matrix at each control update to calculate the least-squares set of coil currents necessary to generate the forces and torques required for stable levitation. The range of motion in the horizontal plane can easily be increased by employing more coils in the array. On the other hand, improving rotation in roll and pitch angles requires extended analysis of the coil and magnet setup. As the levitation height increases, it becomes more difficult to generate sufficient levitation forces and torques due to the rapid attenuation and complexity of the magnetic field. Thus, the analysis was done at a position close to the coils, allowing the translation in the z axis to be reduced to fulfill the goal. As mentioned earlier, the wireless setup requires extra mass to be levitated compared to the wired one. Since stable levitation is easier to achieve with a less massive body, this research focuses on the wired setup. Prior related work published by others and the authors is reviewed in the following sections. Section 2 describes our modeling and design methods in more detail and Section 3 presents the development of the magnet and coil design for levitation with unlimited controlled rotation. Sections 4 and 5 describe position sensing and feedback control for our implementation, and Sections 6, 7 and Section 8 present results, discussion, and conclusions. 1.1. Related work Magnetic levitation has been often applied to fine manipulation and positioning. A magnetic levitation system applied to a micro-robotic system for hazardous environments is described in [1]. This micro-robot can be remotely operated with three degrees of freedom (DOF) by using eight electromagnets and it is capable of manipulating objects up to 1.5 g within a 29 29 26 mm translation range with a precision of 0.05 mm. Another magnetic levitation micromanipulation system investigated in [2] consists of seven electromagnets and provides 3 DOF motion of a micro-robot. The system demonstrated repositioning of objects with 0.1 mm height and 1 mm diameter within a 4 5 4 mm motion range. A 6 DOF micro-positioning system that employs three horizontally and three vertically arranged coils successfully levitated a stage consisting of six permanent magnets with a 10 lm resolution [3]. In addition, [4] discusses various 6 DOF magnetically levitated micro-positioning stages with fine precisions developed in different institutions and analyzes methods for developments of those systems. A Lorentz force magnetic levitation system for haptic interactions described in [5] levitated a hemispherical shell containing six thin coils arranged in two layers and connected to a handle.
The system achieved 6 DOF motion with 50 mm translation and ±30° rotation ranges in all directions. The most closely related work that focuses on large air gap levitation and large rotation ranges was carried out in [6–8]. Levitation of a cylindrical permanent magnet was investigated using five, six, seven, and eight air core electromagnets in planar array configurations and extensive analysis of analytical model of the system and control methods were presented. They reported 0.1 m air gap levitation with 360° yaw rotation over five electromagnets in a circular arrangement with 5 DOF. Furthermore, the research was extended to develop analytical models for different orientations of a cylindrical permanent magnet [9]. Our work in unlimited rotation levitation has similarities also to research and development in spherical motors [10,11], however with the critical difference that in magnetic levitation systems, position must be stably controlled as well as orientation throughout the rotation range. 1.2. Previous work Our initial magnetic levitation research system performed levitation of a single 37.5 mm diameter and 12.5 mm thick disk permanent magnet with a mass of approximately 125 g, using a set of five 28 mm wide by 28 mm long cylindrical coils [12]. Due to rotational symmetry, the yaw rotation of a single levitated disk magnet cannot be controlled. This degree of freedom therefore rotates freely, and the remaining 5 DOF of the magnet motion can be controlled using five coils in an appropriate configuration. In the research, the actuation force and torque between a single coil and magnet were measured experimentally over a range of vertical and horizontal separation distances, performances of different coil configurations with five coils were investigated, and the basic feasibility of the magnetic levitation control methods was demonstrated. A setup that employed ten actuator coils was tested with two different levitation methods; selection of coil subsets and redundant actuation, as described in [13]. The subset selection method chooses the optimal set of five coils for levitation from the full set of 10 depending on the position and orientation of the magnets, and the redundant actuation method uses the full set of 10 coils together. As a result, a redundant actuation method using the Moore–Penrose pseudoinverse [14,15] to calculate the least square sum of currents to generate necessary force and torque for levitation control showed better performance. The setup was able to provide an 80 60 25 mm motion range with rotations about x and y axes using a single 37.5 mm diameter and 12.5 mm thick disk permanent magnet in horizontal orientation. Levitation of the same single disk permanent magnet in horizontal and vertical orientations was successfully demonstrated
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Fig. 2. Previous levitation coil dimensions and arrangement.
(0.5 in.). Two magnets were arranged horizontally with a separation of 48 mm as shown in Fig. 3. The total mass of the levitated object was 375 g including the mass of two magnets, the magnet frame, three LED markers, a wireless strober, and a battery. Since the setup had redundant actuation inputs, the leastsquares minimum coil current was determined based on a static electromagnetic actuation model obtained using Mathematica [18] and Radia [19] analysis software. In the paper, it was revealed that the calculated results agreed closely with the measured data and the details about the formulation of transform matrix, inverting current to force and torque transformation, and levitated body motion control are explained. 2. Electromagnetic modeling methods Fig. 3. Previous magnet dimensions and arrangement.
with a setup of 16 12.5 mm id 25 mm od 30 mm thick coils in [16]. Also, a two disk permanent magnet platform that provided a rotation about the vertical axis was levitated with both coil setups. The research confirmed that a system with redundant coil actuation was able to provide 6 DOF motion control of a levitated body, and adding more coils to the array reduced the overall maximum coil current for levitation and increased the vertical and horizontal ranges. 1.3. Most recent work Our most recent previous levitation system employs a two disk permanent magnet platform with a sixteen coil setup and is described in [17]. This system uses a Linux 2.6 operating system kernel running on a 1.8 GHz Intel Core 2 Duo processor PC using real time, preemption, multithreaded, multipriority programming for levitation control and its user interface. An optical motion tracker with a 0.01 mm resolution and a maximum 860 Hz update rate was used to provide real-time feedback of the position and orientation of the levitated object. Each coil has an outer diameter of 25.4 mm and an inner diameter of 12.5 mm, with a height of 27 mm. The coils are arranged in a hexagonal array with 35 mm and 30 mm separation in x and y directions. The dimensions and arrangement of the coils are shown in Fig. 2. Copper was used for the spool of the coils for avoiding magnetization and obtaining effective heat dissipation. Magnet wire with diameter of 0.402 mm (AWG 26) and thermal grade of 155 °C was wound 1000 times for each coil. The levitated object contained two axially magnetized NdFeB (Neodymium-Iron-Boron) permanent disk magnets with grade N50 (Brmax: 14,700 Gauss, BHmax: 50 MGOe). Diameter of each magnet was 38.1 mm (1.5 in.) with thickness of 12.7 mm
In order to evaluate the performance of a coil and magnet setup, forces and torques generated on the magnet setup from a single coil need to be calculated as a function of the horizontal and vertical distances from the coil to the magnet setup, the orientation of the levitated object, and the coil current. Based on the force and torque models, information that determines the performance of the system such as the required current in each coil to levitate a magnet setup and the controllability of a system is evaluated. 2.1. Force and torque calculation Forces and torques can be obtained from experimental measurement or finite element modeling (FEM) methods. The main problem with experimental measurement is that it requires a high precision device that can hold various sizes and shapes of magnet in place and capable of changing the position and orientation of a magnet with a small increment. Finite element methods eliminate the problem of having such a device, however FEM normally requires long calculation times and since we need to take a large number of position and orientation samples to find a desired setup, FEM is not appropriate. Hence, the best way for force and torque calculation is to employ Radia and Mathematica software. Radia uses a boundary integral method and analytical expressions to calculate fields. Due to its efficient computation approach, the CPU time for the computation is reduced significantly compared to FEM [19]. For this reason, Mathematica and Radia were used to do all modeling and other computations in this research. 2.2. Coil current calculation Magnetization within the setup is assumed to be zero because non-magnetic materials are employed near the setup. Also, self and mutual magnetic inductions in the coils are negligibly small. Hence, the forces and torques acting on a magnet due to the magnetic field generated by a coil are proportional to the amount of
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current applied. Also, the resultant forces and torques from all the coils are calculated by the sum of the forces and torques from each coil. As a result, when N coils are employed, the resultant forces and torques of the system from all the coils as a function of current in each coil (I) are represented by the transformation F ¼ AI, where
3 2 2 3 Fx aFx ðr 1 ; z;h;/; wÞ I1 6F 7 6 a ðr ; z;h;/; wÞ 6 7 Fy 1 6 y7 6 6 I2 7 6 7 6 6 7 6 Fz 7 6 a Fz ðr 1 ; z;h;/; wÞ 6 I3 7 7 F ¼6 7; A ¼ 6 6 s 7; I ¼ 6 6 a ðr ; z;h;/; wÞ 7 6 6 x7 6 sx 1 6 .. 7 6 7 6 4. 5 4 sy 5 4 asy ðr 1 ; z;h;/; wÞ I N a ðr ; z;h;/; wÞ s 2
z
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given position and orientation of the levitated object, the condition number is calculated by the highest singular value of A divided by the smallest singular value of A. In such a way that the magnitude of forces and torques approximately matches, torques are measured in [N-cm] while forces are in [N]. A system is wellconditioned if the condition number is close to 1 and it is illconditioned if the condition number is high. When the condition number is infinite, a system is singular and has loss of at least one degree of freedom and stable levitation cannot be performed. 3. Unlimited rotation design
.. . asz ðr N ; z; h; /;wÞ ð1Þ
so that A is a 6xN matrix. By knowing each coil current, the forces and torques on the levitated object can be calculated for any position and orientation of the levitated object. Also, by setting F z equal to the weight of the levitated object and the other forces and torques equal to zero and solving the equation above for I, the currents in each coil to levitate the magnet setup for any arbitrary position and orientation can be obtained. Then, the absolute value of each coil current is checked to verify that the system does not require too much current for levitation.
To achieve magnetic levitation of a magnet platform in any orientation, we evaluated the stability and current requirements various combinations of magnet and coil arrangements, calculated over a sampled set of positions and orientations. The general design criteria to be satisfied were that maximum currents required for levitation would be under 5 A to avoid coil overheating, and transformation condition numbers would be 10 or less so that feedback control in all directions would be stable and robust with regards to sensing, actuation, and model errors. The goal of this design is to find a coil and magnet arrangement which satisfies these criteria for the levitated body in any orientation.
2.3. Evaluation of controllability 3.1. Coil configuration Depending on the position and orientation of the levitated object with respect to the array of coils, there may be some situations, where the system loses one or more degrees of freedom. This happens because the system cannot generate required forces or torques to control one or more of translations in x, y and z axes and rotations about x, y and z axes independently. In order to evaluate whether the system is controllable or not, the condition number of the transform matrix is calculated. The condition number of a system measures the sensitivity of the solution to small perturbations in the data and thus, it indicates the relative magnitudes of currents in each coil that are required to cancel out disturbances in unnecessary translation and rotation. For a linear system, the condition number is given by the ratio of maximal singular value and minimal singular value. Our system is represented as Eq. 1 and at a
As an initial design candidate, levitation of an object containing six 25.4 mm (1 in.) square permanent magnets with a thickness of 9.53 mm (3/8 in.) in a cube configuration over the existing coil setup was investigated. The grade of the magnets was N52, with axial magnetization and magnetic fields with the same pole facing outward. The total mass of the levitated object was approximately 300 g including the magnets, frame, and three LED markers. To evaluate the performance of the setup, the maximum coil currents required to levitate the object and the current to force–torque transformation condition numbers were calculated for two different orientations of the levitated object. The orientations include no rotation (orientation 1) and 45° and 50° rotations in yaw and roll (orientation 2) as illustrated in Fig. 4.
Fig. 4. Sample orientations of first magnet design.
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Fig. 5 shows the values of the transformation condition numbers as a function of horizontal position, with a 10 mm vertical air gap, and Table 1 shows each coil current required for levitation measured at the x ¼ 52 mm and y ¼ 30 mm position (with the coordinate origin at the center of the bottom left coil) where they have excellent performances in the work plane. The smallest condition numbers for both orientations 1 and 2 are 10 which are sufficiently low for stable control. However, since the highest absolute current required for orientation 1 is 39.0 A and for orientation 2 is 14.6 A, this configuration of coils and magnets cannot be used for successful levitation. To make the magnetic field denser and reduce the levitation coil currents, the coils are arranged closer together with 27 mm spacing leaving 2 mm gap for the current supply wiring. The difference in the original and packed arrangements can be seen in Fig. 6. To make the result from the packed coil arrangement comparable to the results from the original coil setup, the data were taken at the same height and at right above the second coil from the left in the second row from the bottom (x ¼ 52 mm, y ¼ 30 mm for the original and x ¼ 39 mm, y ¼ 22 mm for the closely packed coil arrangement). The currents required for levitation with the packed coil arrangement are also shown in Table 1. As a result, for orientation 1, the required maximum absolute current was improved from 39.0 A to 5.2 A, and from 14.6 to 9.8 A for orientation
2. Therefore, all the following analysis was performed with the packed coil arrangement. 3.2. Magnet configuration When operated at a low levitation height, the fact that the magnetic fields are directed outward causes the bottom magnet to be repelled while the top magnet is attracted. Therefore, the effect of the separation distance between magnets to the levitation performance was examined. In addition, we explored the best magnet shape and size. Due to the limited commercial availability of N52 magnets, research was limited to rectangular and disk shaped magnets. To investigate six magnet designs with different separation distances, the maximum current and condition number were plotted at different combinations of yaw and pitch orientations. We tested the previous block magnet design and a design with 25.4 mm diameter and 9.53 mm thick disk magnets. The total mass of the six disk magnet setup is approximately 260 g. The position of the magnets was fixed at x ¼ 47 mm and y ¼ 40 mm with a 5 mm air gap, where the maximum current and transformation condition numbers were measured over the orientation angles. Figs. 7 and 8 show the results for square and disk shaped magnet designs with 15 mm separations, which produced the lowest numbers for both cases evaluated within a range of 0–30 mm of magnet separation.
Fig. 5. Transformation matrix condition numbers for initial magnet design across planar range in two orientations.
Table 1 Coil currents for 16-coil levitation with six-magnet design for two different orientations and coil arrangements. Coil number
Orientation 1 original coils current (A)
Orientation 2 original coils current (A)
Orientation 1 packed coils current (A)
Orientation 2 packed coils current (A)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
22.9 2.4 12.2 12.4 10.6 2.0 39.0 12.6 6.6 18.8 7.1 5.1 6.4 20.8 9.2 5.7
2.7 3.4 2.7 0.9 0.9 6.7 2.5 2.7 1.1 1.0 1.8 2.4 7.0 14.6 5.0 1.6
2.0 3.4 4.0 0.2 5.2 3.1 0.5 0.2 2.0 0.2 1.0 3.2 3.3 1.7 1.3 0.3
2.2 5.5 3.9 1.1 2.8 1.2 7.3 1.8 1.4 5.7 4.0 0.7 2.1 9.8 2.3 0.2
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Fig. 6. Original and packed coil arrangements.
Fig. 7. Square magnet levitation maximum currents and condition numbers at x, y, z = (47, 40, 5) mm with 16 coils.
Fig. 8. Disk magnet levitation maximum currents and condition numbers at x, y, z = (47, 40, 5) mm with 16 coils.
The square magnet design has better condition numbers, but the disk magnet design required less maximum currents. However, with both candidate designs, the system is not capable of producing sufficient force, torque, and stability for successful levitation at every orientation because the required current exceeds 5 A for the
square magnet design and the transformation condition number goes up to 37 for the disk magnet design at select yaw and pitch angles. We have tested various single magnet designs and designs with combinations of 2–12 magnets with different dimensions. As a
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Fig. 9. Nineteen coil and 27 coil arrangements.
result, we have confirmed that the designs tested above (the six 25.4 25.4 9.53 mm block magnet design and the six 25.4 mm diameter and 9.53 mm thick disk magnet designs) are the best among those for the magnet diameters and lengths of 1 ± 0.5 inch and thicknesses of 0.5 ± 0.25 inch. 3.3. Additional coils Due to the large size of the design with 15 mm magnet separation, the levitated object covers most of the area over the packed coil setup and some orientations result in magnets being at the edge of the workspace. Therefore to increase the motion workspace in translation, improve levitation stability and provide stronger maximum forces and torques, more coils were added. In particular, we tested levitation using the 19 and 27 coil setups as shown in Fig. 9. Figs. 10 and 11 show the maximum current and the condition number plots for rectangle and disk magnet designs over the 19 and 27 coil setups together with the original 16 coil setup at their optimal positions. On the 16, 19, and 27 coil setups, the maximum currents of the rectangular magnet design range between 2.4 and 6.8 A for 16 coils, 2.4 and 5.2 A for 19 coils, and 2.3 and 4.5 A for 27 coils. The condition numbers are between 14 and 36 for 16 coils, 13 and 34 for 19 coils, and 13 and 27 for 27 coils. Also, the maximum currents for the disk magnet design range between 2 and 4.9 A for 16 coils, 1.9 and 4.5 A for 19 coils, and 1.6 and 3.7 A for 27 coils, and the condition numbers are between 13 and 44 for 16 coils, 12 and 42 for 19 coils, and 13 and 28 for 27 coils respectively. The reduction in the maximum coil current from 4.5 A to 3.7 A reduces the maximum coil heating by 1/3, as coil heating is proportional to current squared. Thus, it was concluded that the system with more coils provided significantly better performance, and it is clear that the disk magnet design performs better than the block magnet design using the 27 coil setup. 4. Position and orientation sensing Position and orientation sensing for feedback levitation control is provided by an optical motion tracking system1 which senses the locations of infrared LED position markers fixed to a rigid body to 1
Northern digital OptoTrak certus.
calculate its spatial position and orientation. At least three LED markers must be visible to the tracking sensor to allow both the position and orientation of the rigid body to be calculated. Each LED position marker is visible to the sensor within an angle of approximately 70° to the normal of the surface on which it is mounted. Accordingly, three LED position markers are sufficient for motion tracking of a rigid body only up to a tilt angle of 70° from its initial orientation, and additional markers with different orientations are necessary in order to track a rigid body with arbitrary spatial orientation. There is however a disadvantage to adding more LED markers in that the update rate of the motion tracker decreases when the number of LEDs increases, as each LED is individually activated in sequence by the motion tracker. The motion tracker maximum update rate is given by the manufacturer as f ¼ 4500=ðn þ 2Þ, where f is the update rate in Hz and n the number of LEDs. The reduction of the sensing update rate negatively affects the stability and performance of levitated motion control. For optimal performance of the omnidirectional levitation system, the minimum number of LEDs should be used which ensures that at least three are visible to the motion tracker in all orientations of the rigid body. Symmetry considerations suggest that a regular distribution of LED positions and orientations on a spherical surface would be the most efficient arrangement to minimize the number of LEDs for omnidirectional motion tracking. For example, if 8 LEDs are arranged on the surface of a sphere so that each one is located at the center of one of the faces of the regular octahedron enclosing the sphere, then the normal vector of each LED forms a 70.5° angle with the normals of each of its three neighbors, as the dihedral angle between adjoining faces of the octahedron is 109.5°. This LED arrangement is not suitable for omnidirectional motion tracking, because if the normal vector of one LED points directly vertical, then none of its neighbors will be visible to the sensor. If 12 LEDs are arranged with three on each face of a tetrahedron, then at least one face of the tetrahedron and its 3 LEDs will be visible to the motion tracking sensors at all times. Yet if one of the vertices of the tetrahedron points directly upward, then the three visible faces will each be at a 70.5° angle to the horizontal plane, corresponding to the tetrahedron dihedral angle. As this angle is beyond the angular limit for visibility, this LED arrangement is not suitable for motion tracking. An LED arrangement on the 12 vertices of a regular icosahedron is equivalent to the arrangement on the centers of the 12 faces of a dodecahedron, as the dodecahedron is the dual polyhedron of the icosahedron. If 12 LEDs are used in a regular dodecahedron
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Fig. 10. Best square magnet design maximum currents and condition numbers with 19 and 27 coil arrangements.
arrangement, then all five neighboring LEDs will still be visible to the sensor when one LED points directly upwards, as the dihedral angle of the dodecahedron is 116.5°, producing an angle between neighboring LED normals of 63.5°. However the geometry of the dodecahedron LED configuration would be difficult to fabricate on a common frame with the six magnet configuration described in Section 3.2. An LED arrangement on the 12 faces of a rhombic dodecahedron (or vertices of a cuboctahedron) is also feasible for omnirotational motion tracking. With a dihedral angle of 120°, the four neighboring LEDs to an LED oriented directly upwards will be oriented at a 60° angle and visible to the motion tracking sensors. This LED arrangement is especially suitable for use with the six-magnet arrangement, as the LEDs can be placed on the beams of the frame supporting the magnets as shown in Fig. 12. 5. Feedback control for levitation At a sufficiently high control update rate, the motion of the levitated body can be controlled by independent proportionalderivative (PD) digital controllers for orientation and for each direction in translation. A constant feedforward term is added to the vertical position controller for gravity compensation. With 3
LED emitter position markers, position and orientation can be sensed and calculated by the motion tracker at a 860 Hz rate and these simple digital controllers work well for levitation and position control. With the addition of more emitters to extend the detectable rotation range of the motion tracker, the update rate of the feedback controller must be reduced, to 440 Hz for eight emitters, 375 Hz for 10 emitters, or 320 Hz for 12 emitters. At the slower update rates, these simple digital PD controllers cannot be made stable with any combination of gains. To realize stable levitation and motion control at these lower control update rates, discrete-time modeling and state-space methods [20] were used to implement feedback control while accounting for discrete-time sampling, sensor noise, and velocity estimation in the controller. Second-order discrete-time controllers for rotation and translation of the levitated rigid body were formulated in Matlab and its Control System Toolbox using the acker function for estimator and controller pole placement and the c2d (‘tustin’) function to convert continuous time models to discrete time. Passive damping may also be introduced into the magnetic levitation system to improve stability. When a conductive non-ferrous plate is placed above the coil array, then the motion of any levitated magnets above the plate generates viscous damping forces
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Fig. 11. Best disk magnet design maximum currents and condition numbers with 19 and 27 coil arrangements.
Fig. 12. Ball-shaped levitated platform with magnets and LED position markers.
on the levitated body due to the generation of eddy currents in the plate from the moving magnetic field. The degree of passive damping can be modulated by varying the thickness of the conductive plate. Due to the large rotation range of this levitation setup, it is unsuitable to represent orientation with combinations of Euler or fixed axis rotation angles such as ‘‘roll–pitch–yaw’’, as any of these representations contain distortions, discontinuities, and singularities at certain spatial orientations. Instead, 3 3 rotation matrices
are used in the motion tracking and control calculations to calculate the angular error and velocity vectors used in rotational control. 6. Experimental results The coil configuration and magnet assembly designs from Section 3 were fabricated and used with our existing magnetic levitation setup shown in Fig. 1. Large scale rotational motions with
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rotations of over 180° about each axis were recorded to demonstrate the rotational capabilities of the levitation platform. 6.1. Setup The dimensions of the coils were kept unchanged except for an inner diameter which is 5 mm less than earlier setups and the magnet wire for some coils which was changed to one having better thermal resistance with the same diameter of 0.402 mm. The previous setup had a 9.5 mm (3/8 in.) thick aluminum base plate and a 3.21 mm (1/8 in.) thick aluminum top plate, but to improve heat dissipation the thickness of the base plate was increased to 12.7 mm and the top plate was taken off to operate the levitated object as close to the coils as possible. The levitated object design contains six axially magnetized 25.4 mm diameter, 9.53 mm thick disk magnets with a grade of N52. The magnets are placed in a cube configuration with 15 mm separation with common poles facing out as shown in Fig. 12. The total mass of the six magnets is 217 g. The frame for the magnets was designed using CAD software and printed with a rapid prototyping 3D printer using ABS plastic as model material. The actual picture is shown in Fig. 13. In order to track the object motion at any orientations, 12 LED markers are employed and placed on each joint of the frame. The mass of the frame is 30 g and total mass of 12 markers and the wires are 45 g, resulting in a levitated object mass of 292 g. The control system was implemented with a Linux 2.6 operating system low latency kernel running on a quad core 3.4 GHz Intel i7-2600 processor PC using real time, preemption, multithreaded, and multipriority programming to realize stable levitation control. Real-time feedback of the position and orientation of the levitated object is provided by the same optical motion tracker with 0.01 mm resolution at a maximum 860 Hz sampling rate. 6.2. Experiments The actuation force and torque models were obtained using Mathematica and Radia analysis software with sampled resolutions of 1 mm in translation and 10° in rotation. Multidimensional linear interpolation of these sampled models is used to calculate the coil current to force and torque vector transformation at the sensed position and orientation at each control update of the levitation control. Steady levitation is more successful when the levitated object is close to the coils, however it is prudent to levitate the magnets somewhat higher to reduce the possibility of magnet-coil collisions during levitated motions. To demonstrate stable levitation over large rotation angles in all directions, two rotational motion command trajectories were generated to be executed by the levitation control system. During both
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trajectories, the center of the levitated object was fixed at x ¼ 56 mm, y ¼ 24 mm, and z ¼ 40 mm (with a 4 mm minimum gap between coils and magnets) during both trajectories. The origin of the coordinate system is at the center of top surface of the first coil in the second row from the bottom with the x and y axes on the plane of the array of the coils and the z axis in the axial direction of the coils. Coil currents were limited to ±5 A. The first trajectory consisted of ±100° rotations commanded about the x and y horizontal axes, followed by ±360° rotations about the z vertical axis, over a total time interval of 18 s. Horizontal axis rotational velocities were approximately 60°/s and vertical axis rotations were 180°/s, as vertical rotations are more stable with the levitation system. The rotations in this first commanded trajectory necessitated the use of eight position markers and a control update rate of 440 Hz. Fig. 14 shows commanded motion (dotted lines) and measured responses (solid lines) during the first motion trajectory. The nine elements of the rotation matrices are shown on top, the center position of the levitated object is shown in the middle, and the axis rotation angles calculated from the rotation matrices are shown below. This rotational motion sequence is also shown in the attached Video segment. The second trajectory consisted of sequences 360° rotations in both directions, first about the vertical z or ½0; 0; 1 axis, then about h pffiffiffi. pffiffiffi. pffiffiffi. i the 3 3; 3 3; 3 3 axis, and finally about the h pffiffiffi. pffiffiffi. i 2 2; 2 2; 0 axis, all executed an angular velocity of approximately 100°/s, over a time interval of approximately 20 s. This sequence of rotations required 10 position markers to be used, producing a control update rate of 370 Hz. The results of this rotation trajectory command are shown in Fig. 15. As before, the commanded (dotted lines) and measured (solid lines) elements of the rotation matrix are shown in the top plot, with the position results shown directly underneath. The bottom two plots show the rotational motion results in the axis-angle representation, in which the orientation of a rigid body can be described by a single rotation angle about a unit vector which defines the rotation axis. This representation shows the rotation results more clearly than any roll–pitch–yaw representations about fixed or body axes, as these representations have distortions, singularities, and discontinuities over 360° rotations. For the axis-angle representation, the rotation axis is undefined when the rotation angle is exactly zero; in the result plot the zero vector is displayed in this case. Vertical lines can be seen in the top two plots of Fig. 15 after 8 and 12 s, defining two brief periods with no position and orientation results. During these periods, the rotation of the levitated body caused one or more of the position markers to be occluded by the marker wiring, and the motion tracking data signal was lost. Both of these periods were sufficiently brief that stable levitation was recovered afterwards.
Fig. 13. Support frame for levitation magnets and LED position markers.
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Rotation Matrix Elements
0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1
0
2
4
6
8
10
12
14
16
18
20
4
6
8
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14
16
18
20
16
18
20
60 55
X position
Position (mm)
50 45 40
Z position
35 30
Y position
25 20
0
Axis Rotation (degrees)
150
2
X axis rotation
Y axis rotation
Z axis rotation
100 50 0 −50 −100 −150 0
2
4
6
8
10
12
14
Time (sec) Fig. 14. Rotation trajectory following results.
6.3. Discussion Compared to previous magnetic levitation setups, higher derivative gain was required for the rotation control and the system became more sensitive to noise. Additionally, since more LED markers were employed and they needed to be flashed individually in sequence, the frequency for the motion tracking needed to be reduced from 860 Hz to 440 or 375 Hz resulting in reduced stability in the levitation control. In general, the rotational motion responses closely followed the command trajectory, however the rotational motions produced error motions in translation and in the other rotational axes commanded to remain constant. These errors were most prevalent during the more rapid axis rotations, in which position errors exceeded 2 mm and rotation errors exceeded 10° for brief periods. The possible sources of these errors in positions and orientations are variations in the coils, especially in the inner diameters, the windings of the magnet wire, variations in the magnetization of the magnets, the resolution and accuracy of the actuation models, and the non-uniform mass distribution of the levitated object. One or more of the coil amplifiers are also likely to have reached their
±5 A limits during this period, producing errors in control force and torque generation. Position errors and vibration are more pronounced in the Fig. 15 results. This is due to the slower sensing and control update rate during this trajectory, as 10 position markers were used instead of eight as was the case during the first trajectory. The slower 375 Hz update rate reduces the stability and motion accuracy of the system compared to the 440 Hz rate, as the response speed of the controller is reduced with the increased latency in the position sensing.
7. Conclusions and future work The setup demonstrates the feasibility of a magnetic levitation system with unlimited pitch, roll, and yaw rotation capabilities. However, the fluctuations of the sensed positions and orientations of the levitated object by the motion tracker are not negligible and the overall levitation current is higher than what is expected. The errors in the results could be reduced by having a more accurate actuation model with higher precision and selecting the optimal
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Rotation Matrix Elements
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0.5 0
−0.5 −1
0
2
4
6
8
10
12
14
16
18
20
2
4
6
8
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0
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0
2
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20
60 55
X position
Position (mm)
50 45 40 Z position
35 30
Y position
25
Rotation Axis Unit Vector
20
0
1 Kz
0.5 Ky
0 Kx
−0.5
Rotation Angle (degrees)
−1
150 100 50 0
Time (sec) Fig. 15. Three hundred and sixty degree rotation results.
set of PD gains or implementing more sophisticated control methods. Also, incorporating a dynamic model of the levitated object as a rigid body could reduce overshoots and increase the stability. In order to reduce the levitation coil current further and make the levitation control easier, the coils can be designed to be longer. Furthermore, increasing the current density by winding more turns in the same space using thinner magnet wire could generate a stronger magnetic field, however it overheat the coils much faster as the coil wire resistance would be increased due to both the increased wire length, and the reduced thickness. If it is possible to perform intensive calculations of actuation models and to process the data in the controller with sufficiently high frequency, the most effective way to minimize the levitation coil current would to employ iron cores for all the coils. This approach would require recalculation of the magnetization of all the coil cores at each control update. The performance of this levitation system is limited by the update rate and position signal noise in the position sensing motion tracking system. The current optical motion tracker needs to flash all the LED markers in sequence although only three are necessary for tracking motion. As a result, the control rate of the levitation system must be reduced as more markers are added so that at least three are visible to the sensor across the rotation range of the levitated object. Thus, employing a motion tracker that runs with
higher frequency and better accuracy, or using a motion tracker that can be modified to activate only three visible markers based on a predicted orientation of the levitated object, would definitely improve the stability of the system. It may be possible to sense and calculate levitated magnet positions and orientations using an array of gaussmeters rather than the optical motion tracker, reducing the cost and complexity of the system and eliminating the visibility issues of the LED position markers. Acknowledgment The authors gratefully acknowledge support from NSF awards CNS05-51515 and IIS08-46172, and the University of Hawaii College of Engineering. Sebastian Bozlee contributed to the programming, calibration, and tuning of the magnetic levitation control system.
Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.mechatronics. 2014.02.001.
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