Zero-power tip-tilt control of a magnetically levitated platform by lateral displacement of hybrid-electromagnets

Abstract

This papers discusses a method of magnetic levitation which allows a levitated platform to remain both level and zero-power under arbitrary eccentric loading conditions. A hardware system is proposed and developed which laterally displaces hybrid electromagnets (HEMs) to counteract the moments caused by eccentric loads. In this system, the displaced HEMs are the same HEMs which are used for the levitation of the platform. A mathematical formulation is provided to explain the control methodology, and data from successful experimental trials of the developed hardware is discussed.

Keywords

Zero-power levitation magnetic levitation tip-tilt control

1. Introduction

The technique of magnetic levitation of platforms by the use of hybrid electromagnets (HEMs) consisting of permanent magnets and electromagnets has many industrial applications where non-contact handling is desired. In the handling of disks or plates where delicate surface finishing exists, non-contact helps to prevent surface damage from scratching and contamination. In conveyances, non-contact levitation eliminates friction, which is especially important in clean room environments to prevent dust generation.

In any application, it is desirable to reduce power consumption. In magnetic levitation systems which use HEMs, power reduction allows energy saving, system size reduction, and reduces the generation of unwanted heat. In many zero-power systems, to reduce power, the passive attractive force caused by the permanent magnet in an HEM is used to offset the force due to gravity acting on the levitated object [1–7]. Ideally, a system which levitates an object using only passive magnetic forces is called ‘zero-power’ [3].

In literatures, many studies have proposed zero-power levitation platforms [1–7]. However, these studies have not addressed zero-power levitation with active tip-tilt control. In past studies, the vertical air-gaps at the HEMs are adjusted to control the forces at each HEM. In this study, instead, a novel method of laterally moving HEMs is presented to actively allow zero-power tip-tilt control. Comparisons between the presented methodology and past methodologies is discussed in the following section.

2. Basic principle

Zero-power magnetic levitation aims to levitate an object by suspending it in an equilibrium position where the force acting on the object due to gravity is counteracted by a passive magnetic force [3]. Take for example the single degree-of-freedom (DoF) system depicted in Fig. 1. When the air-gap between the HEM and the magnetic target is at its equilibrium value, no additional force is required to keep the system at rest. However, since this equilibrium position is unstable, zero-power control is necessary to actively control the currents through the HEM’s electromagnetic coils to maintain the proper air-gap [3,6].

Fig. 1.

Schematic of a hybrid-electromagnet (HEM) typically used in 1 degree of freedom (DoF) zero-power magnetic levitation [4].

When eccentrically loaded zero-power systems in more than one dimension are considered, additionally, the tip-tilt of the system becomes a consideration. Figure 2 depicts various equilibrium conditions for an eccentrically loaded platform in two dimensions. In Fig. 2(a), an active current is controlled to account for an eccentric load, therefore the system is not zero-power. In Fig. 2(b), the tilt of the platform is controlled to allow an air-gap differential to account for an eccentric load, causing the platform to not be level. Past zero-power levitation platforms can largely be described as one of these two system types [1–5].

Fig. 2.

An eccentrically loaded platform is balanced by: (a) an active current, (b) an air-gap differential, or (c) relocating the centroid of the passive magnets. Case (a) is not zero-power, case (b) is not orientation-controlled, and case (c) is both orientation-controlled and zero-power in the steady-state.

In one specific case [1], a system of a type similar to the one depicted in Fig. 2(b) was able to achieve passive zero-power level levitation by using a spring system which deformed in such a way that it compensated for the change in air-gaps at the HEMs when the system was loaded. However, this system relied on a congruence between the stiffness of the physical springs in the system and the negative stiffness of the zero-power aspect of control used. This strict requirement poses potential issues for loads with large mass variances as a physical spring typically exhibits a roughly linear stiffness while magnetic attractive forces roughly follow an inverse square trend. As the congruence between a physical spring stiffness and a HEM negative spring stiffness varies either through differing relationships or through poor calibration, it is expected that the passive tip-tilt performance in this type of system would degrade.

The novel zero-power levitation methodology presented in this research is depicted in Fig. 2(c) and would allow for active tip-tilt control without the strict calibration required by a physical spring system. Further, if the HEMs are displaced laterally in the direction orthogonal to their attractive magnetic force, no force is required to hold the HEMs laterally in place. For these reasons, it is anticipated the proposed method of zero-power level levitation is better suited for eccentric loads with high variance in mass.

Fig. 3.

(a) Photogrpah of hardware from slightly elevated view with key components indicated. (b) Photograph of hardware from slightly lowered view with key components indicated.

3. Hardware

A photograph of the hardware developed is shown in Fig. 3 with the key components indicated. As the system is only controlled in 3 DoF (tip, tilt, and elevation) a lateral guide is passed through the center of the platform. The lateral guide consists of a rod passing through a hole and has enough tolerance to allow the platform to freely tip and tilt. Rotations in the z direction are not constrained and the guide allows a lateral motion of 2 mm in any direction. Although it was considered that magnetic lateral self-alignment [4,8–10] could be used in lieu of a lateral guide, it was not implemented in this system due to concerns over the irregular shape of the magnetic elements. Other relevant hardware parameters are tabulated in Table 1.

A detailed schematic is shown in Fig. 4 with key components and reference frame indicated. In Fig. 4(a) an elevated view of the fixed top platform can be seen. Here, two HEMs are constrained to move about a circular guide rail while the third HEM is fixed. Two moving HEMs is the minimum number required to achieve zero-power level levitation under the proposed method. The active portion of the system depicted in Fig. 4(a) is again depicted in a simplified sketch in Fig. 4(b). The HEMs in the active portion of the system form electromagnetic circuits, such as the one depicted in Fig. 1, with the passive magnetically permeable targets on the levitated platform.

Table 1
Hardware parameters

Parameter Value

Permanent magnet dimensions 18 × 39 × 25.4 (mm)

Magnetization direction dimension 25.4 (mm)

Permanent magnet material Neodymium N50

Total number of coil windings 720 turns

HEM pole material SS400

Minimum stepper resolution 9/320 (degrees/step)

Platform mass 1.75 (Kg)

Platform diameter 0.3 (m)

Eccentric load mass 0.35 (Kg)

Parameter	Value
Permanent magnet dimensions	18 × 39 × 25.4 (mm)
Magnetization direction dimension	25.4 (mm)
Permanent magnet material	Neodymium N50
Total number of coil windings	720 turns
HEM pole material	SS400
Minimum stepper resolution	9/320 (degrees/step)
Platform mass	1.75 (Kg)
Platform diameter	0.3 (m)
Eccentric load mass	0.35 (Kg)

Fig. 4.

(a) Schematic drawing of the active portion of the levitation system with key components and variables indicated. F ₁, F ₂, and F ₃ are the magnetic forces at HEMs 1, 2 and 3 respectively. (b) Schematic drawing of full system with key components, key variables, and coordinate system indicated. The depicted passive target geometry is arbitrary and was changed in hardware.

For the edification of the reader, a force profile for the HEMs in the developed hardware system is provided as $\begin{eqnarray}\displaystyle F_{f}=45\left({\displaystyle \frac{i_{f}+6.8}{z_{f}+3.4}}\right)^{2}. & & \displaystyle\end{eqnarray}$ (1) Where F _f is the fitted force (N) for a given current i _f (A) and z-height z _f (mm). The curve fitting from which this profile was created is shown in Fig. 5.

Fig. 5.

Curve fitting for the force profile as a function of current and air-gap. The measured data points are indicated as black dots. The zero-power force curve is shown as a black line.

The provided force profile provide does not account for flux leakages and inconsistences in materials and components. As is explained later, an accurate force profile is not necessary for the empirical tuning of the system.

Also for the edification of the reader, the equations of motion for the levitated platform depicted in Fig. 4, linearized about the unloaded zero-power equilibrium where the passive magnetic forces counter act the forces due to gravity are given as $$\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle m\ddot{z}=f_{1}+f_{2}+f_{3}-m_{L}g\end{eqnarray}$$(2) $$\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle I_{x}\ddot{{\alpha}}=f_{1}r\sin{\theta}_{1}+f_{2}r\sin{\theta}_{2}-m_{L}gy_{L} \end{eqnarray}$$(3) $$\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle I_{y}\ddot{{\beta}}=-f_{3}r-f_{1}r\cos{\theta}_{1}-f_{2}r\cos{\theta}_{2}+m_{L}gx_{L}, \end{eqnarray}$$(4) where

m is the mass of the unloaded platform

z is the height of the center of mass of the unloaded platform

f _n is the total magnetic force at HEM ‘n’

I _x and I _y are the moments of inertia about the x and y axis defined in Fig. 4, respectively

𝛼 and 𝛽 are rotations about the x and y axis, respectively

m _L is the total mass of the load affixed to the system

x _L and y _L are the x and y locations of the of m _L

g is the acceleration due to gravity.

These equations of motion are also not strictly necessary for the empirical tuning of the system.

4. Control strategy

The subsystem interactions of the levitation system are depicted in Fig. 6. The control system consists of two major components, an air-gap controller and an HEM angle controller. The air-gap controller is responsible for locally controlling the air-gap at each HEM and does not consider changes in the HEM angles. The HEM angle controller is responsible for adjusting the HEM angles using current feedback from the air-gap controller. In general, it can be interpreted that the air-gap controller is responsible for minimizing the average current of all HEMs while the HEM angle controller is responsible for minimizing the current variance between HEMs. The zero-power state is achieved when and only when the average and the variance both approach zero.

Fig. 6.

Block diagram showing interactions between subsystems. i _n and z _n denote the current and z-height of the nth HEM respectively.

4.1. Air-gap controller

A block diagram for the air-gap controller used locally at each HEM is shown in Fig. 7. Similar controllers are typically used for 1 DoF levitation systems [3,6] and it can be seen that the controller consists of a positional-differential (PD) controller with two integral feedback loops. The first integrator is a zero-power feedback loop and it uses the local HEM current as feedback to adjust the air-gap to minimize current. The second integrator is a feedback loop which tends the local HEM air-gap to the average air-gap of all three HEMs, ensuring that the system remains level. For simplicity, the same local controller and controller gains are used for all three HEMs. The discrete PD controller used in experimental trials takes the form $\begin{eqnarray}\displaystyle PD(z)=K_{P}+K_{D}{\displaystyle \frac{N}{1+NT_{s}{\displaystyle \frac{1}{z-1}}}}, & & \displaystyle\end{eqnarray}$ (5) where K _P, K _D, N, and T _s are the proportional gain, derivative gain, filtering coefficient, and sampling time respectively.

Fig. 7.

Block diagram detailing local air-gap controller at the nth HEM. Starting from the left, an initial target z-height z _T,0 is given. The first integrator is a zero-power loop which adjusts the z-height target using the average current of all HEMs $\bar{i}$ . The second integrator maintains the level of the platform by adjusting the z-height target using the difference between the z-height at the nth HEM and the average z height of all HEMs $\bar{z}$ . Finally a PD controller uses the error between the z-target z _T and the z-height z _n to output the current command i _n. K _ZP and K _M are the gains for each respective integrator.

4.2. HEM angle control

A block diagram for the HEM angle controller is shown in Fig. 8. The inputs to the controller are the currents though the HEMs and the output is a command as to where the HEM centroid should be located. To realize the centroid command, a relationship between the centroid of the HEMs and the angular locations of the HEMs is required $$\begin{eqnarray} \displaystyle \left\{ \begin{array}{@{} c@{}}\mathit{centroid}_{x}\\ \mathit{centroid}_{y} \end{array} \right\}=\left\{ \begin{array}{@{} c@{}}{\displaystyle \frac{r}{3}}(\cos{\theta}_{1}+\cos{\theta}_{2}+1)\\{\displaystyle \frac{r}{3}}(\sin{\theta}_{1}-\sin{\theta}_{2}) \end{array} \right\}=F_{1}({\theta}_{1},{\theta}_{2}). & & \displaystyle \end{eqnarray}$$(6) The function F ₁ is a one-to-one function for values of θ₁ and θ₂ between 0 and π radians. Using $F_{1}^{-1}$ , the Cartesian HEM centroid can be realized by the angular displacement of the HEMs by the system’s stepper motors.

Fig. 8.

A block diagram detailing the HEM angle controller where k _i and k _z are stiffnesses defined in (12) and (14) respectively, T is a rotation transformation defined in (13), c _x and c _y are the coordinates of the HEM centroid that counteracts the moments caused by the load on the platform as defined in (15), K _θ is a constant with the unit 1/s, C _x and C _y are the centroid commands which loosely equate to the running time averaged values of c _x and c _y over a period of 1∕K _θ, and $F_{1}^{-1}$ is a mapping between the Cartesian coordinates of the HEM centroid and the HEM angles θ₁ and θ₂. K _θ is functionally used in operation with the integrator to control the response time of angle changes.

For a single HEM, it can be determined by (6) that the maximum positional errors for the centroid for a given angular error θ_error can be given by $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathit{centroi}d_{x,\mathit{error}}={\displaystyle \frac{r}{3}}\left(\cos \left({\displaystyle \frac{{\pi}}{2}}+{\theta}_{\mathit{error}}\right)\right)\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathit{centroi}d_{y,\mathit{error}}={\displaystyle \frac{r}{3}}(\sin ({\theta}_{\mathit{error}})).\end{eqnarray}$ (8) The maximum error occurs for centroid _x when an HEM’s angle is at $\frac{{\pi}}{2}$ radians and for centroid _y when an HEM’s angle is at π radians. At all other angles, the error in the centroid location will be less than that of the maximum. If we consider θ_error to be the minimum resolution of the stepper-gear system, then (7) and (8) can be used to determine the maximum centroid location error that can be expected in the system due to resolution limitations.

Further, for a single HEM, the maximum torque error resulting from the angular resolution limitation θ_res of the mechanical stepper can be given by: $$\begin{eqnarray} \displaystyle {\tau}_{{\theta},\mathit{error},max}={\displaystyle \frac{1}{2}}r\sin{\theta}_{res}z_{n}K_{z}. & & \displaystyle \end{eqnarray}$$(9) This can be directly compared to the maximum torque error resulting from resolution limitations of the air-gap sensor z _res which is given by: $$\begin{eqnarray}\displaystyle {\tau}_{z,\mathit{error},max}={\displaystyle \frac{1}{2}}rK_{z}z_{res}. & & \displaystyle\end{eqnarray}$$(10) In cases where the error resulting form ((9)) is comparable to that from ((10)), the application of a mechanical stepper is justified as not contributing excessive error compared to the air-gap sensor.

The principle by which the centroid command is determined is based on the goal of laterally displacing the HEMs as to create an equal and opposite moments M _x and M _y to the moments caused by the active currents through the HEMs as stated in (11). As the net moment acting on the platform in the steady state approaches zero, the variance in the current through the HEMs approaches zero. $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}c@{}}M_{x}\\ M_{y}\end{array}\right\}_{HEM\mathit{Displacement}}=-\left\{\begin{array}{@{}c@{}}M_{x}\\ M_{y}\end{array}\right\}_{\mathit{Currents}}. & & \displaystyle\end{eqnarray}$ (11) The moment caused by the active currents about the geometric center of the platform is given by $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}c@{}}M_{x}\\ M_{y}\end{array}\right\}_{\mathit{Currents}}=k_{i}T\left\{\begin{array}{@{}c@{}}i_{1}\\ i_{2}\\ i_{3}\end{array}\right\}, & & \displaystyle\end{eqnarray}$ (12) where k _i is a linearized force-current stiffness and the rotation transformation T is $\begin{eqnarray}\displaystyle T=\left[\begin{array}{@{}ccc@{}}r\sin {\theta}_{1} & -r\sin {\theta}_{2} & 0\\ -r\cos {\theta}_{1} & -r\cos {\theta}_{2} & -r\end{array}\right]. & & \displaystyle\end{eqnarray}$ (13) Next, the moment caused by the angular locations of the HEMs is given by $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}c@{}}M_{x}\\ M_{y}\end{array}\right\}_{HEM\mathit{Displacement}}=\left[\begin{array}{@{}cc@{}}0 & z_{c}K_{z}\\ -z_{c}K_{z} & 0\end{array}\right]\left\{\begin{array}{@{}c@{}}c_{x}\\ c_{y}\end{array}\right\}, & & \displaystyle\end{eqnarray}$ (14) where z _c is the height of the center of the platform and K _z is a linearized force height stiffness.

Finally, it can then be formulated that the HEM centroid location that results in an equal and opposite moment to the moment caused by the currents through the HEMs is $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}c@{}}c_{x}\\ c_{y}\end{array}\right\}=-k_{i}\left[\begin{array}{@{}cc@{}}0 & k_{z}\\ -k_{z} & 0\end{array}\right]T\left\{\begin{array}{@{}c@{}}i_{1}\\ i_{2}\\ i_{3}\end{array}\right\} & & \displaystyle\end{eqnarray}$ (15) where a variable k _z is defined for simplicity as 1∕z _c K _z. For robust control, the centroid location is passed through an integrator which outputs the final centroid command as is shown in Fig. 8. This Cartesian command is then angularly actuated by the system’s stepper motors using $F_{1}^{-1}$ . It should be noted that the numeric values for k _i, z _c K _z, z and r in (13) and (15) are not strictly necessary as they can all be compensated by the integrator gain K _θ.

5. Experiment

The relevant controller parameters used during the experimental trials are tableted in Table 2.

Table 2
Control parameters

Symbol Parameter Value

K _P Proportional gain −2000

K _D Derivative gain −20

N Filtering coefficient 20000

K _M Mean air-gap feedback gain 5

K _ZP Zero-power feedback gain 0.025

m Mass of unloaded system 2 (Kg)

I _x Unloaded moment of inertia, x 0.5 (Kg m² )

I _y Unloaded moment of inertia, y 0.5 (Kg m² )

r Radius of HEMs 1 (m)

k _i Force-current stiffness 10 (N/A)

K _z Force-height stiffness 10,000 (N/m)

Symbol	Parameter	Value
K _P	Proportional gain	−2000
K _D	Derivative gain	−20
N	Filtering coefficient	20000
K _M	Mean air-gap feedback gain	5
K _ZP	Zero-power feedback gain	0.025
m	Mass of unloaded system	2 (Kg)
I _x	Unloaded moment of inertia, x	0.5 (Kg m² )
I _y	Unloaded moment of inertia, y	0.5 (Kg m² )
r	Radius of HEMs	1 (m)
k _i	Force-current stiffness	10 (N/A)
K _z	Force-height stiffness	10,000 (N/m)

5.1. HEM angle control gain tuning

The HEM angle gain K _θ in the block diagram shown in Fig. 8 regulates the speed of the HEM centroid in the x and y directions relative to the moments caused about the y and x axes by the currents through the HEMs. In other words, if the HEM angle gain is increased, the x coordinate of the HEM centroid will change faster in response to changes in the moment about the y axis caused by currents through the HEMs.

The HEM angle gain tuning was empirically conducted for an eccentric mass of 350 g placed at 0 mm in the x direction and −150 mm in the y direction. To conduct this tuning, the platform was first levitated to its unloaded zero-power equilibrium. Then, the eccentric mass, which can be seen in Fig. 4(a), was then placed by hand on the platform after roughly 0.5 seconds and the system was allowed to reach steady-state. This procedure was repeated three times with HEM angle gains of 0.01, 0.025, and 0.05. The results are shown in Fig. 9. The HEM angle gain of 0.025 was then selected for its low overshoot and fast response.

Fig. 9.

Step response of system with different HEM angle controller gains. A gain of 0.025 was empirically selected.

5.2. Step response

The step response for z-height and current for the 0.025 angle gain experiment from Fig. 9 are shown in Fig. 10 and Fig. 11 respectively. In Fig. 10, it can be seen that in the steady state the platform is level as is indicated by the identical z-heights at each HEM. As expected, it can be observed that the platform exhibits negative stiffness characteristics since the air-gaps decrease when the load is added [2,3].

Fig. 10.

z-height step response to an eccentric mass placed after 0.5 seconds. The platform is confirmed to operate levelly.

In Fig. 11, it can be verified that zero-power is achieved since it can be seen that there is a similar baseline current noise before and after the placement of the eccentric mass, and it can be observed that the currents are centered about zero. The range of HEM currents in the steady state of approximately ±0.25 A is roughly identical to that of another similarly sized zero-power system developed previously [4] and is largely attributed to sensor noise.

Fig. 11.

Current step response to an eccentric mass placed after 0.5 seconds. Zero-power control is confirmed.

5.3. Further testing

The developed levitation system was subjected to further testing with varying eccentric load conditions to verify that the system can achieve zero-power operation under arbitrary eccentric loading conditions. In this experiment, the same 350 g eccentric load was placed 150 mm away from the center of the platform at angular locations as measured from the x axis from 180 to 360 degrees in 22.5 degree increments. First, the platform was levitated in its zero-power unloaded state, then after 10 seconds, the load was applied by hand at the 180 degree location. The load was removed by hand 10 seconds later and placed again after an additional 10 seconds in the location 22.5 degrees greater. This procedure was repeated until finally the load was at the 360 degree location. Angular locations in only a half circle were used as the system is symmetric about the x-axis.

The results from this experiment are shown in Fig. 12. As expected due to geometry, a clear sinusoidal trend can be seen in the HEM centroid position in both C _x and C _y. A small phase shift can be observed in the sinusoidal trend of the HEM centroid graphs in Fig. 12. As the system is not constrained against rotations in the z direction, it is suspected that this phase shift was caused by an unwanted rotation of the platform with respect to the base.

Fig. 12.

An eccentric mass is placed for 10 seconds, removed for 10 seconds, and placed again for 10 seconds at angular locations from 0 to 180 degrees in 22.5 degree increments. A clear sinusoidal shape can be seen in C _x and C _y. Zero-power control for arbitrary eccentric loading is confirmed.

In the bottom graph in Fig. 12, the summation of the absolute value of current in each HEM is shown. It can be seen that the loaded and unloaded current is nearly uniformly at the same baseline level of noise confirming good performance at all angular load positions. The non-uniform magnitude of the current spikes which resulted from the placing and removing of the eccentric mass can be attributed to human variance in handling.

6. Conclusion

A method of magnetic levitation by lateral displacement of HEMs was proposed which allows for a platform to actively maintain zero-power level levitation under arbitrary eccentric loading. A hardware prototype was produced, tested, and validated and mathematical descriptions of system components were formulated.

An empirical tuning of the novel HEM angle controller was described and conducted, resulting in a settling time of approximately 2 seconds during experimental testing. Experimental results showed that the system is able to maintain zero-power level levitation while keeping the baseline current consumption caused by noise to a range comparable to other magnetic levitation systems. Further, experimental results showed that the system performed similarly well for eccentric loads placed at different angular locations relative to the center of the platform.

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