[American Institute of Aeronautics and Astronautics AIAA Guidance, Navigation and Control Conference...

American Institute of Aeronautics and Astronautics

1

A Flux-Pinned Magnet-Superconductor Pair for Close-Proximity Station Keeping and Self-Assembly of Spacecraft

Joseph P. Shoer1 and Mason A. Peck2 Cornell University, Ithaca, NY, 14853

We propose that spacecraft modules be constructed with interfaces consisting of combinations of magnets and Type II superconductors, establishing a non-contacting interaction between the modules thanks to magnetic flux pinning. This stable action-at-a-distance interaction overcomes the limit Earnshaw’s Theorem places on other spacecraft positioning strategies involving electromagnetic fields, allowing fractionated or modular spacecraft to fix their relative positions and orientations without any mechanical connection, active control, or power expenditure. We report two experiments investigating the mechanical properties of the magnetostatic interaction to evaluate its utility. First is an experiment to find the 6DOF linear restoring forces and torques on a flux-pinned magnet and superconductor for small displacements from a set of static equilibria. Second is an transient experiment that permits the use of system identification techniques to characterize the modal damping and stiffness. Our results indicate that flux pinning is promising for modular spacecraft assembly and station-keeping applications, providing mechanical stiffnesses over 200 N/m at small (5 mm) magnet-superconductor separations and potentially useful nonzero stiffnesses at larger (over 3 cm) separations, with significant damping. We find that increasing the magnetic flux density at the superconductor surface strengthens the flux pinning forces, suggesting the possibility that higher stiffness can be obtained over larger distances by increasing or focusing the magnetostatic field.

Nomenclature

ia = unit vectors aligned with tracking camera view frame (i = 1, 2) β = tilt angle of a permanent magnet dipole axis from the vertical d = separation distance between the facing surfaces of a superconducting plate and permanent magnet D = flux pinning damping (6×6 matrix) Dxx = 3×3 damping matrix partition coupling translations into forces Dθθ = 3×3 damping matrix partition coupling rotations into torques Dxθ, Dθx = 3×3 damping matrix partitions coupling translations into torques or rotations into forces δr = small deflection of a magnet position vector from flux pinned equilibrium (6×1 matrix) F = generalized force (6×1 matrix) f = position vector of tracking flag on levitated magnet I = inertia matrix of a levitating magnet (3×3 matrix) K = flux pinning stiffness (6×6 matrix) Kxx = 3×3 stiffness matrix partition coupling translations into forces Kθθ = 3×3 stiffness matrix partition coupling rotations into torques Kxθ, Kθx = 3×3 stiffness matrix partitions coupling translations into torques or rotations into forces m = mass of a single permanent magnet N = number of identical permanent magnets stacked together to form magnets of differing strengths MQC = rotation matrix relating a camera’s coordinate basis to a pinned magnet’s basis

1 Graduate Research Assistant, Department of Mechanical and Aerospace Engineering, 127 Upson Hall, AIAA Student Member. 2 Assistant Professor, Department of Mechanical and Aerospace Engineering, 212 Upson Hall, AIAA Member.

AIAA Guidance, Navigation and Control Conference and Exhibit20 - 23 August 2007, Hilton Head, South Carolina

AIAA 2007-6352

Copyright © 2007 by Cornell University. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


2

Tc = superconductor critical temperature τ = applied torque on a levitating magnet (6×1 matrix) r = displacement of a magnet from flux pinned equilibrium (6×1 matrix) r0 = position vector of a flux pinning equilibrium (6×1 matrix) u = input force and torque on a flux pinned magnet (6×1 matrix)

I. Introduction he assembly of large spacecraft systems in orbit presents many challenges. Some common approaches to these challenges depend on the adaptation of terrestrial construction techniques to the space environment.1 However,

an intriguing magnetostatic phenomenon may enable a new in-orbit construction paradigm: both small and large space systems could be assembled from components that find one another and settle into place without any material connection. The modules maintain a fixed separation distance from one another through the non-contacting interactions characteristic of magnetic flux pinning. Because it features passive stability, this approach to self-assembly requires no active control and no appreciable power; and yet neighboring modules are stiffly positioned and oriented through action at a distance.

The proposed approach to conjoining mechanical parts without physical contact offers the promise of revolutionizing in-orbit construction. The non-contacting interface not only solves a host of technological issues, it also opens up a new way of thinking about modular spacecraft.2, 3 This concept blurs the distinction between modular spacecraft and formation flying, and between spacecraft bus and payload. Articulated payloads, reconfigurable space stations, and adaptable satellite architectures are possible without the risk, mass and power typically associated with maintaining relative position and mechanically rebuilding structures. Technological applications of familiar action-at-a-distance forces (such as gravity and electromagnetic attraction) are limited by Earnshaw’s Theorem, which states that no combination of such forces can result in stable separation of objects in all six degrees of freedom (DOF). Active control is typically required, as in the case of magnetic bearings. Our proposed approach gets around Earnshaw’s theorem by taking advantage of the unfamiliar quantum physics of type II high-temperature superconductors (HTSCs). When a magnetic field of sufficiently high strength is brought near these materials, magnetic flux lines penetrate the superconductor and are frozen or “pinned” in place.4, 5 Flux pinning manifests itself as a force on the origin of the magnetic field (for instance, a permanent magnet). A convenient physical model for this interaction for small variations in relative position is a linear spring-mass-damper system. Many researchers have concentrated on flux-pinning stiffness as a mechanism to achieve magnetic levitation,6, 7 as demonstrated by the permanent magnet levitated over a slab of superconducting yttrium barium copper oxide (YBCO) in Fig. 1. However, a particularly interesting property of this interaction is that flux pinning acts in any direction along which there is a magnetic field gradient. Thus, the magnet in Fig. 1, which has an axially symmetric field, is fixed in every rigid-body degree of freedom except for rotation about its axis of symmetry (its dipole axis), not merely in vertical translation. Flux pinning establishes a stable equilibrium. When flux is frozen into a superconductor (for instance, by cooling a HTSC below its critical temperature Tc with a permanent magnet nearby), the magnet feels a linear restoring force under small perturbations from its original pinned position.8 A flux-pinning equilibrium is not unique: a levitated magnet can be moved away from the superconductor, brought back, and pinned in a new orientation.5 The forces involved are highly hysteretic, and a magnet experiences different forces depending on the direction of its relative motion with respect to a superconductor and its distance from the superconductor.9 Our proposed approach to in-orbit assembly seeks to take advantage of these multi-dimensional flux pinning equilibria for space applications. We envision fractionated spacecraft mated together by interfaces composed of

T

Figure 1. A cylindrical permanent magnet (1.9 cm dia.) flux pinned with its center about 2 cm above the surface of a YBCO superconductor at approximately 77K.


3

flux-pinned pairs of magnets and superconductors rather than material connections. These non-contacting interfaces offer several advantages over their mechanical analogues (mating adapters, trusses, certain applications of tethers, et cetera). First, the risk of electrostatic discharge between differentially charged modules would be reduced or eliminated if the inter-module distance is large enough to preclude arcing. In addition, flux pinning can be activated and deactivated by simply raising or lowering the temperature of the HTSC above or below Tc. In space, such thermal control may simply mean shielding the superconductors from or exposing them to sunlight. The modules’ relative position and orientation before cooling become passive equilibria after cooling, with a range of possible separation distances between modules. In-space construction of modules with flux-pinned interfaces may therefore be much easier than assembling modules mechanically. Instead of precisely maneuvering two modules into place under active control and making a mechanical connection (which might involve spacewalks, robot arm manipulations, or risks of mechanical failures), modules only need be moved within pinning range, and their HTSC elements cooled. Flux pinning requires no power input as long as the superconductors are below their transition temperature, Tc. Therefore, the many possible failure modes of a spacecraft that can lead to loss of power and/or loss of command and data handling do not threaten the structural integrity of a system held together by this effect.

Another application is possible with flux pinning forces that act at a scale distance of meters to tens of meters: passive formation flight in which the formation’s three-dimensional geometry persists indefinitely. Unlike general formations that depend on CW behaviors, a flux-pinned formation retains its shape throughout the orbit. A large system of modules, such as a sparse-aperture telescope, could be assembled without the need for active control. Figure 2 is meant to suggest a reconfigurable sparse-aperture telescope reflector that has been autonomously self-assembled from reflective modules held in place by flux pinning. In addition, it may be possible to establish a flux-pinning

equilibrium at some initial separation distance and then change the properties of the pinned magnetic field to alter the equilibrium without breaking the non-contacting connection. This possibility may enable control of the relative motion of flux pinned modules, leading to articulated spacecraft.10 In the case of the self-assembled large-aperture dish, this articulation suggests a path to adaptive optics.

With appropriate magnet and superconductor geometry, flux pinning may provide the mechanical stiffness necessary for in-orbit construction applications. Also, if moving flux lines travel across the superconductor, against the pinning effect, mechanical energy is damped in a powerful manner reminiscent of eddy-current damping. Thus, the flux pinning connection is heavily damped against oscillation despite the low temperatures. In order to design a “virtual coupling” system, we must first answer a number of questions about these physical interactions and then move on to devise some scaling principles. Other researchers, with magnetic levitation and magnetic bearings in mind as the primary applications of this effect, have investigated the mechanical stiffness of a flux pinned magnet for translations vertically away from a superconductor surface9 and laterally along that surface.11 This paper is concerned with the more general problem of simultaneously characterizing these values, rotational stiffnesses, damping coefficients, and other parameters that determine coupling among degrees of freedom, as functions of the relative positions of a magnet and superconductor. Our results derive from two types of experiments whose objective was to determine the six-dimensional stiffnesses of a pinned magnet-superconductor system: one type treating the magnet-superconductor pair statically, and the other treating the pinned pair as a dynamic system. The process we have developed in these experiments might be extended to performance-verification testing of flux-pinned modules for eventual space application.

Figure 2. Reconfigurable optical mirrors assembled from modules with non-contacting interfaces.


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II. Testing the Mechanical Properties of Flux Pinning

A. Static Test of the Stiffness of a Flux Pinned Magnet 1. Experiment

The purpose of the static experiment is to measure flux-pinning stiffness as the proportionality between small displacements of the magnet relative to the HTSC and the restoring force exerted by flux pinning. The HTSC material consisted of 19 hexagonal tiles of bulk melt-textured YBCO superconductor forming a single roughly hexagonal plate approximately 11 cm in diameter and 1 cm thick. These tiles were epoxied in place on a slate tile (a material with a coefficient of thermal expansion similar to YBCO). During the experiments, the superconductors and tile were immersed in a bath of liquid nitrogen in an insulated container. An Eshed Robotek Scorbot ER-V robot provided a means to translate a stack of one or more permanent magnets with some precision relative to the superconductor as shown in Fig. 3. The end effector of the robot can displace in five degrees of freedom (translation in three dimensions along with roll and pitch rotations) and has position repeatability below roughly 0.5 mm.12 The robot gripper held a non-ferromagnetic ATI Gamma load cell capable of measuring forces and torques in six dimensions with resolution better than 1/320 N. A rigid plastic enclosure mounted an NdFeB permanent magnet(s) to this load cell but stood it off far enough to separate the temperature-sensitive load cell from the nitrogen bath. In addition to the special load cell, we used nonmagnetic components wherever possible to prevent the test equipment from interacting with the magnet and the superconducting plate.

The robot arm ran a program that took the magnet through a series of small deflections in each of its five degrees of freedom from this equilibrium point. Every effort was made to align the uncontrollable sixth degree of freedom, end-effector yaw, with the magnet’s axis of symmetry. Since the setup used a cylindrical magnet with an axisymmetric dipole field, flux pinning constrained all degrees of freedom of the magnet except rotations about this dipole axis. Nevertheless, the six-vector of the magnet’s deflection was recorded for each measurement. The robot carried out ten displacements (n = 10): translations of 2 mm in both the positive and negative directions along x, y, and z, followed by positive and negative rotations of 3° about the x and y axes. Forces exerted by the superconductor on the magnet were transferred to the load cell, which resolved them into a 6D measurement of the force and torque experienced by the magnet.

x

z

y

YBCO in LN2 bath

Magnet

Plastic mount

Robot arm

Load cell

Figure 3. Experimental setup with robot arm, load cell, permanent magnet, and superconductor.


5

This experiment was repeated to measure variation in the 6D stiffness as a function of three parameters: separation between the magnet and the superconductor surfaces, d,. the tilt angle β of the magnet with respect to the superconducting plane, and the number of magnets N mounted on the load cell. Each point in this (d, β, N) parameter space has a unique stiffness matrix associated with it. To prevent hysteretic effects when changing the height or tilt angle, we first warmed the YBCO above its critical temperature, adjusted d and β by having the robot move the magnet, and then cooled the YBCO below Tc again, establishing the magnet’s new position as a field-cooled flux pinning equilibrium.

2. Static Model

When cooled in the presence of a permanent magnet, the system is at equilibrium and the superconductor applies no appreciable force to the magnet. Small relative motions of the magnet and YBCO provided data from which stiffnesses were estimated. For small displacements of the magnet from this equilibrium, the flux pinning force followed a non-hysteretic minor loop and appeared as a linear restoring force4, 8, 9. The linear behavior of this force for small values of displacement |r| justifies our use of a 6DOF Hooke’s Law model:

166616 ×××

−= rKF (1)

The six-dimensional generalized position and force vectors are, in Cartesian coordinates with small rotation angles,

[ ][ ] .T

T

zyxzyx

zyx

FFFF

zyxr

τττ

θθθ

=

= (2)

The stiffness matrix comprises four 3×3 translational, rotational, and cross-coupling partitions:

⎥⎦

⎤⎢⎣

⎡=

θθθx

xθxx

KKKK

K (3)

Cooling the superconductor with the magnet in place flux-pins the magnet to the superconductor without

hysteretic effects induced by, for example, repeatedly moving the magnet towards and away from the superconducting plate, as supercurrents induced in the YBCO persist indefinitely. Some authors refer to this situation as field cooling of the superconductor (FC), contrasted with zero-field cooling (ZFC) in which the superconductor is chilled in the absence of magnetic field and the magnet is subsequently brought in from a large distance away.13 The magnet’s position at the time of cooling defines the origin of the coordinate system for r so that both r and F are zero at equilibrium.

Displacement and force data was used in a least-squares algorithm to solve for the stiffness matrix K in Eq. 1. With the scalar elements of these matrices identified with subscripts, the equation takes the form

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

6

1

6661

1611

6

1

r

r

kk

kk

F

FM

L

MOM

L

M . (4)

One approach to solve for the components of K is to include n ≥ 6 pairs of displacements and forces as column vectors in this equation and multiply both sides by the right pseudoinverse of the matrix of position vectors:

nn

n

nn

n

r

r

r

r

kk

kk

F

F

F

F

×××

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

66

1

61

11

666661

1611

66

1

61

11

MLM

L

MOM

L

MLM (5)


6

+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−=

n

n

n

n

r

r

r

r

F

F

F

FK

6

1

61

11

6

1

61

11

MLMMLM (6)

A MATLAB realization of Eq. 6 gives the flux-pinning stiffness matrix for the n = 10 corresponding displacements carried out by the robot and force/torque measurements from the load cell at each (d, β, N) point.

B. Dynamic Test of Flux Pinning Stiffness and Damping 1. Experiment

The static test measured stiffness. Successfully representing the dynamics of a “virtual structure” of flux-pinned modules also requires some model of damping. The second test was meant to capture those velocity-dependent forces. Exciting and measuring the time-history of deflections in a flux-pinned system provided inputs to a system-identification analysis to determine the natural frequencies, stiffnesses, and damping ratios. We refer to this approach as the dynamic test.

A system consisting of a cylindrical magnet flux-pinned to a superconducting plate, neglecting external forces, exhibits six fundamental modes of oscillation: simple harmonic motion in each dimension, rocking oscillations around the x and y axes (or some linear combination thereof), and rotation about the z axis. Flux pinning does not constrain this last mode for a single dipole magnet since rotation about the magnet’s axis of symmetry does not change the distribution of magnetic flux in the HTSC. Therefore, its frequency is expected to be zero (a rigid-body mode). Some natural frequencies may be repeated if the magnet is perfectly symmetrical. All these modes can be excited and measured dynamically.

The dynamic experiment apparatus is illustrated in Fig. 4. A permanent magnet was suspended from a roughly 3.5m long, lightweight monofilament pendulum. The goal was to separate the pendulum mode in frequency from the flux-pinning modes to such an extent that it approximated rigid-body behavior. The total mass of the pendulum bob was 22 g. The pendulum prevented magnet translation in the y direction (the direction of gravity) and rotation about the z axis (the axis perpendicular to the YBCO surface), leaving four DOFs to be measured. The long pendulum allowed swinging motions of the magnet to appear nearly as translations in the x and z directions. We set up an optical tracking system to follow the motions of a flag attached to the magnet, and we placed an electromagnet coil near the pinned magnet. The magnetic interactions between the current in the electromagnet coil and the

Figure 4. Dynamic experiment setup. (Left) Photo showing the pendulum, coil, and superconductor. (Right) Top-down schematic of the dynamic test setup. The magnet is brought into position by a stage (not shown) that is removed at the time of the experiment. Note that the coordinate system has been oriented with z perpendicular to the YBCO surface for consistency with the static experiments.

Electromagnet coil(s)

Input from computer

Flag

YBCO slab in N2 bath

x

y z

Pinned magnet on pendulum

Tracking

camera

Output to computer


7

permanent magnet served as excitations. In order to flux pin the magnet at known d and β and to avoid hysteretic effects, we set the magnet in place near the superconductor with a stage before chilling the YBCO with liquid nitrogen. After the HTSC reached its critical temperature, establishing flux pinning, we removed the stage. The constraint preventing rotation about the z axis does not limit the relevance of these tests to our proposed application since a physical system built from flux-pinned modules will likely use several flux-pinned pairs to stiffen relative rotation. In addition to enforcing θz = 0 for all time, the pendulum also restricts y = 0 for small pendulum motion. This additional constraint is acceptable because the stiffness resulting from lateral displacements of the magnet ought to be identical in both the x and y directions. Our dynamic system thus has only four degrees of freedom, but the interaction in the two constrained DOFs are either redundant or irrelevant.

The pendulum setup largely removes gravity from the dynamic model. Other flux pinning experiments treating the pinned pair as a dynamic system14, 15 have been performed, but with the application of levitating systems in mind. In those situations, the forces on the magnet represent a balance between gravity and interactions with the superconductor. These interactions may be due to both flux pinning and Meissner effect repulsion of flux as the magnet seeks the potential well defined by gravity and interactions with the superconductor. Since we envision an application of flux pinning—which can exert an attractive force, unlike the Meissner effect—in microgravity environments, we sought to characterize the flux pinning interaction in the absence of external forces. The long pendulum makes this characterization possible without introducing significant additional dynamics and without overly constraining the magnet.

2. Dynamic Model

An assumed state-space model provides the basis for extracting the damping and stiffness matrices from the input and output series. A derivation follows. Magnets subject to flux pinning are in a stable equilibrium. Others have shown that, for small translations, the restoring force on a flux pinned magnet is linearly proportional to displacement in directions both perpendicular to the surface of the superconductor9 and parallel to the surface.8, 11 The restoring torque for small tilt angles is also expected to be linear. Retaining the notation established in Section IIA, a linear system model in terms of an 12×1 state defined by small perturbations from the flux pinning equilibrium at r = 0 is

⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡=

rδδr

rr

x&&0

0 . (7)

Let rx and rθ denote the translational and rotational components of r, respectively. Then the linearized rigid-body equations of motion of a magnet pinned to a HTSC in free space in matrix form are

( )( ) τIrδIrrIIrδ

F/mrδ

θθθθ

x

100

1 −××− +−=

=

&&&&&

&& (8)

for some applied force F and torque τ, where m is the mass of the magnet and flag, I is the inertia matrix of the magnet and flag, and 0θr& is the equilibrium angular velocity of the magnet (e.g. rotation about the unpinned symmetry axis of the dipole). In our pendulum-based system, the combination of flux pinning and the pendulum constrain the magnet so that its equilibrium angular velocity is zero. So, we set 0θr& to zero.

We model the flux-pinning force as a six-dimensional spring-mass-damper system that is linear for small perturbations of the magnet position vector. The applied force on the magnet is then the sum of flux pinning forces and pendulum forces along with applied forces ux and uz and torques uθx and uθy from the electromagnet coils:


8

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ ⋅=⎥

⎦

⎤⎢⎣

⎡

0

0

0

0sin

0sin

0013

y

x

z

x

yp

z

x

θ

x

θθθx

xθxx

θ

x

θθθx

xθxx

θ

x

uuu

u

k

T

T

δrδr

K

KKKK

rδrδ

D

DDDD

rδrδ

M

Im

τF

θ

θ

θ

φ

φ

43421&

&

43421&&

&&

43421

(9)

M, D, and K are the 6×6 mass, damping, and stiffness matrices. The pendulum properties are given by T, the tension in the monofilament line; kp, the torsional stiffness of the pendulum; and the angles φx and φz that the pendulum swings towards the x and z axes. We use the small-angle approximation for both these angles so that sinφx,z ≈ φx,z and take T = mg. Equations 8 and 9 with these approximations give the A and B matrices of the state space system. With the state vector rewritten to include only the four unconstrained degrees of freedom,

[ ]Tzxyx zxzx θδθδδδδθδθδδχ &&&&= , the state evolution equation is

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

′−′−′−′−′−

′−′−′−−′−=

−−−−−y

x

z

x

m

θθθxpθθθx

xθmxxmxθmxxm

uuuu

B

I

A

DIDIkKIKIDDKpK

θ

θ

χχ

443442144444444444 344444444444 21

&

12

1

1111

11112

2

0010000

10000100

, (10)

expressed here in terms of 2×2 partitions. Primes on K and D matrices indicate the deletion of the rows and columns corresponding to forces and motions motions in y and θz to match the state vector. Perturbations due to the pendulum are given by

⎥⎦

⎤⎢⎣

⎡=′

⎥⎦

⎤⎢⎣

⎡=

000

1001

pp

kk

lgp

. (11)

The A matrix resembles that of a damped oscillator, with frequencies shifted from the flux-pinned frequencies by the pendulum dynamics. The parameters of interest appear only in the system matrix. So, an experimentally determined A matrix is all that is needed to find the stiffness and damping matrices. The B matrix does not offer useful insight into the flux-pinning parameters and falls out as a byproduct of the identification process. It is written to accept an arbitrary force and torque input. A transformation can be formulated to convert these arbitrary forces and torques into the currents through the electromagnet coils. The tracking camera outputs two measurements corresponding to the two-dimensional position of a corner of the flag in its field of view. These measurements can be represented as the projection of the position vector of the flag onto two perpendicular unit vectors in the direction of the camera view frame’s axes, 1a and 2a . The position vector to the flag is equal to the vector sum of the position of the magnet center of mass rx and a vector from the magnet center to the flag f as suggested in Fig. 5. This latter vector is fixed in the magnet frame. Thus, the measurement vector can be represented as

Figure 5. Camera measurements.

2a

1a

rx

f

C

M


9

( )( )⎥⎦

⎤⎢⎣

⎡+⋅+⋅

=frafra

x

xy2

1

ˆˆ

, (12)

neglecting a constant offset term. The flag vector f is fixed in the magnet frame M but not in the camera frame C. A coordinate system fixed in the magnet frame is related to one fixed in the camera frame by a direction-cosine matrix MQC, which, for small angular perturbations of the magnet, can be expressed by

[ ] .δr

δrQ

θ

θ×

×

−≈

−= expCM

(13)

The flag vector appears in camera coordinates as Cf = CQM Mf. We transpose the skew-symmetric matrix MQC by negating it and reversing the order of the cross product frθ

M×δ to obtain the 2×12 C matrix of the state space model:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡=

×

×

θ

x

θ

x

rδrδδrδr

C

faafaa

y

&

&444444 3444444 210000

MT2

CT2

C

MT1

CT1

C

(14)

The C matrix can be reduced by deletion of appropriate columns to match an 8×1 state vector instead of the above 12×1 state. The system defined by Eqs. (10) and (14) is both controllable and observable (as long as 1a and 2a are not aligned parallel or perpendicular to the superconductor surface, the xy plane). Using system-identification techniques, we can now experimentally determine the A matrix and extract stiffness and damping parameters by comparing elements of the identified A matrix with Eq. (10).

III. Results and Discussion

A. Static Test Results We performed the static experiment with N = 1 at (d, β) points in the range from d = 5 mm to 30 mm and β = 0°

to 20°, with N = 2 and 3 at three selected (d, β) points. Each point in this three-dimensional parameter space yielded a stiffness matrix. For example, one representative stiffness matrix obtained from the static experiment is

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−−−−

−−−−−−−−−

=

⋅⋅

=°=

=

m/radNm/mN

N/radN/m

1,2

5mm,

001.0001.001.006.0001.002.023.016.038.00007.006.086.056.0009.068.11963.469.3401.131.461.84.7596.803.372.035.61.18101

N

dK

β

. (15)

The column of zeroes on the right is a result of the MATLAB postprocessing script, which assumes zero stiffness for rotations about the magnet axis of symmetry since flux was not pinned about this axis and the robot did not execute displacements about this axis.

Several features of this matrix are readily apparent. First, It is neither diagonal nor symmetric. Had the robot coordinate axes been perfectly lined up with the load cell axes, the plane of the superconductor, and the magnet’s dipole axis, we would have expected a diagonal K matrix. That is, we would not have expected to find coupling


10

terms between orthogonal displacements and forces. Regardless of the particular geometry of the experiment, we did expect to find that approximately K = KT. The asymmetry of the measured K matrices may indicate that the motions of the robot are not consistent along all axes (e.g., commands to move 1 mm along x and 1 mm along y might result in the end effector traveling 1.1 mm in x and 0.9 mm in y) or that load cell measurements are similarly not consistent across axes. The robot is the more likely source of such error since its rated position repeatability is about 25% of the 2mm displacements used.

Second, the first three elements along the main diagonal are large compared to the other elements in the matrix. These elements correspond to the proportionality between x, y, and z displacements to x, y, and z forces, respectively. We expected these elements of the matrix to dominate the other elements in the Kxx partition if the robot axes, superconductor surface, magnet axis, and load cell axes were well-aligned. In addition, these elements are consistent with the order of the nonzero elements of the Kθθ partition if we model the rotational stiffness about x or y as due to two springs of stiffness K33 separated by the diameter of the magnet. In a physical system of spacecraft modules bound by several pairs of flux pinned magnets and superconductors, the elements on the diagonal of Kxx will likely be the most important design parameters since rotational stiffness can be achieved by a combination of translational stiffnesses.

Third, the element K11 is not close to K22 in magnitude. We expected these elements to be equal: the restoring force resulting from a lateral displacement of the magnet over the superconductor should not depend on the direction of the lateral displacement if the superconductor behaves uniformly. To check the uniformity of the superconductor, we repeated an experiment with the YBCO plate rotated 90° from its orientation in all other trials, expecting the values of K11 and K22 to switch if the YBCO was not uniform in its flux pinning ability, but found no change in the measured K matrix. The stiffness matrix similarly did not depend on the orientation of the magnet with respect to the load cell or the load cell with respect to the robot arm. Therefore, we suspect that the asymmetry between K11 and K22 is due to some nonuniformity in small displacements of the robot arm in the

0 5 10 15 20 25 30 3510

-1

100

101

102

Magnet-superconductor separation (mm)

Stif

fnes

s (N

/m)

0° tilt

2° tilt

5° tilt

10° tilt

20° tilt

Figure 6. K11 (mapping x → Fx) versus separation distance d for several magnet tilt angles β. Error bars reflect the stated ±0.5 mm position repeatability of the robot arm.

0 5 10 15 20 25 30 3510

-1

100

101

102


Stif

fnes

s (N

/m)

0° tilt

2° tilt

5° tilt

10° tilt

20° tilt

Figure 7. K22 (mapping y → Fy) versus separation distance d for several magnet tilt angles β. Error bars reflect the stated ±0.5 mm position repeatability of the robot arm.


11

x and y directions or some bias in the load cell calibration. Other investigators have found, both theoretically and experimentally, that K33 = 2K22 = 2K11.13, 16 In our experiments, K11 ≈ K33 /2 while K22 is much lower, which leads us to take K11 as a more accurate measure of the lateral stiffness.

The stiffness matrix elements showed some clear trends as we varied the experimental parameters. According to the theory worked out by Johansen and Bratsberg, the restoring force for displacements in a lateral direction (i.e., along x or y) should decay approximately exponentially as the separation d increases.17 Johansen et al. also conducted experiments that showed that the stiffness for small displacements in z falls off as the height of a magnet

over a superconducting thin film increases.18 Our data is consistent with these previous observations. We expected rotational stiffnesses also to fall off with increasing d. High rotational stiffness is very desirable if non-contacting modules are to remain fixed relative to one another. Additionally, we expected to find that increasing the permanent magnet’s dipole moment (by increasing N) would scale up the stiffness matrices.

Figures 6 through 10 reveal the dependence of the diagonal elements of K on separation d for several values of tilt angle β. Consistent with the aforementioned investigations, both the lateral and translational stiffnesses fall off as the magnet-superconductor separation increases. K11 through K33 appear to exhibit exponential behavior. The calculations in Johansen et al. suggest that K33 should depend on h in a more complex way,18 but the discrepancy is most significant

at smaller values of d than we measured.

The drop in flux pinning stiffness with separation distance imposes a limit on the possible separation between two flux pinned modules in a space system. If fitted to an exponentially decaying function

0/ d-diii e k K = , all the curves in

Figs. 6 through 8 have 1/e decay lengths on the same order, ranging from 2.8 mm to 6.5 mm. Each of the translational stiffnesses in Kxx thus falls off by a similar factor for a given increase in separation. In microgravity, a relatively small stiffness may be all that is necessary to pin together two modules if agility is not required. An interface comparable to the magnet and YBCO used in our experiment might allow inter-module separations of two to three centimeters or more. However, a larger stiffness is desirable

0 5 10 15 20 25 30 35-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1


Stif

fnes

s (N⋅m

/rad)

0° tilt

2° tilt

5° tilt

10° tilt

20° tilt

Figure 9. K44 (mapping θx → τx) versus separation distance d for several magnet tilt angles β. Error bars reflect a ±1° angular repeatability of the robot arm.

0 5 10 15 20 25 30 35100

101

102


Stif

fnes

s (N

/m)

0° tilt

2° tilt

5° tilt

10° tilt

20° tilt

Figure 8. K33 (mapping z → Fz) versus separation distance d for several magnet tilt angles β. Error bars reflect the stated ±0.5 mm position repeatability of the robot arm.


12

for robustness and may require module separations under a centimeter. These figures suggest a possibility for increasing the stiffness of a connection involving a single magnet and

superconductor. We generally observe that the stiffness increases with increasing tilt angle. It appears that β scales the amplitude of the exponential decay. Unfortunately, due to limitations of the experimental setup and our desire to keep the tilt and separation parameters as independent as possible, we do not have a full data set at high tilt angles.

The measured rotational stiffness coefficients K44 and K55 do not exhibit clear trends with separation distance. In fact, it appears that the measured rotational stiffness data has too much scatter to usefully characterize Kθθ(d, β). We attribute this scatter to angular imprecision in the robot. However, the elements of Kθθ may not be as crucial as the elements of Kxx, since a system of modules linked by magnets and superconductors will likely involve several flux pinned pairs and clever arrangements of these pairs could produce much larger rotational stiffness than a single magnet could.

0 5 10 15 20 25 30 35-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02


Stif

fnes

s (N⋅m

/rad)

0° tilt

2° tilt

5° tilt

10° tilt

20° tilt

Figure 10. K55 (mapping θy → τy) versus separation distance d for several magnet tilt angles β. Error bars reflect a ±1° angular repeatability of the robot arm.

5 10 15 20 25 300

20

40

60

80

100

120

140

160

180


Stif

fnes

s (N

/m)

One magnet (full data)One magnet (key points)Two magnets stuck togetherThree magnets stuck together

Figure 11. K33 (mapping z → Fz) versus separation distance d when a number of identical permanent magnets was stacked into a longer cylinder (N = 1, 2, and 3) at tilt β = 0°.

N = 1

N = 2

N = 3


13

We found that increasing the dipole moment of the magnet by increasing N tended to scale up the stiffness coefficients. Figure 11 shows data from Fig. 8 with points for N = 1, 2, and 3 at d = 10, 18, and 30 mm superimposed. The points at these three heights sketch the decaying curve for each value of N and clearly show that an increase in magnetic field strength will increase the stiffness K33. The 1/e decay length of an exponential fit to these data remains approximately the same with varying N, ranging from 5.8 mm to 7.7 mm. A similar relationship holds for both K11 and K22. We note that as N increases much beyond 3, the permanent magnet can be less readily modeled as a dipole.

Since the flux pinning interaction we wish to exploit depends on the magnetic field penetrating the superconductor volume, a possible metric for evaluating the combined effect of d, β, and N on K is the magnetic flux density in the superconductor. We used a laboratory gauss meter to measure the flux density at a point on the YBCO surface directly beneath the magnet for each (d, β, N) set in the experimental parameter space. When all our data for, e.g., K33, is plotted against the corresponding flux density, it becomes apparent that there is a definite correlation (Fig. 12). This relationship, if well characterized, could indeed provide a useful metric for evaluating the stiffness of a given arrangement of permanent magnets. It is particularly useful for simulation and design: if stiffness is quantifiably related to flux density, we can use existing magnetic modeling software to optimize a flux pinned interface without the need to simulate microscopic current flow or quantum phenomena in the superconductor. A linear fit seems to match this data quite well for fluxes below ~1 kG, but at higher flux densities, our data exhibits much more scatter. Further investigations may allow us to better characterize flux pinning stiffness at higher magnetic fields.

As a step toward establishing design criteria for flux pinned modular spacecraft, we used this flux-to-stiffness relationship to estimate how massive a magnet would be necessary to maintain two modules at some arbitrary separation distance and stiffness. A conversion from flux density into magnet mass for a magnet of fixed radius (0.95 cm, matching the magnet in our experiments) and arbitrary height was obtained from a computer simulation.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

50

100

150

200

250

300

Flux density at YBCO surface (kG)

Stif

fnes

s (N

/m)

N = 1; h = 5-30 mm, β = 0°

N = 1; h = 5-30 mm, β = 2°

N = 1; h = 5-30 mm, β = 5°

N = 1; h = 5-10 mm, β = 10°

N = 1; h = 5-10 mm, β = 20°

N = 1-3; h = 10 mmN = 1-3; h= 18 mmN = 1-3; h = 30 mmLinear fit

Figure 12. Stiffness K33 (z → Fz) versus measured flux density in kilogauss over all experimental trials. Note that the leftmost marker in all three series with fixed d repeats a data point; this was accounted for in the fit. Error bars reflect the stated ±0.5 mm position repeatability of the robot arm.


14

Figure 13 shows the mass of the magnet needed to maintain two modules at a desired separation distance for several arbitrary stiffnesses, holding β = 0° constant such that only d and m·N determine the flux density. This figure is promising for the concept of flux pinned modules: at small separations (5 mm), stiffnesses as high as 150 N/m can be attained with less than 10 g on the spacecraft bus taken up by flux-pinning interface hardware. Figure 13 also suggests that separation distances of 2 cm or more could easily be achieved using a magnet twice the size of the one in our experiments; that connection could have a stiffness of about 50 N/m. It is unclear from the figure whether or not there is a limit to the range of a flux pinned connection: the K33 ≈ 150 N/m contour may be approaching an asymptote near d = 2 cm, but the K33 ≈ 50 N/m line is not obviously approaching a particular value of h. The assumptions about magnet size and shape that led to Fig. 13 are particularly intriguing to consider. That chart applies only to a single magnet of radius 0.95 cm, where “magnet mass” and thus flux density is increased by increasing the length of the magnet. This arrangement is likely not the optimal one for maximizing flux density in the superconductor volume for the indicated separation distances. A more complicated array of magnets might allow much higher stiffnesses for fixed separation and magnet mass than Fig. 13 suggests. (Alternatively, a distributed configuration of magnets might allow much larger separation distances than those displayed in the figure for fixed stiffness and magnet mass.) If there are asymptotic limits to the separation distance achievable for a given magnet size and stiffness, we might be able to optimize the magnet and superconductor shapes to increase those limits.

B. Dynamic Test Results We were able to excite the four free modes of the flux-pinned pendulum system by driving the electromagnet coil with Gaussian white noise, periodic impulses, or a sine sweep. The Fourier transforms of the position output fit a linear combination of second-order polynomial transfer functions of the form

( ) ( )200

211

ωω

ωωζ −+ i

, (16)

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14

16

18

20

Fixed-radius cylindrical magnet mass (g)

Mag

net-Y

BC

O s

epar

atio

n (m

m)

K33 ~ 50 N/m

K33 ~ 100 N/m

K33 ~ 150 N/m

Figure 13. Contours of fixed stiffness for desired module separation as a function of the mass of the magnet necessary to achieve that stiffness.


15

giving frequencies and damping ratios for each mode of the flux-pinned system. A position time history where d ≈ 2.5 cm appears in Figure 14, along with a Fourier transform averaged over several impulse responses of the pendulum during a single YBCO cooling cycle. Due to the pendulum constraint, we expected to find a minimum of three unique frequency peaks corresponding to translation perpendicular to the YBCO surface, translation laterally along the YBCO surface, and tilting rotations. The tilting mode might be further resolved into two close frequency peaks, corresponding to tilt perturbed by swinging of the pendulum and tilt perturbed by twisting of the pendulum. The Fourier transform in Figure 14 exhibits five recognizable frequency peaks, indicating that an additional mode of some sort is present. The frequencies of both the pendulum swing and twist modes are near 0.25 Hz, well below all the frequencies found in this data, so the extraneous peak possibly represents a higher-order mode of the monofilament pendulum line. The fitting function from Eq. 16 gives richer information than a static experiment at a given test point. The resonant frequencies and damping ratios appear directly in the fit, and 95% confidence bounds provide an error estimate. Table 1 displays the results of the fit along with the equivalent translational stiffness ktrans and rotational stiffness krot for each frequency, assuming that mktrans /=ω and Ik rot /=ω , respectively, using the mass

0 1 2 3 4 5 6 7 8 9 100

100

200

300

400

500

6000.79254 Hz

2.4679 Hz

3.1445 Hz

5.7106 Hz 6.3177 Hz

Frequency (Hz)

FFT

mag

nitu

de

Figure 14. (Top) Time history of the pendulum response (tracking camera x voltage) to an impulse of current through the coil. (Bottom) Averaged Fourier spectrum of several impulse responses, with fit.


16

(22 g) and inertia (227 g·m2 about the unconstrained tilt axes) of the pendulum bob. The equivalent stiffnesses for each mode should correspond to the appropriate elements in the full K matrix. The translational stiffnesses for the second and third peaks are consistent with the stiffnesses measured in the static experiment for displacements lateral and perpendicular to the YBCO surface at a distance of 2.5 cm (compare 5.4029 N/m and 8.7714 N/m with Figure 6 and Figure 8). These two peaks also best obey the relation that K11 = K33/2. Therefore, we interpret peak 2 as the result of translational stiffness for displacements along the YBCO surface and peak 3 as a result of stiffness for displacements towards or away from the superconductor. Furthermore, the close spacing of peaks 4 and 5 suggest that they are the result of the magnet tilting relative to the superconductor. One of the two tilt modes, which should have identical behaviors, is perturbed by the pendulum twist mode, separating the two into different modes. However, the rotational stiffnesses suggested by these frequencies are several orders of magnitude higher than those measured in our static experiments. The damping ratios for this system, extracted from the fit to the Fourier peaks, indicate substantial energy loss. Damping ratios from 0.02 to 0.04 correspond to oscillations that damp out with a time constant between four and eight oscillation periods after the impulse was delivered. This decay is shown in the upper half of Figure 14. This level of damping is present in both the translation and rotation degrees of freedom. Such a rapid decay will be very beneficial if a flux-pinned interface is used to maintain relative separations in a modular spacecraft under active control, where damping in the open-loop system would likely enhance closed-loop stability. Using the framework built up in Section IIB, these results can be expressed as the stiffness and damping matrices

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

=

⋅⋅⋅⋅

⋅⋅

⋅⋅

s/radmNs/mmN

s/radNs/mN

m/radNm/mN

N/radN/m

4042.000004306.000000149.00

0000113.0

and

3580000292000077.80

00040.5

D

K

(17)

for h = 2.5 cm, β = 0°, and N = 1. Asterisks represent elements that our pendulum technique cannot identify. However, they are not of great interest: the second row (and column) of each matrix should be redundant with the first; the sixth row (and column) corresponds to a DOF that is not affected by flux pinning. Since this analysis is directly fitted to the normal modes exhibited in the Fourier transform, these matrices are diagonal by definition.

Peak 1 Peak 2 Peak 3 Peak 4 Peak 5 Peak f, Hz 0.7925 2.4679 1.5530 5.7106 6.3177 ± 0.0023 ± 0.0649 ± 0.0256 ± 0.0563 ± 0.0059 Damping ζ 0.0411 0.0162 0.0340 0.0264 0.0224 ± 0.0031 ± 0.0344 ± 0.0084 ± 0.0131 ± 0.0131 Equivalent ktrans, N/m 0.5572 5.4029 8.7714 28.9283 35.4058 Equivalent krot, N·m/rad 5.6333 54.6290 21.6327 292.5038 358.0025

Table 1. Frequencies and damping ratios extracted from the Fourier transform fit in Figure 14. Peaks are labeled left to right from the figure.


17

IV. Conclusions Magnetic flux pinning exhibits many desirable properties for application in the space environment. It is an

action-at-a-distance force that is not subject to the limits of Earnshaw’s Theorem. Therefore, modular or formation flying spacecraft systems coupled by magnet-superconductor pairs can be subject to restoring forces and damping without any active control or mechanical contact. This interaction is a property of the bulk material of type II superconductors. While actuation of such interfaces is possible, it is not necessary to maintain the relative separations of a close-proximity formation.

We have observed trends in the properties of magnetic flux pinning that are consistent with previous investigations. The stiffness of the pinning interaction decays exponentially with the separation between the two components. Furthermore, translational stiffnesses parallel and perpendicular to the plane of the superconductor obey the simple relationship that stiffness for displacements normal to the superconductor surface is twice that for displacements parallel to it. An important distinction must be drawn between our work and that of other authors: many other researchers are interested in flux pinning for its application to superconducting bearings and magnetic levitation systems, both of which are viewed as levitation effects in the influence of gravity, while we envision flux pinning used to maintain modules in zero gravity in a 6DOF equilibrium. Our dynamic experiment is particularly important to this work in that it stresses this distinction: not only does it allow experimentation on a pinned system with as many degrees of freedom as possible without experimenting in microgravity, but it allows us to set up the pinned pair such that the magnet is not levitating in an equilibrium between flux pinning and gravity. It allows us to consider flux pinning outside of the earthbound levitation paradigm.

This dynamic technique shows great promise when compared against our static experiments. In future investigations, we will be focusing on the dynamic apparatus. We will refine our data collection and analysis to conduct more extensive system ID, and we will build up a space of experimental test points that span a range of separations, orientations, and magnet strengths, as in the static experiments. In addition, we will better characterize the rotational stiffnesses. The pendulum setup will allow us to extend the test space to include much larger separations than the robot arm allowed in our static experiments, and provide a means to investigate dynamic interactions between the magnet and superconductor (for example, using flux pinning to “capture” a magnet on the swinging pendulum).

Our data indicate that the interaction may be effective across an inter-module separation of at least several centimeters. For example, maintaining a formation in geosynchronous orbit consisting of two 1 kg modules separated in the out-of-plane direction by 3 cm requires balancing gravitational forces with a force of approximately 2×10-8 N. Introducing a modest stiffness of 1 N/m—on the order of the stiffness we have measured for a 3 cm separation—gives a restoring force of this magnitude when the modules deviate from the desired separation by less than one part in 107. This miniscule steady-state error could easily be compensated by either integral control or choosing a different equilibrium separation for the flux-pinned “spring” binding the formation together.

While not suitable for the meters- to kilometers-wide separations involved in traditional formation flight systems, flux pinning provides an ideal platform for what we refer to as non-contacting modular systems. Modules could be assembled and connected, as if by a truss or mating adapter, without coming into physical contact. Simply maneuvering the modules into coarse proximity with one another will result in forming a mechanical configuration without the need for power, active control, or environmental interactions normally associated with attitude control and propulsion. The concept eliminates plume impingement, momentum build-up, and control/structure interactions that plague large systems to be assembled in orbit, such as the International Space Station. These effects make fractionated spacecraft with flux pinning connectors easier to boost into space and mate than modular spacecraft connected in orbit by mechanical means.

Future research will concentrate on designing a viable space system using this effect. The results shown in Fig. 13 inspire two lines of inquiry. First, we will consider optimizing the shape and distribution of the magnets in a flux-pinning interface to maximize the flux density in the corresponding superconductor. We will also optimize the shape of the superconductor to enclose a volume of maximum flux. Second, experimental investigations of stiffness at high flux densities will shed light on the behavior of the contours in Fig. 13 as magnet strength is increased. These investigations will allow us to determine how far we can extend the effective range of flux pinning, which has strong implications for the feasibility of this technology for spacecraft architectures such as sparse-aperture telescopes or missions involving formation flight. Another potentially useful line of investigation is to explore the effects of electromagnets on flux pinning. For instance, an electromagnet near a flux-pinned pair might be used to actuate the interface by affecting the distribution of magnetic fields in the system. In addition, an electromagnet of sufficiently high strength could itself be flux pinned rather than a permanent magnet. Altering the field of a previously pinned electromagnet might produce


18

effects in the stiffness, damping, and equilibrium separation of the flux pinned interface, allowing another means of actuation. Coupled with the hysteretic nature of flux pinning, this provides a possible avenue for reconfiguration of non-contacting modular systems. Reconfiguration could also be enabled by exploiting the thermal requirements of superconductivity to turn a flux-pinned interface “on” and “off” by allowing the temperature of the superconductor to vary above and below its critical temperature. We envision flux-pinned interfaces becoming a standard component of future spacecraft architectures. The scope of potential applications ranges from next-generation mating adapters that align modular components in a robust and safe way prior to final docking to large, segmented arrays of telescopes or instruments held in place ten centimeters apart by a flux-pinned “structure.” In the far field, it is even possible that arrays of flux-pinned modules could form a truss-like substrate for even larger assemblies, allowing flux pinning to become a major structural element of orbiting satellites and stations. Manipulator arms that grasp payloads without contacting them, eliminating the need to find or design any mechanical attachment points on the physical structure of the payload, may be a near-term result of this research. With the mechanical properties of flux-pinned interfaces in hand, we can begin to develop design parameters for such next-generation systems.

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Conference and Exhibit, 28-30 September 2004, San Diego, California, AIAA Paper 2004-5878. 2 Brown, O. & Eremenko, P., “The Value Proposition for Fractionated Space Architectures,” AIAA-2006-7506, AIAA

Space 2006, San Jose, CA, 2006. 3 Brown, O. & Eremenko, P., “Fractionated Space Architectures: A Vision for Responsive Space,” AIAA-RS4-2006-1002,

4th AIAA Responsive Space Conference, Los Angeles, CA, 2006. 4 Kramer, E. J., “Scaling laws for flux pinning in hard superconductors,” Journal of Applied Physics., Vol. 44, No. 3, 1973,

pp. 1360-1370. 5 Brandt, E. H., “Rigid levitation and suspension of high-temperature superconductors by magnets,” American Journal of

Physics, Vol. 58, No. 1, 1990, pp. 43-49. 6 Brandt, E. H., “Levitation in Physics,” Science, Vol. 243, No. 4889, 1989, pp. 349-355. 7 Moon, F. C., Superconducting Levitation, Wiley, New York, 1994. 8 Davis, L. C., “Lateral restoring force on a magnet levitated above a superconductor,” Journal of Applied Physics, Vol. 67,

No. 5, 1990, pp. 2631-2636. 9 Schonhuber, P. and Moon, P. C., “Levitation forces, stiffness, and force-creep in YBCO high-Tc superconducting thin

films,” Applied Superconductivity, Vol. 2, No. 7, 1994, pp. 523-534. 10 Norman, M. C. and Peck, M., “Characterization and modeling of a modular satellite network,” AAS/AIAA Astrodynamics

Specialist Conference, Washington, D.C., 2007 (to appear). 11 Johansen, T. H., et al., “Lateral force on a magnet placed above a planar YBa2Cu3Ox superconductor,” Applied Physics

Letters, Vol. 179, No. 2, 1991, pp. 179-181. 12 Eshed Robotec Ltd., “SCORBOT-ER Vplus User’s Manual,” 1998, p. 2-2. 13 Hull, J. R. and Cansiz, A., “Vertical and lateral forces between a permanent magnet and a high-temperature

superconductor,” Journal of Applied Physics, Vol. 86, No. 11, 1999, pp. 6396-6404. 14 Cansiz, A., “Correlation between free oscillation frequency and stiffness in high temperature superconducting bearings,”

Physica C, Vol. 390, 2003, pp. 356-362. 15 Sugiura, T., et al., “Parametrically excited horizontal and rolling motion of a levitated body above a High-Tc

Superconductor,” IEEE Transactions on Applied Superconductivity, Vol. 13, No. 2, 2003, pp.2247-2250. 16 Basinger, S. A. et al., “Amplitude dependence of magnetic stiffness in bulk high-temperature superconductors,” Applied

Physics Letters, Vol. 57, No. 27, 1990, pp. 2492-2494. 17 Johansen, T. H. and Bratsberg, H., “Theory for lateral stability and magnetic stiffness in a high-Tc superconductor-magnet

levitation system,” Journal of Applied Physics, Vol. 74, No. 6, 1993, pp. 4060-4065. 18 Johansen, T. H. et al., “Magnetic levitation with high-Tc superconducting thin films,” Journal of Superconductivity, Vol.

11, No. 5, 1998, pp. 519-524.

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