2013 IEEE International Conference on Robotics and Automation (ICRA) Karlsruhe, Germany, May 6-10, 2013 An Omnidirectional Electromagnet for Remote Manipulation Andrew J. Petruska and Jake J. Abbott Abstract— An Omnimagnet is an omnidirectional electro- Outer Solenoid magnet comprised of a ferromagnetic core inside of three Middle Solenoid L orthogonal nested solenoids. It generates a magnetic dipole-field J with both a variable dipole-moment magnitude and orientation with no moving parts. The design of an Omnimagnet, in which W+2T each solenoid has the same dipole moment and minimizes the Core differences between each of the solenoid’s fields, is provided and W optimized for strength by tailoring the size of the spherical core R W used to amplify the solenoids’ field. This design is then analyzed c W+2T using FEA tools and shown to be dipole-like in nature. Various magnetic control methods are then motivated by providing the necessary equations relating the three applied currents to x applied field, torque, or force on an adjacent magnetic device. y z Finally, the optimal design is constructed and its utility is demonstrated by driving a helical capsule endoscope mockup through a transparent lumen. Inner Solenoid Assembled Omnimagnet I. INTRODUCTION Fig. 1. The assembled Omnimagnet forms a cube constructed of three Magnetic microscale and mesoscale devices (both tethered nested orthogonal solenoids surrounding a spherical core of ferromagnetic material. Each square-cross-section solenoid has a different inner width W , and untethered) can be manipulated with an externally gen- winding thickness T , length L, and associated current density J. erated magnetic field, which applies a combination of force and torque to the device without any mechanical connection. Although a combination of permanent magnets and electro- associated with traditional electromagnets and the control of magnets can be used to produce the magnetic field required dipole orientation associated with rotating permanent mag- for a manipulation task, some tasks seem better suited to nets. In this paper, we introduce a class of omnidirectional either permanent magnet or electromagnet systems. Because electromagnet that we call an Omnimagnet, comprised of a they have more direct real-time control of the applied mag- ferromagnetic core surrounded by three orthogonal solenoids, netic field, electromagnet systems have been used for multi- which is able to create a dipole-like field in any orienta- degree-of-freedom levitation and position/orientation control tion. Specifically, we consider an Omnimagnet comprised [1]–[4]. Permanent magnets, which require no electrical of a spherical core and square-cross-section solenoids (Fig. power to generate a strong field, have been used for pulling 1), but other design variations could be considered. An and rolling tasks in which the environment provides some Omnimagnet contains no moving parts, and when powered structure [5]–[8], as well as for quasistatic pointing tasks down becomes inert, reducing the safety concerns associated of tethered devices such as magnetic catheters [9]. Because with permanent-magnet field sources. The concept of three both attractive and lateral forces can be generated between nested solenoids has been explored as a method of inductive a rotating dipole source and a sympathetically rotating mag- power coupling [11], but never as a dipole-like magnetic- netic device, a rotating dipole field could be more effective field source, and never with a spherical core. for rolling/screwing propulsion than the rotating uniform field generated by many electromagnet systems [10]. Finally, The paper is structured as follows. First, the general design it is challenging to scale many laboratory electromagnetic problem for an Omnimagnet is presented. Next, the magnetic systems that surround their workspace (e.g., Helmholtz coils) fields generated by the three solenoids are described using to a size that would be required for medical applications, a multipole expansion of the magnetostatic equations, and whereas manipulation systems that utilize dipole fields can the contribution of the ferromagnetic core is quantified. The be located adjacent to their workspace. optimization of a specific Omnimagnet follows, and the An omnidirectional electromagnet, formed by any set of design is described. The field generated by this design is then collocated electromagnets that have dipole moments span- compared to a dipole-field approximation, and the inverse 3 solution for determining the dipole moment, and thus the ning R , combines the real-time control of field strength currents, required to produce a desired static or rotating field, This material is based upon work supported by the National Science torque, or force at a given location are provided. Finally, Foundation under Grant Nos. 0952718 and 0654414. The authors are with the Department of Mechanical Engineering, the capability of the Omnimagnet for the control of capsule University of Utah, Salt Lake City, 84112. fandrew.petruska, endoscopes is then demonstrated, and future research plans [email protected] are discussed. 978-1-4673-5642-8/13/$31.00 ©2013 IEEE 814 II. OMNIMAGNET DESIGN AND OPTIMIZATION A. Solenoid Multipole Field Expansion For positions outside of the Omnimagnet’s minimum- The general concept of an Omnimagnet is broad, con- bounding sphere (i.e., the smallest sphere that the Omnimag- sisting of three orthogonal nested solenoids surrounding a net can fit within), the solenoid fields can be represented by ferromagnetic core; however, design choices must be made a multipole expansion of a vector potential [12]: to realize and optimize a physical Omnimagnet. First, we B (p) = r × Ψ (p) (1) chose the shape of the solenoids to be square-cross-sectional sleeves to result in a dense packing (see Fig. 1). Next, we where 1 Z chose the core to be a sphere because a spherical core has µ0 X 1 n Ψ (p) = J (r) krk P (p^ · ^r) dV (2) three desirable properties: 4π n+1 n n=0 kpk Vs • A sphere does not have a preferential magnetization −7 −1 direction. where µ0 = 4π × 10 T·m·A is the magnetic perme- ability of free space, p is the vector (with associated unit • When placed in a uniform field (similar to the field in the center of a solenoid), a sphere produces a pure vector p^) from the center of the Omnimagnet to the point point-dipole field [12]. of interest in units fmg, r is the vector (with associated unit vector ^r) from the center of the Omnimagnet to the • The average applied magnetic field within a sphere is equal to the applied magnetic field at the center of the point in the solenoid being integrated, J (r) is the current sphere [12]. density vector that points in the direction of the current flow, Vs represents the solenoid’s volume, and Pn () are the We chose that the dipole moment generated in each direction, Legendre polynomials. Since the divergence of a magnetic which consists of the contribution of both an individual field through a closed surface must be zero, all of the even solenoid and the magnetization of the core due to that terms (those corresponding to P0, P2, :::) must be zero, solenoid, should be the same when an equal electrical current leaving only the odd terms. The first non-zero term in the density is applied through each solenoid. Other geometric multipole expansion (corresponding to P1) is the dipole field, design choices (e.g., cylindrical solenoids or a cubic core) or which can be expressed in a coordinate-free form as: dipole-moment relationships (e.g., scaling the dipole moment µ0 T B (p) = 3p^p^ − I m (3) of each solenoid with its heat-transfer capability), could also 4π kpk3 be pursued using the general framework for Omnimagnet where I is a 3×3 identity matrix and m is the dipole moment design outlined below. Finally, we constrain our design to 2 use a single wire gauge for all solenoids, which means that in units fA·m g. “the same current density” is synonymous with “the same The dipole moment for a current density of any configu- current”; current and current density are related by the cross- ration is [12]: 1 Z sectional area of the wire used. Throughout this paper, I will m = r × J (r) dV (4) be used to refer to currents in unitsfAg and J will be used to 2 Vs refer to current density in units fA·m−2g. Because current The dipole moment for a square-cross-section solenoid as density is invariant to wire selection, the optimization for shown in Fig. 1 with uniform current density (i.e., the current shape is performed using J; general discussion, however, density does not vary along the thickness or length of the will use I, as it is the more natural parameter from a control solenoid) is: perspective. The final design of the Omnimagnet shown in JL4 m = β3 − β3^l (5) Fig. 1 requires ten total constraints (the length, width, and 6 2 1 thickness of each solenoid, and the radius of the core). where J is the magnitude of J, L is the axial length of the ^ The magnetic field generated by the Omnimagnet can solenoid (with associated axial unit vector l), and β1 = W=L be represented by the field contributed by the magnetized and β2 = (W + 2T ) =L respectively describe the inner- spherical core superimposed with the field contributed by the width-to-length and outer-width-to-length aspect ratios. solenoids. Modeling the total field can be performed using The maximum dipole moment that any electromagnet FEA tools with a resolution limited by the number of ele- with a bounding cube of edge length L containing no ments used. Alternatively, an analytical dipole approximation ferromagnetic material could generate in one direction can 4 can be used to model the field.