Magnetic equilibrium postures of a multibody magnetic microrobot composed of a spherical magnet chain

Transcription

1 Microsyst Technol : DOI /s Technical Paper Magnetic equilibrium postures of a multibody magnetic microrobot composed of a spherical magnet chain Seungmun Jeon Gunhee Jang Jaewang Nam Received: 10 October 2013 / Accepted: 8 March 2014 / Published online: 6 April 2014 Springer-Verlag Berlin Heidelberg 2014 Abstract Multibody magnetic microrobots composed of multiple magnetic bodies have been widely investigated to achieve complex mechanical motions in various biomedical applications. This research proposes a methodology to simulate the magnetic equilibrium postures of a microrobot composed of a spherical magnet chain MSMC under various external magnetic fields, non-magnetic torques and forces. An equivalent point-dipole model is used to precisely and effectively calculate the magnetic interactions within the MSMC, and geometric and mechanical characteristics of the MSMC are investigated to describe and calculate the geometric deformations of the MSMC with minimum variables. We also formulate nonlinear constraint equations for the equilibrium posture of the MSMC, and then develop an optimized solution procedure for those equations. We finally examine the simulated and experimental results of the magnetic equilibrium postures of MSMCs under different conditions to verify the proposed method. 1 Introduction The use of microrobots has been widely investigated as a promising alternative technology in various biomedical applications Nelson et al. 2010, Peyer et al Microrobots are expected to navigate through complex environments, such as the pulsatile flows in human blood vessels, and to perform various tass that cannot be performed with S. Jeon G. Jang * J. Nam PREM, Department of Mechanical Engineering, Hanyang University, 17 Haengdang dong, Seongdong gu, Seoul , Republic of Korea ghjang@hanyang.ac.r conventional technologies, such as unclogging clogged blood vessels or targeted drug delivery Hosseini et al. 2011; Zhang et al Magnetic microrobots composed of magnetic materials have especially received attention because of their battery-free and wireless manipulation capabilities. Since the manipulating power is transferred from the external magnetic field generated by a magnetic navigation system MNS, magnetic microrobots can be simplified and miniaturized to be applied in complex environments. Various mechanisms have been investigated to achieve different mechanical abilities of the magnetic microrobot. Single-body magnetic microrobots SMMs composed of a single magnet have been widely investigated because of their structural and mathematical simplicity. Jeon et al showed that a SMM with a transverse magnetization and screw thread can helically swim or translate in a fluidic environment by using a rotating magnetic field and linear magnetic gradient. Hou et al showed that a SMM with an axial magnetization can move along a surface with a rolling locomotive mechanism generated by a three-dimensional rotating magnetic field. Since these SMMs are limited to simple mechanical motions, several researchers have investigated multibody magnetic microrobots MMMs composed of multiple magnetic bodies to generate more complex mechanical motions. Benosi et al showed that a MMM composed of self-assembled ferromagnetic particles can beat lie artificial cilia, which can be driven by an external magnetic field. Kim et al showed that a MMM composed of several magnets and connecting structures can generate snae-lie crawling locomotion under a rotating magnetic field. These motions are generated by maneuvering magnetic equilibrium postures of the MMM under various external magnetic fields. However, previous researchers

2 1472 Microsyst Technol : Fig. 1 a Schematic view of the MSMC composed of 5 spherical magnets. Equilibrium postures of the MSMC under b gravitational effects and c an oscillating magnetic field only roughly and experimentally predicted the magnetic equilibrium postures of MMMs; effective methodologies to simulate the equilibrium postures have not been investigated thoroughly due to the complex magnetic interactions within the structure. Since the size of general MMMs ranges from nanometers to millimeters Nelson et al. 2010; Dreyfus et al. 2005, an effective methodology to analyze the magnetic equilibrium postures of the MMMs would be useful for the design and manipulation of MMMs in various applications. We propose an effective methodology to simulate the equilibrium postures of a microrobot composed of a spherical magnet chain MSMC such as that shown in Fig. 1. We use an equivalent point-dipole model to precisely and efficiently calculate the magnetic torques and forces of the MSMC. Geometric and mechanical characteristics of the MSMC are also investigated to describe and calculate its geometric deformations with a minimum number of variables. We then formulate nonlinear constraint equations for the equilibrium posture of the MSMC, and develop an optimized solution procedure for those equations. We finally conduct several experiments and compare the experimental results with those of the simulations under various conditions to verify the proposed methods. 2 Basic characteristics of the MSMC 2.1 Mechanically continuous structure A MMM is a mechanical structure composed of multiple magnetic bodies, and the MSMC is one type of MMM. If a MMM is composed of permanent magnetic materials, it can self-assemble and form a magnetic equilibrium posture due to the inherent internal magnetic interactions. The equilibrium posture of the MMM may be changed under external forces or magnetic fields. By controlling external magnetic fields, MMMs can be maneuvered with multi-degree-offreedom motions such as a cilia-lie beating, fish tail-lie oscillating, or snae-lie crawling locomotion Benosi et al. 2010; Guo et al. 2008; Kim et al This research investigates the motion of the MSMC. Although different shapes of magnets such as cubes, bars, spheres, etc. can form a chain, spherical magnets are especially useful from geometric and mechanical points of view, as follows. The relative distance between each adjacent magnet of the MSMC can always remain constant without mechanical linages due to the geometry of the sphere, and the MSMC can deform its equilibrium postures continuously under varying conditions. Thus, the equilibrium postures and mechanical characteristics of the MSMC can be analytically determined. 2.2 Simplified magnetic dipole model Generally, a magnet is composed of numerous atomic magnetic dipoles, and each magnetic dipole generates a magnetic field that can be specified by the point-dipole model as follows Cullity and Graham 2009: B point-dipole = µ 0 4π 3rm r r 5 m r 3 where μ 0, m, r, and r are the permeability of free space, the magnetic moment of the dipole, the vector from the dipole to a point, and the magnitude of that vector, respectively. When the field of interest is relatively far, a magnet can be idealized as a single dipole regardless of its shape Cullity and Graham This idealization cannot be simply applied to the case where multiple magnetic bodies are placed close to each other, such as in the MSMC. Assuming that a magnet is composed of infinite numbers of magnetic dipoles as shown in Fig. 2a, the magnetic field generated by the magnet at point q can be expressed as follows: B magnet = µ 0 4π V where M p, V, r pq, and r pq are the magnetization of the pth dipole, the volume of the magnet, the vector from the pth dipole to the point q, and the magnitude of that vector, respectively, as shown in Fig. 2. Although the magnetic field generated by a magnet with an arbitrary shape can be precisely calculated by using Eq. 2 with a volumetric integral, it is computationally inefficient to use Eq. 2 in calculating the magnetic interactions within a MSMC 3r pq M p r pq rpq 5 M p rpq 3 dv 1 2

3 Microsyst Technol : Fig. 2 a A spherical magnet composed of infinite numbers of dipoles and b its equivalent point-dipole model Fig. 3 Identical magnetic fields generated by a a spherical magnet with a unit radius and b its equivalent point-dipole considering the same magnetic moment composed of multiple magnets with arbitrary shapes. However, a spherical magnet with uniform magnitude of magnetization throughout the body can be idealized as a single dipole, as shown in Fig. 2b, due to the spatial geometric symmetry of the sphere regardless of the distance from the interested field Cullity and Graham Figure 3 shows the almost identical magnetic fields calculated by Eqs. 1 and 2. Therefore, magnetic interactions within the MSMC can be simply analyzed by using the point-dipole model, provided that the spherical magnets are uniformly magnetized throughout the body. 3 Magnetic equilibrium postures of the MSMC 3.1 Geometric configuration of the MSMC The magnetic torque and force of a magnet in a magnetic field can be expressed as follows Cullity and Graham 2009: T m = m B F m = m B where B is the applied magnetic field. We define an initial equilibrium posture that is only determined by the magnetic interactions within the MSMC. It is physically straightforward that the initial equilibrium posture of the 3 4 MSMC is similar to a straight chain, as shown in Fig. 1a. From Eqs. 1, 3, and 4, the MSMC at the initial equilibrium posture undergoes zero magnetic aligning torque and the strongest attractive magnetic force. Assuming that the attractive magnetic forces within the MSMC are relatively strong compared to external torques and forces, it can be assumed that the magnets of the MSMC roll along the surface of adjacent magnets without slipping when the MSMC deforms, as shown in Fig. 4a. Under this assumption, the geometric deformation of the MSMC can be simply expressed with a minimum number of variables. When a MSMC is composed of n spherical magnets, general equilibrium postures of the MSMC can be expressed with n variables which are the deflection angles ϕ between the magnetization direction of the 1th magnet and the line between the centers of the 1th and the th magnets, as shown in Fig. 4a. The magnetization vector of the th magnet and the position vector from the jth magnet to the th magnet of the MSMC can be expressed as follows: m = m [ cos Ψ sin Ψ 0 ] T for Ψ = 2 ϕ i i=1 r j = r r j for r [ = ρj 1 + ρ j cos ] T Ψ j 1 + ϕ j sin Ψj 1 + ϕ j 0 j=1 6 5

4 1474 Microsyst Technol : Fig. 4 a Magnetization vector and position vector of the th magnet of the MSMC. b Magnetic, non-magnetic, and reaction forces of the th magnet of the MSMC where m, r j, and ρ j are the magnitude of the magnetic moment of the th magnet, the vector from the jth magnet to the th magnet, and the radius of the jth magnet, respectively. 3.2 Formulation of constraint equations From Eqs. 1, 3, and 4, the expanded equations of magnetic torque and force exerted on the th magnet of the MSMC in an external magnetic field can be expressed as follows, after some mathematical manipulations: T m = n j=1, j = µ 0 3m r j m j r j 4π rj 5 m m j rj m B e n F m = 3µ 0 mj r j m + m r j mj + m j m rj 5 m j r j m r j r j +m B e j=1, j = 4πr 5 j F centerline = N N +1 cos ϕ +ϕ +1 R +1 sin ϕ +ϕ +1 + F m + Fnm ˆr 1 = 0 T contact point where T center, T contact point, F centerline, T nm, F nm, R, N, and ˆr 1 are the net torques of the th magnet with respect to the center and the contact point between the 1th and th magnets, the net force of the th magnet along the magnet centerline between the 1th and th magnets, non-magnetic torque and force exerted on the th magnet, tangential and normal reaction forces at the contact point r 2 j = ρ ˆr 1 F m + z Fnm z + T m ρ R cos ϕ +ϕ +1 +N +1 sin ϕ +ϕ +1 = 0 10 z 11 8 where r j and B e are the magnitude of r j and the applied external magnetic field, respectively. When external magnetic fields or non-magnetic torques and forces such as a friction torque or gravitational force are applied to the MSMC, it deforms into a new equilibrium posture where the net torque and force of each magnet of the MSMC are zero. Assuming that the magnetization vectors of the MSMC lie in the xy-plane, the following constraint equations should always be satisfied for the equilibrium posture of the MSMC. T center z = ρ R + R +1 + T m + Tnm z = 0 9 between the 1th and th magnets, and the unit vector of r -1, respectively. z represents the z-directional component of the vector inside the parentheses. Assuming that the magnet at the right end of the MSMC is the nth magnet, Eqs. 9 and 10 can be explicitly expressed as follows: { T m R = + T nm z /ρ for = n R T m + Tnm z /ρ for = n F m + Fnm ˆr 1 for = n N = N +1 cos ϕ +ϕ +1 + R +1 sin ϕ +ϕ +1 F m + Fnm ˆr 1 for = n 13

5 Microsyst Technol : From Eq. 12, all the values of R can be sequentially calculated bacward R n, R n-1, R n-2, and so on by first calculating the value of R n. From Eq. 13, all the values of N can be sequentially calculated in a similar manner by calculating the value of N n using the values of R. Thus, the nonlinear constraints equations in Eqs composed of 3n equations and 3n unnown variables ϕ, R, and N can be effectively reduced to the following Eq. 14 composed of only n equations and n unnown variables ϕ of an x-directional Helmholtz coil, a y-directional uniform saddle coil, and a z-directional uniform saddle coil so that three-dimensional uniform magnetic fields can be effectively generated and controlled Jeon et al. 2010, Two types of spherical magnets with radii of 2 and 2.5 mm are used for the MSMC. The magnetization of the magnets is 955,000 A/m, and the mass density of the magnets is 7.5 g/cm 3. T contact point = z { ρ ˆr 1 F m + Fnm z + T m z = 0 for = n ρ ˆr 1 F m + Fnm z + T m z ρ R cos ϕ +ϕ +1 + N +1 sin ϕ +ϕ +1 = 0 for = n 14 Therefore, the nonlinear problem for the magnetic equilibrium postures of the MSMC can be formulated with the minimum number of constraint equations and unnown variables. 3.3 Optimized solution procedure Since the obtained constraint equation is a strongly coupled nonlinear problem with respect to ϕ, an optimized solution technique can be used to effectively solve the problem. We developed a numerical procedure to solve the nonlinear constraint equations, as shown in Fig. 5. This procedure can be divided into two processes: an equilibrium constraint problem solving process P1 and an optimized nonlinear problem solving process P2. During P1, when the external magnetic field, non-magnetic torques and forces, and the initially assumed values of Φ initial = [ ϕ 1 ϕ 2... ϕ n ] T are given, the variables for the constraint equations in Eq. 11 are calculated, and the values of the constraint equations are compared to the given termination limit of error e. If the values of the constraint equations are greater than e, P1 forwards the values to P2. During P2, an optimization algorithm that effectively reduces the number of iterations and computation time calculates the values of Φ using the forwarded values, and then returns the values to P1. P1 and P2 can be repeated until the calculated values of the constraints equations become smaller than the given termination limit of error. 4 Results and discussion 4.1 experimental setup In this research we constructed an experimental setup to measure the deflections of the MSMC under various conditions in order to verify the proposed methodology, as shown in Fig. 6. The MNS shown in Fig. 6 is composed 4.2 Deflections of the MSMCs under gravitational forces We performed simulations and experiments on various equilibrium postures of the MSMC. Since the MSMC can be deflected by gravitational forces, we first measured the deflections of the MSMCs under gravitational forces only. The deflections of the MSMCs are measured assuming a fixed boundary condition of the left end of the MSMC, as shown in Fig. 7. We also simulated the deflections of the MSMCs by using the technique shown in Sect In the simulation, we assumed that the gravitational forces act vertically along the central axis of each magnet, and the trust-region dogleg method is used for the optimized nonlinear solver from MATLAB Moré The termination limit of errors was set to 1e 20, and the initially assumed values of Φ initial were set to zero. Figure 7 shows the overlapped images of the measured and simulated deflections of the MSMCs composed of 7 and 9 magnets, and the results match well within a reasonable range. The differences between the measured and simulated results may result from the unequal magnetizations of each magnet and uneven magnetizations throughout the magnet. As shown in Fig. 7, the deflection of the MSMCs tends to increase when the number of magnets increases. The MSMCs seem to stay relatively flexible as the number of magnets increases. This result is analogous to the structural beam, in which the deflection increases as its longitudinal length increases. Additionally, the MSMCs with a radius of 2.5 mm show relatively large deflections compared to the MSMCs with a radius of 2 mm. This result can be explained by Eqs. 7 and 8, which state that the interactive magnetic torques within the MSMCs are proportional to the cube of its size, while the magnetic forces are proportional to the square of the size once the magnetization eeps the same value. Considering that the gravitational force is proportional to the cube of the size, the attractive magnetic forces within the MSMCs become relatively wea as the size of the MSMCs increases. Therefore, we conclude that the relative magnitude of the deflection of a MSMC increases under the

6 1476 Microsyst Technol : Fig. 5 Nonlinear problem solving procedure to calculate the equilibrium posture of the MSMC Fig. 6 MNS to test the deflections of the MSMCs under various external magnetic fields Fig. 7 a MSMC with a fixed boundary condition of the left end. Measured and simulated deflections of the MSMCs composed of b 7 magnets with radius of 2 mm, c 7 magnets with radius of 2.5 mm, d 7 magnets with radius of 2 mm, and e 9 magnets with radius of 2.5 mm under gravitational forces only

7 Microsyst Technol : Fig. 8 Measured and simulated deflections of the MSMCs composed of 7 magnets under gravitational forces and the uniform magnetic field applied along a 90, b 45, c 0, d 45, and e 90 in the xy-plane with a magnitude of 3 mt Fig. 9 Measured and simulated deflections of the MSMCs composed of 7 magnets under gravitational forces and the uniform magnetic field applied along a 90, b 45, c 0, d 45, and e 90 in the xy-plane with a magnitude of 14 mt

8 1478 Microsyst Technol : gravitational effect as the size and number of magnets in the MSMC increase. 4.3 Deflections of the MSMCs under gravitational forces and uniform magnetic fields When an external magnetic field is applied, MSMCs can deform along the applied field. We measured the variations of the deflections of the MSMCs under various uniform magnetic fields with different directions and magnitudes generated by the MNS shown in Fig. 6. Figures 8 and 9 show overlapped images of the measured and simulated deflections of the MSMC composed of 7 magnets under gravitational forces and external magnetic fields with magnitudes of 3 and 14 mt varying within the xy-plane, respectively. The figures show that the proposed simulation method is also valid when non-magnetic forces gravitational force and external magnetic fields are applied to the MSMC at the same time. They also show that the deflection geometry of the MSMCs changes with the direction and magnitude of the external magnetic field. Figures 8e and 9d show almost identical deflections of the MSMC, even though the direction and magnitude of the applied field are different. This demonstrates that the same equilibrium postures can be obtained by controlling either the magnitude or direction of the external magnetic field. When an external magnetic field oscillating in the xy-plane is applied to the MSMCs, the MSMC can generate the repeated deflection motions shown in Figs. 8 and 9, which can be used to generate beating motion lie a fish tail or crawling motion lie a snae. We observed that the MSMC can generate repeated oscillating motion synchronized to the external oscillating magnetic field used in Figs. 9a e within the oscillating frequency range of 0 10 Hz. The inertial effect due to the oscillatory motions of the MSMC at relatively high frequencies may be responsible for the non-synchronization. Therefore, the magnetic equilibrium postures and mechanical motions of the MSMCs under various conditions can be effectively utilized by using the proposed simulation method. 5 Conclusions In this study, we proposed a method to effectively simulate the magnetic equilibrium posture of a MSMC. The pointdipole model was used to precisely and effectively calculate the magnetic interactions within the MSMC. We investigated the geometric and mechanical characteristics of the MSMC to describe and calculate its deformation with minimum variables. We also formulated a nonlinear problem for the equilibrium postures of the MSMC, and developed an optimized nonlinear solution procedure to solve the problem under various external magnetic fields, torques, and forces exerted on the MSMC. We then conducted several experiments and simulations to verify the proposed method. This investigation can contribute to the multi-scale and multi-degree-of-freedom utilization of the MSMC in various biomedical applications. The findings of this study can be further extended to the study of structural stiffness characteristics of MSMCs composed of various numbers and sizes of spherical magnets, so that the static, dynamic, and vibrational characteristics of the MSMC can be utilized for various micro-robotic applications. Acnowledgments This wor was supported by a National Research Foundation of Korea NRF grant funded by the Korean government MEST No. 2012R1A2A1A01. References Benosi JJ, Deacon RM, Land HB et al 2010 Dipolar assembly of ferromagnetic nanoparticles into magnetically driven artificial cilia. Soft Matter 6:602 Cullity BD, Graham CD 2009 Introduction to magnetic materials. Wiley, New Jersey Dreyfus R, Baudry J, Roper ML et al 2005 Microscopic artificial swimmers. 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