Eight-Degrees-of-Freedom Remote Actuation of Small Magnetic Mechanisms

Transcription

1 Eight-Degrees-of-Freedom Remote Actuation of Small Magnetic Mechanisms Sajad Salmanipour and Eric Diller Abstract Magneticall-driven micrometer to millimeterscale robotic devices have recentl shown great capabilities for remote applications in medical procedures, in microfluidic tools and in microfactories. Significant effort recentl has been on the creation of mobile or stationar devices with multiple independentl-controllable degrees of freedom (DOF) for multiagent or comple mechanism motions. In most applications of magnetic microrobots, however, the relativel large distance from the field generation source and the microscale devices results in controlling magnetic field signals which are applied uniforml over all agents. While some progress has been made in this area allowing up to si independent DOF to be individuall commanded, there has been no rigorous effort in determining the maimum achievable number of DOF for sstems with uniform magnetic field input. In this work, we show that this maimum is eight and we introduce the theoretical basis for this conclusion, reling on the number of independent usable components in a magnetic field at a point. In order to verif the claim eperimentall, we develop a simple demonstration mechanism with 8 DOF designed specificall to show independent actuation. Using this mechanism with 5 µm magnetic elements, we demonstrate eight independent motions of.6 mm with 8.6 % coupling using an eight coil sstem. These results will enable the creation of richer outputs in future microrobotic devices. I. INTRODUCTION Progress in the wireless control of microrobots has demonstrated the potential of the field in applications such as microfluidics, drug deliver and minimal invasive surger, where the challenge is to remotel control and actuate microscale devices without phsical access to the workspace. Recentl, there has been a wide variet of innovative and successful methods developed for remote actuation and control in microscale, with actuation based on magnetism 1], 2], electrostatic forces 3], chemical reactions 4], thermal activation 5], optical 6] and even bacteria motilit-based 7] techniques. At small sie scales, one particular challenge is in providing power for remote agents in order to generate relativel large forces and torques. Another major challenge is to individuall address different motions of remote agents and control them independentl for team or multi-degree of freedom (DOF) motions within one workspace. Considering both challenges, magnetic-based actuation is commonl used, where in addition to its abilit to generate relativel large forces and torques, magnetic actuation does not require *This work is supported b the NSERC Discover Grants Program. S. Salmanipour and E. Diller are with the department of Mechanical and Industrial Engineering, Universit of Toronto, 5 Kings College Road, Toronto, M5S 3G8, Canada. ediller@mie.utoronto.ca on-board selection modules to achieve multiple controllable DOFs. Wireless magnetic actuation would allow for multi DOF sstems that is not possible with other techniques without a compromise in their sie or actuation forces 8], 9]. Magnetic-based approaches featuring multi-agent or multi- DOF control can be categoried b their control with assumptions of uniform or non-uniform magnetic fields over the operating workspace. Uniform fields are assumed in applications where the displacements of remote agents are negligible compared to distance of magnetic sources from the workspace 2] and thus the eternal magnetic field can be considered uniform over the workspace. This assumption is usuall considered valid for most medical applications where the field sources must be located outside the bod. On the other hand, when agents displacements are large compared with the distance to the field source, it should be assumed that the eternal magnetic field varies over the workspace 1], a fact which could be used for independent control 11]. In this paper we focus on the former case where a uniform magnetic field assumption is made; in other words, we anale magnetic field components at a single point in workspace. This makes it a more comprehensive stud, where analing magnetic fields at a single point, determines capabilities of both, uniform and non-uniform fields. One approach to achieve multiple DOFs in the uniform field case is through time-encoded signals. Becker et al., for eample, suggest a mechanical decoding sstem for modulated magnetic control signals 12]. However, this method requires relativel large agents in an MRI which is not feasible in microscale and can not be generalied to other microrobotic sstems. In this work, on the other hand, we consider stead magnetic actuation; in other words, where magnetic fields are in stead state and actuation forces and torques can be seen as linearl dependent on field and field gradient inputs. Thus the results of this stud can be applied to more comple actuation tpes including both stead and time-dependent signals. In a stead uniform magnetic field, actuation power is transmitted to remote magnetic devices b the application of magnetic torque and magnetic force. The magnetic field, denoted b the vector B B B induces a rigid-bod torque, while the spatial gradient of the field generates a force. There has been significant effort in eploiting magnetic torque capabilities in order to actuate micro-robots, including helical-based 1], 13] or traveling-wave propulsion methods 14], 15]. However, torque-actuated mechanisms have at most three independent control inputs, one each for the

2 independent components of the magnetic field vector at a point (B,B,B ). As a result, the number of independent DOFs can not eceed three for torque-onl actuation. Therefore, in order to achieve more than 3 DOFs, the gradient of magnetic field should also be used. B developing sstems capable of controlling both magnetic field and its gradient to appl force in addition to torques, current designs in the literature have achieved 5 DOF 2], 16] (3 DOF positioning and 2 DOF orientation control of a rigid bod microdevice) and up to 6 DOF magnetic sstems 17], 18]. (3 DOF positioning and 3 DOF orientation control). While it has been noted that the magnetic field and its gradient contain eight independent components at an point in space (which has served as justification for the use of a minimum of eight actuating magnetic coils in man of these sstems 19]), microsstems with onl up to 6 DOF have been developed in previous designs. A single rigid bod has onl 6 DOF, however, more comple sstems such as multi-agent sstems or those containing jointed or fleible mechanisms 2] can have man more free DOF which would be useful to control. In this work, we first investigate the magnetic field and its gradient to determine the theoretical maimum number of independent controllable inputs for a generic magnetic sstem operating at a point in space under the influence of a uniform field input. Based on the available independent parameters of the field and its gradient at a point, we show that this number of independent inputs is eight. To demonstrate this independent actuation, we design a magneticall-driven sstem in a wa that those eight independentl controllable inputs result in eight independentl controllable outputs. The proposed 8 DOF magnetic mechanism consists of two parts, a magnetic field generation sstem and remote mm-scale magnetic agents (which could be generalied to an smallscale magnetic remote devices). Since a rigid bod cannot have more than 6 DOF, a sstem with eight outputs requires at least two bodies. For a simple demonstration, we use seven agents (possessing total of 8 DOF), requiring onl one camera to measure all of the eight outputs. We show calibration and validation of the eperimental sstem, and demonstrate 8 DOF actuation, where all of its 8 DOFs can be individuall addressed and actuated with a maimum coupling of 8.6%. Thus, for the first time, we investigate the capabilit of stead uniform magnetic fields to achieve a wireless mechanism with maimum number of DOFs which is eight. This work serves as a proof of concept which will enable more deterous actuation of future wireless magnetic mechanisms with maimum deterit. II. MAGNETIC ACTUATION In this section, magnetic actuation principles are introduced, with force and torque equations written in matri form as the basis for discussing our scheme for full-dof actuation. These equations are utilied to investigate the theoretical maimum number of DOFs, and we limit discussion to sstems utiliing stead uniform magnetic fields onl. A. Background on Magnetic Field Actuation Mawell s equations describe a magnetic field in a freecurrent workspace as: B = (1) B = (2) where B 3 1 is the magnetic field vector. This magnetic field results in a force F 3 1 and a torque vector τ 3 1 acting on the magnetic agent (the remote micro-device), calculated as: F = ( m ) B (3) τ = m B (4) where m = m m m is the dipole moment of the magnetic agent. Using cross product identities, the torque equation (4) can be rewritten in matri form as: τ = τ τ = m m m m B B (5) τ m m B In order to also write the force equation (3) in a matri form, Mawell s equations are utilied. Equation (1) implies that gradient of the magnetic field vector has a ero trace and (2) constrains it to be smmetric. As a result, equation (3) can be rewritten as: B B B F = B m (6) B ( + B ) m m As it can be seen from (6), the gradient matri has five independent components 19]. B rearranging them into a 5 1 vector, equation (6) can be simplified to: F = f f = m m m m m m f m m m m B. Multi-DOF Magnetic Actuation Using equations (5) and (7), eerted force and torque on a magnetic ] device with magnetiation vector m =, m m m can be summaried in a matri form as: f m m m f m m m f τ = m m m m m m U τ m m τ m m (8) U = B B B B B B B B B A combination of forces and torques can be used to actuate remote magnetic agents. Sstem output depends on this (7)

3 agent 4 agent 7 agent 5 f & τ M7 f & τ d 7 f & τ f & τ M1 d 8 d 1 f & τ r M4 M5 d 4 d 5 f & τ M3 d 6 M2 d 3 M6 d 2 f & τ agent 3 agent 1 agent 2 agent 6 f & τ Fig. 1: 8 DOF magnetic mechanism used to demonstrate independent actuation. Remote magnetic agents are cubic permanent magnets, phsicall constrained to eperience deflections along one (agents 1-6) or two aes (agent 7). Considering relativel small deflections, orientation of the net magnetiation vector for each magnet is assumed to be fied. combination, whereas sstem input is alwas the same and we choose it be an 8 1 vector, named U. In order to investigate what is the maimum number of independent outputs in a local magnetic field, the input/output vectors must be studied. Based on equation (8), maimum number of independent outputs can not be greater than number of inputs which is eight. To validate the possibilit of this maimum of eight, we designed a simple sstem with seven small magnets possessing eight free DOFs. As illustrated in Fig. 1, magnets are denoted b M i (i : 1 7) and output displacements as d i (i : 1 8). In this sstem, remote magnetic devices are simpl cubic permanent magnets that are attached to fleible arms. As described in Table I, these fleible arms are phsicall constrained in a wa that the can onl eperience deflections in one or two directions (each agent has 1 or 2 DOFs). TABLE I: Specifications of the proposed 8 DOF magnetic mechanism. Agent # Number of DOFs Output d 1 d 2 d 3 d 4 d 5 d 6 d 7,d 8 Motion Ais, Actuation Force f f f f f f f,f Actuation Torque τ τ τ τ τ τ τ,τ Orientation m i m i There are 8 DOFs in total where each one of them is actuated b a force element (f or f ) and a torque element (τ or τ or τ ). B choosing d 1 d 8 as outputs, sstem output vector Y is calculated as: Y = d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 (9) d 1 = +αf 1 βτ 1 d 2 = +αf 2 βτ 2 d 3 = +αf 3 βτ 3 d 4 = αf 4 βτ 4 d 5 = αf 5 βτ 5 d 6 = αf 6 βτ 6 d 7 = +αf 7 βτ 7 d 8 = αf 7 βτ 7 where α and β are constant scalars mapping from force and torque inputs to deflection outputs, respectivel. Using (8), equation (9) can be written in matri form as: Y = m 1 m 1 m 1 m 1 m 1 m 2 m 2 m 2 m 2 m 2 m 3 m 3 m 3 m 3 m 3 m 4 m 4 m 4 m 4 m 4 m 5 m 5 m 5 m 5 m 5 λ U m 6 m 6 m 6 m 6 m 6 m 7 m 7 m 7 m 7 m 7 m 7 m 7 m 7 m 7 m 7 Y = K λ U Y = S U (1) S 8 8 = K 8 8 λ 8 8 λ 8 8 = diag(α,α,α,α,α,β,β,β) Here, S represents the sstem matri which maps from the input vector U to output Y. In order to be able to have full control over all of the eight outputs, sstem matri S must be full rank. As equation (1) shows, for a full rank matri, magnetiation vectors of agents which have the same motion ais (agents 1-3 and 4-6) must be linearl independent, and the magnetiation vector for the last agent must have a non-ero component in plane. Therefore, we choose magnetiation vectors for the agents as: m 1 = m 4 = γ m 3 = m 6 = γ m 2 = m 5 = γ m 7 = γ (11) where γ is a scalar representing the magnitude of magnetiation vectors (same for all). B inserting (11) into (1), sstem matri S is calculated as: αγ β γ αγ βγ αγ S = αγ β γ αγ β γ (12) αγ αγ βγ αγ where its determinant is equal to: S = α 5 β 3 γ 8 (13) As it can be seen from (13), if α, β and γ are non-ero, sstem matri is full rank and all of the outputs d 1 d 8

4 can be controlled independentl through the input vector U. It should also be noted that this sstem matri represents a linearied model of the sstem with fied magnetiation vectors m 1 m 7, which is onl valid when deflections are small enough. This criteria can be epressed as: sin(θ) θ = d i /l i (14) where l i is the ith arm s length. In order to meet this condition, we choose our desired output deflections to be 2 times smaller than arms lengths. III. GENERATING MAGNETIC FIELD AND GRADIENT Having a full rank sstem matri guarantees that sstem outputs can be controlled independentl if we have full control over sstem inputs. In other words, we need to be able to control all of the eight terms in the U vector. To do so, we need a magnetic field generation sstem capable of simultaneousl generating three field terms (B,B,B ), and five gradient terms ( B, B, B,, ). As a result, at least eight independent magnetic field sources are needed. These sources can either be electromagnetic coils, permanent magnets or a combination of both. In this work, we chose to have eight electromagnetic coils as our controllable input sources. Using the Bio-Savart law, magnetic field vector B and its gradient matri G, for a coil pointing toward direction at a distance of D on its ais, can be calculated as: B = ρ 1 G = σ = B B B B µ Nia 2 ρ = 2(a 2 + D 2 ) 1.5 (15) B = σ (16) ( + B ) µ Nia 2 D 4(a 2 + D 2 ) 2.5 µ = 4π 1 7 T.m.A 1 where a is the coil radius, i is the current, N represents number of turns and µ is the permeabilit of free space. Here we assumed the coil is pointing toward ais, but using a rotation matri R, these calculations can be generalied for a coil with a desired ais n, as: B = R ρ (17) G = R σ σ R (18) 2σ n = R 1 Having eight coils positioned around workspace with different orientations, magnetic field and gradient terms can be calculated as: U = L 8 8 I 8 1 (19) U = B B B B B B where L is the sstem matri mapping from eight coil currents to the eight field and gradient terms. B utiliing workspace cm Fig. 2: Magnetic field generation prototpe, consisting of four inner coils (1 4) and four outer coils (5 8). The sstem is capable of independentl generating three magnetic field components as large as 15 mt, and five gradient components as large as.55 T/m. equations (17) and (18) for each coil individuall, columns of L matri are calculated; where ith column maps from ith coil current I i to U. The sstem is able to generate all eight field and gradient terms independentl, if and onl if L is full rank. In order to optimie coil configurations, MATLAB fmincon function (interior-point algorithm) was utilied. The parameters to be optimied were coils aes, their radius and their distance to workspace. Constraints were power limitations (maimum current of 15 A for each coil), workspace sie (a circle with minimum radius of 3 cm, maimum of 4 cm) and L matri rank (must be eight). The following cost function was also defined to maimie generated field and gradient components: f = (η 1 B B + η 2 g g) 1 (2) B = B B B g = B where η 1 and η 2 are positive constant scalars. The optimied solution has four outer coils with their centers in plane and four inner coils positioned below plane (3 degrees tilted). Figure 2, shows the final developed coil sstem and coils specifications can be found in Table II. In order to calibrate the coil sstem, a test current of 1 A was sent to each coil separatel and resulted data (Table III) were used to calculate the L matri (equation (19)). Calculated matri is full rank and considering our 15 A current limit, the sstem can reliabl generate three magnetic field components as large as 15 mt, and five gradient components as large as.55 T/m. B B

5 TABLE II: Coils specifications; including radius a, distance to workspace center D, number of wire turns N, wire gauges and coils aes n. a (cm) D (cm) N Gauge Coils aes ( n) Coil Coil Coil Coil Coil Coil Coil Coil TABLE III: Coils characteriation data; sending 1 A to each coil individuall. Field components are measured using 3- ais gauss-meter (46, Lake Shore Crotronics), and gradient components are estimated based on dipole model and measured field data. B (mt) B (mt) B (mt) B (T/m) B (T/m) B (T/m) (T/m) (T/m) Coil Coil Coil Coil Coil Coil Coil Coil M 4 M 3 M 7 M 6 M 1 M 2 M 5 (a) 3 mm M 4 M 3 M 7 M 6 M 1 M 2 Fig. 3: a) Prototpe of the 8 DOF magnetic mechanism. b) Top view camera recording displacements. Video is available in supplementar materials. M 5 (b) 2 mm TABLE IV: Test inputs to calibrate the 8 DOF magnetic mechanism. B (T/m) B (T/m) B (T/m) (T/m) (T/m) B (mt) B (mt) B (mt) Input Input Input Input Input Input Input Input IV. EXPERIMENTAL RESULTS As shown in Fig. 3a, the proposed 8 DOF magnetic mechanism was built using seven pieces of Nitinol wire (2.5 cm length, 75 µm diameter) as fleible arms attached to seven cubic permanent magnets (NdFeB, 5 µm). One stationar camera (FO134TC, Foculus) provided position measurements (Fig. 3b), with a LED plate placed beneath the workspace as its light source. The magnetic field generation sstem and the 8 DOF magnetic mechanism can be considered as two sstems working in series. B combining equations (1) and (19), we will have: Y = S L I = H I (21) H 8 8 = S 8 8 L 8 8 where H is the overall sstem matri mapping from coil currents I to sstem output Y. In order to determine this matri, S must be calculated first. To do so, eight different input sets were applied and agents deflections were measured. The information specifing input sets are listed in Table IV, and the resulted measured deflections can be found in Table V. Calibration results (Table V) were utilied to calculate S and consequentl H matri. The calculated H matri is full rank, and considering the 15 A current limit for our coils, the sstem was able to actuate each one of the eight outputs with a maimum deflection of 8 µm. TABLE V: Calibration results for the 8 DOF magnetic mechanism. d 1 (µm) d 2 (µm) d 3 (µm) d 4 (µm) d 5 (µm) d 6 (µm) d 7 (µm) d 8 (µm) Input Input Input Input Input Input Input Input To verif the capabilit of the sstem in individuall actuating each DOF, eight open-loop eperiments were conducted; duration of each 5 seconds and designed to displace d i from to 6 µm (linear ramp profile). As plotted data in Fig. 4 shows, during each eperiment, the targeted output followed the desired trajector and the rest of seven outputs remained close to ero. In order to investigate sstem performance in terms of independent actuation of DOFs, we defined cross-talk for each DOF, as maimum deflection occurred during eperiments which ideall should not affect d i, divided b its maimum deflection during its designated eperiment. Cross-talk for each one of DOFs are reported in Table VI, where the maimum belongs to d 1 which is 8.6%.

6 displacement (7m) d 1 d 2 d 3 d 4 d 5 d 6 d 7 d time (sec) Fig. 4: Validating 8 DOF magnetic mechanism through eight sets of open-loop eperiments; duration of each is 5 s, and is designed to displace d i from to 6 µm (linear ramp profile). Video is available in supplementar materials. TABLE VI: 8 DOF Magnetic Sstem cross-talk; calculated for each DOF as maimum non-diagonal element divided b corresponding diagonal element. d 1 (µm) d 2 (µm) d 3 (µm) d 4 (µm) d 5 (µm) d 6 (µm) d 7 (µm) d 8 (µm) -5 (s) (s) (s) (s) (s) (s) (s) (s) Cross- Talk 8.6% 8% 4% 8.3% 8% 6.6% 7.7% 7.1% V. CONCLUSIONS This paper studies, for the first time, how to utilie maimum number of magnetic field parameters at a single point in the workspace to allow for maimum possible DOFs, which is eight. In this work, we have developed an 8 DOF millimeter-scale magnetic mechanism as a proof of concept to demonstrate maimum capabilities of uniform stead magnetic fields in independentl controlling remote magnetic agents. These results serve as a general framework which will potentiall allow for developing future magnetic sstems with maimum deterit, specificall in applications such as multi-agent or fleible mechanisms where more DOFs are needed to be controlled. Our future work will be focused on developing a generalied framework to appl this theor on an mechanism with desired constraints and DOFs, and we will use this framework to design functional microrobotic devices. 2] M. P. Kummer, J. J. Abbott, B. E. Kratochvil, R. Borer, A. Sengul, and B. J. Nelson, Octomag: An electromagnetic sstem for 5-dof wireless micromanipulation, IEEE Transactions on Robotics, vol. 26, no. 6, pp , 21. 3] B. R. Donald, C. G. Leve, and I. Paprotn, Planar microassembl b parallel actuation of mems microrobots, Journal of Microelectromechanical Sstems, vol. 17, no. 4, pp , 28. 4] A. A. Solovev, Y. Mei, E. Bermúde Ureña, G. Huang, and O. G. Schmidt, Cataltic microtubular jet engines self-propelled b accumulated gas bubbles, Small, vol. 5, no. 14, pp , 29. 5] O. Sul, M. Falvo, R. Talor, S. Washburn, and R. Superfine, Thermall actuated untethered impact-driven locomotive microdevices, Applied Phsics Letters, vol. 89, no. 2, p , 26. 6] H. Maruama, T. Fukuda, and F. Arai, Laser manipulation and optical adhesion control of functional gel-microtool for on-chip cell manipulation, in Intelligent Robots and Sstems, 29. IROS 29. IEEE/RSJ International Conference on. IEEE, 29, pp ] B. Behkam and M. Sitti, Bacterial flagella-based propulsion and on/off motion control of microscale objects, Applied Phsics Letters, vol. 9, no. 2, p. 2392, 27. 8] E. B. Steager, M. S. Sakar, C. Magee, M. Kenned, A. Cowle, and V. Kumar, Automated biomanipulation of single cells using magnetic microrobots, The International Journal of Robotics Research, vol. 32, no. 3, pp , ] C. Pawashe, S. Flod, and M. Sitti, Modeling and eperimental characteriation of an untethered magnetic micro-robot, The International Journal of Robotics Research, vol. 28, no. 8, pp , 29. 1] P. Berkelman and M. Dadovsk, Magnetic levitation over large translation and rotation ranges in all directions, IEEE/ASME Transactions on Mechatronics, vol. 18, no. 1, pp , ] D. Wong, E. B. Steager, and V. Kumar, Independent control of identical magnetic robots in a plane, IEEE Robotics and Automation Letters, vol. 1, no. 1, pp , ] A. Becker, O. Felfoul, and P. E. Dupont, Simultaneousl powering and controlling man actuators with a clinical mri scanner, in 214 IEEE/RSJ International Conference on Intelligent Robots and Sstems. IEEE, 214, pp ] A. Ghosh and P. Fischer, Controlled propulsion of artificial magnetic nanostructured propellers, Nano letters, vol. 9, no. 6, pp , ] J. Zhang, P. Jain, and E. Diller, Independent control of two millimeterscale soft-bodied magnetic robotic swimmers, in Robotics and Automation (ICRA), 216 IEEE International Conference on. IEEE, 216, pp ] J. Zhang and E. Diller, Millimeter-scale magnetic swimmers using elastomeric undulations, in Intelligent Robots and Sstems (IROS), 215 IEEE/RSJ International Conference on. IEEE, 215, pp ] P. Ran and E. Diller, Five-degree-of-freedom magnetic control of micro-robots using rotating permanent magnets, in Robotics and Automation (ICRA), 216 IEEE International Conference on. IEEE, 216, pp ] E. Diller, J. Giltinan, and M. Sitti, Independent control of multiple magnetic microrobots in three dimensions, The International Journal of Robotics Research, vol. 32, no. 5, pp , ] E. Diller, J. Giltinan, G. Z. Lum, Z. Ye, and M. Sitti, Si-degreesof-freedom remote actuation of magnetic microrobots, in Robotics science and sstems, ] A. J. Petruska and B. J. Nelson, Minimum bounds on the number of electromagnets required for remote magnetic manipulation, IEEE Transactions on Robotics, vol. 31, no. 3, pp , ] J. Zhang, O. Onaiah, K. Middleton, L. You, and E. Diller, Reliable grasping of three-dimensional untethered mobile magnetic microgripper for autonomous pick-and-place, IEEE Robotics and Automation Letters, vol. 2, no. 2, pp , 217. REFERENCES 1] T. Honda, K. Arai, and K. Ishiama, Micro swimming mechanisms propelled b eternal magnetic fields, IEEE Transactions on Magnetics, vol. 32, no. 5, pp , 1996.