Controlling Multiple Microrobots using Homogeneous Magnetic Fields

Transcription

1 Controlling Multiple Microrobots using Homogeneous Magnetic Fields Mohammad Salehiadeh, Sajad Salmanipour, Jiachen Zhang, and Eric Diller Abstract This paper overviews two different approaches in individuall addressing multiple motions for small-scale magnetic sstems where the controlling signals are provided through a generic quasistatic homogeneous magnetic field. The first approach relies on direct independent magnetic actuation of multiple DOFs to make deterous microdevices. The second approach eploits inter-agent magnetic forces to accomplish desired formations between a team of mobile magnetic microrobots. This paper presents a comparison of the two methods, focusing on capablities, prospects and potential applications. Kewords: mutli-dof, multi-agent control, soft robotics, micro-robotics, targeted cargo deliver I. INTRODUCTION Multi-agent control has started to pla an essential role in the control of untethered microrobots at small scales. The abilit to eert independent control over a team of microrobots working together on a task has potential to increase task speed and capabilit to carr out parallel operations with a broad range of potential applications in minimall invasive surger [1], targeted drug deliver [2], microassembl [3], and cell manipulation [4]. Among man proposed techniques [5], remote actuation using a magnetic field is a common option because it can remotel generate relativel large forces and torques on magnetic materials, penetrate most materials, and is eas and safe to generate and manipulate. However, team control of magnetic micro-agents remains an open-ended problem as in most actuation sstems, all magnetic micro-agents share a global driving magnetic signal. As such, all agents receive identical control inputs and thus it is difficult to steer independentl for comple task completion [6]. In the microrobotics field, a variet of methods have been eplored toward the team control of magnetic microrobots. Becker et al. [7] utilied differences in cell population to steer cells to goal positions using ensemble control. Martel et al. [8] investigated swarm control of bacterial actuators in the human microvasculature trackable b a clinical MRI sstem. However, these methods are usuall limited to a small number of independentl-controllable degrees of freedom (DOFs), has no control over the formation orientation, cannot be generalied to microrobots moving in three dimensions, and is onl applicable to groups of agents which are This work is supported b the NSERC Discover Grants Program. M. Salehiadeh, S. Salmanipour, J. Zhang, and E. Diller are with the Department of Mechanical and Industrial Engineering, Universit of Toronto, 5 King s College Road, Toronto M5S 3G8, Canada ediller@mie.utoronto.ca each magneticall unique [9]. Besides, controlling multiple magnetic microrobots close to each other is difficult due to magnetic interactions between the agents. Most work in the field of magnetic micro-agents assume that inter-agent magnetic fields are small compared with the driving actuation field strength, with the associated requirement that the agents be kept far apart from each other. Another major challenge is to separatel address different motions of remote agents and control them independentl for team or multi-dof motions within one workspace. Magnetic-based approaches emploing multi-dof or multi-agent control can be classified b their control with assumptions of homogeneous or non-homogeneous magnetic fields over the operating workspace. Homogeneous fields are assumed in platforms where the displacements of remote agents are insignificant compared to distance of magnetic sources from the workspace [10] and thus the eternal magnetic field can be considered uniform over the workspace. This assumption is generall considered valid for most medical applications where the field sources must be placed outside the bod. On the other hand, when agents displacements are large compared with the distance to the field source, it should be supposed that the eternal magnetic field changes over the workspace, a fact which could be used for independent control [11]. One wa to master 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 [1]. 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. The paper presents a comparison of two different published approaches that can be used as a tool to control multiple microrobots under the assumption of quasistatic, homogeneous magnetic field. Degrees of freedom is the number of independent aes along which a sstem of multi-agent can eperience motion. The first actuation approach presents a framework to eploit the maimum number of magnetic field parameters at a single point in the workspace to allow for maimum possible DOFs using stead magnetic actuation. One can appl this method on an mechanism with desired constraints and DOFs, following the fact that to have full control over all the outputs of sstem, the sstem matri mapping inputs to outputs must be full rank. The second approach shows a principle on how to use a single global magnetic field input to accomplish desired formations between multiple mobile magnetic micro-agents. The method relies on

2 displacement (7m) inter-agent forces in close proimit and acts to prevent agents from touching each other. The paper is structured as follows. The two methods are reviewed briefl from [12] [15] in sections II and III, respectivel, with the novel contribution of this work being the comparison and discussion of section IV. II. APPROACH 1; DIRECT INDEPENDENT MAGNETIC ACTUATION OF MULTIPLE DOFS A. Background In a quasistatic homogeneous magnetic field, actuation power is transmitted to remote magnetic devices b the application of magnetic torque and magnetic force. The magnetic field, induces a rigid bod torque, while the spatial gradient of the field generates a force. B developing magnetic field generation sstems capable of controlling both magnetic field and its gradient, current designs in the literature have achieved up to 6 DOF magnetic sstems [16], [17] (3 DOF positioning and 3 DOF orientation control). The maimum number of independentl controllable magnetic parameters, in a homogeneous magnetic field, is eight [18]; the consist of three field elements and five gradient elements. To demonstrate an 8 DOF independent actuation, we designed a magneticall-driven sstem converting those eight independent inputs to eight controllable outputs. B. Proof of Concept Mechanism Since a rigid bod cannot have more than 6 DOFs, 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. As illustrated in Fig. 1a, 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. 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). 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 8 1 = S 8 8 U 8 1 (1) U = [ B B B B B B B B ] where U is the sstem input and S represents the sstem matri which maps from inputs to outputs. In order to be able to have full control over all of the eight outputs, sstem matri S must be full rank. The specific orientations of magnetic agents, as illustrated in Fig. 1a, results in a non-ero determinant for S matri. Thus, sstem matri is full rank and all of the outputs d 1 d 8 can be controlled independentl through the input vector U. Having a full rank sstem matri guarantees that sstem outputs can be controlled independentl if we have full control over sstem inputs. To do so, an eight-coils magnetic field M7 d 7 M1 d 8 d 1 M4 M5 (a) d 4 d 5 M3 d 6 M2 d 3 M6 d 2 3 mm M 4 M 3 M 7 M 6 M 1 M 2 Fig. 1: (a) 8 DOF magnetic mechanism used to demonstrate independent actuation. (b) Prototpe of the mechanism time (sec) M 5 (b) d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 Fig. 2: Validating 8 DOF magnetic mechanism through eight sets of open-loop eperiments. generation sstem [12], capable of independentl generating all of the eight terms in in the U vector (three magnetic field and five gradient components) was used. C. Results As shown in Fig. 1b, 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, 500 µm), and one stationar camera (FO134TC, Foculus) provided position measurements. 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 0 to 600 µm in a linear ramp profile. As plotted data in Fig. 2 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. The maimum cross-talk belongs to d 8 which is 10.7%. With another set of eperiments reported in [12], we

3 achieved a better performance where the maimum cross-talk was 8.6%. III. APPROACH 2; TEAM FORMATION CONTROL OF MAGNETIC MICROROBOTS VIA INTER-AGENT FORCES A. Magnetic actuation and inter-agent kinematics Here, we use an eternal homogeneous magnetic field to regulate pairwise separations and orientations among the mobile micro-agents. Optionall, one can superimpose a weak magnetic field gradient to control the center-of-mass position of the team as well. We base our analsis on the assumption that the global field dominates the local fields such that all agents alwas align with the actuation field as shown in Fig. 3(a). As the applied field is uniform over space, no eternal magnetic forces are generated ( = 0). This figure shows onl two agents for simplicit. From a general point of view, for a team of n agents, let i F tot denote the total force vector created at the location of agent i b the rest of agents of the set, then i F tot = k i F ki whereb k {1, 2,..., n} with n as the number of agents, and F ki (r ki, ψ, α) = 3µ 0 4πr ki 5 [(m k r ki )m i + (m i r ki )m k +(m k m i )r ki 5(m k r ki )(m i r ki ) r rki 2 ki ] (2) is the pairwise magnetic force eerted at the location of agent i b agent k [15]. Here µ 0 is the permeabilit of free space, m k = m i is the magnetic moment vector, r ki is the separation vector connecting agent k to agent i, and r ki is the norm of this vector. The net radial and transverse components of the total magnetic force eerted on agent j b the rest of agents, linked to pair i j are represented b j F ri j and j F ti j in Fig. 3(a) and can be calculated in local Cartesian coordinates defined eclusivel for each pair of agents (ê ri j,ê ti j,ê i j ) (see [15]). B. Two-agent Configuration Control 1) Two agents moving in a 2D plane: In 2D plane, two agents can possess 2 DOFs in the relative coordinate (separation r and pair heading φ). In the constrained scenario magnetiation is constrained to horiontal plane of motion characteried b in-plane rotational angle ψ available as the single global control input. This leads the sstem to be underactuated [13]. The task in designing a controller is to choose the input magnetic field angle(s) to push the relative spacing and pair heading angle toward the goal state. The basis for producing the associated radial and transverse forces is illustrated in Fig. 3(b). When two agents are too close with respect to the desired separation (r < r des, note that r des can change in eperiment), the controller will be saturated b pointing the field orientation perpendicular to the pair s separation vector r so that agents repel each other with full radial force, see (i). If two agents are too far (r > r des ), the controller will be saturated b orienting the field parallel to r so that agents attract each other with full radial force, see (iv). Importantl, at ψ = ±54.74 the radial force becomes ero, but a transverse force is created which rotates the pair. If the space between two agents is around the goal spacing (r r des ), global frame (i). repulsion too close at goal e 12 e 12 m 1 e t12 2 F t12 2 F 12 local frame m 2 r agent 1 12 agent 2 φ interface r < r des r > r des ψ = 90 ψ = 0 ψ = ψ = r r des φ < φ des (ii) and (iii). ero-radial force r r des φ > φ des α ψ 2 F r12 counterclockwise clockwise oil water e r12 (a) (iv). attraction too far at goal Fig. 3: Two-agent configuration control principle. (a) 3D pairwise orientation parameters defined in global and local coordinates for agents in close proimit with magnetic moments m 1 and m 2 aligned with the actuation field. The pairwise distance vector connecting agent 2 to agent 1, and the pair orientation are denoted b r 12 and φ, respectivel. The radial and transverse coordinates are shown b e r12 and e t12. The 2D motion of the agents occurs at the interface between water and oil. The out-of-plane angle that magnetiation M makes with the -ais within the purple plane is denoted b α and the in-plane angle that the projection of magnetiation makes with the radial ais in the motion plane (in can) is shown b ψ. The radial, transverse, and out-of-plane forces are represented b 2 F r12, 2 F t12, and 2 F 12, respectivel, eerted on the second agent b the first agent (corresponding forces acting on agent 1 are not shown). (b) States of agent attraction or repulsion as well as pair heading rotation are determined b the direction and sign of the global magnetic field, respectivel: i) repulsion at ψ = 90, ii) ero radial-force at ψ = with counter-clockwise rotation, iii) ero radial-force at ψ = with clockwise rotation, iv) attraction at ψ = 0. the controller would choose intermediate angles between 0 and 90 centered around the setpoint angle ψ s = ±54.74, leading to the P-control law: ψ = ψ s K ε r where the radial separation error is denoted b ε r = r r des and K represents the control gain, see (ii) and (iii). In free magnetiation scenario where the agents are capable (b)

4 to magneticall rotate in 3D, the sstem can be full actuated. In other words, there eists a one-to-one relation between two inputs and two outputs of the sstem. As such both out-of-plane and in-plane rotations are feasible, and magnetiation orientation can be controlled freel in 3D b 2 DOF variables ψ and α [14]. 2) Two agents moving in 3D space: For 3D manipulation of two agents, one can still frame the sstem model as an underactuated problem and etend the same approach. We will still have two control input angles ψ and α, however, one additional state, elevation angle θ, will be required besides the separation r and aimuth angle φ to describe the relative position of the second agent with respect to the first agent. C. Generic Multiple-agent Configuration Control As presented in [15], for more than two agents, the problem becomes generic for which we presented a solution using gradient descent optimiation method. The controller searches for ψ and α angles solution that minimies a weighted L 2 -norm fitness function so that the relative spacings and angles of the pairs are pushed toward the desired ones between a set of 3 or more magnetic agents. D. Results As an application demonstration in Fig. 4, we applied the second control strateg to independentl and simultaneousl position two microgrippers on a horiontal plane for parallel targeted cargo deliver in [19]. The separation and orientation of the two-microgripper pair are modulated b the local magnetic interactions between the two microgrippers, which are governed b a global homogeneous magnetic field. The microgripper action of grasping or releasing cargoes is full controlled b the global magnetic field without requiring additional chemical, thermal, or other stimuli. As a result, the proposed strateg onl requires a single input, i.e., a global magnetic field, to control two microgrippers and therefore is eas and cheap to implement. IV. USAGE AND DISCUSSION A. Comparison between two approaches The ke capabilities and differences of the two presented actuation methods are summaried in Table. I. The first method demands the orientation of the net magnetiation vector for each magnet to be fied, hence the approach is valid when the deflections are small enough. To meet this criteria the desired output deflections are chosen to be 20 times smaller than arms lengths. Despite the first method, the second method dnamicall reorients the agents magnetiation to modulate the attraction and repulsion forces between the agents. In the first method a minimum distance between magnets is imposed to make inter-agent forces negligible, whereas the second method benefits from inter-agent forces as the control basis. The second approach is well suited for mobile agents, although limited to the close proimit range to ensure agents would remain connected magneticall. As estimated in [13], this close proimit range spans between 2.5 and 20 of agents (a) cargoes 1 mm (b) reach threshold (c) fl over cargoes position goals grippers 0 sec 18.0 sec 20.3 sec (d) grasp (e) move to destinations (f) release 24.3 sec 28.5 sec 62.3 sec Fig. 4: Pick-and-place cargoes using two microgrippers. bod radius. The second method is applicable to an magnetic micro/milli-robotic sstems regardless of using hard or soft magnets, either identical or slightl dissimilar magnets, and possible to be combined with other control methods so that agents motion can still be manipulated even when the are far from each other. B. Prospects and usage In contrast to the first method, where DOFs of the sstem can be controlled independentl, in the second method the DOFs are coupled and the sstem becomes underactuated for more than two DOFs. Therefore, the second approach still remains as an open research problem. In fact, the deploed formation controller in [15] which is based on Lapunov fitness function is a generic approach proposed to handle a team of mobile microrobots onl for a few number of configuration states. Moreover, the fitness function which describes the kinematics of magnetic microrobot team is non-conve and high-dimensional. Therefore, a more advanced actuation and control solution is needed for higher dimensional sstems. To this end, our future work is to re-pose this underactuated sstem as a first-order kinematic path planning problem. Path planning tool would allow us to avoid obstacle-avoidance and can fit the change of configuration space over time as the number of agents increases. Due to the maimum deterit that the first approach dedicates, it can be widel applied in the design of fleible mechanism. We are working to appl this method to fleible mechanisms with up to eight DOFs. On the other hand, the second approach is well-suited for mobile agents navigation and localiation to be applied to targeted drug deliver, or can be further developed to be utilied in medical devices. V. CONCLUSION In this paper, we eplored the contrasts of two control principles toward solving the problem of manipulating

5 Benchmarks Control methods Direct independent magnetic actuation of multiple DOFs Team formation control via inter-agent forces Consider inter-agent forces? No Yes (used as the control basis) Minimum and maimum distance between agents Not too close, maimum determined b coil workspace Close proimit 2.5R r 20R Scalable Yes Yes Deterit Maimum 8 DOFs possible (proved) So far shown for 2 DOFs Agents are constrained? Yes (fied) No (mobile) DOFs are independent? Yes Underactuated and coupled for n>2 Agents drift when input field is off? No Yes (alwas) Open-loop actuation feasible? Yes Under investigation Good for deterous mechanism design Good for mobile agents navigation and localiation Table. I: Comparison between the two control methods. multiple agents, to complete comple tasks which are not feasible using a single agent. The results suggest that the multi-dof approach can be etensivel used to make deterous medical devices where it is necessar to satisf the eisting constraints and DOFs. Compared to that, the multi-agent control via inter-agent forces can be a complementar approach to be applied in the motion control of multiple mobile microrobots with the focus on targeted cargo deliver and localiation. REFERENCES [1] A. Becker, O. Felfoul, and P. E. Dupont, Simultaneousl powering and controlling man actuators with a clinical mri scanner, in Intelligent Robots and Sstems, IEEE/RSJ International Conference on, 2014, pp [2] B. J. Nelson, I. K. Kaliakatsos, and J. J. Abbott, Microrobots for minimall invasive medicine, Annual Review of Biomedical Engineering, vol. 12, pp , [3] K. Choi, M. Salehiadeh, R. B. Da Silva, N. Hakimi, E. Diller, and D. K. Hwang, 3d shape evolution of microparticles and 3d enabled applications using non-uniform uv flow lithograph (nufl), Soft matter, vol. 13, no. 40, pp , [4] 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 , [5] M. A. Rahman, J. Cheng, Z. Wang, and A. T. Ohta, Cooperative micromanipulation using the independent actuation of fift microrobots in parallel, Scientific Reports, vol. 7, no. 1, p. 3278, [6] S. Chowdhur, W. Jing, and D. J. Cappelleri, Controlling multiple microrobots: recent progress and future challenges, Journal of Micro-Bio Robotics, vol. 10, no. 1-4, pp. 1 11, [7] A. Becker, Y. Ou, P. Kim, M. J. Kim, and A. Julius, Feedback control of man magnetied: Tetrahmena priformis cells b eploiting phase inhomogeneit, in Intelligent Robots and Sstems, 2013 IEEE/RSJ International Conference on, 2013, pp [8] S. Martel, M. Mohammadi, O. Felfoul, Z. Lu, and P. Pouponneau, Flagellated magnetotactic bacteria as controlled mri-trackable propulsion and steering sstems for medical nanorobots operating in the human microvasculature, The International Journal of Robotics Research, vol. 28, no. 4, pp , [9] L. Mellal, D. Folio, K. Belharet, and A. Ferreira, Optimal control of multiple magnetic microbeads navigating in microfluidic channels, in IEEE Int. Conf. Robot. Autom., [10] 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 , [11] 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 , [12] S. Salmanipour and E. Diller, Eight-degrees-of-freedom remote actuation of small magnetic mechanisms, in IEEE Int. Conf. on Rob. and Autom., Accepted for publication, [13] M. Salehiadeh and E. Diller, Two-agent formation control of magnetic microrobots, in Int. Conf. Manipulation, Automation and Robotics at Small Scales, 2016, pp [14] M. Salehiadeh and E. Diller, Two-agent formation control of magnetic microrobots in two dimensions, J. Micro-Bio Robot., vol. 12, no. 1, pp. 9 19, [15] M. Salehiadeh and E. Diller, Optimiation-based formation control of underactuated magnetic microrobots via inter-agent forces, in Int. Conf. Manipulation, Automation and Robotics at Small Scales, 2017, pp [16] 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 , [17] E. Diller, J. Giltinan, G. Z. Lum, Z. Ye, and M. Sitti, Si-degrees-of-freedom remote actuation of magnetic microrobots, in Robotics: Science and Sstems, [18] 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 , [19] J. Zhang, M. Salehiadeh, and E. Diller, Parallel Pick and Place Using Two Independent Untethered Mobile Magnetic Microgrippers, in IEEE Int. Conf. Robot. Autom., 2018, pp. 1 6.