Design of a Rotating Permanent Magnet System for 5-DOF Control of Micro-Robots

Transcription

1 Design of a Rotating Permanent Magnet System for 5-DOF Control of Micro-Robots by Patrick Stanley Ryan A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Department of Mechanical and Industrial Engineering University of Toronto Copyright by Patrick Ryan 2016

2 Design of a Rotating Permanent Magnet System for 5-DOF Control of Micro-Robots Abstract Patrick Ryan Master of Applied Science Department of Mechanical and Industrial Engineering University of Toronto 2016 Recent work in magnetically-actuated micro-scale robots for biomedical and microfluidic applications has resulted in magnetic actuation systems which can remotely command precise five-degree-of-freedom control of magnetic devices. The objective of this work is to evaluate the capabilities and limitations of these existing systems and to provide a more complete understanding regarding the limits of field generation. As part of this study, a novel actuation system composed of an array of rotating permanent magnets with the potential for increased field and gradient strength, and minimal heat generation is presented. The nonlinear control inputoutput relationship is modeled, a technique to determine the control inputs is developed, and an optimization framework for designing system configurations for targeted applications is shown. A proof-of-concept prototype system is used to demonstrate the feasibility of this type of actuation by performing three standard microrobotic locomotion methods requiring independent control over the applied magnetic fields and forces in three dimensions. ii

3 Acknowledgements I would like to thank my advisor and committee chair Eric Diller for all his help during this project and for providing a truly enjoyable master s experience during these last two years. Also, a special thanks to Professors Ridha Ben-Mrad and Goldie Nejat for serving on my thesis committee. To my fellow members of the Microrobotics Lab: Jiachen, Zhe, Mohammad, Onaizah, and Sajad, I appreciate all the advice and feedback I have received while working on this project and thank you for making the lab such a pleasant in which to work. Thanks to Ahmed Ujjainwala for adding the magnetic encoders and creating the CAD model of the prototype system and to Dongsub Shim for this thorough thermal analysis of this system. Lastly, thanks to my family and friends for their support and perpetual interest regarding the details of this project. iii

4 Table of Contents Abstract... ii Acknowledgements... iii List of Tables... vii List of Figures... viii Chapter 1 Introduction Motivation Research Objectives Contributions... 2 Chapter 2 Magnetic Actuation Systems Literature Review Electromagnetic Coil Systems Permanent Magnet Systems Rotating Permanent Magnet System Chapter Organization Chapter 3 Control Using Rotatable Permanent Magnets Magnetic Actuation Background Determining Required Control Inputs Minimizing a Nonlinear Objective Function iv

5 3.2.2 Inverting the Rate Jacobian Control Capability Metrics for System Optimization Strength and Isotropy of Field and Force Production Minimum Singular Value of the Rate Jacobian Minimum Number of Actuator Magnets Required for 5-DOF Control System Parameter Optimization for Improved Control General Considerations for System Optimization Maximizing Combined Field and Force Strength and Isotropy Maximizing the Smallest Singular Value Chapter 4 Case Study Proof of Concept Prototype Device Design Considerations and System Details Physical Components Motor Set Point Driving System Configuration and Capabilities Heat Generation Experimental Results Magnetic Model Verification Experimental Control Results Calibration of Magnet Positions and Rotational Axes v

6 Chapter 5 Considerations for Future Systems Maximum Field and Field Gradient Strength Forces and Torques for Typical Magnetic Implements Reducing Heat Generation Chapter 6 Conclusions and Future Work Summary of Contributions Future Work References Appendix A Appendix B vi

7 List of Tables Table 3.1: Performance Comparison for Five-, Six-, Eight-, Ten-, and Twelve-Magnet Systems Table 3.2: Strength and Isotropy of the Control Outputs Pre- and Post- Optimization of the System Parameters Table 3.3: Strength and Isotropy of the Control Outputs Pre- and Post- Optimization of the System Parameters for Systems with Constraints Related to a Pseudo-Medical Application Table 3.4: Strength and Isotropy of Control Outputs Pre- and Post-Optimization For Highly Constrained System Parameters Table 3.5: Minimum Singular Value Pre- and Post- Optimization of the System Parameters Table 4.1: Positions and Rotational Axes Defined using Spherical Coordinates for the Eight Actuator Magnets in the Prototype Setup Table 4.2: Comparison of the Measured Field to the Desired Field of 0 mt and 30 mt in Eight Different Directions Table 5.1: Magnet Radii and Workspace Separations that yield a 40% Area Ratio for Three Different Workspace Sizes and the Corresponding Values of the Field, Gradient, and Inter- Magnetic Torque vii

8 List of Figures Figure 1.1: CAD model of the rotating magnet prototype system... 4 Figure 2.1: Control capability and field density comparison for magnetic actuation systems... 6 Figure 2.2: OctoMag: an electromagnetic system capable of 5-DOF magnetic control... 7 Figure 2.3: The Niobe Stereotaxis catheter steering system... 9 Figure 2.4: Microrobot actuation methods that require applied rotational fields... 9 Figure 2.5: Permanent magnet systems for the production of rotating fields Figure 2.6: Permanent magnet systems for 4-DOF control of capsule endoscopes Figure 2.7: Single, robotically-manipulated, permanent magnet system that is capable of full 5- DOF control of a mock-up capsule endoscope Figure 3.1: Schematic image of the rotating magnet system Figure 3.2: Smallest singular value of the control Jacobian vs. maximum motor rate Figure 3.3: System fitness values at the conclusion of each coordinate descent iteration during the optimization starting from the prototype system parameters Figure 3.4: Constraint boundaries representing the patient trunk and imaging system field of view for a pseudo-medical application Figure 4.1: Photo of the rotating magnet prototype system Figure 4.2: Physical components required to mount each actuator magnet Figure 4.3: Swimming speed vs. rotational frequency for a helical swimmer Figure 4.4: Two-dimensional path following results for a micromagnet being rolled using rotating fields viii

9 Figure 4.5: Path deviation and speed vs. desired force magnitude and path following gain for gradient pulling of a micromagnet in 3D Figure 4.6: Three-dimensional path following results for a micromagnet being pulled using magnetic field gradients Figure 5.1: Relative magnet spacing of the baseline system configuration for area ratios between 9 and 40% Figure 5.2: Maximum field strength, gradient strength, inter-magnetic torque, and magnet radii for a rotating magnet system with eight magnets Figure A.1: Pictorial diagram of the electronic components in the prototype system ix

10 1 Chapter 1 Introduction 1.1 Motivation An intriguing area of research is the precise, untethered control of small-scale robotic devices. Such devices can be wirelessly actuated within fully enclosed spaces for a wide range of applications including minimally-invasive medical procedures and biomedical experiments in microfluidic channels. The small size-scale of these devices makes actuation using onboard power and control systems difficult. An established technique for driving these devices involves externally-generated magnetic fields. Magnetic fields are able to penetrate most materials and can be used to adjust the heading and position of small, magnetically-active devices without any physical tethers. Externally-generated magnetic fields have been shown to be capable of driving magnetic devices of many different sizes from micrometer-scale microrobots [1, 2], to centimeter-scale medical devices including capsule endoscopes [3, 4], ophthalmic implements [5], steerable needles [6] and catheters [7, 8]. These magnetic tools can be considered the end-effectors of a larger robotic system consisting of the external field source, the camera or other feedback device, and the computer that controls the tool s motion. Two different types of external magnetic field source have been proposed: 1) electromagnetic coils systems and 2) permanent magnet systems. Actuation systems using either type of field source have been shown to be capable of controlling the position and heading of a magnetic tool, however, with some notable limitations. Substantial temperature increases within the coils of electromagnetic systems can result in workspace heating and these systems are difficult to scale up in order to produce magnetic fields with clinically-relevant magnitudes. There currently exists only one permanent magnet system with the ability to control the position and heading of a magnetic tool. This system, however, requires an expensive robotic manipulator that exhibits potentially dangerous translational movement and has less robust heading control at low force applications. 1

11 2 1.2 Research Objectives To accomplish positional and heading control of small-scale robotic devices while addressing the limitations of existing magnetic actuation devices we propose a new magnetic actuation system. This system uses multiple permanent magnets that can be rotated in place. The use of permanent magnets allows the production of clinically-relevant magnetic fields and forces with no heat generation. We present a control algorithm for calculating the desired control inputs in real time during operation. We demonstrate experimentally the capabilities of this type of system by performing three standard microrobotic locomotion methods. The main objectives of this thesis are: Evaluating existing magnetic actuation systems in terms of their abilities and weaknesses. Developing a novel magnetic actuation platform with an equivalent level of micro-device control. Demonstrating the dexterity of this novel system in a number of experimental tests. Measuring the upper limits of system performance for this type of magnetic actuation and how these limits compare to those of existing systems. 1.3 Contributions The major contribution of this work is the development of a novel permanent magnet actuation system that is capable of wirelessly driving small-scale robotic devices. This is the first permanent magnet system that does not suffer from less robust heading control during small force applications. The feasibility of using this system for untethered device actuation is explored though characterization and feedback control experiments. In summary, this thesis provides the following contributions to the field of magnetic manipulators:

12 3 Development of a novel permanent magnet actuation system that is relatively low cost, has no translating components, with no intrinsic heat generation near the workspace, and is capable of untethered position and heading control of small-scale robotic devices. Development of a technique for determining the nonlinear system inputs that produce a desired field and force output. Generalized optimization technique to determine the optimal magnet positions and rotational axes for a targeted application. Implementation of a preliminary calibration method for calculating the magnet positions using experimental field measurements. Design and construction of a prototype device for experimentally demonstrating the control abilities of this type of system. Characterization of the prototype device through the completion of static field and force production trials. Open loop helical swimming using a rotating field in one plane. Automated path following in 2D for a micro-device using rotating fields in two planes. Feedback-controlled path following in 3D for a micro-device using magnetic gradient pulling. Analysis of the upper limits for this type of magnetic control relative to existing systems.

13 Figure 1.1: CAD model of the rotating magnet prototype system presented in Chapter 4. Only six of the eight actuator magnets are shown here for clarity. 4

14 5 2.1 Literature Review Chapter 2 Magnetic Actuation Systems In general, magnet actuation systems must be able to position or orient the magnetic tool with a high level of precision. The orientation of a magnetic tool can be adjusted by applying a magnetic field, which interacts with the moment of the magnetic tool to produce a torque. Similarly, the position of a magnetic tool can be adjusted by applying a magnetic field gradient (i.e. a field that varies with distance) that interacts with the tool moment to produce a force. An implement containing a single magnetic dipole can be driven with a maximum of five-degreesof-freedom (DOF), consisting of three translational DOF and two rotational DOF (the third rotational DOF requires a torque to be applied about the magnetization axis which is not possible for a single dipole). Existing magnetic control systems can be divided into two broad categories based on the field source: 1) electromagnets that produce a field when current is applied to a coiled wire and 2) permanent magnets that produce a constant field due to the alignment of the domains in their internal microcrystalline structure. A rudimentary comparison between electromagnetic coil and permanent magnet systems is presented in Figure 2.1. This figure shows the demonstrated control DOF of each system plotted versus the field density, defined as the maximum field strength the device can produce for a 10 cm workspace normalized by the volume of the field source. There are many electromagnetic devices with wildly varying control capabilities that have been presented in literature, but only a few are included here. This figure, however, does show the majority of permanent magnet system that has been presented to date. Two important conclusions to draw are: 1) many electromagnetic systems have been shown to be capable of 5-DOF control and only recently have a small number of permanent magnet systems been demonstrated with equivalent abilities and 2) the field density for permanent magnet systems is much greater than coil devices. The rotating magnet system being presented in this paper is represented by point [32] in this figure and is the first permanent magnet system with equal control to that of electromagnetic systems. 5

15 6 Figure 2.1: Control capability and field density comparison for select electromagnetic devices and almost all permanent magnet systems Electromagnetic Coil Systems For the majority of applications involving magnetic tool control, electromagnetic devices are the preferred method for field generation. This is likely due to two characteristics of coil systems: 1) the magnetic field that an electromagnetic coil produces is proportional to the applied current and 2) the size and spacing of the coils can be varied in order to produce different types of fields. The first characteristic allows the magnetic field strength to be easily varied and even turned off completely. Also, high frequency field modulation is limited only by the maximum frequency of the coil current (approximately Hz depending on the coil inductance). The second characteristic allows systems to be easily designed for specific applications including Helmholtz

16 7 style coils which produce a uniform field, Maxwell style coils which produce a fixed fieldgradient, and coils which produce an approximately dipole-shaped field. Coil systems have been shown to be capable of driving a magnetic tool containing a single magnetic dipole with 5-DOF [5, 9]. One such system is shown in Figure 2.2. If the magnetic device has a more complicated magnetization profile, full 6-DOF has been demonstrated [10], although the utility of these devices for practical applications is limited. Electromagnetic systems have also been used to simultaneously control multiple magnetic implements in two or threedimensions [11, 12]. A potential issue with electromagnetic systems is that the high current required for strong field generation results in a significant temperature rise within the coils due to Joule heating. This heating often requires active cooling solutions, and can result in increased workplace temperature, making this type of system undesirable for heat-sensitive applications such as biomedical procedures involving cells. Figure 2.2: OctoMag: an electromagnetic system that uses eight coil inputs to generate fields and field gradients for 5-DOF untethered magnetic control. Image from [5].

17 Permanent Magnet Systems As an alternative to electromagnetic coils, permanent magnets can be used as the field source. Permanent magnets produce a magnetic field using no input power, resulting in no heat generation near the workspace. Additionally, relative to electromagnetic devices, permanent magnet systems are able to generate stronger fields and field gradients by a factor of approximately 10 to 20, and 2 to 3, respectively, depending on the workspace size [25]. The static dipole-shaped field produced by a permanent magnet introduces some limitations for this type of control relative to electromagnets, however. The field can never be turned off, necessitating additional safety precautions; the permanent magnets must be physically manipulated in order to vary the applied field which limits the maximum field frequency; and the relationship between the control inputs and outputs is more complicated. Therefore permanent magnet systems are best suited for applications that are heat-sensitive, or with large field and field gradient requirements although with relatively lower frequencies. There is currently a permanent magnet system used for catheter steering that is in clinical use whereas many of the other permanent magnet systems proposed by researchers can be divided into two general categories: 1) systems composed of one or more rotating magnets that are used to generate a rotating magnetic field and 2) systems containing a single permanent magnet that is manipulated in order to control the position and orientation of a capsule endoscope within the body. The Niobe Stereotaxis system, shown in Figure 2.3, is used to generate a uniform magnetic field for catheter steering in clinical cardiac procedures [7] using two enormous NdFeB permanent magnets positioned beside the patient. The two opposing magnets can be positioned in order to produce a uniform field in any direction with magnitude of approximately 80 to 100 mt inside a spherical workspace of size 20 cm diameter. This field is used to steer a magnetically tipped catheter inside the blood vessels and heart. Since the field is uniform, no magnetic force is induced on the catheter tip so the catheter is advanced manually or with the use of an automatic advancement system. Compared to traditional catheter procedures in which the catheter is manhandled through the blood vessels, magnetic catheter steering has the potential to reduce tissue trauma, allow more precise probe placement, and shorten procedure times.

18 9 Figure 2.3: The Niobe Stereotaxis system which consists of (a, b) two large permanent magnets positioned adjacent to (c) the patient s table. Additional features include (d) a fluoroscopic scanner for positional feedback and (e) the automatic catheter advancement system. Image from [7]. A commonly proposed use for permanent magnetic actuation systems is the production of a rotating magnetic field in order to drive micrometer- or millimeter-scale helical swimming and rolling robots, shown in Figure 2.4. Helical swimmers are spiral-shaped devices that can be driven through a liquid environment in 3D by applying a rotating magnetic field perpendicularly to the desired direction of movement. Rolling robots are typically cube- or sphere-shaped and can be rolled end-over-end on a horizontal surface by applying a rotating magnetic field that is parallel to the desired motion. Figure 2.4: Microrobot actuation methods that use applied rotational fields B to produce motion in the direction of v: (a) helical swimming and (b) planar rolling.

19 10 An effective way to generate this rotating magnetic field, with clinically-relevant magnitudes, is by using a single rotating permanent magnet [13-16]. The complex shape of the dipole field produced by a permanent magnet leads to a non-intuitive relationship between the pose of the rotating permanent magnet, and the plane in which the rotating field is produced. Due to this complex relationship, the placement of the permanent magnet relative to the workspace is typically constrained to be perpendicular or parallel to the rotational axis, although a method to determine more general arrangements has been developed [17]. In addition to producing a rotating field, a single rotating magnet will also induce a magnetic force on the micro-device. Techniques to manage this force include setting the rotational speed of the magnet such that the force applied on the micro-device is in a favorable direction (i.e. aligned with the direction of micro-device motion) [18] or using two paired magnets that rotate in sync to produce an approximately uniform field at the position of the micro-device with double the strength of a single magnet system [19]. A slightly more complex system with the ability to generate a rotating field that is approximately uniform across the entire workspace has been presented by Zhang et al. [20]. This system uses multiple rotating permanent magnets that are spaced evenly in a 2D ring surrounding the workspace. Figure 2.5: Examples of systems with (a) one permanent magnet, (b) dual permanent magnets, and (c) multiple permanent magnets, all of which produce rotating magnetic fields. One magnet from each system has been circled in red. Images for (a), (b), and (c) are from [17], [19], and [20], respectively.

20 11 Another potential application for permanent magnetic actuation systems is improving the effectiveness of wireless capsule endoscopy by manipulating a single magnet outside of the body. Wireless capsule endoscopy is a diagnostic procedure for disorders in the gastrointestinal tract performed using a small pill-shaped camera that is swallowed by the patient. Traditional wireless capsule endoscopy, in which the camera moves passively through the body, is limited by the random orientation of the camera during its journey as well as the slow completion time. The non-uniform field created by a single permanent magnet is able to both adjust the heading of, and induce a magnetic force on a magnetic capsule endoscope. Therefore a single permanent magnetic source that is maneuvered outside the body can be used to orient the camera to ensure that targeted images are obtained, as well as pull the camera in order to repeatedly image the same area or complete the procedure more quickly. The simplest way of realizing this control is by maneuvering the permanent magnet by hand. An investigation into this method by Keller et al. [9] found that driving the capsule endoscope in this manner has a learning curve, but ultimately results in a more thorough set of images. Another system uses a single permanent magnet that is positioned by hand but can be rotated mechanically to more easily orient the capsule within the body [21]. Alternatively, the permanent magnet can be mounted to the end of a multi-degree-of-freedom robotic manipulator and maneuvered automatically to increase the precision and reliability of the capsule movement relative to a manually positioned magnet, although at the cost of additional procedure time [22, 23]. The methods described here are able to achieve 4-DOF control of the capsule endoscope (2- DOF heading control, and only 2-DOF position control since the capsule is always in contact with a tissue surface).

21 12 Figure 2.6: (a) Handheld and (b) robotically-manipulated permanent magnet systems for 4-DOF control of a capsule endoscope. The permanent magnet actuator from each system has been circled in red. Images for (a) and (b) are from [9], and [22], respectively. Full 5-DOF control of a mock-up capsule endoscope containing a single magnetic dipole has been shown by Mahoney and Abbott in [3] using a permanent magnet mounted to the end of a 6-DOF robotic manipulator. The added capability of this system compared to the 4-DOF capsule positioning systems described above is control over the vertical position of the capsule above the tissue surface which is achieved using feedback of the capsule position obtained by optical cameras. This 5-DOF control is the highest of any existing permanent magnet control system, although this system does have some potential limitations. The device requires an expensive 6-DOF robotic manipulator which exhibits potentially hazardous translating movement (i.e. the robotic arm may harmfully collide with the workspace). Another limitation is that the field magnitude scales with the applied magnetic force, and therefore small force applications (i.e. to move the capsule downwards) result in less robust heading control. For situations that require small applications of magnetic force, the corresponding susceptibility to heading disturbances may invalidate the fundamental assumption that the capsule heading is aligned with the field. In this situation, both the heading and position of the capsule may become unstable since the capsule is no longer aligned with the field and the applied force varies with the capsule heading.

22 13 Figure 2.7: Single, robotically-manipulated, permanent magnet system that is capable of full 5-DOF control of a mock-up capsule endoscope. The permanent magnet has been circled in red. Image from [3]. 2.2 Rotating Permanent Magnet System To address the limitations of these existing systems, we propose a new method to achieve full 5- DOF control which uses permanent magnets that rotate in place. Unlike a roboticallymanipulated single magnet system such as that shown in [3], the proposed system is composed of multiple permanent magnets, each with the ability to be rotated about its own fixed axis, independently of the other magnets. This system configuration is similar to [20] but 3D magnet positioning combined with nonparallel rotational axes and independent magnet rotation improve the control output from 2D uniform fields to 3D fields and field gradients. Specifically, we show that this new system can be used to generate magnetic fields and gradients in any direction with

23 14 strengths comparable to or exceeding those of existing electromagnetic and permanent magnet systems. Each magnet rotates about its volumetric center, hence the new system contains no translating components and the rotational motion can be realized using inexpensive DC or stepper motors. Thus this system is a simple, low-cost option for untethered magnetic control Chapter Organization In this thesis, Chapter 3 presents the details required to operate and construct a general rotating magnet system. Techniques for determining the rotational positions of the magnets that produce a desired field and force are described. An optimization framework for designing systems for specific applications is presented. In Chapter 4, the design and capabilities of the prototype system are given. System characterization and experimental trials demonstrating three typical methods of microrobotic motion are presented. Also, a preliminary calibration technique for determining magnet positions and rotational axes is shown. Chapter 5 provides details on the upper limits of the field and force output for this type of system. Finally, conclusions and future work directions are given in Chapter 6. Results from these chapters are adapted from published work [32, 38].

24 15 Chapter 3 Control Using Rotatable Permanent Magnets 3.1 Magnetic Actuation Background The untethered micro-device that is to be controlled is assumed to contain a permanent magnet with moment m d and to be located at position p d in the workspace. The torque T exerted on this magnetic moment when subjected to an applied magnetic field with flux density B at point p d is given by T = m d B (p d ). (3.1) This magnetic torque, when unopposed, will orient the magnetic moment in the direction of the applied magnetic field. For device applications in a liquid environment at low rotational speeds, the magnetic moment is able to quickly align with the field. In these cases, the magnetic moment can be assumed to be always aligned with the field, and therefore the device heading can be controlled simply by adjusting the direction of the applied field. The rotatable permanent magnets that are used for device actuation (henceforth referred to as actuator magnets) are approximated as point dipole sources located at the volumetric center of the magnets. The error associated with this approximation is less than 1% for cubic magnets located at least two side lengths from the workplace [24]. The volumetric center of the i th permanent magnet p i is defined using spherical coordinates: D i cos(α i ) sin(φ i ) p i (D i, α i, φ i ) = [ D i sin(α i ) sin(φ i )] (3.2) D i cos(φ i ) where D i is the distance of the i th permanent magnet from the workspace center, and α i and φ i denote the azimuth and inclination angles, respectively. 15

25 16 Figure 3.1: (a) Schematic image showing N = 3 actuator magnets, as well as the magnetic moment of the i th magnet m i, the direction of which is defined by the motor spin angle θ i about its rotational axis ω i. The magnet center points are defined using spherical coordinates (D i, α i, φ i ). (b) The position of the microdevice p d relative to the position of the i th magnet p i is denoted r i. The magnetic field B at point p d in the workplace is given by the linear addition of the dipole fields from all N actuator magnets as N B = μ 0 m i 4π r i 3 (3r i r i T I ) m i (3.3) i=1 where μ 0 = 4π 10 7 Tm A -1 is the permeability of free-space, I is the 3x3 identity matrix, r i is position of the micro-device relative to the center of the i th permanent magnet r i = p d p i, m i is the magnetic moment of the i th magnet, and r i and m i are unit vectors such that r i = r i r i and m i = m i m i. The actuator magnetic moment unit vector m i can be parameterized by the rotational position of the i th magnet θ i (henceforth referred to as motor angle) as

26 17 cos(θ i ) m i(θ i ) = R zyi [ sin(θ i )] (3.4) 0 where R zyi is a zy Euler angle rotation matrix defined by two rotation angles β i and ζ i which correspond to rotations around the z and y axes, respectively. by The change in the field with respect to the position, known as the field gradient G, is given G = B = [ B x N B y T B z ] = 3μ o m i 4π r i 4 (m ir i T + r im it [5r ir i T I](m i r i)). i=1 (3.5) The force F exerted on the magnetic device with moment m d at location p d from the field gradient produced by the actuator magnets, assuming no current flowing in the workspace, is given by F = Gm d N = ( 3μ o m i 4π r i 4 (m ir i T + r im it [5r ir i T I](m i r i)) ) m d. i=1 (3.6) 3.2 Determining Required Control Inputs In order to control a device with 5-DOF, the orientation and position of the micro-device are adjusted by changing the magnetic field and force, respectively. As shown in equations (3.3) and (3.6), the field and force that are applied to the micro-device are a function of the magnetic moment direction m i of each actuator magnet, which in turn varies with the motor angle θ i as

27 18 described in equation (3.4). This framework provides the forward relationship between the motor angle control inputs θ = {θ 1 θ 2 θ N } T and the field and force control outputs (i.e. for a given permanent magnet configuration, if the device position and motor angles are known, then the field and force that act upon the micro-device are straightforward to determine). For control purposes the reverse relationship needs to be determined in as little time as possible (i.e. what is the set of motor angles that produces a desired field and force). Due to the nonlinear relationship between the motor angles and the field and force outputs, linear algebra techniques cannot be used to determine the required inputs as they can be with electromagnetic systems. Instead the motor angles that produce a desired field and force can be found using one of three methods: as the solution to a nonlinear objective function, as a solution to a linear objective function with nonlinear Lagrangian terms, and by considering the pseudoinverse of the rate Jacobian. The three methods produce similar results and have similar limitations and therefore the first method will be presented in detail and the others discussed more briefly Minimizing a Nonlinear Objective Function The motor angles that produce a desired field B 0 can be found by minimizing the error between the field vector produced for a given set of motor angles B (θ) and the desired field. Similarly, a desired force F 0 can be determined by minimizing the error between the force vector F(θ) and the desired force. Since a set of motor angles is required that simultaneously produces a desired field and force, this problem can be considered a multi-objective optimization which can be formulated using a linear scalarization of the two component functions: min f = K B (θ) B (1 K) F(θ) F 0 2 (3.7) where K is used to equally weigh the field and force error by accounting for the difference in the units of measurement, 0 < K < 1.

28 19 For an arbitrary permanent magnet configuration and arbitrary desired field and force vectors B 0 and F 0, this optimization problem is non-convex with one or more local minima. The existence of multiple local minima increases the difficulty of finding the globally-optimal solution. For magnet configurations with few actuator magnets, a brute force search over every combination of magnet angles can be used to approximately determine the motor angles that produce a global minimum (depending on the coarseness of the search). However, as the number of actuator magnets increases, the time required for this approach becomes prohibitively long because the number of function evaluations increases exponentially with the number of magnets. Even considering a modest search coarseness of 10 increments of the motor angles, one full rotation of each magnet would correspond to 36 function evaluations and therefore a brute force search for a system with N actuator magnets would require 36 N function evaluations. Instead, starting from some initial motor angle guess a gradient descent method is performed which uses the gradient of equation (3.7) to determine a small change in motor angles that reduce the function value, and these small motor angle steps are performed iteratively until a local minimum is reached. The gradient of equation (3.7) gives the change of the objective function with each motor angle at the current motor angle state: f θ = 2 [ K(B (θ) B 0 ) T K(F(θ) F 0 ) T ] [ J B(θ) J F (θ) ] (3.8) where J B (θ) and J F (θ) are the rate Jacobians that locally relate a small change in motor angles to a small change in the field and force that are produced, respectively. The 1 N gradient given by equation (3.8) is equivalent to the N-dimensional motor angle directions in which the maximum increase in the objective function occurs at the current motor angle state. Therefore, iteratively considering small changes in motor angle corresponding to the negative direction of the gradient can be used to find a local minimum of the function. For many convex optimization problems, the 1 st order function approximation provided by the function gradient results in an inefficient search near the function minimum because the

29 20 gradient may point almost orthogonally to the shortest direction to the minimum point (i.e. the search zigzags instead of travelling directly toward the minimum). Therefore, a commonly used technique is the Newton-Raphson method which uses the function hessian to determine a 2 nd order approximation of the function. The increased time required to calculate the hessian at each iteration is offset by the improved step direction which reduces the total number of steps required (and therefore the time). An issue with non-convex optimization problems, such as the one being considered here, is that the step direction provided by the Newton-Raphson method is not necessarily directed towards a local minimum but instead towards the closest 2 nd order derivative zero (i.e. the closest local minimum, local maximum, or saddle point). In these cases, the direction provided by the gradient is used and the time required to calculate the hessian is wasted. Therefore, the Newton-Raphson method when applied to the non-convex optimization problem given by (3.7) results in an increase in the optimization time compared to the gradient descent method and hence the gradient method is preferred. The function gradient provides the direction of the change in motor angles and the step size is calculated using the backtracking line search method. Once the direction is determined, this backtracking line search is used to find a step size that roughly minimizes the objective function in the given direction. An initially large step size is iteratively reduced until a step magnitude is found that adequately reduces the objective function based on the magnitude of the local function gradient. The gradient descent continues until the gradient magnitude is smaller than some tolerance value indicating a local minimum has been reached. Gradient descent iterations can be repeated from multiple starting points in order to find potentially superior local minima. As more local minima are found, the likelihood of finding a set of motor angles that exactly produce the desired field and force increases, however, for use in a feedback controller, control outputs are needed quickly to ensure control over the device is not lost, and therefore in general there will be insufficient time to find an arbitrarily small error for the motor angles. Instead, the search is halted once an acceptably-accurate field and force are obtained. The acceptability of the result is determined by comparing the magnitude error and angle error between both B (θ) and B 0 as well as F(θ) and F 0 to a user-controlled threshold error

30 21 value (in practice, several percent of the full magnitude and within a few degrees of the desired angle). A further consideration for feedback control is that the field and force applied to the device will fluctuate as the magnets are rotated from one set of motor angles to the next. For systems using motors with limited speed, this phenomenon can have a large effect on the position and orientation of the device during these transitions. To minimize this effect, a Δθ term could be added to (3.7), with the purpose of reducing the change in motor angles relative to the previous set of angles at each instance of the control update. A similar formulation proposed by Andrew Petruska involves reformulating the problem as a constrained quadratic optimization [37]. The error between the desired field and desired gradient is minimized instead of the desired force. Each permanent magnetic is treated as two linear and orthogonal dipole sources and therefore the field and gradient error formulation is linear, however, nonlinear Lagrangian multipliers must be introduced to incorporate the constraints. Ultimately, this method performs similarly to the nonlinear optimization described above Inverting the Rate Jacobian An alternate way to determine the motor angles that produce a desired field and force is by using the pseudoinverse of the rate Jacobian at a particular motor angle state. If the magnetic microdevice is assumed to be aligned with the field, (3.4) can be substituted into (3.3) and (3.6) to produce the nonlinear formula that gives the magnetic field and force as purely a function of the motor angles for a known arrangement of actuator magnets and micro-device location [ B F ] = [B (θ) ] = BF(θ). (3.9) F(θ) Taking the time derivative of (3.9) yields

31 22 [ B ] = [ J B(θ) F J F (θ) ] θ = J BF (θ)θ (3.10) where B, F, and θ are the rate of change of the field, force, and motor angles, respectively, and J BF is the 6 N Jacobian matrix computed by differentiating (3.9) with respect to θ. The Jacobian is a function of the fixed actuator magnet configuration, as well as the current state of the motor angles and micro-device position. The matrix J BF can also be used to approximately map small changes in the motor angles to small changes in the field and force: [ δb δf ] J BF(θ)δθ. (3.11) Assuming that J BF (θ) is full rank (determining the Jacobian rank will be discussed in Section 3.3.2), the Moore-Penrose pseudoinverse of J BF (θ) can be used to determine the required change in the motor angles that reduce the error between the current field and force relative to the desired field and force: δθ J BF (θ) + [ δ (B 0 B (θ)) ]. (3.12) δ (F 0 F(θ)) All three motor angle determination methods have a common limitation; the resultant motor angles that are found depend on the starting angle guess from which the search is initialized. An ideal set of motor angles may be found if the search is started from one initial guess but not another (although conducting multiple searches from many different starting guesses reduces this likelihood). The use of any of these three methods cannot guarantee with certainty that an arbitrary, desired field and force can be exactly produced. However, the nonlinear objective function method is able to reliably find motor angles that produce adequate fields and forces for feedback control as exhibited in Section A more rigorous method to exactly determine the motor angles independently of the starting guess is ongoing work.

32 Control Capability Metrics for System Optimization There are a number of ways to quantify the control capabilities of an arbitrary configuration of actuator magnets. Here we present several metrics that can be used to define an optimal system for a given application. The strength of the magnetic fields and forces that can be produced within the workspace is an important consideration for most applications. The ability to produce isotropic fields and forces ensures that control of the micro-device is not limited in some directions. The smallest singular value of the rate Jacobian relating motor angle speed to the time rate of change of the field and force output gives an approximate measure of the maximum motor rates required. More specific application-dependent system fitness measurements can be defined as well such as the region of uniform workspace size or the maximum inter-magnetic torque between the actuator magnets. In this thesis, two separate fitness metrics will be considered for measurement and optimization of the control capabilities of a given magnet configuration: 1) a combined weighting of the strength and isotropy of the force and field generation, and 2) the minimum singular value of the rate Jacobian Strength and Isotropy of Field and Force Production The isotropy and strength of the fields and forces that a system is able to produce can be calculated based on the maximum field and force that can be generated in a number of sample directions. For each sample direction, the maximum field that the system can generate while simultaneously applying a zero magnitude force is determined, as well as the maximum force that can be generated for a number of microrobot orientations while simultaneously applying a low strength field aligned with the microrobot heading. These maximum field and force samples are denoted B sm and F sm respectively. The strength of the field production capability of the system can be taken as the average of the field sample magnitudes and similarly, the force production capability is equal to the average magnitude of the of the force samples: B str = AVG(B sm ), (3.13)

33 24 F str = AVG(F sm ) (3.14) where AVG(B sm ) and AVG(F sm ) denote the average of the sampled field and force magnitudes, respectively. There is no simple formula to calculate the isotropy of the sampled magnitudes (B sm and F sm ) that is suitable for use as an optimization metric. An ideal isotropy measurement would have the following characteristics: 1) Be unaffected by uniform scaling (i.e. the isotropy of B sm = [1 2 3] should be equal to that of B sm = [2 4 6]). 2) Have a lower bound corresponding to 0% isotropy. 3) Have an upper bound corresponding to 100% isotropy which occurs when every sampled magnitude is identical. 4) Provide a semi-intuitive relationship for the variance between the sampled magnitudes. With these criteria in mind, an initial candidate considered as an isotropy metric was one minus the average percent difference between the sampled magnitudes and the mean sample magnitude: isotropy = 1 1 n ( v i AVG(v) ) AVG(v) n i (3.15) where v is the array of sampled field or force magnitudes, AVG(v) is the mean of v, and n is the number of elements of v. This metric fulfills criteria one, three and four (an isotropy value of 0.85 corresponds to an average percent difference of 15% between the sampled magnitudes and the mean sampled magnitude). When used with relatively capable systems, where the number of control inputs is greater to or equal than the number of desired control outputs, every sampled magnitude is likely to be greater than zero (i.e. the system can make a field or force in every sampled direction) and this isotropy measure performs adequately. However, when used for less capable systems that are not able to produce a desired output in every sampled direction, the average percent difference between the sampled magnitudes and the mean magnitude can be

34 25 greater than one and therefore the isotropy measure becomes negative. As the number of sampled magnitudes n increases, the lower bound on this isotropy measurement goes to negative infinity. Therefore this isotropy measure does not satisfy criteria two. This problem was solved by designing an isotropy measure that is bounded between 0% and 100% and is approximately equal to the isotropy given by equation (3.15) at isotropy values greater than 0.7 (which is a rough lower bound on the isotropy of a relatively capable system). The isotropy measures are given as: B iso = (1 SD(B 1.6n B 0.4 sm ) B str n B 1 ) (3.16) F iso = (1 SD(F 1.6n F 0.4 sm) F str n F 1 ) (3.17) where SD(B sm ) and SD(F sm ) denote the standard deviation of the sampled field and force magnitudes, respectively, and n B and n F are the number of samples contained in B sm and F sm, respectively. One metric to quantify the overall performance of a desired configuration of rotating magnets is the weighted sum Q of the field and force strength and isotropy: Q = K 1 B str + K 2 B iso + K 3 F str + K 4 F iso (3.18) where K 1, K 2, K 3, and K 4 are used to weigh the constituent fitness metrics Minimum Singular Value of the Rate Jacobian Another way to quantify the control capability is to consider the smallest singular value of the rate Jacobian J BF as given in equation (3.10). A full rank Jacobian at every motor angle state and micro-device position indicates that singularity-free control over the field and force per unit time is possible. The rank of the Jacobian at each state can be determined using a singular value decomposition; J BF is full rank if the smallest singular value is larger than zero. However, as the

35 26 smallest singular value approaches zero, the maximum required motor angle speed goes to infinity so a full rank Jacobian at every motor angle state and micro-device position would theoretically imply singularity-free control can be achieved, but in practice the required motor angle rates may be unachievable. To determine the minimum singular value, the columns and rows of J BF are scaled to produce a non-dimensional Jacobian J BF that maps changes in motor angle speed to non-dimensional changes in field and force per unit time J BF (θ) = [ 1 B max I 0 0 J 1 BF (θ) (3.19) F I max ] where I is the 3x3 identity matrix, 0 is a 3x3 matrix of zeros, and B max and F max are equal to the maximum field and force, respectively, that the system would be able to produce if the total magnetic volume of all N actuator magnets was concentrated at a single point p distance from the workspace center. (The formulation of B max and F max is given in Appendix A.) The nondimensional Jacobian J BF has the same rank as J BF and a singular value decomposition of J BF yields unit-consistent singular values [6]. The smallest unit-consistent singular value for many motor angle states and micro-device positions also provides a measure of system fitness because this value is an indicator of the maximum motor rotation speed that is required in worst-case control scenarios near singularities. The relationship between maximum motor speed and minimum singular value is determined using the following method. The pseudoinverse of J BF can be used to find the motor rotational rates that are required to produce a desired non-dimensional field and force per unit time at the current motor angle state and microrobot position. For a random system state (the system configuration being considered consists of six permanent magnets spaced evenly around the workspace), the pseudoinverse of J BF is used to calculate the maximum motor angle rate required to produce a unit magnitude, non-dimensional field and force per unit time in a random direction. The minimum singular value of J BF is also determined for this state. This process is repeated for thousands of different system states and unit magnitude, non-dimensional field and

36 27 force rates and the relationship between maximum required motor speed and minimum singular value is shown in Figure 3.2. The maximum required motor speed (measured in radians per second) roughly scales with the reciprocal of the minimum singular value. Repeating this test for system configurations with more than six magnets produces similar results. Figure 3.2: Smallest singular value of the control Jacobian and corresponding maximum motor rate required to produce unit magnitude field and force rates plotted for 100,000 random motor angle states and micro-device positions. The maximum motor rate is roughly equal to the reciprocal of the smallest singular value. 3.4 Minimum Number of Actuator Magnets Required for 5- DOF Control The minimum number of rotating permanent magnets required to achieve the desired control DOF is a valuable piece of information when designing a new system. The minimum number of control inputs for different levels of untethered magnetic control has been well-defined for electromagnetic systems but is less clear when using permanent magnets as the field source. The minimum number of statically positioned electromagnetic coils required for different levels of untethered magnetic control has been thoroughly investigated [26]. Full 5-DOF control

37 28 requires a minimum of eight coil inputs. In some special cases, 5-DOF control can be achieved using only seven coil inputs but such systems may not have robust heading control during small force applications, and a non-magnetic restoring force (i.e. gravity) is required for stability. The minimum number of inputs required for permanent magnet systems is less clear. Permanent magnet systems are briefly discussed by Petruska and Nelson in [26], ending with the conclusion that the nonlinear relationship between the non-static pose of the permanent actuator magnet(s) and the resultant field and field gradients reduces the minimum number of control inputs by two relative to an electromagnetic device for force related applications. This claim is substantiated by the robotically-actuated, single permanent magnet system developed by Mahoney and Abbott in [3] that is able to achieve 5-DOF control with only five control inputs instead of the seven required by static electromagnetic systems (although similarly exhibiting the potential heading robustness issue and non-magnetic restoring force requirement). Therefore a rotating magnet setup should require a minimum of six actuator magnets in order to achieve 5- DOF actuation with full heading control at any force magnitude, and no restoring force requirement (compared to the eight inputs required by electromagnetic systems). A preliminary investigation into the control capability based on the number of actuator magnets supports this claim. This investigation was conducted for setups with five, six, eight, ten, and twelve magnets, and the system fitness was measured using the two methods described in the previous section. Typical system fitness values for non-optimized arrangements of the actuator magnets are shown in Table 3.1. Although the number of actuator magnets is varied in these cases, the magnet-workspace separation distance and total magnetic volume is held constant (and is equal to that of the prototype system). These system fitness measurements were made using 20 sample field directions, 144 sample force directions (for a micro-device with dipole moment of 10-6 Am 2 and varying orientation) and 1500 combined motor angle states and micro-device positions for finding the minimum unit-consistent singular value.

38 29 Table 3.1: Performance Comparison for Five-, Six-, Eight-, Ten-, and Twelve- Magnet Systems Number of Actuator Magnets Bstr (mt) Fstr (µn) Biso (%) Fiso (%) Smallest Singular Value The Jacobian for any configuration with N 5 actuator magnets has fewer than six singular values, which means that singularity-free control of the field and force per unit time is never possible. Correspondingly, the N = 5 configurations were unable to produce a force in every sample direction, resulting in low force isotropy for these systems. The minimum singular value of the Jacobian for the N 6 configurations is non-zero (for the 1500 test cases), although the motor speeds required at some states may be undesirably high. A more rigorous examination of the minimum number of actuator magnets required for singularity-free control at every system state is ongoing work. The smallest singular value, as well as the strength and isotropy of the outputs increase dramatically with the number of actuator magnets from N = 5 to N = 8, but increasing N greater than eight has a diminishing effect on further improvements. An advantage, however, to using more actuator magnets is that the size of the solution set for a particular desired field and force is increased, i.e. a field and force can be generated using a larger number of different actuator motor angles. This additional solution space makes it easier to minimize the change in motor angles between control updates.

39 System Parameter Optimization for Improved Control The system fitness, as defined either by the combined strength and isotropy of the field and force production (given in Section 3.3.1) or the minimum singular value of the rate Jacobian (given in Section 3.3.2), can be used in an objective function to optimize the system parameters in order to design a rotating magnet system for a targeted application. Alternative fitness metrics can also be considered but will not be discussed here. In this section, a general optimization framework for rotating magnet systems will be presented as well as more specific methods for each of the two fitness metrics. Optimization results will be given for multiple unconstrained rotating magnet setups and for two types of constrained systems: 1) a pseudo-medical application in which the magnets are restricted by the position of the patient and feedback device and 2) a more extreme limitation on the allowable magnet positions representing applications where the actuator space is largely inaccessible General Considerations for System Optimization In an optimization of the system control capability, potential choices for the optimization variables include the positions of the centers of the actuator magnets, the direction of the rotational axes of the magnets, the number of magnets, and the magnitude of the dipole moments of the magnets, which is proportional to the magnet volume. The magnitude of the field and force produced by each magnet scale linearly with the dipole moment as given in equations (3.3) and (3.6), respectively, and therefore any increase in dipole moment magnitude will result in an increase in field and force strength. In practice, however, the dipole moment will be limited by the size of the actuator magnets that are available. Therefore the dipole magnitude is an unsuitable variable when considering the field and force strength as the optimization metric. Also, the non-dimensional Jacobian J BF is normalized using the B max and F max terms as shown in equation (3.19), so the minimum unit-consistent singular value is independent of the dipole

40 31 magnitude. The relationship between control capability and the number of actuator magnets was analyzed in Section 3.4. The optimization results discussed hereafter will be for system configurations with N = 8 actuator magnets with fixed dipole moment magnitudes but with variable positions and rotational axes. A major consideration for this optimization is the large size of the remaining parameter search space. The position and rotational axis of each actuator magnet can be defined using three and two parameters, respectively, for a total of five variables per magnet. Placing the magnets closer to the workspace increases the magnitude of the field and force generation, as described in equations (3.3) and (3.6), respectively. If all the magnets are placed at an equal separation distance from the workspace (the minimum distance is typically necessitated by the physical workspace constraints and the dipole approximation spacing), each magnet position can be defined using two spherical coordinates reducing the number of variables from five to four. This constraint was implemented for the optimization trials resulting in a search space for an eight magnet configuration with 32 dimensions. The optimization trials were performed using the MATLAB fminsearch algorithm. This algorithm uses the Nelder-Mead simplex method which is inefficient when optimizing over a large number of variables [27]. One way to reduce the search complexity is to use a coordinate descent algorithm to iteratively optimize over a smaller search space until convergence is achieved for the full optimization problem. In practice this was done by optimizing over the four free parameters of a single magnet at one time while holding the parameters of the other seven magnets constant (hereafter referred to as a coordinate descent iteration). A coordinate descent iteration was performed for each of the eight magnets in sequence repeatedly until convergence was reached. This method has similar convergence properties to a steepest descent algorithm performed over all the variables simultaneously [28] and therefore is suitable for finding a local optimal solution near the starting configuration. Due to the nonlinearity of the fitness functions and the large search space, it is unlikely that the globally-optimal solution will be found, so the search is ended when a local optimum is reached. Four non-optimized system configurations were considered as the initial setups for the optimizations. These initial setups include: 1) the prototype system described in Section 4.1.3; 2) magnet centers equally spaced on cube vertices;

41 32 3) magnet centers randomly positioned but equally spaced; 4) magnet centers arbitrarily positioned. The initial rotational axes for setups 2, 3, and 4 were arbitrarily selected Maximizing Combined Field and Force Strength and Isotropy For system applications requiring large magnetic fields and forces, one way to design a suitable rotating magnet system is to optimize the system parameters in order to maximize the weighted sum of the strength and isotropy of the field and force production given by equation (3.18). Although equation (3.18) can be optimized using the fminsearch function, it requires the maximum force to be sampled for many different robot directions to form the F sm vector and this step represents the majority of the calculation time required to measure the fitness. The total optimization time can be greatly reduced if F sm is not calculated at all and the system fitness is calculated solely on the ability to produce strong and isotropic fields by considering only the portion of the total system fitness given in the objective function Q B = K 1 B str + K 2 B iso. (3.20) In most of the cases analyzed, as the field production capabilities are improved, so too are the force production capabilities. In other words, optimizing the fitness of a system as given by (3.20) usually improves the isotropy and strength of both the fields and forces that can be generated. The objective function given in equation (3.20) was calculated using ten representative field samples which are indicative of the system s ability to produce a field in every direction. To ensure that the system fitness given by equation (3.18) was monotonically increasing during the optimization of (3.20), a more thorough check of the total system fitness was completed using 20 sample field directions and 144 combined force and microrobot orientation directions after each coordinate descent iteration. In instances where the total system fitness failed to increase after a coordinate descent iteration, this configuration change was discarded and the optimization was

42 33 continued using the parameters of the next magnet in the sequence. The optimization was considered to have reached convergence once the total fitness failed to increase for five of the eight coordinate descent iterations in a given sequence. The system fitness, normalized field and force strengths, and field and force isotropies are shown in Figure 3.3 for each coordinate descent iteration of the manually optimized prototype setup. Figure 3.3: System fitness values at the conclusion of each coordinate descent iteration during the optimization starting from the prototype system parameters. The symbol represents instances where the optimization failed to find an improvement in total system fitness. The field and force strength have been normalized as a percentage of the maximum field and force that can be produced by the total actuator magnetic volume placed at a single point R distance from the workspace.

43 34 The numerical values of the system fitness components before and after the optimization are given in Table 3.2 for the four starting configurations that were considered. The field and force strength are improved by approximately 10 to 50% depending on the initial fitness values while the isotropy increase is capped at about ten percentage points. Table 3.2: Strength and Isotropy of the Control Outputs Pre- and Post- Optimization of the System Parameters Q Bstr (mt) Fstr (µn) Biso (%) Fiso (%) Starting Configuration Initial Final Initial Final Initial Final Initial Final Initial Final In situations where the desired application requires strict constraints on the positions of the magnets, the optimization technique described above has the ability to find non-intuitive setups with relatively high fitness that are not likely to be found through manual manipulation of the system parameters. For example in a medical application, the actuator magnets would need to be located some minimum distance from the patient and be positioned such that the imaging system is unobscured. Optimization trials with pseudo-medical application constraints are considered here; specifically the magnets centers cannot be placed within an ellipse with major and minor axis of 4.5 cm and 3.5 cm, respectively (to approximately represent a scaled-down human trunk)

44 35 nor placed within a truncated cone with radius of 4 cm (to represent the field of view required by the imaging system. These two constraints are shown graphically in Figure 3.4. For these trials, the workspace separation D can be varied for each of the eight actuator magnets individually because the cylindrical constraint provides the minimum separation distance. Figure 3.4: Constraint boundaries representing the patient trunk and imaging system field of view for a pseudo-medical application. Three starting arrangements are considered, each with the magnets placed as close to the workspace as allowed by the constraints. The optimization results are shown in Table 3.3. The fitness values cannot be directly compared to the results given in Table 3.2 since the workspace separation D is not constant for each magnet. The increase in workspace distance results in a

45 36 decrease in optimized field and force strength by approximately 30%. Field and force isotropy results are similar to those shown above. Table 3.3: Strength and Isotropy of the Control Outputs Pre- and Post- Optimization of the System Parameters for Systems with Constraints Related to a Pseudo-Medical Application Q Bstr (mt) Fstr (µn) Biso (%) Fiso (%) Starting Configuration A B C Initial Final Initial Final Initial Final Initial Final Initial Final To represent an application that requires extreme restrictions on the magnet positions, consider constraining the azimuth angle of the magnet centers to be between 0 and 90 (i.e. a bird s eye view of the setup would show all magnets placed in the first quadrant). The initial configuration consists of actuator magnets evenly spaced within the first quadrant. All magnets are the same distance from the workspace and therefore the results of this trial can be directly compared to the non-constrained cases. Performing the automated optimization with this constraint yields a system with fitness components given in Table 3.4. The control capabilities of this highly constrained system are approximately equal to those of the prototype system despite the extreme limitation on the magnet positions. This result also demonstrates the rotating magnet

46 37 system concept presented in this paper has the ability to achieve a high level of control in applications where a significant portion of the actuator space is inaccessible. Table 3.4: Strength and Isotropy of Control Outputs Pre- and Post-Optimization For Highly Constrained System Parameters Constrained Optimization Initial Final Prototype System Q Bstr (mt) Fstr (µn) Biso (%) Fiso (%) Maximizing the Smallest Singular Value For system applications in which the maximum motor speed is a more important consideration compared to the field and force strength, the system parameters can be optimized in order to increase the minimum unit-consistent singular value of the non-dimension Jacobian J BF given in equation (3.19) in order to reduce the maximum motor rates required during operation. The smallest singular value was calculated for 100 motor angle states at 15 micro-device locations consisting of the workspace center as well as 14 equally spaced points that define a sphere of radius 5 mm. The optimization metric was taken as the smallest minimum singular value from these 1500 states. The coordinate descent method described above was used to optimize over the parameters of each of the magnets individually in order to reduce the size of the search space. A

47 38 non-constrained optimization was performed on the same four initial setups as the nonconstrained optimizations performed previously. The minimum unit-consistent singular values for the systems before and after the optimization are shown in Table 3.5. The minimum singular values are improved by a factor of roughly 1.5 to 4. Table 3.5: Minimum Singular Value Pre- and Post- Optimization of the System Parameters Minimum Singular Value Starting Configuration Initial Final The analysis completed in Section shows that the maximum required motor speed roughly scales with the reciprocal of the minimum singular value. Therefore improving the minimum singular value by a factor of approximately 1.5 to 4 results in a decrease of the maximum required motor speed by roughly 40 to 70% in worst-case scenarios near singularities.

48 39 Chapter 4 Case Study Proof of Concept Prototype Device A prototype device was constructed in order to demonstrate the feasibility of using rotating permanent magnets to generate fields and forces for 5-DOF control of microrobots. The analysis conducted in Section 3.4 indicates that 5-DOF control is theoretically achievable using only six actuator magnets, however the prototype system was designed with eight magnets for better conditioned control capabilities and greatly reduced motor speed requirements for worst-case scenario control conditions. The magnets are rotated using stepper motors which are fixed in place using a custom-made framework of high density fibreboard (hardboard). This section will present the design considerations for the prototype system, the basic system capabilities, verification of the control model, and experimental results showcasing the different methods of microrobot locomotion that can be achieved. Figure 4.1: Photo of the rotating magnet prototype system. 39

49 Design Considerations and System Details The following section explains the design choices made for the prototype system, including the actuator magnets, motors, structural frame and encoders. The components required to operate the motors are given and the motor set point driving algorithm is explained. Finally the system parameters defining the position and rotational axes of the magnets in the prototype system are given along with the corresponding control capabilities Physical Components Potential choices of permanent magnet type for the prototype system include ferrite, alnico, samarium-cobalt, and neodymium. Neodymium magnets were selected due to the stronger field production per unit volume compared to the other types. Neodymium magnets have a high coercivity and the field produced by adjacent magnets is not strong enough for demagnetization and therefore the magnetic hysteresis in this application is non-existent. The ideal shape of the actuator magnets is spherical because the magnetic model used to derive the control algorithms uses the assumption that each magnet produces a perfect dipole field, which is true for spherical magnets. Spherical magnets, however, are impossible to mount to the motor shafts in a secure but non-permanent way. The field produced by non-spherical magnets can be calculated exactly using a more complicated multipole expansion [24], but can also be approximated as a simple dipole field at sufficient magnet separation distances. Cubic magnets were chosen for this application since cubes have the lowest dipole approximation error of the commonly available magnet shapes (consisting of rectangular prisms and cylinders with different aspect ratios) and can be mechanically fixed to the motor shafts. The magnet volume is proportional both to the magnitude of the field and field gradients that are produced as well as the cost of the magnet. Relatively large cubic magnets of size 2.54 cm 2.54 cm 2.54 cm with dipole moments of 16.6 Am 2 were selected because they can produce fields and gradients with application-relevant strengths at a sufficient distance from

50 41 the workspace (15 mt field and 0.75 T/m gradient at 6 cm separation) which allows the motors to be more easily positioned without physical interference. Larger magnets are available but would require more powerful motors to overcome the increased inter-magnetic torque and would be more expensive. Potential motor choices were evaluated based on torque output, speed, and cost. Stepper motors were chosen over DC motors for three reasons: 1) a static power input will cause stepper motors to hold a constant angular position with maximum torque, 2) the intrinsic positional control that stepper motors exhibit is more suited to the motor angle determination algorithm compared to the velocity control that DC motors use, and 3) keeping track of the step inputs would allow the motors to be driven without encoders to reduce the system complexity (this assumption turned out to be false). NEMA 23-size, two-phase, hybrid stepper motors capable of 0.39 Nm stall torque and no-load speed of 600 RPM were selected for their moderately high torque and speed capabilities and low cost. The magnets are mounted to the motors using custom-built enclosures make of hardboard. The magnet position is fully constrained within the enclosure. The opposite end of the enclosure is attached to a motor hub which is used to fix the enclosure onto the motor shaft. The face of each motor is mounted to a hardboard faceplate that is attached to the base of the system using two cantilever support pieces (also custom designed hardboard). The rotational axis of each magnet is defined using two spherical coordinates: an azimuth angle β, and an inclination angle ζ, which are set by the angle between the support pieces and the base, and the angle between the faceplate and the support pieces, respectively. The height of the support pieces and length of the magnet enclosure are chosen to position the magnet center correctly in the workspace. This structural configuration is easy to design and assemble but results in somewhat limited magnet positions in order to ensure no physical interference between the structural components.

51 42 Figure 4.2: Physical components required for each actuator magnet of the prototype system. The actuator magnet (a) is attached to the stepper motor (b) using a magnet enclosure (c). The stepper motor is bolted to the faceplate (d) which connects to the base of the system using two cantilever support pieces (e). One cantilever support piece is transparent in this image so the inner details can be seen. During initial control experiments, the stepper motors were operated with open loop control based on the current step count. A major issue with this technique is that inter-magnetic torques between actuator magnets can be greater than the torque supplied by the motor causing the motor to rotate away from its set point position (referred to as motor diversion). Due to the open loop control of the angular position, any instance of motor diversion results in loss of the motor position and hence ruins the experimental trial. This problem was initially mitigated by running the motors at a small fraction of their top speed in order to increase the torque output, however, the low motor speed severely limited the control update rate. A more effective solution involved operating the motors at higher speeds and accounting for motor diversion using encoders that provide feedback of the angular motor position. Since the encoders were added after the system had been designed and built, the encoder choice needed to be compatible with the existing system. Finding a suitable encoder was difficult

52 43 because the motor shaft on the front of the motor is largely inaccessible due to the closely positioned structural faceplate and magnet enclosure while the motor shaft on the rear of the motor is recessed and can be accessed only through a cut-out in the motor casing. A suitable encoder was found in the AS5040 Rotary Sensor produced by AMS. This encoder requires a small disc shaped magnet to be attached to the end of the motor shaft and four field sensors positioned above this magnet are used to determine the absolute angular position. This encoder has a resolution of 10 bits which corresponds to angular position errors of ± 0.2 which is less than the 0.9 stepper motor increments. The encoder signal is robust to magnetic interference from alternate sources (such as the larger actuator magnets located in the workspace). Finally, these encoders are small enough to be mounted onto the rear side of the motors, are not prohibitively expensive, and are a fitting way to determine angular position feedback for this rotating magnet system Motor Set Point Driving The magnetic control algorithm given in Section 3.2 requires the magnets to be rotated to specific angular positions in order to produce the desired fields and forces. The power signals provided to the stepper motors to achieve this rotational motion are generated by two Sparkfun Quadstepper motor driver boards. Each Quadstepper board is able to provide the power signals for four stepper motors simultaneously and draws power from an AC/DC power converter. The required voltage and current outputs from the AC/DC power converter can be determined by looking at the required inputs of the stepper motor. The rated power inputs for each two-phase stepper motor, provided by the manufacturer, are a rated voltage and current of 5.7 V and 1 A per phase (for a total of 2 A total), respectively, while a maximum voltage of 30V can be supplied to each Quadstepper. Although providing the motors with current greater than the rated current will cause damage, a higher than rated voltage will not cause damage and will allow the current to ramp up faster in order to achieve higher step rates compared to using the rated voltage. At higher voltages the motors also draw less current. Therefore a power converter with a high voltage output (but less than 30V) and adequate current output is required. A suitable

53 44 power supply was found in the PMT-24V150W1AA converter which is capable of 24 V and 6.5 A output. At 24 V input voltage, each motor draws roughly 0.87 A of current and therefore the current output required to drive the four stepper motors from a single Quadstepper board is 3.48 A. Two power converters are required, one to provide the power to each of the two Quadstepper boards used. The Quadstepper board requires three digital input signals, (denoted ENB, DIR, and STP) in order to control the motor stepping of each motor. The ENB signal acts as an on/off switch to allow power to the coils, the DIR signal indicates the rotational direction (clockwise or counter clockwise), and the STP signal is a square-wave that corresponds to individual motor steps. The angular displacement of each motor step is also set by the Quadstepper board. The default amount is 1.8 per step but this amount can be iteratively halved in order to get steps equal to 0.9, 0.45, 0.225, etc. Smaller steps result in slower motor rotations but increased motor torque. The prototype system uses 0.9 steps for slightly increased torque. Twenty-four digital output signals are required to control the motor stepping of eight stepper motors simultaneously, and these digital signals are provided by a digital input/output (DIO) module that is controlled using a computer. The DIO module is used to both send the command signals to the Quadstepper boards and to read the angular position provided by the encoders. The AS5040 encoder output mode used for this application is the synchronous serial interface. This method requires two digital output signals and one digital input signal, denoted SEL, CLK and DAT, respectively, from the DIO module for each encoder. The process of reading the encoder position is started by setting SEL low. CLK is set low and then high which shifts out one bit of data which can be read on the DAT signal. CLK is repeatedly set low and high in order to read the ten bits of absolute angular position data given by the DAT signal. Once DAT has been read ten times, SEL is set high and the process can be begun again to read the next angular position. Determining the angular position of the eight motors requires 16 digital output signals and 8 digital input signals to be provided by the DIO module.

54 45 The DIO module chosen for this application is the Accessio USBDIO-48 which has 48 channels that are independently selectable for digital inputs or outputs (separated into six ports of eight channels each) and is capable of processing approximately 4000 commands per second for the six channels on a single port. Twenty-four of these channels are set as digital outputs in order to provide the ENB, DIR, and STP signals for each of the eight motors. Every motor step requires a minimum of three commands since a single motor step is generated when the STP pin is set low, then high, then low. Therefore at 4000 commands per second, and considering 0.9 motor steps, the maximum rotational speed for a single motor is approximately 3.3 rotations per second or 200 RPM. The remaining twenty four channels are split into 16 digital output channels and 8 digital input channels in order to read the eight encoder values. Determining a single encoder position requires 32 total input/output commands to be completed and therefore the encoder positions can theoretically be updated at a rate of approximately 125 Hz. The digital input and output signals that are required to run the motors and read the encoders are generated by a PC running Ubuntu Linus using custom C++ code. The frequency of each square wave STP signal provided to the Quadstepper board sets the frequency of the corresponding motor steps. If the motors are stopped and a square wave frequency corresponding to the fastest motor speed is applied, the motors will not have sufficient inertia to match this top speed and will not move. The fastest motor speed can only be achieved if the motors are accelerated from rest up to this top speed by linearly increasing the frequency of the square wave STP signal. This motor acceleration is handled using the AccelStepper Arduino library that has been modified for compatibility with the custom C++ code used to interface with the DIO module. When given a desired motor angular position, the AccelStepper library calculates the square wave STP signal profile that will accelerate the motor to the desired set point and sends the low and high STP outputs at the required times. The AccelStepper library is able to control the eight stepper motors simultaneously.

55 System Configuration and Capabilities The prototype system was designed by manually varying the positions and rotational axes of the eight actuator magnets in order to improve the combined field and force strength and isotropy system fitness given by equation (3.18). The separation distance between the magnet centers and the workspace center was set at a constant value of D = 7.5 cm. This distance was chosen as a compromise between maximizing the magnitude of the field and force generation while limiting the inter-magnetic torque that would have to be overcome by the motors, allowing for the components to be placed without physical interference, and ensuring a sufficient workspace separation to justify the dipole approximation which in this case has an error of less than 0.2% [24]. Since the workspace separation distance is the same for each of the eight actuator magnets, the positions and rotational axes can be more concisely defined using spherical coordinates: the azimuth and inclination angles for the magnet positions and rotational axes are denoted by (α, φ) and (β, ζ) respectively, and are given in Table 4.1.

56 47 Table 4.1: Positions and Rotational Axes Defined using Spherical Coordinates for the Eight Actuator Magnets in the Prototype Setup Magnet Positions (deg) Rotational Axes (deg) α φ β ζ A moderately good system fitness was achieved despite a number of configuration constraints, most notably the limited motor placement positions that result from the simple structural pieces used to mount the motors to the base. The fitness metrics for the prototype device are B str = 31.2 mt, F str = 0.85 µn, B iso = 90.7%, F iso = 82.9%, and a minimum singular value of The prototype is able to produce magnetic fields and field gradients in every direction with magnitudes of at least 30 mt and 0.83 T/m, respectively. For a spherical workspace of approximately 5 mm diameter, the field is uniform within 10% and 2 of the nominal magnitude and orientation, respectively, when a gradient of zero magnitude is requested. Stronger gradients reduce the volume over which the field is uniform. For applications requiring a larger workspace, the position of the micro-device must be tracked in order to determine the field and force at the correct location. The theoretical top motor speed and encoder update rate using the DIO module is 3.3 rotations per second and 125 Hz, respectively, although

57 48 in practice simultaneously sending motor pulses and reading the encoder values reduces these capabilities by approximately 50% to 1.5 rotations per second and 67 Hz, respectively. The inter-magnetic torque between the i th and j th actuator magnets T ij can be found by considering a modified version of equation (3.1): T ij = m i B j (p i ) (4.1) where m i is the moment of the i th actuator magnet and B j (p i ) is the field produced by the j th magnet at position p i. The maximum inter-magnetic torque between any two magnets can be found using equation (4.1) by considering every combination of the two motor angles. Only the component of the torque that is aligned with the rotational axis will have to be overcome by the motor torque. For any actuator magnet, this process can be repeated for each of the seven other magnets in the workspace and the sum of these seven inter-magnetic torques can be used to determine the upper bound on the total torque that the corresponding motor will experience during operation. The largest upper bound on inter-magnetic torque for the eight actuator magnets is 0.34 Nm Heat Generation One of the advantages of using permanent magnets instead of electromagnets is that permanent magnets do not generate heat. The prototype system, however, uses stepper motors to rotate the magnets and these stepper motors are positioned relatively close to the workspace and do generate heat. Dongsub Shim, a fourth-year student, conducted an investigation regarding the heat generation of the prototype system as part of his thesis project [36]. The purpose of the investigation was to compare the heat generation for two magnetic control devices in the Microrobotics Lab at the University of Toronto: the rotating magnet system shown here, and the 3-axis Helmholtz electromagnetic coil system presented by Zhang and Diller in [33]. The results of this analysis will be summarized below.

58 49 The thermal analysis was conducted for stepper motors that have full voltage delivered to both coil phases and therefore exhibit a maximum amount of heat generation. The steady state temperature in the workspace was measured both experimentally and numerically using SOLIDWORKS Flow Simulator. The steady state workspace temperature increased from ambient 22 C to approximately 28 C. The majority of this heat is generated by a single motor that is positioned almost directly beneath the workspace. This temperature increase is small compared to that of the coil system with full current input which results in a workspace temperature of around 83 C. The report found that introducing a modest amount of forced convection from air applied by a fan at 1 cm/s reduces the workspace temperature of the rotating magnet system to 22 C. Techniques for reducing the heat transfer in future versions of this system with more powerful motors will be discussed in Section Experimental Results The magnetic model presented in Section 3.1 that relates the motor angles of the eight actuator magnets to the field and force that are produced is verified though the demonstration of static field and force production in two separate tests. The effectiveness of the control input determination method presented in Section in which the motor angles are taken as the solution to a non-convex optimization is demonstrated using several proof-of-concept field and force-application experiments Magnetic Model Verification The static field generated by the system is verified by requesting a 30 mt field in eight directions as well as a field of zero magnitude and comparing this desired field to the field produced by the system at the center of the workspace, measured using a single-axis gaussmeter (model 425, Lakeshore) in the x, y, and z directions. Table 4.2 shows the desired field, the average measured field for two trials, the magnitude ratio of desired field to measured field, and the angle between

59 50 the desired and measured field. The misalignment and magnitude difference between the desired field and measured field is small, less than 3.35 and 4.6%, respectively. These errors are likely due to fabrication and position errors in the laser-cut prototype frame since the errors predicted from the model are less than 0.09 and 0.3%. Table 4.2: Comparison of the Measured Field to the Desired Field of 0 mt and 30 mt in Eight Different Directions The static force produced by the system is characterized by measuring the heading of a small magnetic device as it is subjected to a desired force. The device used for this test is a cubic neodymium magnet with side length equal to 250 µm and the test is conducted in a horizontal container filled with silicone oil with a viscosity of 350 cst. The position of the device is detected using a stationary camera (FO134TB, Foculus) and the opencv library. The device appears dark against the backlight and the grayscale image is passed through a threshold function in order to isolate the device. A contour finding algorithm is used to detect the outside edge of the device within the image range of interest and the center of this contour is taken to be the position of the device. This tracking is done at 60 fps. The micromagnet is maneuvered away from the walls of the workspace and then held stationary by applying a zero force. Once stationary, a set of motor angles is found that result in a desired force direction and magnitude. After the motors have completed the rotation to the set of motor angles, the heading of the micromagnet over time is measured and compared to the

60 51 requested force direction. Three different force directions were tested (x, y, and xy) and the heading error averaged over five trials per heading direction was found to be 4.6, 5.2, and 5.3, respectively. The speed of the micromagnet during each experiment varies between trials, an effect likely due to changing friction and viscous drag from dragging the magnet along the bottom of the container Experimental Control Results The feasibility of using a rotating permanent magnet system for control of small-scale robotic devices is shown for several control experiments that showcase different modes of microrobot locomotion including 1D open loop helical swimming, 2D feedback-controlled rolling pathfollowing, and 3D feedback-controlled path following using gradient pulling. The ability to dynamically generate magnetic fields for micro-device control is demonstrated by two experiments. For the first, a helical, millimeter-scale swimmer was driven using a rotating field. The swimmer was assembled using a steel spring (length 6.4 mm, diameter 3 mm, period 1.3 mm) attached to a spherical NdFeB magnet head (diameter 1.9 mm) with magnetic moment oriented perpendicular to the spring axis. The helical swimmer was immersed in 350 cst viscosity silicone oil inside a tube with inner diameter 4.1 mm. A rotational field was applied in the plane perpendicular to the tube axis causing the swimmer to rotate and screw through the liquid. A magnetic force of zero magnitude was requested during the experiment. The linear speed of the swimmer was measured for field rotation frequencies of 0 to 1.6 Hz, as shown in Figure 4.3. For applied field frequencies up to roughly 1.5 Hz, the swimming speed increases with frequency at approximately 0.02 body lengths per second times the frequency in hertz, although the relationship is not exactly linear most likely due to intermittent contact between the swimmer and the tube wall. This approximately linear relationship between speed and field frequency is similar to the results given by [29] and [30] of and body lengths per second times frequency in hertz for millimetre-scale swimmers actuated in 500 and 1000 cst silicone oil, respectively. At frequencies above 1.5 Hz, the magnitude of the magnetic torque is not sufficient to keep the swimmer synchronized with the field frequency so the

61 52 swimmer velocity decreases. This is known as the step-out frequency [29]. The maximum rotating field frequency that the prototype system is able to generate is approximately 2 Hz while a highly engineered system with high torque, DC motors would be capable of producing rotational fields with frequency greater than 100 Hz. Figure 4.3: (a) Image of the 6.4 mm helical swimmer used to demonstrate rotational field production. (b) Swimmer speed as a function of applied field frequency from 0 to 1.6 Hz. The second proof of concept experiment involved rolling a 250 µm cubic magnet in a 2D path-following demonstration by applying rotational fields with no magnetic force application. The average path deviation and speed for five trials was 102 μm and 149 μm/s, respectively. The outcome of a typical trial is shown in Figure 4.4. The rotational frequency of the applied field was around 0.2 Hz.

62 53 Figure 4.4: A typical result for rolling a 250 µm magnet on a horizontal surface. The micromagnet position has been low-pass filtered. (a) Path of the micromagnet in black and the goal points and desired path in red. Elapsed time at each goal point is indicated. The micromagnet deviation from the path (b) and speed (c). The dynamic force generation capabilities of the prototype are demonstrated in a 3D feedback control experiment. The task was to pull the 250 µm micromagnet along a predetermined path defined by seven goal points using magnetic forces. The position of the micromagnet was obtained from the top and side-view cameras at a rate of 60 Hz. The required change in motor angles at each control update was reduced by limiting the change between

63 54 consecutive desired force vectors. For example, the large change in desired force vector direction after a goal point was reached would require a large change in motor angles. Instead, the desired force was decreased to zero as each goal was approached, then increased in the direction of the next goal point. This approach reduced the average change in motor angle between control updates to less than three degrees. In addition, a constant, vertical, magnetic force offset was applied to counteract the weight of the micro-device. The magnitude of this vertical offset force was found by driving the micro-device to the center of the workspace and manually tuning the gain value until there was no vertical motion. The heading of the micromagnet was set by requesting a field constant in magnitude (7 mt) and direction but with allowable error up to 5 mt and 12 in order to increase the speed of finding a suitable solution to equation (3.7) in the shortest amount of time. For each control update, the allowable force error is initially set at 5% and 5 relative to the magnitude and direction of the desired force and the maximum change in motor angle is set to 10. If a suitable solution to equation (3.7) is not found in s, the allowable force error is iteratively increased to a maximum of 10% and 7.5 and the maximum change in motor angle is increased to 60. Using the simple gradient-descent search algorithm, the average computation time for one control update was s. The direction of the requested force during each control update is determined using a simple path following algorithm. The desired force is chosen such that the micro-device is driven along the path towards to the next goal point and perpendicularly back to the path to reduce the deviation error. The direction is given by d F = (1 K 1 )(K 2 )d G + (K 1 )(1 K 2 )d P (4.2) where d F is a unit vector in the direction of the desired force; d G is a unit vector in the direction of the next goal point; d P is a unit vector from the microrobot back to the path; K 1 is gain value that increases as the perpendicular distance from the micromagnet to the path increases, 0 < K 1 < 1; and K 2 is a gain value that can be used to tune the relative amount of path following, 0 < K 2 < 1. The magnitude of the desired force can also be modified to affect the micromagnet

64 55 motion. In order to tune the values of K 2 and the desired force magnitude, multiple trials of a short 3D path were completed using a range of gain parameters. During these trials, the average micromagnet path deviation and speed along the path, defined as the perpendicular distance between the agent and the path, and the path length divided by the completion time, respectively, were analyzed. For 3D path following, random instances of motor diversion, as described in Section 4.1.1, can cause a large deviation of the micromagnet from the path. After the motor diversion is detected by the encoder, the motor will be driven back to its set point and the micromagnet will return to the path, however, the deviation during this interval can be quite large compared to the rest of the trial. The large path deviations that result from randomly occurring motor diversions (which, for identical trial parameters, may occur multiple times in a single trial or not at all) can produce wildly varying average path deviations and path speeds between trials despite identical gain values. Therefore in order to determine a clear relationship between the gain parameters and the path deviation and speed, any portion of each trial in which a motor diverted was omitted from the analysis. The results of this experiment are shown in Figure 4.5.

65 56 Figure 4.5: Path deviation and speed as a function of desired force magnitude and path following gain K 2 for gradient pulling of a micromagnet in 3D. The results for ten trials of each parameter value are shown as open circles along with the average of the ten trials with filled circles. Points marked by * represent a single trial. This analysis omits any section of the trials during which a motor diverted. As the K 2 gain is decreased from a value of 1, path following is weighed more heavily over waypoint following and the average deviation decreases. At K 2 values lower than 0.7, however, the micromagnet starts to overshoot the path resulting in an increase in average deviation and a decrease in path speed. For values of K 2 much smaller than 0.5, the motor speed is not fast enough to achieve the rapid changes in desired force direction and the micromagnet oscillates around the path making no progress. Additionally the incidence of motor diversion increases as K 2 is decreased; this result is omitted from this set of tests but will have an effect on the comprehensive path following results given below. The magnitude of the desired force has a negligible effect on the path deviation but shows an approximately linear relationship with path speed. The results indicate that choosing K 2 to be 0.7 and force magnitude to be 0.5 µn will produce results with minimal deviation and the quickest path speed.

66 57 For the full 3D path following demonstration, the path deviation and speed were determined for the entirety of each trial even in the presence of motor diversion. Path following was conducted using two different silicone oil viscosities: 350 cst and 1000 cst. In the 350 cst trials, instances of motor diversion can cause large path deviations and therefore the K 2 gain was set at 0.85 to reduce the diversion frequency. The average deviation across ten trials was 38 µm and the average velocity 580 µm s -1. In the 1000 cst trials, motor diversion causes smaller deviations so the K 2 gain was set at 0.7. The average deviation across ten trials was 25 µm and the average velocity 310 µm s -1. The outcome of a typical feedback control test conducted in 1000 cst silicone oil is shown in Figure 4.6. The average deviation for this single trial is 22 µm.

67 58 Figure 4.6: A typical feedback result for a 250 µm magnet performing path following in the three dimensions. The micromagnet position has been low-pass filtered. (a) Path of the micromagnet in black and the goal points and desired path in red. Elapsed time at each goal point is indicated. The micromagnet deviation from the path (b) and speed (c). 4.3 Calibration of Magnet Positions and Rotational Axes When physically constructing a rotating magnet system, small errors in the positions and rotational axes of the magnets can cause the field and force at the micro-device position to differ from the expected value. Positional deviations of 5 mm for the volumetric centres of the actuator magnets result in average errors in the expected field direction and magnitude of approximately 6 and 7%, respectively, and average errors in the expected force direction and magnitude of 13

68 59 and 12%, respectively. Similarly, rotational axis deviations of 10 result in average field errors of 6 and 5% and force errors 8 and 7% for the directions and magnitudes, respectively. Therefore, if the construction process used to make the rotating magnet system is likely to introduce errors in the positions and rotational axes of the magnets, a calibration technique to determine the true system parameters is needed in order to reduce the field and force error during operation. A preliminary calibration of the prototype system was conducted using the following method. The position and rotational axis of each magnet was calibrated individually by removing the other seven magnets from their fixtures in the workspace. A gaussmeter was used to measure the field produced in the x, y, and z directions at the center of the workspace by the magnet during three full rotations. An optimization was performed over the magnet position and rotational axis in order to reduce the root mean squared error between the expected field and the measured field. This optimization was conducted using the MATLAB fminsearch algorithm using the expected magnet parameters as the initial guess for the optimization variables. The average difference between the expected field and measured field for the non-calibrated prototype setup is 0.16 mt. Performing this calibration for the prototype system yields relatively small changes in the magnet positions and rotational axes: approximately 1 mm and 5 on average. However, this small change in the calibrated parameters reduces the average field error from 0.16 mt to 0.08 mt and the maximum field error from 0.41 mt to 0.22 mt. This basic calibration method performs adequately for calibrating the parameters of the prototype system, however, it can be improved. The method presented here is only able to find very local solutions to the optimization that is performed, and therefore requires the starting parameter guess to be very close to the actual parameters. Taking field measurements from a single position in the workspace does not provide any information about the field gradient which is required to more accurately determine the positions of the magnets. Also, calibrating each of the actuator magnets requires the other magnets to be removed from the setup which is time consuming and may reintroduce errors when reattached. Development of a calibration method that can be completed for all the magnets simultaneously using field measurements from multiple positions is ongoing work.

69 60 Chapter 5 Considerations for Future Systems 5.1 Maximum Field and Field Gradient Strength The optimization results in Section 3.5 show that the field and force production capabilities of a system can be moderately improved by adjusting the positions and rotational axis of the actuator magnets. These optimization trials, however, do not consider the effects of varying the actuator magnet size or workspace separation distance. In this section, the upper limit on the output capabilities of the system that can be achieved by modifying the magnet shape, magnet volume, workspace distance, and motor choice, will be determined. Specifically, the system output will be quantified by the minimum field and field gradient magnitude that can be produced in every direction. As a baseline for the field and gradient strength, consider the optimized prototype system configuration given in Table 3.2 with no constraints on the magnet positions. This system is capable of generating a field and field gradient with magnitude of at least 37.5 mt and 1.01 T/m, respectively, in every direction. This field and gradient strength are an approximate upper bound on the capabilities for a rotating magnet system that uses eight magnets of volume 16.4 cm 3 with 7.5 cm workspace separation (i.e. the magnet volume and spacing of the physical prototype system). Adjusting the magnet volume and spacing will also change the magnitude of the intermagnetic torque, potentially necessitating the use of more powerful motors. For this baseline system, the upper bound on the inter-magnetic torque is 0.35 Nm. The scaling relationship between the system parameters and the field, field gradient, and inter-magnetic torque magnitude can be analyzed in order to determine the upper limits on the system outputs. This analysis assumes that the relative positions of the magnets remain unchanged. If all the actuator magnets have the same moment m and same workspace 60

70 61 separation D, equations (3.3), (3.5), and (4.1) can be used to determine the following proportionalities: B m m, G D3 D 4, and T m m (5.1) D 3 where B is the field strength, G is the gradient strength, and T is the inter-magnetic torque. The magnetic moment m is proportional to the magnet volume V and therefore the relationships given in equation (5.1) can be simplified to B V D 3, G V D 4, and T V2 D 3. (5.2) If the system is scaled uniformly by a constant C (i.e. the distance D increases proportionally to C and the volume V increases relative to C 3 ) then the proportionalities given in equation (5.2) reduce to: B 1, G 1 C, and T C 3. (5.3) As an example, consider a system that has been uniformly scaled by a factor of C = 2. The magnets would have eight times the volume, the workspace distance would be doubled, the field output would remain unchanged, the gradient output would be halved and the inter-magnetic torque would increase by a factor of eight. Conversely consider a system uniformly scaled by a factor of C = 1. The field output remains unchanged, the gradient output is doubled and the inter- 2 magnetic torque is decreased by a factor of eight. It is clear that for uniformly scaled systems, it is advantageous to consider smaller magnet volumes positioned closer to the workspace. Analyzing the effect of uniform scaling shows the importance of positioning the magnets as close to the workspace as possible even as the magnet volume is decreased. The system parameters, however, do not need to be uniformed scaled; the workspace separation can be decreased and the magnet volume increased in order to increase the field and gradient production at the expense of increasing the inter-magnetic torque. The upper limits of the system ability

71 62 occur when the magnets are positioned as close to the workspace as possible and the magnetic volume increased to the maximum physically realizable amount. As discussed in Section 4.1.1, the dipole error associated with non-spherical magnets increases as the magnets are positioned more closely to the workspace. Therefore a system with large magnets that are positioned close to the workspace should use spherical magnets because they exactly produce the field predicted by the dipole model. The size of the workspace has a large effect on the magnitudes of the system outputs that can be produced. The workspace is defined as a spherical volume inside which no portion of the actuator magnets can be positioned. Three workspace sizes of interest have been defined in the literature [25] with radii of 5 cm, 10 cm, and 40 cm, corresponding to proposed applications of in vitro manipulation, manipulation in the brains and eyes, and applications in the heart and intestines, respectively. Note that the workspace separation distance D is defined as the distance between the magnet center and the workspace center and is therefore equal to the sum of the workspace radius and the magnet radius (because the entirety of the magnet must lie outside the workspace). This means that increasing the magnet volume, and therefore the magnet radius, also results in an increase in workspace separation. The maximum physically realizable volume of the actuator magnets for a given workspace size can be approximately determined by considering the ratio between the total cross-sectional area of the N magnets and the surface area of the sphere defined by the workspace separation D (this ratio will henceforth referred to as the area ratio). As the area ratio increases, it becomes increasingly hard to position the magnets without physical interference. The area ratio for the baseline system is 9%. A rough upper limit of 40% area ratio can be seen by analyzing CAD models of the baseline system configuration with different magnet sizes and the corresponding area ratio as shown in Figure 5.1.

72 63 Figure 5.1: Relative magnet spacing of the baseline system configuration for area ratios of (a) 9%, (b) 20%, (c) 35%, and (d) 45%. For each of the three workspace sizes, the magnet radius and minimum workspace separation distance that result in an area ratio of 40% can be calculated. These magnet radii and separation distances can be used with the relationships given in equation (5.2) to scale the baseline field, gradient, and inter-magnetic torque magnitude relative to the new system parameters. The results are given in Table 5.1. Table 5.1: Magnet Radii and Workspace Separations that yield a 40% Area Ratio for Three Different Workspace Sizes and the Corresponding Values of the Field, Gradient, and Inter-Magnetic Torque Magnet Radius (cm) Workspace Separation D (cm) Field Strength B (mt) Gradient Strength G (T/m) Torque T (Nm) Workspace Radius (cm)

73 64 These field and gradient magnitudes are an upper limit on the capabilities of a rotating magnet system with eight permanent magnets. The maximum field and gradient that can be produced for an equivalent electromagnetic system is difficult to determine because the field and gradient strength are dependent on many factors including the shape of the coils, the number of windings, the core material, and the maximum current. Based on the relative output strength for existing permanent magnet and electromagnetic systems, it s reasonable to assume that the maximum field and gradient magnitude that can be generated by an electromagnetic system would be weaker by a factor of 10 to 100 compared to the values given in Table 5.1. For example, consider the abilities of the coil system presented in [6]. This system is similar to the Octomag configuration but scaled-up for a cm 3 workspace (which sets the spherical workspace size to somewhere between the small 5 cm radius and medium 10 cm radius workspaces described above). This system is capable of producing fields and gradients in every direction with magnitudes of 90 mt and 0.1 T/m. A more rigorous comparison can be made between rotating magnet systems and systems that use a single permanent magnet robotically-manipulated above the workspace such as the one presented by Mahoney and Abbott in [3]. If both systems have an equivalent magnetic volume and workspace separation, the field and gradient output of a rotating magnet setup with eight magnets is approximately 40% and 60% weaker, respectively, compared to a single magnet. Each of the eight magnets, however, is only one eighth the volume of the single magnet and therefore can be positioned closer to the workspace in order to increase the field and gradient output. The field, gradient, inter-magnetic torque, and magnet size of the rotating magnet system are shown as a function of area ratio in Figure 5.2 along with the values for a single magnet system with equal magnetic volume and minimum workspace separation (assuming the single permanent magnet is also spherical in shape).

74 Figure 5.2: Maximum field strength, gradient strength, and magnet radii for a rotating magnet system with eight magnets for area ratios between 0 and 40% as well as the upper bound on inter-magnetic torque. System outputs for a single, robotically-manipulated permanent magnet with equivalent magnetic volume is also shown. 65

75 66 The field and gradient strength for rotating magnet systems with area ratios less than 15% are approximately equal to those of single magnet systems. At higher area ratios the rotating magnet output is superior. In reality, the inter-magnetic torque between actuator magnets for the medium and especially the large workspace sizes would pose significant design challenges. This type of system is unquestionably suitable for magnetic actuation in small workspaces due to the high field and gradient strength and surmountable inter-magnetic torques. For medium sized workspaces, some applications requiring large system outputs would motivate the use of this type of system although the system design would have to account for the large inter-magnetic torques and forces. The colossal inter-magnetic torques required in applications with large workspaces would make system design extremely challenging. For large workspaces, a more suitable system choice is likely a robotically-manipulated, single permanent magnet, despite the somewhat limited field and gradient strength capabilities relative to a rotating magnet system. 5.2 Forces and Torques for Typical Magnetic Implements The magnitude of the forces and torques that can be applied to a magnetic tool by the actuation system is dependent on the magnetic moment of the tool and the field and gradient strength of the system. Table 5.2 shows the torque and force that can be applied to four different magnetic implements for the small, medium, and large workspace sizes based on the field and field gradient capabilities for rotating magnets corresponding to an area ratio of 40%. The four implements include the capsule endoscope discussed in [3], the steerable needle discussed in [6], the catheter discussed in [8], and the microparticles discussed in [1]. The soft magnetic implements are assumed to be at saturation magnetization. The torque is calculated assuming the tool moment and the field and are perpendicular, while the force is calculated assuming the tool moment is aligned favorably with the gradient to produce the maximum force.

76 67 Magnetic Implement Capsule Endoscope Table 5.2: Maximum Applicable Forces and Torques on Four Magnetic Implements by a Rotating Magnet System with Small, Medium, and Large Workspace Sizes and 40% Area Ratio of the Actuator Magnets Magnetic Moment (Am 2 ) 5 cm Workspace 10 cm Workspace 40 cm Workspace Torque Torque Force Torque Force Force (N) (Nm) (Nm) (N) (Nm) (N) Steerable Needle Catheter Microparticle Reducing Heat Generation The increased inter-magnetic torque between larger and more closely positioned permanent magnets will necessitate more powerful motors. More powerful motors will result in an increase in heat transferred to the workspace, and therefore heat generation may pose to be a more significant issue compared the small amount of heat generated by the prototype system. However, since the motors do not necessarily have to be positioned near the workspace (unlike the magnetic coils in electromagnetic systems) there are some ways to mitigate this heat transfer. The thermal analysis conducted in [36] found that the majority of the heat transferred to the workspace by the prototype system was generated by a single motor positioned under the workspace. Therefore positioning each of the motors above the workspace will lower the amount of heat transfer. In addition to positioning the motors above the workspace, the motors can be placed further from the workspace though the use of extended motor shafts or drive belts to further reduce the heat transfer. Lastly, as suggested in the thermal analysis conducted in [36], forced convection using a fan is also an effective method for reducing the workspace temperature.

77 68 Chapter 6 Conclusions and Future Work 6.1 Summary of Contributions This thesis presents a novel magnetic actuation system that is capable of untethered control of small-scale robotic devices with 5-DOF. Although this level of control has been demonstrated by existing magnetic actuation systems, the system described here has a number of advantages. The use of permanent magnets as the field source allows for stronger fields and field gradients to be produced relative to electromagnetic systems with no intrinsic heat generation near the workspace. The rotating motion of the permanent magnets used in this system can be realized using a simpler and less expensive setup compared to existing permanent magnet systems. Also the system presented here has no issues regarding heading robustness at low force applications. A number of considerations regarding the design and operation of a general rotating magnet system are discussed. We present a control strategy for calculating the motor angle control inputs that produce a desired field and force output. We show that the magnet configuration can be optimized for a high level of control even in the presence of strict constraints on the positions of the magnets. Finally, we present a calibration method that uses experimental field measurements to adjust the magnet positions and rotational axes in the magnetic model to be more consistent with the physical device. A prototype system consisting of eight permanent magnets is shown to be capable of independent field and force control in a number of experiments involving different methods of microrobotic locomotion. The upper limits on the strength of the field and gradient production for this type of system are analyzed in detail for small, medium, and large workspace sizes. The strong fields and gradients that can be produced for small- and medium-sized workspaces make this type of system superior to existing permanent magnet and electromagnetic devices. For large workspaces, however, finding suitable mechatronic and structural components that can withstand 68

78 69 the massive inter-magnetic torques and forces would be challenging. The use of a rotating magnet system for magnetic actuation is ill-suited to tasks requiring the use of high frequency or uniform magnetic fields, situations where the field must be turned off over the entire space around the system, and applications with a workspace size much larger than 10 cm. This type of system, however, is particularly capable for heat-sensitive procedures requiring strong magnetic fields and forces for full 5-DOF control in small and medium sized workspaces. Potential applications include laboratory experiments such as the manipulation of single cells as well as medical procedures involving larger magnetic implements such as steerable needles inside the brain. 6.2 Future Work One major question to be answered definitively is: What is the minimum number of rotating permanent magnets required to achieve singularity-free 5-DOF control of a micro-device? The minimum number of control inputs required by existing magnetic actuation systems is well described. The linear input-output relationship of electromagnetic systems makes this analysis relatively straightforward [26] whereas the linearized rate Jacobian for the single, roboticallymanipulated permanent magnet system can be shown to be always invertible due to the axial symmetry of the dipole field [3]. The numerical analysis of the rate Jacobian for rotating magnet systems conducted in Section 3.4 indicates that a minimum of six magnets is sufficient, however, this analysis of 1500 combined motor angle states and micro-device positions is not conclusive proof. Ideally, an analytical investigation of the rate Jacobian for a general rotating magnet system with a variable number of actuator magnets could be used to determine the minimum number required. However, due to the complex relationship between the control outputs and the motor angles, and the lack of symmetry to simplify the analysis, a suitably thorough numerical analysis may have to suffice. Other future work involves the development of techniques to improve the precision of the control results. Instead of determining the control inputs as the solution to a non-convex

79 70 optimization problem, a dynamic system formulation can be considered. The nonlinear system state will have nine components (three field, three force, and three micro-device position) and the motor velocities will be taken as the control input. For micro-device actuation at low Reynold s number, the micro-device velocity is proportional to the force. Therefore a series of ordinary differential equations can be used to define the A and B matrices that relate the change in system state x to the current state x and the control input u: x = Ax + Bu. (6.1) The motor velocities u that drive the current system state to the desired system state at every control update can be found using a suitable gain matrix K. A number of test cases can be used to find a K matrix that is always stable. This system formulation may result in the control input calculation being completed in a shorter time and yield more precise fields and forces. This dynamic formulation requires an accurate representation of how the field and force vary with the micro-device position and the motor angles in order to form the A and B matrices in equation (6.1). An analytical relationship between these variables can be formulated, however, it requires the error for physical magnet positions and rotational axes to be low. This motivates the use of a robust calibration technique to exactly determine parameters of the physical setup. Although a preliminary calibration method is presented in Section 4.3, an improved method that is able to precisely calibrate the magnet parameters for all actuator magnets simultaneously using field measurements taken from a number of different positions should be developed.

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84 75 Appendix A Electrical Diagram Figure A.1: Pictorial diagram of the electronic components required to rotate a single magnet including the (a) PMT- 24V150W1AA AC/DC converter, (b) SparkFun Quadstepper motor driver board, (c) Pololu NEMA-23 stepper motor (item number 1476), (d) ACCESS USB-DIO-48 DAQ board, (e) ACCESS spring-cage terminal board, and (f) AS5040 magnetic encoder with custom PCB board.