Uncalibrated trajectory following with magnetically controlled microrobots

Microrobot systems have demonstrated potential in the areas of manipulation, assembly, and therapeutic tasks at the microscale. This paper investigates uncalibrated visual servoing of microrobot devices in a fluidic environment. Planar actuation is achieved with four electromagnets, and both point-to-point motion and trajectory following are demonstrated. The unique contribution of this paper is the implementation of a dynamic recursive least squares estimation algorithm that is used to control the device as it follows a desired path without any calibration or training steps. Tracking accuracy on the order of 1.6 pixels (10.5μm) is demonstrated for a 200μm diameter microrobot. Initial trials successfully achieved micromanipulation of both 50 and 200μm glass beads.


I. INTRODUCTION
Tetherless microrobot technology is advancing to new levels and adapting to an increasing number of fields. A challenge facing microrobots is providing power and control. A range of actuation methods, extending from optical [1] to electrostatic [2] and magnetic, have all been demonstrated. An excellent review and comparison of various magnetically actuated microrobot systems can be found in [3], but a selected number of magnetically controlled systems are reviewed here.
Using electromagnetic actuators with ferromagnetic-corebased, Tamaz et al. [4] developed a proportional-integralderivative (PID) controller capable of navigating a 1.5 mm spherical bead along a predefined path using waypoints. They concluded that an adaptive controller would significantly decrease complications in the system and allow for more robust uses in the biomedical field.
The OctoMag, developed by Kummer et al. [5], is a five degree-of-freedom microrobot controller that has successfully controlled microrobot navigation along preplanned paths while maintaining specific orientations within the workspace.
A 2 mm long microrobot was successfully navigated along a figure-eight path while maintaining its orientation toward the center of the sphere. Another microrobot, 500 µm in length, was successfully controlled along a spiral path defined via waypoints, constantly orienting itself toward the vertex of the spiral. In Keuning et al. [7] used magnetic actuation to control the path of paramagnetic microparticles with diameters ranging from 60 µm to 110 µm. A proportional-integral (PI) controller was implemented and circular and figure-eight paths were demonstrated. When the microparticle was within 10 pixels of a particular waypoint, it would automatically change course to the next programmed waypoint. Average errors ranged from 4.7 µm to7.0 µm with standard deviations on the order of 2.0 µm.
Belharet et al., [8] used an MRI-based predictive controller to navigate a ferromagnetic robot along a path. With the goal of performing minimally invasive cardiovascular tasks, these microrobots used a linear model to eliminate the need for predefined waypoints. The 250 µm microrobot was successful in following these systemgenerated paths. Two dimensional (2D) navigation models with noise resulted in paths following that exhibited a average tracking errors on the order of 230-300µm and 1.2 pixels.
In [9], Pawashe et al. demonstrated model-based learning controllers for micro-object manipulation using side-pushing with a magnetic microrobot operating in a fluid. Modeling included flow velocities induced by the microrobot and equations of motion for the microsphere being manipulated.
All these works required system models and camera calibration. In this paper, we describe a microrobot control scheme that uses uncalibrated visual feedback for an unmodeled system consisting of a microrobot device operating in a fluidic environment observed via a microscope and controlled by four electromagnets. The only a priori information that is required is the number of actuation signals. This controller is developed in Section II.B and demonstrated experimentally in Section IV with both pointto-point motion and path following.

A. System Model for a Microrobot in a Fluid
Consider a ferromagnetic microrobot device surrounded by two electromagnet pairs and suspended in between two fluid layers as shown in Fig. 1. The device is subject to a variety of forces including the magnetic control forces and viscous drag. The position of the microrobot is observed in a microscope and measured within the image plane at some position . If the microrobot device is operating in a low Reynolds number regime, then inertial forces are negligible compared to the drag forces as explained in [10]. The Reynolds number is a dimensionless quantity that relates inertial forces to viscous forces in the Navier-Stokes equations for a body moving in an incompressible Newtonian fluid.

Uncalibrated Trajectory Following with Magnetically Controlled
Microrobots* Jenelle Armstrong Piepmeier, Samara Louise Firebaugh, Caitlin S. Olsen Experimental results presented in [11] and a linear relationship between actuation and for the type of system considered here. Thus, we assume as system such t signal, , the state velocity of the mi observed in the image plane can be expresse where is similar to the composite Jacob in traditional image based robotic control.
Computing a closed form solution for J doing so requires accurate models of the field, the drag coefficients and magnetic microrobot, the vision system, etc. and init training step. Any changes to the phys system, components, device geometry, flui would require a repeated system calibration In lieu of careful modeling and calibratio recursive least squares estimation met estimation of the J matrix. Such unca methods have been successfully implemente manipulators and mobile robots for a variet with more complex nonlinear system mo degrees of freedoms (DOF) [13][14] [15].

B. Recursive Least Squares Estimation
Consider a microrobot system with an obse will vary when the control signal (t) i actuators. The image error between the obse position, , is given by Let represent the cost function, a image error squared as 1 2 Note that may be a constant or a mo on a desired motion trajectory. The control [12] demonstrate device velocities that the control icrorobot as ed as (1) bian matrix used J is possible, but applied magnetic properties of the tial calibration or sical position of id properties, etc. n. on, we propose a thod for online librated adaptive ed in macro-scale ty of applications odels and higher dic environment with icroscope and camera erved state that is applied to the erved and desired (2) a function of the (3) oving target point algorithm should compute a control signal that will m microrobot to the target point . As shown in [15], a quasi-New dynamic Broyden's method will dr zero by servoing the robot to follo The Jacobian estimation scheme us each iteration.
Let Δf represent the change in be the change in the desir actuation signal consist of the a electromagnets. Then the recursive l the Jacobian and the desired computed at the kth iteration by the

End for End
The matrix is the estimate of the actuation signal, and λ is a weigh memory and reduces the effects o the Jacobian estimation. Values of longer memory.
Given a non-pathological probl described by of achievable velo adaptively learns the relationship signal and the robot velocities and desired position. For planar 2DOF p data implies the m=2. For an electro of opposing pairs of magnets, n= calibrate the vision system, com coefficients, or the magnetic field gr calibration or training is required, control various microrobot device s zoom settings.
Such uncalibrated adaptive successfully implemented in macro mobile robots with more complex n and higher degrees of freedoms (DO minimize (8) and drive the . wton method utilizing a rive the cost function to owing the moving target. ses a rank one update at image error fkfk-1, let red position, and let u k actuation signal for the least squares estimate for d control signal , is following algorithm.
, , the covariance matrix of hting factor that effects a of measurement noise on f λ closer to 1 effect a lem (i.e. a desired path ocities), this algorithm between the actuation d drives the robot to the position control the image omagnet array consisting 2. There is no need to mpute the viscous drag radients. Since no system the same algorithm will hapes at arbitrary optical methods have been o-scale manipulators and nonlinear system models OF) [13][14] [15].

C. Practical Implementation
One final consideration regarding the 2 computed in (13) is necessary for imple physical system. It is possible in the first the initial magnitude of the control signal m physical limitations of the electromagnets. signal is scaled such that its magnitud system's capabilities but that the direction a maximum allowable scalar magnitude actuation signal is used.
max This is similar to the trust region metho Jagersand [14] to prevent large motions estimated model's current area of validity. While no modeling is necessary for implementation, there are certain practical are necessary. It is therefore assumed tha been thoughtfully designed such that the strength is sufficient to pull the microrob viscous drag and other fluid interface reactio Furthermore, while a 2DOF system here as described in the following section, extendable to higher degrees of freedom a size of the measurements vector (denote algorithm) is larger than the size of th (denoted as n). For instance, the measurem consist of position and orientation (m=3 vector included an actuation term that w microrobot's orientation.

III. EXPERIMENTAL SYSTEM
The uncalibrated control developed section has been implemented on a system for the Microrobot Challenge held at International Conference on Robotics (ICRA).
A variety of microrobot mo designed for another application) have bee previous work [11]. The microrobot device resembles a throwing star, is 20 μm thick, 200 μm diameter circle as shown in Fig. 2. nickle device was generated by t MetalMUMPS process 1 .
The microrobot operates at the in vegetable oil, and a solution consisting of and sodium bicarbonate dissolved in water, 3. The microrobot device will float suspe (even without fluid A). This provides workspace while fluid A (oil) increases th the microrobot and results in a more con The Reynolds number for this system is ∼10 assumption made to ignore inertial terms is The robots and fluid are in a 20 placed at the center of four cy arrayed along the four points of the is driven with a pulse of amplitude Hz. The duty cycle of the control s 50%. The robots have no permane placed in a magnetic field they magnetization.
A simple actuation scheme w cardinal direction is utilized. Two o (E-28-150-24 model from Magnet used to actuate one of the two con the amplitude or duty cycle of squar each electromagnet, the varying ma the microrobot induces varying microrobot through its workspace. One in each pair is used for po the other is used for negative terms only one magnet from each of the pull the robot in the desired direction array of electromagnets and the imag Visual feedback is used to mea microrobot device using a 740×48 experiments presented in the next was set to 20× with a single pixel System integration is achieved in a with an 8Hz vision update rate. Sim to distinguish the microrobot objec and the centroid of the object is used The magnets are pulse width m cycle determined by the gain term a b 0 × 20 × 15 mm chamber lindrical electromagnets e compass. Each magnet 11 V and frequency 100 signal is varied from 0 to ent magnetization. When y develop an induced with a magnet for each opposing electromagnets tic Sensor Systems) are ntrol signals. By varying re wave input voltages to agnetic field imposed on forces that propel the designs derived from the mask the microrobot as captured by control. system used for visual or actuation.. ositive control terms and . At any given time step, two pairs is actuated to n. Fig. 3 shows the ging system. asure the position of the 80 USB camera. For the section, the microscope corresponding to 6.5µm. a LabVIEW environment mple thresholding is used ct from the background, d as the robot position. modulated with their duty ms adaptively calculated within LabVIEW through the algorithm described in equations (4) through (8).

A. Static Target
First, simple point-to-point motion is demonstrated. The algorithm drives the microrobot to the goal position with a velocity term given as 0. Fig. 4 shows the results for a goal 200 pixels (1.300 mm) away from the starting point. There is no training or calibration, and and are arbitrarily set equal the identity matrix. The term λ is set to 0.99 reducing the effects of system or measurement noise. Fig. 5 demonstrates convergence to the desired position with the error norm in pixels.
The control effort , normalized to a maximum value of 1, is presented in Fig. 6 with each subplot providing the scaled control signals sent to each electromagnet pair (where a scaled effort of 1 represents a 50% duty cycle signal sent to the electromagnet). Notice in this figure that even after the microrobot has reached the goal position, the robot may drift prompting a response from the controller to maintain the desired position. Fig. 7 shows the individual elements of the estimated Jacobian as they are updated over time. As the robot moves through the workspace, both a sign change and magnitude change are observed. This is not surprising since the initial terms are arbitrary and since the magnetic field varies as the robot moves relative to the electromagnets. Once the robot has achieved the goal position, there is little to no change to the Jacobian.

B. Moving Target
To demonstrate the ability to follow a desired path or trajectory, the target was prescribed as follows 100 0.035 0.035 (10) This results in an average velocity of 3.5 pixels/s or approximately 22.8 µm/s. Fig. 8 Shows the trajectory path (superimposed over an image of the robot at the final location) including a portion of the initial approach to the circular path. Once the robot reaches the desired path, it maintains tracking with a steady state error of 1.6 pixels. At this magnification, that corresponds to an average 10.5µm error as shown in Fig. 9.
The scaled control effort is shown in Fig. 10. Compared to the static target case presented in Fig. 6, this control effort varies a great deal as the controller seeks to make small moves keeping the microrobot on the desired path. The target velocity is such that the change in the goal position is on the order of the resolution of the position measurement and the signal is noise dominated. However, the underlying sinusoidal effort is observed with an expected 90° phase shift between the N/S and E/W electromagnet pairs. The variation in the estimated Jacobian elements can be seen in Fig. 11 demonstrating how the algorithm is essentially adapting the control gains as the robot moves throughout the workspace.

C. Bead Manipulation
Potential for small object manipulation led to the development of a trial testing the uncalibrated controller with disturbances. Polystyrene beads with diameters of 50 and 200 µm were placed in the workspace of the star shaped microrobot. The beads naturally rest at the interface between the oil and saline solutions, eliminating the need to control the microrobot along the z-axis. Using the manual control system, the star microrobot was capable of pushing and maneuvering both the 50 and 200 µm beads. The beads do not stick to the microrobot but instead are dragged primarily by the fluid forces surrounding the microrobot. Because of this, even after leaving the immediate vicinity of the microrobot, the beads continued to travel in the general direction of microrobot motion.
A final experiment was conducted where many 50 µm beads were placed in the workspace with the star microrobot. The microrobot was programmed to move in a circle with a 100 pixel radius at a speed of 3 deg/sec. The microrobot ran into a clump of the beads and pushed the beads approximately a quarter of the way along its path. Fig. 12 shows the Jacobian elements as the microrobot encountered the beads.
The figure clearly shows a significant change in the Jacobian elements of the controller when the robot encounters the beads . This shows that the uncalibrated controller successfully adjusted for this unforeseen interference.

V. CONCLUSION
A vision-based control method is applied to successfully drive a microrobot to a desired target. Both point-to-point motion and path following are demonstrated with a steadystate tracking error of 1.6 pixels (10.5 μm). The controller requires no calibration or training phase. This is significant because the same control scheme can be used at different microscopic zoom settings, for different microrobot shapes, and for different viscosity fluids. There is no need for viscous drag modeling or camera calibration which results in a system in which it is extremely easy to test different configurations. This controller was successfully used to push microbeads along a path, highlighting potential for future applications in micromanipulation.