7.2.1 Handedness of achiral microswimmers

Though it is not possible to visually distinguish the geometrical handedness of achiral shapes, the magnetic microswimmers are handed as a result of their magnetic dipole. Fig. 7.2(A) illustrates the handedness of the achiral microswimmers which can be defined based on the direction of their magnetic dipole relative to the orientation of the microswimmers. Much like for the helices, the handedness plays an important role in determining the direction of motion; specifically, whether they will swim forward or backwards due to clockwise or counterclockwise rotation. Unlike for helices, the magnetic handedness of the achiral microswimmers can lead to uncertainties in the swimming direction. For instance, a reversal of rotation direction does not lead to a reversal in swimming direction for some of the microswimmers. This is attributed to geometrical variations due to randomness from the manufacturing process leading to different swimming behavior among microswimmers. As a result, the motion of the achiral microswimmers can be classified into two types of motions, primary motion and secondary motion (Fig. 7.2(B)), where the red arrow indicates the direction of the dipole, the blue arrow is the swimming velocity, and the purple circular arrow is the rotation direction.

7.2.2 Actuation of achiral microswimmers

The achiral microswimmers are actuated via a rotating magnetic field; once actuated the microswimmers convert the rotational motion created by the externally applied torque into translational motion (Fig. 7.1(C)). The translation motion is perpendicular to the magnetic field and is a consequence of the rotation of the microswimmer, not of an externally applied force. The use of magnetic fields includes the benefit of having the ability to permeate over long ranges with minimal health effects and to wirelessly transmit large amounts of power for propulsion, motion control, and localization [10,45].

The exact motion of the achiral microswimmer can be captured by tracking the movement of the individual beads using image processing. The individual beads of an achiral microswimmer have their own distinct helical path. The helical motion of each bead can be expressed as

$\begin{array}{cc}\hfill X(t)& =a_{x}t+b_x,\hfill \\ \hfill Y(t)& =a_y\sin (b_{y}t+c_y)+d_y,\hfill \\ \hfill Z(t)& =a_z\cos (b_{y}t+c_z)+d_z,\hfill \end{array}$

where t is time, and a, b, c, and d are the parameters for the helical trajectory for each bead. These parameters can be determined through curve fitting of experiment data obtained from tracking the individual beads. The equation for $X(t)$ is the same as the forward swimming velocity of the microswimmer and should have the same values for all three beads.

7.2.3 Propulsion of achiral microswimmers

The achiral microswimmer, as the name suggests, is a rigid achiral structure that can be actuated via rotation. When rotated, the microswimmer can swim forward in bulk fluid. Previous studies examined the mechanical properties of the microswimmer; most importantly, the rigidity of the microswimmers was investigated by examining the structural deformation of the microswimmers under stress by the hydrodynamic forces during swimming [40]. Such an examination was done by observing relative positions of the three beads to one another. The examination was done quantitatively using a 3D tracking algorithm that can track the position of each individual bead of any given swimmer. It was confirmed that the deformation of the microswimmers was less than 5% over multiple rotation cycles [40].

The microswimmers' ability to swim can be determined using a symmetry analysis. According to Purcell, a shape that is asymmetrical can swim forward when rotated. This is a simplified rule. A deeper investigation requires an examination of the number of planes of symmetry. According to the paraphrased statement by Purcell, it is intuitive to consider an achiral object to be incapability of swimming due to the existence of planes of symmetry. However, according to Happel and Brenner, an achiral object with two planes of symmetry has been shown to be capable of propulsion [44]. Considering the three-bead achiral microswimmer which has 2 planes of symmetry, the translational velocity v and angular velocity Ω of the swimmer are related to the applied force F and torque N by

$\left(\begin{array}{c}\hfill \mathbf{v}\hfill \\ \hfill \mathbf{\Omega }\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill \mathbf{K}\hfill & \hfill \mathbf{C}\hfill \\ \hfill {\mathbf{C}}^{\mathit{T}}\hfill & \hfill \mathbf{M}\hfill \end{array}\right)\left(\begin{array}{c}\hfill \mathbf{F}\hfill \\ \hfill \mathbf{N}\hfill \end{array}\right)$

where the $3\times 3$ submatrices K, M, and C are the translational, rotational, and coupling resistance tensors, respectively. The resistance tensor depends on the geometry of the swimmer. To actuate the microswimmers, we applied a uniform magnetic field, which means that a torque was applied, not a force. To clarify, only a magnetic field gradient generates a magnetic force that can result in pushing and pulling; a uniform magnetic field will only result in a reorientation of a magnet. This simplifies the relationship to

$\left(\begin{array}{c}\hfill \mathbf{v}\hfill \\ \hfill \mathbf{\Omega }\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \mathbf{C}\hfill \\ \hfill \mathbf{M}\hfill \end{array}\right)\mathbf{N}.$

The torque N is equal to the cross-product of the magnetic dipole m and the magnetic field H. The translational and angular velocity become

$\mathbf{\Omega }=\mathbf{M}(\mathbf{m}\times \mathbf{H})$

and

$\mathbf{v}=\mathbf{C}(\mathbf{m}\times \mathbf{H}),$

respectively. Essentially, if the geometry of the swimmer can produce nonzero rotational and coupling resistance tensors (M and C) then it is possible to yield a nonzero value for the translational velocity v. For the achiral microswimmer with 2 planes of symmetry, the M and C tensors are nonzero and can produce a nonzero translational velocity v if a nonzero torque N is applied [40]. The direction of N will be reflected by rotation axis observed during swimming. The numerical simulations were performed based on the analysis performed by Cheang et al. [40] and Meshkati et al. [46]. The details served as a validation of the swimming capability of the achiral microswimmers and showed how the microswimmers swim.

7.3.1 Approximate Helmholtz coils

The approximate Helmholtz coil system is based on the Helmholtz configuration and, likewise, can generate a near-uniform magnetic field. The purpose of the near-uniform field is to exert a constant torque on the microswimmer without introducing translational force. A pair of true Helmholtz coil requires a distance between two coils of the same size to be the radius of the coils. Given the space constraint of the microscope, the distances between the coils were modified to be equal to the combined dimension of the outer diameter and the thickness for the coil, creating a cube-like configuration (Fig. 7.3). While the approximate Helmholtz coils can generate near-uniform magnetic fields, the field strength is significantly weaker. The strength of the magnetic field generated from a pair of approximate Helmholtz coils is calculated using a modified version of the Biot–Savart law [47]

$B_{\mathit{coils}}=\frac{u_{0}nI{R}^2}{2{(R^2+x^2-2dx+d^2)}^{\frac{3}{2}}}+\frac{u_{0}nI{R}^2}{2{(R^2+x^2+2dx+d^2)}^{\frac{3}{2}}}$

where ${\mu }_0$ is the permeability, n is the number of turns of wires per coil, I is the electrical current passing through the wires, R is the effective radius of the coil, d is the distance between the center of a coil pair, and x is the coil distance to a point. With 1 A applied to the coils, experimental measurements yield a field strength of 5 mT while the calculated value using Eq. (7.6) yields 5.06 mT. The field profile of the coils validates the claim for a near-uniform magnetic field with a 2 mm region at the center. Given the size of the microswimmers, the 2 mm region provides sufficient space for experiments.

7.3.2 Real time image processing

For the implementation of a vision-based feedback controller, a real time imaging processing program was created using LabVIEW. Microswimmers from the camera live feedback images were tracked. Through tracking, the x and y positions of microswimmers were obtained. The x and y positions were then used to calculate velocity and heading angle which were then used for feedback control.

Tracking involved four main steps: (i) image binarization using grayscale thresholding, (ii) size thresholding, (iii) structure definition, and (iv) geometrical centroid calculation. First, in image binarization, each individual frame was converted to a binary black/white image, and the area of objects was defined by setting a grayscale threshold. Then, by removing unwanted objects such as debris using size thresholding, we deleted objects with too few or too many pixels. Next, the structure was defined by filling the interior of the microswimmers and labeling the microswimmers. Finally, the geometrical centroid ($x,y$) was calculated. With the centroid as the position of the microswimmer, the displacement, instantaneous velocity, and heading angle could be calculated.

7.4.1 Environmental disturbances

At low Reynolds number, microswimmers are subjected to environmental disturbances such as Brownian motion and background flows. In cases where environments are highly stochastic, the microswimmer may not be able to reach the destination without compensating for all the environmentally dependent uncertainties. Thus, the external disturbances must be included in the kinematics of the microswimmers. The displacement due to random disturbances can be modeled with Brownian motion

$r_D(t)=P\sqrt{2Dt}$

where $r_D$ is the displacement, P is a normally distributed random number, D is the diffusion coefficient of the microswimmers, and t is time. The theoretical diffusion coefficient was calculated as 0.0546 μm²/s. To increase the stochasticity of the environments for the purpose of evaluating a controller's performance, the coefficient can be artificially increased.

7.4.2 Kinematic model

The achiral microswimmer's kinematics is described by the nonlinear model

$\begin{array}{cc}\hfill \dot{x}(t)& =v(u_1(t))\cos (\theta (t)),\hfill \\ \hfill \dot{y}(t)& =v(u_1(t))\sin (\theta (t)),\hfill \\ \hfill \dot{\theta }(t)& =u_2(t),\hfill \end{array}$

where v is the velocity of the microswimmer, $\theta (t)$ is the heading angle, $u_1(t)$ is the rotation frequency, and $u_2(t)$ is the turning rate of the rotating magnetic field. The velocity v is obtained quantitatively from the linear velocity profile plotted against rotation frequency (i.e., 1, $2,\dots$ , and 8 Hz) from 7 different microswimmers. Using measurement errors due to video resolution, 30 random samples for each microswimmer were generated. The estimated linear velocity v by curve fitting statistic data is written as

$v(u_1)={\hat{p}}_1{u}_1+{\hat{p}}_2=0.428u_1-0.100$

where the coefficient $\hat{p}=\mathrm{E}p={[{\hat{p}}_1{\hat{p}}_2]}^T$ and the estimated covariance matrix is

$\mathrm{\Sigma }=\mathrm{E}[(p-\hat{p}){(p-\hat{p})}^T]=\left[\begin{array}{cc}\hfill 0.0338\hfill & \hfill -0.1523\hfill \\ \hfill -0.1523\hfill & \hfill 0.8629\hfill \end{array}\right].$

Using the estimated coefficients, the following estimated deterministic model can be obtained:

$\begin{array}{cc}\hfill \dot{x}(t)& ={\hat{p}}_1{u}_1(t)\cos (\theta (t)),\hfill \\ \hfill \dot{y}(t)& ={\hat{p}}_1{u}_1(t)\sin (\theta (t)),\hfill \\ \hfill \dot{\theta }(t)& =u_2(t),\hfill \end{array}$

where ${\hat{p}}_1=0.428$. A stochastic model is then created by adding the environmental factors and other uncertainties, represented as the estimated offset ${\hat{p}}_2$,

$\begin{array}{cc}\hfill \dot{x}(t)& ={\hat{p}}_1{u}_1(t)\cos (\theta (t))+a_1(t)+v_1(t),\hfill \\ \hfill \dot{y}(t)& ={\hat{p}}_1{u}_1(t)\sin (\theta (t))+a_2(t)+v_2(t),\hfill \\ \hfill \dot{\theta }(t)& =u_2(t),\hfill \end{array}$

where

$a(t)={\left[\begin{array}{cc}\hfill a_1(t)\hfill & \hfill a_2(t)\hfill \end{array}\right]}^T={\left[\begin{array}{cc}\hfill {\hat{p}}_2\cos (\theta (t))\hfill & \hfill {\hat{p}}_2\sin (\theta (t))\hfill \end{array}\right]}^T$

models the uncontrolled flow and $v={[v_1{v}_2]}^T$ models random disturbances at time t

$v(t)=M(t)(p-\hat{p}),p\sim \mathcal{N}(0,\mathrm{\Sigma }),$

where

$M(t)=\left[\begin{array}{cc}\hfill u_1(t)\cos (\theta (t))\hfill & \hfill \cos (\theta (t))\hfill \\ \hfill u_1(t)\sin (\theta (t))\hfill & \hfill \sin (\theta (t))\hfill \end{array}\right].$

7.5.1 Magnetic control

The motion uncertainties from the microswimmers' handedness can be eliminated by suppressing the secondary motion; this can be achieved using a combination of a static and rotation magnetic fields [39]. The addition of the static magnetic field serves to control the orientation the microswimmer. The achiral microswimmers can be controlled in a 2D plane using the resultant magnetic field generated from three pairs of approximate Helmholtz coils

$\mathbf{B}=\left[\begin{array}{c}\hfill -B_s\cos ({\theta }_s)+B_r\sin ({\theta }_s)\cos (\omega t)\hfill \\ \hfill B_s\sin ({\theta }_s)+B_r\cos ({\theta }_s)\cos (\omega t)\hfill \\ \hfill B_r\sin (\omega t)\hfill \end{array}\right]$

where $B_r$ is the maximum amplitude of the rotating magnetic field, $B_s$ is the magnitude of the static magnetic field, ω is the rotational frequency of the field, ${\theta }_s$ is the direction of rotation, and t is time. The magnitudes of the rotating field $B_s$ and the static field $B_r$ can be calculated using Eq. (7.6). A schematic for resultant field from Eq. (7.16) is shown in Fig. 7.1(C). The swimming direction can be expressed as the vector perpendicular to the rotating field

$\hat{\mathbf{n}}={\left[\begin{array}{ccc}\hfill -\cos ({\theta }_s)\hfill & \hfill \sin ({\theta }_s)\hfill & \hfill 0\hfill \end{array}\right]}^T.$

In experiments, the speed of the microswimmers is controlled using ω from Eq. (7.16) and the swimming direction is controlled using ${\theta }_s$ from Eq. (7.17). For direction control, we expect the heading angle of the microswimmer θ and the direction of the rotating field ${\theta }_s$ to be approximately equal.

7.5.2 Feedback control law

Based on the kinematics (7.11), the nonlinear feedback controller can guide an achiral microswimmer from any initial condition towards a target set of the position and angle by calculating the control variables $u_1$ and $u_2$ given by

$\begin{array}{cc}\hfill u_1(t)& =\gamma e(t)\cos (\alpha (t)),\hfill \\ \hfill u_2(t)& =k\alpha (t)+\gamma \frac{\cos (\alpha (t))\sin (\alpha (t))}{\alpha (t)}(\alpha (t)+h\varphi (t)),\hfill \end{array}$

which are the rotation frequency and turning rate, respectively.

The controller (7.18) was evaluated in both simulation and experiment. Using parameters $k=0.2$, $h=2$, and $\gamma =0.1$, a simulation was performed to evaluate the performance of the controller. The microswimmer started from the initial position at ($x=50\text{ μm}$, $y=25\text{ μm}$) and was guided to reach the final position at (150, 125). The microswimmer's rotation frequency $u_1$ reached a peak at 14 Hz and eventually arrived at the target position in 301 s (Fig. 7.4). The microswimmer's time to reach the target was too long and impractical for applications. The long duration is attributed to the uncertainty from the parameter $p_1$ since this parameter assumes a constant relationship between the swimming speed and the rotation frequency and ignores environmental effects.

7.5.3 Integral controller for swimming velocity

To compensate for the uncertainty from the parameter $p_1$, a nested control loop was implemented to control the swimming speed of the microswimmers. The nonlinear control (7.18) computes the turning rate and relative velocity $v_{\mathrm{ref}}$ while the inner loop compensates the velocity error using an Integral (I) controller. The inner loop controller is expressed as

$\begin{array}{cc}\hfill u_1(t)& =K_I\int (v_{\mathrm{ref}}-v_m(t))dt,\hfill \\ \hfill v_{\mathrm{ref}}& =\gamma e(t)\cos (\alpha (t)),\hfill \end{array}$

where $v_{\mathrm{ref}}$ is the reference velocity, $v_{\mathrm{m}}$ is the measured velocity, and $K_I$ is the integral gain. The I controller (7.19) has a single pole at zero which is the simplest pole for maintaining the reference velocity without introducing additional zeros and poles that can interfere with the unmodeled dynamics of the swimmer at high rotation frequencies. The value of $K_I$ determines a balance between maintaining the reference velocity and the stability of the velocity feedback loop. For instance, a small $K_I$, close to 0, results in a slow increase of the frequency $u_1$ until the actual velocity reaches the reference velocity. A higher $K_I$ results in a faster increase of the frequency $u_1$, but may lead to oscillation.

A simulation was performed using $K_I=1$ using controller (7.19) and is shown in Fig. 7.5. The parameters γ, k, and h were set to 0.1, 0.2, and 1, respectively. Based on the simulation, the time for the microswimmer to reach the target is 44 s. This is around 7 times faster than the previous case with controller (7.18).

7.5.4 Experimental validation under environmental disturbances

Experiments were performed to validate the simulation and to demonstrate the performance of the controller in realistic experimental conditions. In experiments, the microswimmers were subjected to environmental disturbances such as Brownian motion and adverse flow which causes motion uncertainties. In cases when environmental disturbances were too prominent, the use of controller (7.18) resulted in a critically low velocity where the microswimmers could not overcome environmental flows leading to a failure in reaching a destination. By adding in the nested I controller (7.19), the error in velocity due to external disturbances could be compensated over time; in other words, the microswimmer rotated faster in order to swim faster to overcome environmental conditions. A representative experiment using controller (7.19) is showed in Fig. 7.6 which illustrates an achiral microswimmer reaching a target position (red triangle) from an initial position (green triangle). Using real-time image processing, a microswimmer's velocity can be maintained at any reference velocity.

The parameters k, h, γ, and $K_I$ were chosen to be 0.5, 0.65, 0.2, and 0.9, respectively. The parameters k and h serve to give a curvature to the trajectory, while γ controls the swimming velocity with respect to the distance from the target. From the representative experiment in Fig. 7.6, an achiral microswimmer was able to move from the initial position of $x=111\text{ μm}$, $y=45\text{ μm}$, and $\theta =1.04\pi \text{ rad}$ to the target position of $x_T=138\text{ μm}$, $y_T=127\text{ μm}$, and ${\theta }_T=0.18\pi \text{ rad}$ in 24.23 s.