11.3.4.1 Constantly rotating magnetic field

To make multiple cells controllable by the same magnetic field, we must exploit heterogeneity in turning rate. One method is by using a constantly rotating magnetic field $\left(t\right)=ft$, where $f\in {\mathrm{R}}^{+}$ is the frequency of rotation. For $f<Ma$ the cells will reach a steady-state phase lag as they attempt to align with the field. At steady-state the cells are turning at the same speed as the magnetic field

$\begin{array}{c}{\dot{\theta }}_i=M{a}_i\sin ({\theta }_i(t)-\psi (t)),\hfill \\ f=M{a}_i\sin ({\theta }_i(t)-ft),\hfill \\ {\sin }^{-1}\left(\frac{f}{M{a}_i}\right)={\theta }_i(t)-ft.\hfill \end{array}$

This steady-state phase lag is shown in Fig. 11.10. The quantity $\frac{f}{M{a}_i}$ is the step-out frequency, after which the phase lag grows without bound. This growth is approximately linear for $>1.5a$, as shown in Fig. 11.11. The effective period for the cell is

$T_i=\{\begin{array}{cc}\frac{2\pi }{f},\hfill & f<M{a}_i,\hfill \\ \approx 12.9f{(M{a}_i)}^{-2}-2.3{(M{a}_i)}^{-1},\hfill & \text{else}.\hfill \end{array}$

We can also compute the effective radius of the limit cycle the cell follows. For $f<a$, the cell completes a cycle every $2\pi /f$ seconds and the radius is therefore $v/f$. Past the step-out frequency, the cells turn in periodic orbits similar to the hypotrochoids and epitrochoids produced by a Spirograph toy. Representative limit cycles are shown in Fig. 11.12. The radius of rotation is

$r_i=\{\begin{array}{cc}\frac{v_i}{f},\hfill & f<M{a}_i,\hfill \\ \approx 1.45f{\left(M{a}_i\right)}^{-2}-0.3{\left(M{a}_i\right)}^{-1},\hfill & \text{else}.\hfill \end{array}$

11.3.4.2 Arbitrary orientations

If we could control the orientation of each cell independently, the cells could swim directly to the goal. Fig. 11.13 shows two cells with different a parameters. If the rotation frequency f and the a values are coprime, the range of possible ${\theta }_1$ and ${\theta }_2$ values spans $[0,2\pi ]\times [0,2\pi ]$. By increasing f we can control the density we sample these angles. The left side of Fig. 11.13 shows that the time required to span $[0,2\pi ]\times [0,2\pi ]$ increases with f.

11.3.4.3 Straight-line swimming

By turning the magnetic field off, the cell dynamic model simplifies to

$\left[\begin{array}{c}\hfill {\dot{x}}_i\hfill \\ \hfill {\dot{y}}_i\hfill \\ \hfill {\dot{\theta }}_i\hfill \end{array}\right]=\left[\begin{array}{c}\hfill v_i\cos {\theta }_i\hfill \\ \hfill v_i\sin {\theta }_i\hfill \\ \hfill 0\hfill \end{array}\right].$

Without an external magnetic field, the cells swim straight in the direction they were headed when the magnetic field was last on. If we store the orientation of the magnetic field when the magnetic field is turned off at time $t_a$ as ${\psi }_a=f{t}_a$, then when we turn the field back on at time $t_b$ we can resume where we last stopped $\psi (t)={\psi }_a+f(t-t_b)$ and the cells will continue their limit-cycle behavior, but the center of rotation will be translated $v_i(t_b-t_b)$ along the vector ${\theta }_i(t_a)$.

11.3.4.4 System identification

To choose the optimal frequency of the rotation magnetic field requires knowing the a values for the set of cells we want to control. We employ the method of least squares to determine the $a_i$ values. First we discretize the continuous plant model (Eq. (11.2))

$\left[\begin{array}{c}\hfill x_i(k+1)\hfill \\ \hfill y_i(k+1)\hfill \\ \hfill {\theta }_i(k+1)\hfill \end{array}\right]=\left[\begin{array}{c}\hfill v_i\mathrm{\Delta }T\cos {\theta }_i(k)\hfill \\ \hfill v_i\mathrm{\Delta }T\sin {\theta }_i(k)\hfill \\ \hfill M{\alpha }_i\sin (\psi (k)-{\theta }_i(k))\hfill \end{array}\right]$

where ΔT is the sampling time and ${\alpha }_i=a_i\mathrm{\Delta }T$.

To identify the ${\alpha }_i$ parameter for each cell, we record position and orientation measurements under a constantly rotating magnetic field. We record the discrete-time cell orientation information as ${\theta }_i(0)$, ${\theta }_i(1),\dots ,{\theta }_i(k),\dots$, ${\theta }_i(n)$, and the magnetic field orientation as $\psi (0)$, $\psi (1),\dots ,\psi (k),\dots ,\psi (n)$. The following equation is derived from Eq. (11.6):

$\left[\begin{array}{c}\hfill {\theta }_i(1)-{\theta }_i(0)\hfill \\ \hfill {\theta }_i(2)-{\theta }_i(1)\hfill \\ \hfill ⋮\hfill \\ \hfill {\theta }_i(k)-{\theta }_i(k-1)\hfill \\ \hfill ⋮\hfill \\ \hfill {\theta }_i(n)-{\theta }_i(n-1)\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \sin (\psi (0)-{\theta }_i(0))\hfill \\ \hfill \sin (\psi (1)-{\theta }_i(1))\hfill \\ \hfill ⋮\hfill \\ \hfill \sin (\psi (k-1)-{\theta }_i(k-1))\hfill \\ \hfill ⋮\hfill \\ \hfill \sin (\psi (n-1)-{\theta }_i(n-1))\hfill \end{array}\right]\mathbf{\cdot }{\alpha }_i.$

We rewrite this equation as $Y=\mathrm{\Phi }{\alpha }_i$. Then, using the method of least squares, the parameter set with the best fit to the data is given by ${\hat{\alpha }}_i={\mathrm{\Phi }}^{†}Y$, where ${\mathrm{\Phi }}^{†}={({\mathrm{\Phi }}^T\mathrm{\Phi })}^{-1}{\mathrm{\Phi }}^T$ is the pseudoinverse of Φ. The cell's $a_i$ value is derived as $a_i=\frac{{\alpha }_i}{\mathrm{\Delta }T}$.

The $a_i$ value can also be measured directly by inverting (11.3). If the frequency of the rotation magnetic field is below the step-out frequency, the cells turn in a circle with a constant phase lag ${\theta }_{i,\mathit{lag}}$. The turning-rate parameter is then $a_i=-f/\sin ({\theta }_{i,\mathit{lag}})$.

11.3.4.5 Swarm control

Once a rotating magnetic field has been removed, cells continue to swim straight, although in slightly different directions. This difference in orientation may be used to control swarms of cells to congregate or steer them to arbitrary positions. Using a combination of rotating and straight swimming (swimming in the presence and absence of a rotating magnetic field), a scenario such as in Fig. 11.14 may be accomplished with many cells. A system can implement a toggling magnetic field to characterize cells and then calculate the most efficient path for goals.

Our control input consists of an alternating sequence of ORBIT and SWIM-STRAIGHT modes. The oscillation frequency f of the magnetic field is constant for every ORBIT mode. At the beginning of each ORBIT mode, the phase of the magnetic oscillation is resumed from the previous ORBIT mode. During the first ORBIT mode, we identify the centers of rotation $(x_{c,i},y_{c,i})$ of each cell by recording the cell positions for at least one period, calculated by Eq. (11.4), and computing

$\begin{array}{c}\hfill x_{c,i}(t)=\max (x_i(t-T:t))-\min (x_i(t-T:t)),\hfill \\ \hfill y_{c,i}(t)=\max (y_i(t-T:t))-\min (y_i(t-T:t)).\hfill \end{array}$

The center of rotation of each cell translates along with the cell during each SWIM-STRAIGHT mode (see Fig. 11.14). Control laws were designed from a control-Lyapunov function and investigated in [31].