2D Movement

Position transitions of a levitating microrobot using two pairs of serpentine traces

The basic system for 1DOF movement consists of two serpentine traces, individually actuated.^[30][31] Figure shows the schematic of the trace paths and a 3x3 magnet microrobot on top. On position number 1, the magnets are in their equilibrium position where the magnetic flux density is the highest, in between two opposite currents from the same trace path.

On moving from 1 to 2, the first trace path is turned off while the second is turned on. This causes the magnets to move to their new equilibrium, toward the higher magnetic flux density.

Repeating this procedure with opposite currents on the same trace paths, a movement in the desired direction is produced.^[32]

Magnetic flux vectors induced on a magnet cube over a pair of serpentine traces

To find the velocity, the forces on the microrobot must be analyzed (fig. ). The microrobot is supposed to levitate and so no friction forced is produced, other than the air drag which is also not considered.

The force produced by the interaction of the magnetic moments of the microrobot and the flux density of the serpentine traces is:

${\displaystyle F=\nabla ({\overrightarrow {m}}\cdot {\overrightarrow {B}})}$

The magnetic moment vector, given the orientation requirement for the diamagnetic levitation, is:

${\displaystyle {\overrightarrow {m}}={\bigl (}{\begin{smallmatrix}0&0&{\frac {B_{r}V_{m}}{\mu _{0}}}\\\end{smallmatrix}}{\bigr )}}$

Meanwhile, the contribution to the B field by the 2 closest traces is:

${\displaystyle {\vec {B}}={\vec {B}}_{1}+{\vec {B}}_{2}=\ \mu _{0}{\frac {{\vec {I_{1}}}\times {\vec {r}}_{1}}{2\pi \left|{\vec {r}}_{1}\right|^{2}}}+\mu _{0}{\frac {{\vec {I_{2}}}\times {\vec {r}}_{2}}{2\pi \left|{\vec {r}}_{2}\right|^{2}}}}$

Since for this approximation ${\displaystyle {\overrightarrow {m}}\cdot {\overrightarrow {B}}}$ is not dependent on y or z, their derivatives are zero and only force in the x direction is produced:

${\displaystyle F_{x}={\frac {B_{r}V_{m}I}{2\pi }}{\frac {\partial }{\partial x}}\left({\frac {x-\Delta x_{0}}{(x-\Delta x_{0})^{2}+d^{2}}}+{\frac {x+\Delta x_{0}}{(x+\Delta x_{0})^{2}+d^{2}}}\right)}$

This is the only force applied on the magnet, and it can be equated to the robot's mass multiplied by its acceleration. This equation can be integrated to find the velocity of the microrobot:

${\displaystyle F_{x}=mv{\frac {dv}{dx}}}$

${\displaystyle {\frac {B_{r}V_{m}I}{2\pi m}}\left({\frac {2\Delta x_{0}}{4\Delta x_{0}^{2}+d^{2}}}\right)={\frac {1}{2}}v_{max}^{2}}$

Introducing the relation between magnet volume, mass, and density ${\displaystyle V_{m}={\frac {m}{\rho _{m}}}}$ in the previous equation cancels out the mass, which means that if more magnets are added (N number of magnets), force will increase linearly:

${\displaystyle v_{max}={\sqrt {N{\frac {B_{r}I}{\pi \rho _{m}}}\left({\frac {2\Delta x_{0}}{4\Delta x_{0}^{2}+d^{2}}}\right)}}}$

This is the expression for the robot speed as a function of the current.

For a second DOF, more traces must be added. Two more intertwined serpentine traces must be added below the existing ones, rotated 90 degrees, to generate forces in the Y direction. Intensity on these traces will have to be higher to account for the higher distance.

Levitation

Diamagnetically levitated milli- and micro-robots can be controlled and moved with near-zero noise in their force, and they can be made intrinsically stable. In this way there is highly optimized control that uses zone or area control.^[33]

Diamagnetic levitation can produce two effects on a micro robot. The first is reducing the sliding friction and the second is fully levitating the micro robot. The fully levitation system will be the focus. To produce passive levitation a diamagnetic layer (such as graphite) must exist in the presence of a ferromagnet (such as NdFeB).^[34] Diamagnetic materials are characterized by having negative susceptibility, induced magnetic moment opposite to the external magnetic field. For that reason, they are repelled by an external magnetic field and tend to move toward the field minimum. This repulsive force is a result of the diamagnets having a magnetization direction antiparallel to the external magnetic fields.

The magnetizations of diamagnetic materials vary with an applied magnetic field which can be given as:

${\displaystyle M=\chi _{p}H}$

Where ${\displaystyle H}$ is the magnetic field strength and ${\displaystyle \chi _{p}}$ is the dimensionless susceptibility. For an object with volume ${\displaystyle V_{p}}$, the induced magnetic moment m can be given by:

${\displaystyle {\overrightarrow {m}}=\chi _{p}{\frac {V_{p}}{\mu _{0}}}{\overrightarrow {B}}}$

The magnetic force acting on the object is there for described as:

${\displaystyle F_{m}={\frac {\chi _{p}V_{p}}{\mu _{0}}}\nabla ({\overrightarrow {B}}\cdot {\overrightarrow {B}})}$

If the object has density ${\displaystyle \rho _{p}}$ and is levitating in a medium with density ${\displaystyle \rho _{m}}$ and magnetic susceptibility ${\displaystyle \chi _{m}}$ the total energy of the object, with a magnetic and gravitational term, is:

${\displaystyle E={\frac {\chi _{p}-\chi _{m}}{2\mu _{0}}}V_{p}({\overrightarrow {B}}\cdot {\overrightarrow {B}})+(\rho _{p}-\rho _{m})V_{p}gz}$

Such that the resulting force becomes:

${\displaystyle \sum F={\frac {\chi _{p}-\chi _{m}}{2\mu _{0}}}V_{p}\nabla ({\overrightarrow {B}}\cdot {\overrightarrow {B}})+(\rho _{p}-\rho _{m})V_{p}g{\overrightarrow {k}}}$

The necessary condition for stability is:

${\displaystyle \chi _{p}-\chi _{m}<0}$

To calculate the whole diamagnetic force acting on the levitated materials, each single dipole of the diamagnetic material must be considered. The diamagnetic force for the entire volume can be expressed as:

${\displaystyle F_{m}={\frac {\chi _{p}}{\mu _{0}}}\iiint _{V_{p}}\nabla ({\overrightarrow {B}}\cdot {\overrightarrow {B}})dv}$

The diamagnetic repulsion force is proportional to the magnetic susceptibility of diamagnetic materials. To counteract gravity in the magnetic field, materials with strong diamagnetism and lightweight properties are preferred.