Feasibility of generating 90 Hz vibrations in remote implanted magnets

Limb amputation not only reduces the motor abilities of an individual, but also destroys afferent channels that convey essential sensory information to the brain. Significant efforts have been made in the area of upper limb prosthetics to restore sensory feedback, through the stimulation of residual sensory elements. Most of the past research focused on the replacement of tactile functions. On the other hand, the difficulties in eliciting proprioceptive sensations using either haptic or (neural) electrical stimulation, has limited researchers to rely on sensory substitution. Here we propose the myokinetic stimulation interface, that aims at restoring natural proprioceptive sensations by exploiting the so-called tendon illusion, elicited through the vibration of magnets implanted inside residual muscles. We present a prototype which exploits 12 electromagnetic coils to vibrate up to four magnets implanted in a forearm mockup. The results demonstrated that it is possible to generate highly directional and frequency-selective vibrations. The system proved capable of activating a single magnet, out of many. Hence, this interface constitutes a promising approach to restore naturally perceived proprioception after an amputation. Indeed, by implanting several magnets in independent muscles, it would be possible to restore proprioceptive sensations perceived as coming from single digits.


Equations modelling a myokinetic stimulation interface
Assuming that the far-condition is met, a permanent magnet or a coil can be approximated with the magnetic dipole model 1 , which field can be described as: where describes its position in a given reference frame and ⃗⃗ corresponds to the magnetic moment vector. In particular, for a coil, ⃗⃗ can be written as: where is a scaling factor, ( ) is the current flowing through the coil, and corresponds to the direction vector of the magnetic dipole. Equation (4) holds under the quasi-static approximation of the magnetic field 2 , which indicates that the direction of the magnetic field vector can be approximated as that of a static field while its amplitude is modulated by the coil's current 2 . This approximation can be used in our case, considering that the frequencies required to stimulate the muscle spindles are relatively low (70-100 Hz range).
Using this, equation (3) can be rewritten as: Notably, ( ) represents the spatial distribution of the field, and depends only on the position. In turn, the amplitude of the magnetic field produced by the coil depends only on its current ( ) (the time-index is dropped from here onwards for notational simplicity).
If C coils are used, and the current ij of each one is independently controlled, then the compound magnetic field generated in a given point of space can be calculated as: Where (in ℝ ) represents the vector of all coil currents, and matrix ̅ ̅ ( ) (in ℝ 3× ) maps the current vector to the 3D components of the magnetic field at point .
The torque developed by a permanent magnet (applied to its medium) can be calculated if the total magnetic field applied to it, ⃗ , is known. Assuming that ⃗ is produced by C coils flown by current, and by close magnets, the torque can be expressed as 3 : : , Where ⃗ is the disturbance field, generated by the other magnets, and is its associated disturbance torque vector. Using equation (6), can be written as: Where matrix ̅ ̅ describes the torque generated by vector on all magnets. Its dimensions are 3 × (i.e., ℝ 3 × ). Three coils are thus required per magnet, in order to control its full torque vector. Using fewer coils leads to an under-actuated system. Using more than 3N coils leads to an over-actuated system. Matrix ̅ ̅ is always rank deficient, thus meaning that some torque configurations (vectors) are unfeasible. This is due to the fact that a magnet cannot rotate around its central axis, under the influence of an external magnetic field.
The force exerted by C coils on a magnet can be computed using a procedure analogous to the previously described one. The total force can be decomposed into a disturbance force , generated by nearby magnets, and a coil-generated force . If the magnetic dipole model is used to describe the coils magnetic field, then can be computed as 4 : Matrix ̅ ̅ (∈ ℝ 3 × ) describes the mapping from the currents in the coils to the forces of the magnets. Three coils per magnet are required to independently control each force vector.
Using ̅ ̅ and ̅ ̅ , it is possible to calculate the control matrix ̅ ̅ . This matrix is required by the system to calculate the currents necessary to achieve a desired force/torque configuration. The matrix can be constructed in different ways: i) ̅ ̅ and ̅ ̅ can be stacked together, if the aim is to control the force and the torque of each magnet; ii) ̅ ̅ can be constructed using only the rows of ̅ ̅ , or those of ̅ ̅ , in which case only the forces or the torques are controlled, respectively; iii) rows can be arbitrarily mixed together; in such case, e.g. the forces and torques of one magnet might be fully controlled, while only the forces or the torques of the remaining magnets are controlled.
Case i) requires the use of at least 6N coils; this is required if, for instance, it is desired to control their motion across the workspace, as in the works by Diller and colleagues 5 and Chowdhury and team 6 . Case ii) requires at least 3N coils, given that only one of the vectors (forces or torques) is controlled. This can be used if, for instance, the magnet is constrained by the medium and cannot move in a given direction. Case iii) requires as many coils as variables being controlled.
Thus by unifying equations (9) and (11) to construct ̅ ̅ , and using its inverse (or the pseudoinverse), and the vector of reference signals (forces and/or torques), it is possible to compute the required currents as: Where includes the disturbance terms due to the nearby magnets. Its entries are extracted from and , depending on how matrix ̅ ̅ is constructed. Vector can be neglected, if its entries are negligible with respect to those of .