Field-programmable dynamics in a soft magnetic actuator enabling true random number generation and reservoir computing

TL;DR

The paper investigates leveraging complex, chaotic dynamics in untethered magnetic soft actuators to enable soft computing tasks. It demonstrates a fabrication- and tomography-validated NdFeB-PDMS membrane platform whose magnetic dynamics span periodic, quasi-periodic, and chaotic regimes under low-field AC excitation. The authors showcase true random number generation and stochastic computing from chaotic motion, and implement reservoir computing for nonlinear waveform transformation and Mackey--Glass time-series prediction, outperforming non-RC baselines. This work presents a hardware-software co-design pathway for soft robotics to perform secure, low-power computation and real-time neuromorphic processing at the edge.

Abstract

Complex and even chaotic dynamics, though prevalent in many natural and engineered systems, has been largely avoided in the design of electromechanical systems due to concerns about wear and controlability. Here, we demonstrate that complex dynamics might be particularly advantageous in soft robotics, offering new functionalities beyond motion not easily achievable with traditional actuation methods. We designed and realized resilient magnetic soft actuators capable of operating in a tunable dynamic regime for tens of thousands cycles without fatigue. We experimentally demonstrated the application of these actuators for true random number generation and stochastic computing. {W}e validate soft robots as physical reservoirs capable of performing Mackey--Glass time series prediction. These findings show that exploring the complex dynamics in soft robotics would extend the application scenarios in soft computing, human-robot interaction and collaborative robots as we demonstrate with biomimetic blinking and randomized voice modulation.

Figures (4)

• Figure 1: Chaotic behavior in classical systems and its translation to magnetic soft actuators. (a) Schematic of fabrication of magnetic actuators, (b) particularly tailored to actuate in alternating magnetic fields. (c) Young's modulus of the composite for distinct magnetic powder concentration (400-$\mu$m-thick membranes of NdFeB in PDMS, 10:1 ratio base to curing agent) and (d) sample thickness (NdFeB in PDMS, 20:1 ratio of base to curing agent). (e) Magnetic hysteresis curve of the composite. (f) In the periodic dynamics region, the magnetic actuators move at the driving frequency, creating flapping-like movements. (g) Under certain field intensities and frequencies, the actuators show unpredictable chaotic motion. Tracking of the position of a fluorescent marker on the body of the soft actuators (h, i) during (j) periodic and (k) chaotic dynamic modes controlled by the frequency and intensity of the applied magnetic fields. (l) Cross sections and (m) computed tomography images of the resilient composite withstanding (n) 40 000 cycles in the chaotic dynamics regime.
• Figure 2: Dynamic states of a rectangular-shape magnetic soft actuator in alternating magnetic fields along its long axis. Tracking magnetic soft actuator motion over time in alternating magnetic fields of specific intensity and frequency, demonstrating (a) periodic (10 Hz, 5 mT), (b) quasiperiodic (18 Hz, 5 mT), and (c) chaotic behavior (16 Hz, 5 mT) of the actuator schematically shown as an inset in panel (a). For the latter, no clear pattern is observed. Solid (dashed) lines show the time evolution of the $x$ ($y$) coordinates of the arm of the soft robot. (d) Dynamic states diagram across varying frequencies and intensities of the driving magnetic field. Poincaré diagrams and Fourier spectra of the actuator's $x$-position for (e, f) periodic, (g, h) quasiperiodic, and (i, j) chaotic behavior corresponding to panels (a), (b), and (c), respectively.
• Figure 3: Chaotic dynamics of soft robots for stochastic soft computing. (a) Time-tracked motion of a magnetic soft robot driven into chaotic dynamics. (b) Generation of random number sequences using the tracked x-coordinate of the soft robot. Each number is encoded in 7-bit sequences (separated by gray bars) and compared with the multiplicand to create a true random number sequence. (c) Rapidly decaying autocorrelation of the random number sequence used for calculations where lag refers to the distance between numbers in series. The linearity of Q-Q plot in the inset confirms that the generated random number sequence corresponds to a uniform distribution in the range $[0,1]$. (d) The soft-robot-generated sequence passed 14 NIST random tests (P-value threshold $= 0.01$). (e) Schematic of conversion of the given decimal number $B$ into $N$-bit stochastic random number. $X$ is converted to its binary representation $B$ that is compared with the sequence of $N$ random numbers RND. The output sequence is filled with 1 if $\text{RND} < B$, otherwise with 0. (f) Schematic of the stochastic computing algorithm using soft-robot-generated random numbers as input. At each clock cycle, random numbers $\mathrm{RND}_1$, $\mathrm{RND}_2$ and exact numbers $X_1$, $X_2$ are compared. If $\mathrm{RND}_1 < X_1$ or $\mathrm{RND}_2 < X_2$, a stochastic bit "1" is generated. Otherwise, if $\mathrm{RND}_1 \ge X_1$ or $\mathrm{RND}_2 \ge X_2$, a stochastic bit "0" is generated. (g) Stochastic multiplication of $42 \times 54$, demonstrating convergence towards the exact result (dashed line) with increasing the random bit count. (h) Stochastic multiplication results for different multipliers of 42 using varying stochastic bit counts ($N = 10, 100, 1000$). The red line indicates the exact result.
• Figure 4: Magnetic soft robot for reservoir computing (RC). (a) Schematic representation of RC: an input dataset, a time-series data such as a sinusoidal function, is applied to the RC as magnetic field to drive the dynamics of the magnetic soft robot. The RC output consists of the $x(t)$ and $y(t)$ trajectories of a point at the apex of one of the robot’s arms as indicated by the orange point A. At each time step $t$, a set of past observations (lag) is used to transform or predict the time series $\tau$ step-ahead via ridge regression with trained weights W. (b) Examples of waveform transformation using the RC: a sine wave dataset, sin$(t)$, is transformed into a square wave, square$(t)$, (left) or a sawtooth wave, saw$(t)$, (right). The mean squared error (MSE) obtained with and without RC in performing this task is also reported and clearly shows the better performance of the RC. (c) Example of future prediction of Mackey--Glass (MG) time-series at $\tau = 10$ and a lag = 20. The black dashed line is the target signal, red solid line is the prediction obtained with RC, while the blue solid line is the prediction without RC. The MSE is reported for both cases. (d) Complete phase diagram of the MSE (red low, blue high) of the prediction performance for MG time-series as a function of the lag and step-ahead, predictions without RC (top) and with RC (bottom) are shown. The star highlights the prediction illustrated in panel c. (e) Example of MSE as a function of step-ahead predictions for lag = 20, RC predictions (red line) exhibit lower MSE than those without RC (blue line) above $\tau= 9$.