Physics-informed digital twins of brainbots

TL;DR

The paper addresses the challenge of explaining diverse brainbot trajectories in 2D by developing a physics-informed kinematic model that assigns a constant angular velocity $\omega$ and a translational velocity $\bm v_{\rm c}$ (constant in either body-fixed or lab frame) to decouple linear, spinning and orbital motion. It provides analytical, closed-form trajectory descriptions and implements a Crank-Nicolson-based simulator to reproduce experiments and generate long synthetic trajectories, forming a digital twin via statistical matching of $\omega$ and $\eta$ distributions and Fourier-mode augmentation. The key contributions are (i) decoupled motion-mode descriptions, (ii) exact trajectory expressions for circular, orbital and helical paths, (iii) a lightweight simulation pipeline that closely matches experimental data and yields faithful long-time statistics, and (iv) a digital twin framework enabling data-rich methods and physics-informed control. This work enables long-time data generation for diffusion studies and physics-informed control on brainbots, with future extensions to external potentials and many-body interactions.

Abstract

A brainbot is a robotic device powered by a battery-driven motor that induces horizontal vibrations which lead to controlled two-dimensional motion. While the physical design and capabilities of a brainbot have been discussed in previous work, here we present a detailed theoretical analysis of its motion. We show that the various autonomous trajectories executed by a brainbot -- linear, spinning, orbital and helical -- are explained by a kinematic model that ascribes angular and translational velocities to the brainbot's body. This model also uncovers some trajectories that have not so far been observed experimentally. Using this kinematic framework, we present a simulation system that accurately reproduces the experimental trajectories. This can be used to parameterize a digital twin of a brainbot that executes synthetic trajectories that faithfully mimic the required statistical features of the experimental trajectories while being as long as required, such as for machine learning applications.

Figures (5)

• Figure 1: (a) Position vector $\bm r(t)$ and orientation angle $\varphi(t)$ of a brainbot at time $t$, in the global reference frame. (b) Position vectors of the geometric center $\bm r$ of the brainbot and an arbitrary point $\bm r'$ in the laboratory frame, and the instantaneous unit vectors $\bm{\hat{n}_1}$ and $\bm{\hat{n}_2}$ along the major and minor axis, respectively. (c) Semi-major and semi-minor axis lengths $A_1$ and $A_2$, and the coordinates of the geometric center $(0,0)$ and an arbitrary point ($\rho_1 A_1, \rho_2 A_2$) in the body-fixed reference frame.
• Figure 2: Properties of the model. Top row: Deterministic trajectories of the geometric center (shown in red) and the instantaneous center of rotation $\bm r_{\rm c}$ (shown in green) of a brainbot, found using the kinematic model. The black ellipse in each panel indicates the starting position of the brainbot. Middle row: The values of the $\eta$ parameter corresponding to the different trajectories shown in the top row. For purely linear, spinning and orbital trajectories the $\eta$ values are constant, while for other trajectories $\eta$ is sinusoidal. Bottom row: The probability distributions of the different $\eta$ values attained in the corresponding plots in the middle row. To get each distribution the trajectory was calculated for $200$ seconds. For visualization purposes, the plotted trajectories cover only the first $30$ seconds.
• Figure 3: Results of simulation (in red) compared to experimental data (in black). The upper row shows the trajectories, while the lower row shows the corresponding values of the $\eta$ parameter for each trajectory as a function of time. Values of $\eta$ close to $1$ indicate circular trajectories, while values close to $0$ indicate linear trajectories Noirhomme2025. The experimental parameters (leg angle $\alpha_\text{leg}$ and effective motor voltage $V_\text{E}$) in panels (a) to (d) are: (a) $\alpha_\text{leg} = 15 \degree, \ V_\text{E} = 2.7 \ \text{V}$, (b) $\alpha_\text{leg} = 15 \degree, \ V_\text{E} = 3.0 \ \text{V}$, (c) $\alpha_\text{leg} = 25 \degree, \ V_\text{E} = 2.1 \ \text{V}$, (d) $\alpha_\text{leg} = 5 \degree, \ V_\text{E} = 2.1 \ \text{V}$.
• Figure 4: The top row shows synthetic trajectories similar to the four experimental trajectories shown in Fig. \ref{['fig:sim1']}. The second row shows the variation in $\omega$ with time, from the experiments (black curves) and the simulations (red curves). The third row shows histograms (in gray) of $\omega$ values obtained from simulations, and the resulting skew-normal distribution function (shown by the dashed red line), from which $\omega$ values are picked to obtain the trajectories in the top row. The bottom row shows the histograms of $\eta$ values from the experiments (in black) and from the simulations (in red).
• Figure 5: The top row shows synthetic trajectories corresponding to the experimental trajectory in the third column of Fig. \ref{['fig:traj_eta']}, for different values of $\omega$ picked in the simulations. These $\omega$-values are picked by first plotting $\omega$ vs time (black curves in the second row), and then taking Fourier modes to different orders (red curves in the second row). The number of Fourier modes taken are, respectively, $1$, $3$, $11$ and $69$ from the leftmost to the rightmost panel in the second row. The third row shows the resulting variation of $\eta$ with time, as measured from the experiments (in black) and as found in the simulations (in red). The bottom row shows the histograms for $\eta$ values, as found for the experiments (in black) and from the simulations (in red).