Reprogrammable, in-materia matrix-vector multiplication with floppy modes

TL;DR

This work demonstrates a reprogrammable mechanical matrix–vector multiplier built from floppy modes in an elastic metamaterial. By designing a continuously tunable unit cell with well-defined compatibility, a planar tiling computes arbitrary $\vec{y}=A\vec{x}$ while preserving zero-energy deformations in the ideal limit. Numerical FE simulations combined with automatic differentiation address real-world stiffness and constraints, enabling scalable design; experiments with rubber substrates validate 2×2 tiles and programmable coefficients, revealing high fidelity in the linear regime and revealing practical nonidealities like hysteresis. The approach advances in-materia computing, with potential for embodied intelligence, smart MEMS, and edge computing, and suggests feasible scaling to larger matrices via higher aspect ratios or cascaded units.

Abstract

Matrix-vector multiplications are a fundamental building block of artificial intelligence; this essential role has motivated their implementation in a variety of physical substrates, from memristor crossbar arrays to photonic integrated circuits. Yet their realization in soft-matter intelligent systems remains elusive. Here, we experimentally demonstrate a reprogrammable elastic metamaterial that computes matrix-vector multiplications using floppy modes -- deformations with near-zero stored elastic energy. Floppy modes allow us to program complex deformations without being hindered by the natural stiffness of the material; but their practical application is challenging, as their existence depends on global topological properties of the system. To overcome this challenge, we introduce a continuously parameterized unit cell design with well-defined compatibility characteristics. This unit cell is then combined to form arbitrary matrix-vector multiplications that can even be reprogrammed after fabrication. Our results demonstrate that floppy modes can act as key enablers for embodied intelligence, smart MEMS devices and in-sensor edge computing.

Figures (7)

• Figure 1: A metamaterial (red) computes a matrix-vector multiplication by decomposing it into elementary operations (blue), corresponding to individual matrix elements. Each elementary operation (purple) is implemented by a metamaterial unit cell. The input vector is presented as a displacement applied on the input control rods (orange, $x_i$) and the result is deterined by measuring the displacement of the output rods (blue, $y_i$). Each internal output is connected to an input, ensuring that the resulting metamaterial is frustration-free irrespective of the particular matrix being implemented.
• Figure 2: (a) single degree of freedom, the angle of the support beams control the coupling between x and y (b) the unit made of 9 DOFs. The angle at site 3 and 6 (0 and 1) control the coupling between $u_1$ ($u_2$) and $v_1$. (c) the 2 floppy modes of the system
• Figure 3: (a) Evolution of the loss function during the optimization of the design from 1x1 lattices (black) to 6x6 (yellow). For large matrix sizes, the loss function cannot go below a saturation threshold. (b) Minimum aspect ratio required to accurately approximate a set of unitary random matrices. In the red curve, only $\theta_3$ is varied, while for the red curve, 5 angles are allowed to vary. (c) Input/output relation for a single unit cell, as a function of the angle of site 3, from a low (black) to high (yellow) amount of bending stiffness (black). We see that as the the aspect ratio is reduced, both the response at each given angle, and the maximum attainable matrix coefficient are lowered. (d) 3D model of a unit cell
• Figure 4: (a) Complete experimental input/output transfer relation for a single unit cell, the measurements, in blue, are gathered from an optical flow algorithm. The fitted linear response (solid red lines) is within 20$\%$ of the target matrix value. The dashed olive line corresponds to a sigmoidal fit of the output displacement. (b) Picture of the experimental setup during characterization. The blue components are the output of the linear actuators. Videos of the experiment are provided as supplementary information. The red lines represent the deformation of the unit, it is computed using an edge detection algorithm and some edited to remove the edges of holes around the sample.
• Figure 5: (a) input/output relationship of a 2x2 lattice. The matrix of the transformation is within 15% of the target matrix (b) Picture of the 2x2 lattice, with the deformation plotted in red. The first output is zero because the components cancel each other out, while they add for the second output.