Continuous Design and Reprogramming of Totimorphic Structures for Space Applications

Abstract

Recently, a class of mechanical lattices with reconfigurable, zero-stiffness structures has been proposed, called Totimorphic structures. In this work, we introduce a computational framework that allows continuous reprogramming of a Totimorphic lattice's effective properties, such as mechanical and optical properties, via continuous geometric changes alone. Our approach is differentiable and guarantees valid Totimorphic lattice configurations throughout the optimisation process, thus providing not only specific configurations with desired properties but also trajectories through configuration space connecting them. It enables re-programmable structures where actuators are controlled via automatic differentiation on an objective-dependent cost function, altering the lattice structure at all times to achieve a given objective - which is interchangeable to achieve different functionalities. Our main interest lies in deep space applications where harsh, extreme, and resource-constrained environments demand solutions that offer flexibility, resource efficiency, and autonomy. We illustrate our framework through two proofs of concept: a re-programmable metamaterial as well as a space telescope mirror with adjustable focal length, both made from Totimorphic structures. The introduced framework is easily adjustable to a variety of Totimorphic designs and objectives, providing a light-weight model for endowing physical prototypes of Totimorphic structures with autonomous self-configuration and self-repair capabilities.

Figures (10)

• Figure 1: Illustration of Totimorphic lattices. (A) Totimorphic unit cell and elementary cell used for the tile, as defined in this work. (B) Actuation of angles (or lattice dynamics) are defined as gradient descent on an objective-dependent cost function. This way, both a lattice configuration with a desired property is found as well as the trajectory connecting it to the initial configuration. The surface plot represents the cost for different configurations. An example trajectory following the gradient $-\nabla C$ is shown in white, and the corresponding lattice configurations to the right. For illustration purposes, the task here is to move the upper right corner of the lattice (blue dot) to the height of the red line. In more realistic scenarios, the landscape might contain insurmountable barriers, because the lattice needs to break in order to move between certain configurations.
• Figure 2: Totimorphic structures as building blocks of, e.g., space habitats, spacecraft components, and tools with reprogrammable properties. In this work, we specifically look into lattice structures that can alter their mechanical properties, i.e., how they behave under stress. In addition, we explore the concept of using Totimorphic structures as the backbone of large-scale space infrastructures such as space telescopes -- not only to ease their deployment, but to equip them with reprogrammable properties as well. For example, we demonstrate a mirror element that can adjust its surface shape to alter the location of its focal point.
• Figure 3: Parametrizing Totimorphic lattices. (A) When connecting elementary cells to an existing lattice, overlapping elements are shared with already existing cells (shown in red). Thus, the number of degrees of freedom of the newly added cell is reduced. (B) Iterative construction of a lattice made of Totimorphic cells. (C) If all points except one in an elementary cell are predetermined (shown in blue), the remaining point is obtained from the Totimorphic condition. This is illustrated here for the case where $\mathrm{A'}$ and $\mathrm{C}$ are given. Since $\mathrm{P'}$ is connected via a beam and lever to these points, it has to be a distance of $\frac{1}{2}l_\text{b}$ away from both. The solution to this problem is given by the intersection points of two circles, or equivalently, by a quadratic equation. Two points $\mathrm{P'}$ and $\mathrm{P}"$ satisfy the Totimorphic condition, however, the latter would reverse the order of $\mathrm{A'}$ and $\mathrm{B'}$ and is therefore not a valid physical solution.
• Figure 4: Inverse design of mechanical properties of Totimorphic lattices. (A) Schematic of the inverse design pipeline. (B) During optimisation, we obtain a continuous trajectory through configuration space (here represented using the first principal component of the angles representing the lattice) connecting the initial configuration ($\nu = 0$) and the two final configurations ($\nu = \pm 0.5$). Since the lattice always fulfills the Totimorphic condition, it passes through all intermediate Poisson's ratios while morphing. (C) Illustration of the final design for $\nu = 0.5$ (left) and $\nu = - 0.5$ (right), shown without (left) and with load (right, red arrows).
• Figure 5: Deployment of a Totimorphic lattice structure. A Totimorphic sheet reconfigures from its collapsed (left) to its unfolded (right) configuration by raising all levers. The unfolding is shown from two different perspectives.