# Constrained Form-Finding of Tension-Compression Structures using Automatic Differentiation

arXiv: https://arxiv.org/abs/2111.02607 · HTML: https://mlsanity.com/paper/2111.02607/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Nov 2021.

## Built on by (most cited first, 1 of 1)

- [Creativity from Friction: Human-AI Interaction for Exploratory Structural Design](https://mlsanity.com/paper/2607.07521/) (2026)

## Abstract (arXiv)

This paper proposes a computational approach to form-find pin-jointed, bar structures subjected to combinations of tension and compression forces. The generated equilibrium states can meet force and geometric constraints via gradient-based optimization. We achieve this by extending the combinatorial equilibrium modeling (CEM) framework in three important ways. First, we introduce a new topological object, the auxiliary trail, to expand the range of structures that can be form-found with the framework. Then, we leverage automatic differentiation (AD) to obtain an exact value of the gradient of the sequential and iterative calculations of the CEM form-finding algorithm, instead of a numerical approximation. Finally, we encapsulate our research developments into an open-source design tool written in Python that is usable across different CAD platforms and operating systems. After studying four different structures -- a self-stressed planar tensegrity, a tree canopy, a curved bridge, and a spiral staircase -- we demonstrate that our approach enables the solution of constrained form-finding problems on a diverse range of structures more efficiently than in previous work.