⚡ electrical engineering

TERA: A Unified Taylor Model Enabled Reachability Analysis Framework

This paper introduces TERA, a free and open-source Python-native framework that unifies Taylor Model-based reachability analysis for continuous, hybrid, and stochastic systems to provide rigorous, tight enclosures of state trajectories while overcoming the limitations of existing specialized tools.

Original authors: Salma Iraky, Andrew Sogokon

Published 2026-07-02

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CC BY 4.0

Original authors: Salma Iraky, Andrew Sogokon

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict exactly where a car will be in the next hour. If the road is perfectly straight and the driver never makes a mistake, it's easy. But real life is messy: the driver might swerve, the wind might push the car, or the engine might sputter. In the world of engineering and safety, we call these "uncertainties."

To keep things safe, engineers need to draw a "safety bubble" around the car that is guaranteed to contain every possible path it could take, no matter what happens. This is called Reachability Analysis.

The problem is that as time goes on, these safety bubbles tend to get huge and sloppy. Imagine trying to wrap a gift with a piece of paper that keeps getting bigger and bigger with every fold until it's a giant, useless sheet. In math, this is called the "wrapping effect." It makes the safety bubble so big that it's useless for checking if the car will actually crash into a wall.

The Solution: TERA

The paper introduces a new tool called TERA. Think of TERA as a super-smart, flexible wrapping paper that doesn't just get bigger; it actually learns the shape of the gift.

Here is how it works in simple terms:

1. The "Smart Map" (Taylor Models)
Instead of just drawing a big box around the car's possible location, TERA uses something called a Taylor Model.

The Analogy: Imagine you are trying to describe a winding road. A simple box would say, "The car is somewhere in this huge square." A Taylor Model is like a GPS that says, "The car is likely on this specific curve, and if it drifts, it will drift this much."
Why it helps: By keeping track of the shape of the movement (using high-order math) rather than just a box, TERA prevents the "wrapping effect." The safety bubble stays tight and accurate, even over long periods.

2. One Tool for Many Jobs
Before TERA, you needed different tools for different problems: one for smooth roads (continuous systems), one for roads with traffic lights or sudden stops (hybrid systems), and one for roads with unpredictable weather (stochastic systems).

TERA's Claim: TERA is a "Swiss Army Knife" built entirely in Python (a popular, easy-to-use programming language). It can handle all three types of systems in one place. It's free, open-source, and designed so engineers can quickly test new ideas without needing expensive, locked-down software.

3. What It Can Do Right Now
The authors tested TERA on some very difficult puzzles to prove it works:

The Chemical Reaction: They modeled a complex 7-part chemical reaction (like a recipe mixing ingredients). TERA calculated the safe limits of this reaction over 20 seconds in just 13 seconds of computer time, keeping the safety bubble very tight.
The Bouncing Ball: They simulated a ball bouncing on the ground. The ball has two modes: "falling" and "bouncing." TERA successfully tracked the ball's path, switching between these modes without losing accuracy.
The Wobbly Spring: They simulated a spring moving in water with random jiggles (stochastic motion). TERA didn't just guess; it calculated a "probabilistic safety bubble." It guaranteed that 99.9% of the time (specifically, $1 - \delta$ where $\delta = 0.001$ ), the spring would stay inside the green zone they drew. They even checked this against 2,000 random computer simulations, and the spring never escaped the bubble.

The Bottom Line

The paper claims that TERA is the first free, Python-based tool that can rigorously calculate these "safety bubbles" for smooth systems, systems that switch modes, and systems with random noise, all while keeping the bubbles tight enough to be actually useful.

The authors are currently working on making it even better to handle systems that combine randomness with mode-switching (like a self-driving car in a storm) and eventually, systems controlled by artificial intelligence. But for now, they have built a solid, open foundation that solves some very hard math problems for engineers today.

Technical Summary: TERA – A Unified Taylor Model Enabled Reachability Analysis Framework

Problem Statement
The verification of safety-critical systems, defined by dynamics such as $\dot{x} = f(x, u, w)$ where $w$ represents bounded disturbances or uncertainty, requires rigorous enclosures of all possible state trajectories over a finite time horizon. While exact computation of flowpipes is generally infeasible for non-linear systems, set-based over-approximations are necessary for safety verification. However, traditional representations like (hyper-)intervals and zonotopes suffer from the "wrapping effect," where accumulated over-approximations lead to excessively conservative reachable set enclosures. Furthermore, existing Taylor Model (TM)-based tools (e.g., Flow*, CORA, JuliaReach) are often written in compiled languages (C++, MATLAB, Julia), making rapid prototyping difficult, or they lack support for stochastic systems and broad system classes within a unified workflow.

Methodology
TERA addresses these limitations by providing a Python-native framework that integrates with the scientific Python ecosystem and leverages the SageMath computer algebra system. The core methodology relies on Taylor Models (TMs) of the form $P(x_0, t) + I$ , where $P$ is a high-order Taylor polynomial approximation of the solution and $I$ is an interval remainder guaranteeing the true solution is enclosed.

Key methodological components include:

Propagation: Continuous-time dynamics are propagated using validated TM integration schemes grounded in the work of Berz and Makino, alongside flowpipe construction approaches for non-linear systems by Chen.
Wrapping Effect Mitigation: To maintain tight enclosures over long time horizons, TERA implements compositional left-right propagation using the "shrink-wrapping" method.
System Support:
- Non-linear ODEs: Supports systems with non-polynomial nonlinearities (e.g., transcendental functions like $\sin$ and $\cos$ ).
- Hybrid Systems: Computes hybrid automaton guard intersections and discrete transitions following TM-based hybrid reachability semantics.
- Stochastic Systems: For continuous-time stochastic systems, TERA combines deterministic TM flowpipes with probabilistic deviation bounds to compute $\delta$ -probabilistic reachable set ( $\delta$ -PRS) enclosures. This guarantees that trajectories remain within the computed sets with a probability of at least $1-\delta$ .
Rigorous Arithmetic: Correct rounding is achieved via the GNU MPFR library integrated within SageMath to ensure mathematical rigor.

Key Contributions

Unified Framework: TERA is presented as the first free, open-source, Python-native framework capable of handling continuous, hybrid, and stochastic systems within a single symbolic-numeric workflow.
Extensibility: By leveraging Python and SageMath, the framework facilitates rapid prototyping of reachability analysis techniques, addressing the rigidity of existing C++ or proprietary MATLAB tools.
Stochastic Capability: The framework extends TM-based reachability to continuous-time stochastic systems, computing $\delta$ -PRS enclosures.
Open Availability: The software is distributed under the GPLv2 license and is available via GitHub.

Results and Validation
The authors validated TERA against the ARCH competition benchmark problem suite [5]. The results demonstrate the framework's ability to compute tight reachable set over-approximations for difficult benchmark problems:

Non-linear Continuous Systems: TERA successfully computed a tight enclosure for a 7-dimensional non-linear Laub-Loomis biochemical reaction network over $t \in [0, 20]$ in 13 seconds. It also scaled to a 12-dimensional controlled quadrotor problem involving transcendental functions.
Hybrid Systems: The framework generated rigorous enclosures for a bouncing ball hybrid system, correctly projecting flowpipes onto the $(x, t)$ plane while distinguishing between active modes (up/down).
Stochastic Systems: TERA computed a $\delta$ -PRS enclosure ( $\delta = 0.001$ ) for a stochastic spring model over $t \in [0, 20]$ . The resulting enclosure was shown to contain 2000 Euler-Maruyama Monte Carlo sample paths.

Significance and Future Outlook
The paper positions TERA as a foundational step toward a robust, open-source Python infrastructure for rigorous reachability analysis. Its significance lies in democratizing access to high-order TM-based methods for a broad class of systems, including those with stochastic disturbances. The authors state that ongoing work is focused on extending capabilities to stochastic hybrid systems. The long-term goal is to apply this infrastructure to even broader classes of systems, such as those controlled by artificial neural networks. The authors maintain a modest tone, noting that while the current implementation handles difficult benchmarks, the framework is designed to evolve to support more complex system classes.

The Solution: TERA

The Bottom Line

Technical Summary: TERA – A Unified Taylor Model Enabled Reachability Analysis Framework

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