In our recent research, we proposed a powerful alternative to classical stochastic filtering techniques: zonotope-based set-membership filtering, integrated within a Model Predictive Control (MPC) framework. This approach doesn’t just improve estimation—it ensures bounded safety, scalability, and computational efficiency in environments dominated by unknown-but-bounded (UBB) disturbances.

📦 What Are Zonotopes and Why Use Them?

A zonotope is a geometric object—essentially a center point plus a weighted sum of line segments. Mathematically, it’s expressed as:

Z = <c, G> = {c + Gξ | ||ξ||_∞ ≤ 1}

This structure makes zonotopes:

• Closed under linear transformations and Minkowski addition
• Compact and computationally efficient
• Naturally suited for set-based estimation and control

Unlike ellipsoids or Gaussian-based filters like the Kalman Filter, zonotopes make no probabilistic assumptions. Instead, they simply assume noise is bounded—which is often a more realistic model in aerospace.

🔍 Why Not Kalman Filters?

Kalman Filters (and even their nonlinear counterparts like EKF or UKF) assume that:

• Uncertainties are Gaussian
• The system dynamics are either linear or well-approximated

But in real-world aerospace scenarios:

• Disturbances like solar pressure, drag, or model mismatch are non-Gaussian
• Accurate covariance information may be unavailable
• Safety demands guaranteed bounds, not statistical confidence

Zonotope-based Set-Membership Filters (SMFs) overcome these limitations by estimating a set of all possible states consistent with measurements and dynamics.

🚀 Case Study: Rendezvous and Docking in Elliptical Orbits

In our paper, we applied zonotopic SMF to a spacecraft performing rendezvous and proximity operations using the Tschauner-Hempel equations to model relative dynamics in elliptical orbits.

We designed a filter that:

1. Propagates zonotopes forward using time-varying linear dynamics
2. Updates the estimate with new measurements by minimizing the Frobenius norm of the generator matrix
3. Feeds the center of the zonotope into a Model Predictive Controller to guide the spacecraft

Our MPC formulation used a standard quadratic cost function, incorporated real-world constraints like thrust limits, and relied on the set estimate rather than a point.

📊 Results at a Glance

Conclusion: Zonotopic SMF strikes a sweet spot between robustness and real-time feasibility, especially when integrated with MPC.

🛠️ Toolbox Development

We are actively working on a Python-based reachability and estimation toolbox, incorporating:

• Zonotopic operations
• Set-based filtering
• MPC integration
• Visualization utilities

Stay tuned for an open-source release, and feel free to reach out if you’re interested in collaborations or early access.

🛰️ Final Thoughts

Zonotope-based filtering is more than an academic curiosity—it’s a scalable, real-world-ready solution for modern aerospace systems operating under bounded uncertainties. Whether it’s orbital maneuvers, cooperative multi-agent control, or autonomous rover navigation, set-theoretic methods provide rigorous guarantees where classical filters fall short.

If your mission can’t afford uncertainty, it’s time to think in sets, not points.

Want to know more or see the math behind it? Read our full paper for technical details.

Why Reachability Analysis is a Game-Changer for Space Missions

Reachability analysis allows us to predict all possible states a spacecraft might encounter as it re-enters the Earth’s atmosphere, providing invaluable data for ensuring a safe and efficient journey back to Earth. The benefits of employing this method are manifold:

1. Enhanced Safety: This technique ensures that the spacecraft follows a safe trajectory, substantially minimizing the risks of catastrophic failures.
2. Increased Efficiency: By preemptively knowing all possible outcomes, engineers can design the most efficient re-entry paths.
3. Improved Reliability: The method provides a comprehensive map of potential scenarios, preparing mission planners to effectively handle various contingencies.

How We Implement Reachability Analysis

To perform these complex calculations, we use mathematical tools called zonotopes and constrained zonotopes. These tools help simplify the equations needed to accurately model the spacecraft’s trajectory.

Simulations and Real-world Application

Our simulations, which include both simplified models and more complex ones of hypersonic re-entry vehicles, demonstrate the effectiveness of our approach as demonstrated in our work. These simulations are crucial for testing our predictions and refining the algorithms that underpin our models. Below is a figure illustrating the reachable sets for a hypersonic re-entry vehicle, showcasing how the heat rate constraint impacts the vehicle’s trajectory options:

In summary, reachability analysis is not just a theoretical construct but a practical tool that significantly boosts the safety and efficiency of hypersonic re-entry, paving the way for more reliable and ambitious space exploration missions. Through continuous research and development, we aim to extend these techniques to even more challenging missions, potentially including manned missions to other planets.

Stay tuned for more updates!

• Install homebrew for mac, if you don’t have already. Visit brew to install homebrew
• Installing gcc with brew

 brew install gcc 

• Find the location where gcc is being installed. Generally, it should be in /usr/local/bin. However, when calling a compiler your bash will look into /usr/bin by default. Hence, you need to add the new directory to your PATH. Export the path using following command.

 export PATH=/usr/local/bin:$PATH 

• Now compile with OpenMP with following command

 gcc-10 -fopenmp -o prog_name prog_name.c 

Note:- gcc-10 is the current version that I installed. You may have installed gcc-8 or gcc-7 or maybe someother version in future release.