My research is focused on guaranteeing that autonomous systems are safe. In the case of mobile robots, this means that they avoid collisions. I work to bridge the gap between theory and application, to ensure that the beautiful math describing safety actually holds true when we put it on hardware.

During my Ph.D. work at Michigan, I developed Reachability-based Trajectory Design, or RTD. This is a receding-horizon planning method that generates dynamically-feasible, collision-free trajectories for autonomous mobile robots. Check out the tutorial for a walkthrough, or my dissertation talk for a broad overview. Below, you can see the various platforms on which my collaborators and I have implemented RTD.

At Stanford, I'm developing geometric methods for sensing- and perception-based uncertainty modeling. This is critical to enable autonomous systems to adapt and learn their own notions of safety on the fly, and to eventually communicate safety to people.

Note, all of my publications are listed here.

Reachability-Based Trajectory Design


We started our RTD research with a differential-drive Segway. It is able to navigate unforeseen, random environments consistently without crashing. An RTD implementation for the Segway is available on GitHub.

To learn more about RTD, check out my talk from R:SS '19.

We also enabled RTD to guarantee that it finds optimal trajectory plans in each receding-horizon planning iteration, which we call RTD*. To do this, we make use of a Parallel Constrained Bernstein Algorithm.


The Segway demonstrates RTD for unstructured scenarios. We have also shown that this trajectory design method can exploit scenario structure with this Rover on a mini road, making lane changes around random obstacles.

Passenger Sedan

With our friends at Ford, we put RTD on a full-sized passenger sedan in the high-fidelity vehicle simulator, CarSim.

Electric Vehicle

In collaboration with the University of Sydney, we put RTD on their Electric Vehicle (EV) platform.


This work won the ASME Best Student Paper Award at DSCC '19!

To show the flexibility of the RTD way of thinking, we applied to autonomous drone flight. We also used zonotope reachability analysis instead of the sums-of-squares approach from our previous work.

The simulations use a dynamic model of an AscTec Hummingbird, and are entirely created (and rendered) in MATLAB (code here). The hardware demo uses a Parrot Mambo drone with a PhaseSpace motion capture system.

Fetch Manipulator

We have also applied RTD to robotic manipulators -- in particular, a Fetch shown in the video. This leverages our previous zonotope reachability results from the quadrotors, but for the creation (at runtime) of reachable sets of a 6-DOF arm. This enables real-time, safe trajectory planning for arms.

Modeling Uncertainty in Estimators and Learned Models

The Geometry of GNSS Uncertainty

I spent my first year at Stanford as a postdoc with Grace Gao's NAV Lab, which has a strong focus on GNSS (Global Navigation Satellite System) for safe autonomous navigation.

My labmate Ramya and I developed a computationally-efficient method for shadow matching, where one uses a 3-D urban map to identify GNSS shadows, or areas where satellite signals are blocked, to create artificial set-valued measurements that represent uncertain possible receiver positions. Check out Ramya's great talk!

You might notice that our shadow matching method uses polytopes, which can't represent curved convex shapes, such as the Gaussian distribution confidence ellipsoids commonly associated with measurement uncertainty. To solve this challenge, Adam Dai and I created ellipsotopes, a novel set representation that fuses the benefits of polytopes and ellipsoids.

Reachability for Learned Models

Despite the challenges of the COVID-19 pandemic in 2020 and 2021, I've had an amazing opportunity to mentor excellent students.

Simon Shao, Chao Chen, and I used RTD to build a safety layer for a reinforcement learning (RL) agent, which can outperform vanilla RTD. Check out the paper here, and Simon's talk from ICRA 2021!

Edgar Chung, Adam Dai, and Derek Knowles worked with me to compute exact forward reachable sets of feedforward neural networks. Excitingly, this work bridges the gap between verification and training of a neural network. Check out the paper here.

Selected Publications

  1. S. Kousik*, S. Vaskov*, F. Bu, M. Johnson-Roberson, and R. Vasudevan. “Bridging the gap between safety and real-time performance in receding-horizon trajectory design for mobile robots.” The International Journal of Robotics Research, September 2020. Link.

  2. S. Kousik, P. Holmes, and R. Vasudevan. “Safe, aggressive quadrotor flight via reachability-based trajectory design.” DSCC 2019. Link.

  3. P. Holmes, S. Kousik, B. Zhang, D. Raz, C. Barbalata, M. Johnson-Roberson, and R. Vasudevan. "Reachable sets for safe, real-time manipulator trajectory design." RSS 2020. Link.

See my CV or this page for a complete list of publications.