Explainers

PID

Interactive cart-pole demo with tunable gains, live disturbances, and real-time plots.

LQR

Solve the Riccati equation in-browser and watch penalty weights shape the closed-loop response.

MPC

A meta-algorithm that optimizes over planned future trajectories in receding-horizon fashion.

MCTS

The lookahead search algorithm powering AlphaGo, AlphaZero, and modern game AI.

Differential Flatness

Plan trajectories in flat output space and map back to states and controls exactly.

FLASK

Ultrafast sampling-based kinodynamic planning via differential flatness and SIMD parallelism.

Respect the Unstable

Gunter Stein's inaugural Bode Lecture on fundamental limitations in control.

Static output feedback stabilization is NP-hard

NP-hardness of static output-feedback stabilization and implications for control policy structure.

Explorations in Dynamics

Physics in N dimensions, reaction-diffusion systems, and SmoothLife.

Permutation-Invariant Neural Networks for Reinforcement Learning

Architecture-induced robustness to observation permutations, without domain randomization.

Polyak-Łojasiewicz

A walkthrough proof of linear convergence of gradient descent under the PL inequality.

Waymo AI

Waymo's approach to demonstrably safe AI using foundation model distillation.

Adversarial Training & Robot Safety

Why adversarially-trained classifiers can reduce rather than improve safety in robot learning.

Policy Gradient for LQR with Multiplicative Noise

Robust control as a gradient-dominated objective, with global convergence guarantees for policy gradient.

Bootstrapped Multiplicative Noise

Adaptive control that sizes its robustness to the actual estimation uncertainty in finite, noisy data.

Risk-Averse RRT* Planning

Distributionally robust sampling-based planning and tracking for nonlinear robots under uncertainty.