James Bern
Postdoctoral Associate
MIT CSAIL
Distributed Robotics Lab
jbern@mit.edu
he / him / his

Research: Soft-Rigid Robot Design, Modeling, and Control

Draw Robots: Data-Oriented Introduction to Graphics × Robotics

This is a short course to help undergraduate and early graduate researchers pick up the practical skills needed to do novel research at the intersection of computer graphics and robotics.

> MIT IAP 2022 Course Page  click to enlarge sample slides

Optimization Glossary

Notation: E_x = dE / dx, E_xx = d2E / dx2.

Gradient descent: The first order Taylor expansion of the energy E(x + dx) ≈ E(x) + E_x(x) dx is minimized by search direction aligned with dx ← -E_x.

Newton's method: The first order Taylor expansion of the gradient E_x(x + dx) ≈ E_x(x) + E_xx(x) dx is zero for search direction dx ← -inv(E_xx) E_x.

Basic backtracking line search: Given current best guess x and search direction dx, our goal is to find stepsize α such that E(x + αdx) < E(x). First check if α = 1 decreases the objective. If not then try α = 1/2, then 1/4, and so on. Give up after e.g. 15 attempts, either returning some nominal α or terminating the optimization.

Direct sensitivity analysis: Given control u, the resulting state x(u) solves the physical constraint C(u, x) = 0. Taking the total derivative of C(u, x(u)) = 0 yields the linear system C_u + C_x x_u = 0, which can be solved for the sensitivity x_u = -inv(C_x) C_u.