Advances in Numerical Algebraic Geometry with Applications
summary
Robonaut 2 has 42 DOF with a 12‑DOF hand driven by ~16 tendons, requiring an open wrist whose kinematics reduce to polynomial systems; forward/inverse problems yield 4 or 8 solutions in the example. Software Bertini (and Bertini Real) implements path‑tracking with adaptive precision, witness sets to represent components, regeneration to build solutions equation‑by‑equation, and isosingular deflation plus diagonal homotopies for intersections. Concrete results: 8,652 solutions for the 9‑point Burmester case, a degree‑362 3–3 curve; sphere‑packing enumeration uses adjacency graphs and rule‑based continuations.
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chapters (22)
- 0:00Intro
- 1:54Outline
- 2:57Robonaut 2 on ISS
- 7:56Big Picture
- 10:51How do micro-spheres cluster?
- 13:05Homotopy Algorithms (a.k.a. Continuation)
- 19:14Basic Construct: Witness Set
- 20:14The Bertini Package
- 21:49Intersection A
- 27:29Regeneration: Step 1
- 28:48Regeneration: Step 2
- 33:08Projections and Cell Decomposition
- 37:56Real curves and surfaces
- 39:25Real Cell Decomposition
- 41:08Four-Bar Design: Burmester Problems
- 44:51Mixed Burmester family of problems
- 46:27Degree of Solution Set
- 48:00Case 3-3: Curve of degree 362
- 48:39Another 3-3 Burmester curve
- 51:35Sphere Packings
- 57:11Solving Packings
- 58:46Combinatorics of packings
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