# Implementing Rotation Matrix Constraints in Analog VLSI

David B. Kirk, Alan H. Barr

California Institute of Technology
From *Proceedings of SIGGRAPH 1993*, Computer Graphics
Proceedings, Annual Conference Series, New York: Association for
Computing Machinery Special Interest Group on Computer Graphics (ACM
SIGGRAPH), 1993, p. 45-52.

## Abstract

We describe an algorithm for continuously producing a 3x3 rotation matrix from 9
changing
input values that form an approximate rotation matrix, and we describe
the implementation of that constraint in analog VLSI circuits. This constraint
is useful when some source (e.g., sensors, a modeling system, other
analog VLSI circuits), produces a potentially "imperfect" matrix,
to be used as a rotation. The 9 values are continuously
adjusted over time to find the "nearest" true rotation matrix, based
on a least-squares metric.
The constraint solution is implemented in analog VLSI circuitry; with
appropriate design methodology (D. Kirk, Ph.D. thesis, Caltech, 1993),
adaptive analog VLSI is a fast, accurate, and low-power computational
medium. The implementation is potentially interesting to the graphics
community because there is an opportunity to apply adaptive analog
VLSI to many other graphics problems.

CR Categories and Subject Descriptors:

- C.1.2--- [Processor Architectures]: Multiprocessors - parallel
processors;
- C.1.3--- [Processor Architectures]: Other Architecture Styles;
- I.3.1--- [Computer Graphics]: Hardware Architecture - raster display
devices;
- I.3.3--- [Computer Graphics]: Picture/Image Generation;
- I.3.5--- [Computer Graphics]: Computational Geometry and Object
Modeling;
- I.3.7--- [Computer Graphics]: Three-Dimensional Graphics and Realism

General Terms: Algorithms, Graphics, Hardware

Additional Key Words and Phrases:
Animation, rotation, robotics, simulation,
constraint solution, interaction, adaptive, analog, CMOS, VLSI.