Coordinate Transformation in 3D Final Project Presentation ME 535: Computational Techniques in Engineering Michael Searing 4 June 2018

Problem Statement Mating of space vehicle stages Limited choice of control systems Inaccessible interfacing geometry Precision control required

Quantitative Phrasing Control vector c Input value to physical system Measurement vector m Output value from physical system Iterative process All values are differential Represents actual control and measurement of system Transformation matrix D Solves the control problem No closed-form solution for many 3-dimensional systems involving rotations

Addition of Intermediate State Problem Controls and measurements are unrelated, creating messy transformations These should be separated Solution Intermediate, simple state vector s Adds more equations, but equations are much simpler Re-formulation D can be written as combination of A and B

Example in 2 Dimensions Simplified system to illustrate problem Two measurements m1 and m2 Two controls c1 and c2 Two-element state vector of s1 and s2

Example in 2 Dimensions – Continued Reasoning for intermediate state Imagine finding partial derivatives of c with respect to m Now imagine in 3 dimensions with 6 degrees of freedom

Example in 2 Dimensions – Continued Solving in MATLAB Start at s0 = (1, 1) End at sf = (1.5, 0.75) 1st-order solver, rate = 0.5 Iterative solution:

Example in 2 Dimensions – Continued Accelerating solution Similar to successive over-relaxation method No effect on order, just rate

Example in 2 Dimensions – Continued Analytical solution difficult or impossible in multiple dimensions Numerical determination of transformation matrices much easier Assume geometry is well-known Sample points near final state 1st-order (2-point) derivative is good enough

Example in 2 Dimensions – Continued Acceleration is still possible Optimal value of alpha is shifted, but best rate is similar Large error in linearization is not that bad for solution time

Insight into System Using Condition Number What’s wrong with this system? cond(A) = 30 control system is bad cond(B) = 2 measurements are OK Powerful tool to augment engineering intuition Numerical solution (sampled) delta-x of 0.1 cond(A) = 6.5 delta-x of 0.01 cond(A) = 20 Linearization error matters here

Insight into System Using Condition Number – Continued Keep-out zones for control/measurement systems visible on contour plots

Conclusions Simple numerical tools aid understanding of 3-dimensional problems Concepts in 2D map directly to 3D