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

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

3. 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

4. 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

5. 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

6. 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

7. 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:

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

9. 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

10. 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

11. 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

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

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