Engineering Excellence Symposium Azusa, CA

1. Reconfigurable Computing Machine Implementation of Rational Trigonometry Algorithms for Missile Tracking and Prediction Engineering Excellence Symposium Azusa, CA 8 November 2006 Richard Wallace Senior Engineer

2. Rational Trigonometry (RT)What is it? • RT defines a system of construction for triangles without reference to circles • Rational trigonometry does not use any transcendental values, and is entirely solved through algebra and quadratic equations. • It is a recently consolidated form of non-Euclidean geometry that depends on • Separation of points (quadrance) • Separation of lines (spread) • The name “Rational” • Calculations use rational numbers or roots of rational numbers • Reference Text: Divine proportions: rational trigonometry to universal geometry. N.J. Wildberger, Wild Egg, Australia, 2005, ISBN 097574920X More on these later…

3. Brief introduction to RT (1 of 3) • Quadrance (Q) is a measure of separation of points, while spread (s) is a measure of separation of lines. A relation can be shown that quadrance is the square of distance and spread is the square of the sine of an angle • The actual definitions of quadrance and spread in RT are independent of distance, angle, and their trigonometric functions and are based on finite arithmetic and algebra • Spread is a dimensionless number between 0 and 1. The spread between parallel lines is 0. The spread between perpendicular lines is 1. Its definition can be shown by construction. Given that l1 andl2 intersect at the point A as shown choose a point where B  A on one of the lines. For this construction l1 is used. Let C be the foot of the per­pendicular from B to l2 as shown . If quadrance, Q(B,C) = Q and Q(A,B) = R the spread s is the ratio:

4. Brief introduction to RT (2 of 3) Triple Quad Formula Spread Law Cross Law There are four basic laws and a restatement of the Pythagorean theorem in RT Triple Spread Formula

5. Brief introduction to RT (3 of 3) Given the coordinates of two points (x1,y1) and (x2,y2), the quadrance between them is: Given the coordinates of two points on each of two lines (x11,y11), (x12,y12) and (x21,y21) (x22,y22), the spread between them can be calculated as: Spread protractor

6. RT projections to spheres (1 of 2) Rational Trigonometry has spherical projections. Given the sphere x2 + y2 + z2 = 1 and center O = [0, 0, 0] any two non-antipodal points A and B lying on it determine a unique spherical line which is the intersection of the sphere with the plane OAB. Any two such spherical lines intersect at a pair of antipodal points. As shown on the left, a spherical triangle is formed by three spherical points A, B, and C and three spherical lines, and on the right the corresponding projective triangle, consisting of three projective points a, b, and c and the three projective lines that they form.

7. RT projections to spheres (2 of 2) • Projective Thales’ theorem Spread S; Quadrances q & r as shown • There are twelve other theorems which we don’t have time to go through. Please reference the text • Summary: The geometry is complete to cover conic, spherical, and elliptic constructions as well as planar RT does not cover circular or harmonic functions to deal with circular motion, Fourier analysis and the like, but those wave-like functions with no natural zero would be better not called “trigonometric.” They are not related to triangles. Classic Thales’ theorem: If A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle.

8. ECF in Euclidian (ET) and Rational form

9. RT method ½ of the solution • Questions: • Why not use CORDIC algorithms? • Isn’t rational trigonometry is just a reformulation of what we already know in Euclidian geometry? • Not quite… • CORDIC is an approximation for non-HW multiplier FPGAs. We use modern FPGAs. Slower computational speed issues using CORDIC • RT does expresses the planar and solid geometry we know with three improvements: • The calculations are exact rather than approximate, eliminating compounded error • Calculations can use more efficient fixed-point, algebraic operations taking advantage of HPC/FPGA structures/circuits • Few, or no, transcendental calculations will result in faster time to solution and fewer computational resources needed for such calculations. • The other ½ is the machine implementation… COordinate Rotation DIgital Computer

10. Elements Xilinx Virtex-4 XC4VLX160 or XC4VSX55 device Co-processing with AMD Opteron All programmable using Celoxica DK HTX Interface Data transfer up to 3.2GB/s Direct access to entire host system memory space Bridge FPGA manages I/O tasks freeing user FPGA for co-processing Dedicated Memory 24MB QDR SRAM on board 9.6 GB/sec max transfer rates Reconfigurable Computing (RC) Prototype System RCHTX-XV4

11. Programming the RC prototype • Use the appropriate DK design suite components; not all elements needed • Refactor existing C/C++ algorithms in Handel-C • Use Dr. Prasanna, et. al. (USC) method to improve algorithms Celoxica full DK design suite Celoxica DK subset needed Dr. V. Prasanna, USC

12. Reconfigurable Computing Fabric Fixed construction von Neumann machine Where RT and RC synergize • Computational processing made of highly flexible computing “fabrics” of single-purposed circuits that can be reconfigured, reconnected, into new-single purposed circuits using control configurationsdriven by the computational calculation itself. • The principal difference of reconfigurable computing (RC) when compared to using ordinary microprocessors is the ability to make substantial changes to the data path itself in addition to the control flow of the computation. • No instructions • Hardware is configured for a particular application • Parallelism • Multiple functional units of a given type • Better resource utilization than general purpose processor • Can be reconfigured for new application • Memory structure tailored to the application • Good for data-intensive applications

13. Rational Trigonometry (RT)Why use it? • What we know • Knowledge that the majority of engagement angles are acute • Knowledge that the majority of tracking calculations can be represented in fixed-point • Knowledge that the majority of filter calculations are scalar and the calculations can be algebraic rather than trigonometric • Knowledge that fixed-point and scalar operations are best fit to structural calculation rather than temporal calculation in RC devices • What we get • With RT + RC  Accuracy, Precision, and Speed • Flexibility, Size, Weight, Power, and Thermalefficiencies

14. Progress & Improvements • Progress • Testing ET to RT mapped algorithms on GP system. Promising results, > 20% improvement on GP • Discrete filter operations using Bridge as controller • System thermal improvements expected based on code written • Challenges • Assuring all operations are RT, not ET • Proper use of Handel-C pragmas • Balancing use of RC resources USC method Prasanna, et. al. FPGA register Energy Use

15. Questions and Answers Questions?