Mathematics in Orbit

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1 Mathematics in Orbit Dan Kalman American University Slides and refs at

2 Outline Basics: 3D geospacial models Keyhole Problem: Related Rates! GPS: space-time triangulation Sensor Diagnosis: domain dominated max/min Resource Allocation: when existence is enough

3 Basic Geospacial Model Work in 3 Earth is a sphere Equator in xy plane; north pole on z axis Rotate about z axis once per day Inertial Frame: Newton s s laws hold None of these are really true

4 Topic 1: Keyhole Problem

5 Tracking a satellite Antenna moves like desk chair Full rotation about vertical axis A bit more than 90 degrees about horizontal axis If the satellite passes directly overhead, antenna cannot rotate fast enough to maintain track

6 Problem Set Up Known satellite trajectory r(t) Known antenna location s(t) ) (moves with rotating Earth Antenna coordinate frame: east, north, up Pointing angles: α(t) ) about up axis; δ(t) ) about a horizontal axis like spherical coords Find maxima of α, α, δ, δ

7 Local Coord System

8 Up, East, North Antenna location s(t) constant latitude φ longitude λ linear with t Up unit vector parallel to s(t) k = z axis unit vector East unit vector e North unit vector n

9 Pointing Angles Line of sight vector v = r s Use local coords: v = (v e,( v n, v u) = (x,( y, z) v = ( v v ) 1/2 tan α = x / y sin δ = z / v

10 Related Rates Factorial (!) Need to compute α, α, δ, δ Chain of calculations for α and δ Symbolic differentiation is a nightmare Alternative: automatic differentiation That s s a topic for another talk Calculus in Orbit, Math Horizons, November 94

11 Topic 2: GPS Space Time Triangulation

12 Two D Positioning Find location given directions from 2 known locations triangulation Find location given distances from 2 known locations

13 2D Space-Time Location Determine the unknown position as well as the time Receiver correlates signals received at one time from three different sources Each signal gives the source s s location and time at broadcast This determines receiver s s position and time We can simulate the situation

14 Show Simulation

15 Analytic Formulation Let the first signal say its position and time are (x( 1, y 1, t 1 ) Let the unknown location and time be (x,( y, t) Speed of light = c Distance = rate x time (x - x 1 ) 2 + (y( - y 1 ) 2 = c 2 (t - t 1 ) 2

16 Analytic Formulation (x - x 1 ) 2 + (y( - y 1 ) 2 = c 2 (t - t 1 ) 2

17 Analytic Formulation (x - x 1 ) 2 + (y( - y 1 ) 2 = c 2 (t - t 1 ) 2 Similar equations for other 2 sources Each is a cone in (x,( y, t) ) space Solution is a point of intersection of three cones Aside: each cone is light cone (of relativity fame) in 2 x 1 space-time

18 Cone Intersection Geometry

19 For 3x1 Space Time Need 4 sources, rather than 3 Simulation has 4 expanding spheres instead of 3 expanding circles We solve four equations of the form (x - x k ) 2 + (y( - y k ) 2 + (z( - z k ) 2 = c 2 (t - t k ) 2 Each is a light cone in 4 dimensions Pairs intersect in 3D hyperplanes

20 Solving the System (x - x k ) 2 + (y( - y k ) 2 + (z( - z k ) 2 = c 2 (t - t k ) 2 2x k x 2y k y 2z k z + 2 c 2 t k t = c 2 t 2 - x 2 - y 2 - z 2 - c 2 t 2 k - x 2 k - y 2 k - z 2 k Quadratic terms don t t depend on k Subtracting in pairs gives three linear equations in four unknowns Express x, y, z as linear functions of t Substitute in one quadratic equation and solve College Math Journal, Nov 2002

21 Topic 3: Sensor Diagnosis

22 Angle Detection Sensor detects a bright object in field of view Intensity of signal varies with angle θ Sensor gives θ within some error Reading from one sensor puts the bright object within an annulus Direction of bright object given by intersecting annuli from different sensors

23 Intersecting 2 Annuli Favorable geometry: direction vector must lie in one of 2 small regions. Zero in on one with 3 rd annulus Less favorable geometry: greater uncertainty; no ambiguity

24 Intersecting 3 Annuli Consistent readings, star direction found Inconsistent readings, possible indication of one or more sensors malfunctioning

25 Fault Diagnosis via Voting 3 Pairs of sensors with common boresights Bold circles represent consistent sensor pairs Lighter circles show output of individual sensors Numbers on intersections show how many sensors participate Highest count is most reliable output Nonparticipating sensors are suspect

26 Voting Problem Find maximal intersection regions Maximal in terms of number of sensors involved in the intersection Combinatorics: : with n sensors there are 2 n 1 possible subsets Determine which subsets have nonempty intersection

27 McGuire Method Rod McGuire idea Intersect annuli in pairs Find minimal rectangular cell containing each nonempty intersection region Find overlaps between cells by sorting the endpoints Complexity is order n 2 instead of 2 n

29 Algorithm Development Inputs: centers and radii for two annuli on the unit sphere Outputs Number of intersection regions (0,1,2) For each region, bounding cell specs: x min y min z min x max y max z max Three max/min problems for each region Trivial objective functions Domain geometry dominates problem

30 Many Possible Configurations

31 How Many Regions?

32 Extreme Coords in a Region Region is a polygon on sphere Objective function has constant gradient Extreme values occur in the interior of a region, on a boundary arc, or at a vertex, ie,, an intersection of two boundary circles All of these can be found with vector methods As in calculus, we generate a finite candidate set and select max and min values.

33 Topic 4: Resource Allocation

34 Instantaneous Visibility Geometric Models for visibility At any time, positions of satellites and ground stations given by motion models Constraints described in terms of lines, angles, cones Line of sight from station to satellite is computed as a vector Vector methods used to compute angles Communication possible when satellite can see the station

36 Discrete Time Step Model Compute positions of all satellites and stations at one fixed time Determine which satellites can see which stations Advance time by one minute, repeat all calculations Repeat many many times For a 24 hour simulation, repeat 1440 times

38 Design Problem Fixed ground stations Predefined connection time requirements LARGE range of choices for satellite orbits For a given set of orbits, can all connect time requirements be met?

40 Graph Theory Formulation Bi-Partite Graph: Two sets of vertices One vertex for each ground station A separate vertex for each satellite for each time step Edges indicate visibility Visibility graph

Problem Formulation Edge Count = degree At station vertex degree = amount of connect time Assignment Subgraph: Degree 1 at each satellite/time vertex Given:

42 Problem Formulation Edge Count = degree At station vertex degree = amount of connect time Assignment Subgraph: Degree 1 at each satellite/time vertex Given: Visibility graph and required degree at each station vertex To Find: Assignment subgraph that meets all requirements Existence Question: does a solution exist?

44 Satellites: Necessary Conditions ConReq: : connection requirement for a station ConReq for a station must be visibility graph degree for the station Total of all ConReqs must be number of satellite-time time nodes These are two extreme cases of a more general constraint For any subset of stations, the sum of ConReqs must be the number of satellite-time time nodes connected to the subset

46 Necessary and Sufficient Chris Reed approach With n stations, 2 n - 1 necessary conditions Check them all If one fails, no solution What if all conditions are met? A solution must exist! The marriage problem in graph theory We can check whether an assignment graph exists without actually finding one!

47 Minimal Transmission Rate Another Chris Reed Idea Connection requirements inversely proportional to transmission rate Assignment Problem Unsolvable: try over with increased transmission rate Assignment Problem Solvable: try over with decreased transmission rate Find smallest transmission rate that permits assignment problem to be solved Use a computer program to do this for a large number of satellite constellations One way to compare system designs Existence result used for a practical purpose

48 References 1. Calculus in Orbit, Math Horizons,, November 94 n/pdffiles/calcorbit.pdf 2. An Undetermined Linear System for GPS. College Mathematics Journal,, November n/pdffiles/gps.pdf 3. Marriages Made in the Heavens: A Practical Application of Existence. Mathematics Magazine,, April n/pdffiles/mmexistence.pdf 4. When Existence is Enough, Math Horizons,, April 97

An Underdetermined Linear System for GPS

An Underdetermined Linear System for GPS Dan Kalman Dan Kalman (kalman@american.edu) joined the mathematics faculty at American University in 1993, following an eight year stint in the aerospace industry