Solving System of Nonlinear Equations with the Genetic Algorithm - PowerPoint PPT Presentation

Solving System of Nonlinear Equations with the Genetic Algorithm and Newton’s Method Ryuji Koshikawa, Akira Terui, and Masahiko Mikawa University of Tsukuba Tsukuba, Japan International Congress on Mathematical Software 2020 TU Braunschweig, Germany (Online) 1

What’s in this talk • An implementation of a method combining the Genetic Algorithm (GA) and Newton’s method for solving a system of nonlinear equations • Application of the proposed method for solving the positioning problem of multiple rovers for asteroid exploration 2

Application: asteroid explorations 3

Asteroid explorations Asteroid Explorer “OSIRIS-REx” Asteroid Explorer “Hayabusa 2” NASA (USA) Japan Aerospace Exploration Agency (JAXA) https://solarsystem.nasa.gov/missions/osiris-rex/in-depth/ https://global.jaxa.jp/projects/sas/hayabusa2/ 4

筑波大学計算機数学グループ 春の合宿 格納状態 母船から分離後， 離散しつつ降下 1) 複数の小型ローバで構成される探査システム 本研究の提案(1/3) Asteroid exploration with multiple small rovers [3] 30 [mm] 30 [mm] 5

Asteroid exploration with multiple small rovers [3] (a) (b) • Making redundant communication paths with a wireless mesh network • In radio communication with other rovers, use the Received Signal Strength Indicator (RSSI) for estimating each other’s position (relative distance and angle) 6

Estimating each other’s position • It is di ffi cult to control the orientation of the rovers on the asteroid Antenna b Antenna a • The rover has twelve antennas on each side to keep the good condition for communication with other rovers • Antennas and : a pair of the two on 30 [mm] a b the top side (used for communicating with other rovers) 7

The rover coordinate system • For Rover ( ), define a i i = 1,…, n coordinate system Σ i z 15 [mm] 15 [mm] b i a i Antenna Antenna b i • and : antennas and on the top a i b i a b a i y y Σ i Top view x x side of Rover , respectively i z y Rover i Main body Front view of rover and : a coordinate system with • Σ a i Σ b i the origin at and , respectively a i b i 8

The position of the antennas a i p b j = t ( a i x b j , a i y b j , a i z b j ) : the position of antenna w.r.t. • b j Σ a i 1 p a i • : the position of antenna w.r.t. a i Σ 1 1 R a i ∈ ℝ 3 × 3 : the orientation of the antenna w.r.t. • a i Σ 1 9

Position of the antennas Then, by using the relationship b j p a i 1 p a i = 1 p b j + 1 R b j 1 p b j = 1 p a i + 1 R a i a i p b j and , we derive the relative position of the antenna w.r.t. and vice versa, as b j Σ a i b j p a i = ( 1 p a i − 1 p b j ) b j R 1 a i p b j = ( 1 p b j − 1 p a i ) a i R 1 and b j R 1 = ( 1 R b j ) − 1 a i R 1 = ( 1 R a i ) − 1 with and . 10

The horizontal and the elevation angle of the antennas ⎝� ⎠ a i ϕ b j a i θ b j and : the horizontal and the elevation 
 • z Σ z a i b j x θ a i Σ b j angles of the antenna w.r.t. , respectively 
 y b j Σ a i a i b j θ a i φ a i x z y b j φ (see the figure) b j Σ 1 y x a i p b j = t ( a i x b j , a i y b j , a i z b j ) a i ϕ b j a i θ b j • Using , and are derived as 
 a i ϕ b j = arctan ( a i y b j a i x b j ) a i z b j , , respectively a i θ b j = arctan ( a i x b j ) 2 + ( a i y b j ) 2 b j ϕ a i b j θ a i • and are derived similarly 11

The RSSI between the antenna and a i b j : the RSSI between the antenna and r a i _ b j a i b j r a i _ b j ( a i x b j , a i y b j , a i z b j , a i ϕ b j , a i θ b j , b j ϕ a i , b j θ a i ) = r ′ � ( a i x b j , a i y b j , a i z b j ) + r h ( a i ϕ b j ) + r h ( b j ϕ a i ) + r v ( a i θ b j ) + r v ( b j θ a i ) where x 2 + y 2 + z 2 + 0.31) − 49.17 r ′ � ( x , y , z ) = − 14.69 log 10 ( r h ( ϕ ) = 5 2 (cos(2 ϕ ) − 1) cos( 5 2 π )cos( 5 2 π − | θ | ) r v ( θ ) = 25 − 1 sin( 5 2 π − | θ | ) 12

Combining GA and Newton’s method 13

̂ Genetic Algorithm (GA) • Heuristic algorithm used for general problems • An individual chromosome ↔ a solution, 
 Genes ↔ components of the solution • Deriving the optimal solution through crossover and mutation • Evaluation function: ( : the measured RSSI values, : the estimated RSSI r a i _ b j ¯ r a i _ b j values) 
 n n r a i _ a j ) 2 + (¯ r a i _ b j ) 2 + (¯ r b i _ a j ) 2 + (¯ ∑ ∑ r b i _ b j ) 2 } f ( r ) = {(¯ r a i _ a j − ̂ r a i _ b j − ̂ r b i _ a j − ̂ r b i _ b j − ̂ i =1 j = i +1 14

Advantages and disadvantages of GA 👏 👎 • Slow convergence rate compared • Computed solutions are not so to other algorithms sensitive to the initial values as other methods • Accuracy of the solutions may • Can be used for a global vary on each solution search 15

A combination of GA and Newton’s method for solving a system of nonlinear equations [1] 👎 👏 Convergence does not GA Relatively slow convergence depend on the initial values Relatively fast convergence Newton’s Convergence of the roots and high accuracy (for well- method depend on the initial values posed problems) 16

A combination of GA and Newton’s method for solving a system of nonlinear equations [1] Use Newton’s Use GA to obtain method for fast good initial convergence to the approximation for roots the roots 17

Rearrange nonlinear equations for the use of GA and Newton’s method C 2 ( x 2 , y 2 ) (Antenna a 2 ) A 2 C 21 • Assume that all the rovers have the 
 ϕ B 2 (Antenna b 2 ) same direction and placed on the π 2 − ϕ Rover 1 a i θ b j plane, thus are omitted C 1 ( x 1 , y 1 ) xy ϕ (Antenna a 1 ) A 1 C 12 A 12 B 12 B 1 (Antenna b 1 ) 18

Setting up nonlinear equations for the use of GA and Newton’s method ( a 1 x a 2 ) 2 + ( a 1 y a 2 ) 2 + 0.31) − 49.17 + 5(cos 2 φ − 1) r a 1 _ a 2 ( a 1 x a 2 , a 1 y a 2 , a 1 φ a 2 , a 2 φ a 1 ) = − 14.69 log 10 ( ( b 1 x b 2 ) 2 + ( b 1 y b 2 ) 2 + 0.31) − 49.17 + 5(cos 2( π r b 1 _ b 2 ( b 1 x b 2 , b 1 y b 2 , b 1 φ b 2 , b 2 ϕ b 1 ) = − 14.69 log 10 ( 2 − φ ) − 1) By subtracting both sides, the equations are reduced to: r a 1 _ a 2 − r b 1 _ b 2 = 10 cos(2 φ ) φ = arctan ( a 1 x a 2 ) a 1 y a 2 , , ( a 1 x a 2 ) 2 + ( a 1 y a 2 ) 2 + 0.31) − 49.17 + 5(cos 2 φ − 1) r a 1 _ a 2 ( a 1 x a 2 , a 1 y a 2 , φ ) = − 14.69 log 10 ( 19

Experiments 20

Experiments: calculating positions of the rovers • Calculate the relative position of Rover i y ( ) from Rover 0 i = 1,…,8 (50000 . 0 , 70000 . 0) Rover 7 (0 , 60000 . 0) (90000 . 0 , 60000 . 0) • The population: 100 Rover 8 Rover 6 (80000 . 0 , 50000 . 0) Rover 5 (30000 . 0 , 40000 . 0) • The number of generation: 200 Rover 4 (20000 . 0 , 30000 . 0) Rover 1 • Single-point crossover with crossover rate: (10000 . 0 , 20000 . 0) Rover 3 (10000 . 0 , 10000 . 0) 0.9 Rover 2 (0 , 0) x Rover 0 • Roulette wheel selection Fig. 5. Positions of the rovers (unit: mm). • Mutation rate: 0.01 21

Experiments: calculating positions of the rovers • After repeating the GA computation for y times, the best 10 × 10 = 100 (50000 . 0 , 70000 . 0) Rover 7 (0 , 60000 . 0) (90000 . 0 , 60000 . 0) individuals which minimize were f ( r ) Rover 8 Rover 6 (80000 . 0 , 50000 . 0) used as the initial values for Newton’s Rover 5 (30000 . 0 , 40000 . 0) method Rover 4 (20000 . 0 , 30000 . 0) Rover 1 (10000 . 0 , 20000 . 0) • Newton’s method was terminated when: Rover 3 (10000 . 0 , 10000 . 0) Rover 2 (0 , 0) x • |updated value| < 1.0 × 10 − 10 Rover 0 Fig. 5. Positions of the rovers (unit: mm). • # of iterations reached 100 22

Experiments: calculating positions of the rovers y • Computing environment: (50000 . 0 , 70000 . 0) Rover 7 (0 , 60000 . 0) (90000 . 0 , 60000 . 0) • Intel Xeon E5607 2.27 GHz, RAM Rover 8 Rover 6 (80000 . 0 , 50000 . 0) Rover 5 48 GB (30000 . 0 , 40000 . 0) Rover 4 (20000 . 0 , 30000 . 0) Rover 1 • Linux 3.16.0, GCC 9.1.1 (10000 . 0 , 20000 . 0) Rover 3 (10000 . 0 , 10000 . 0) Rover 2 (0 , 0) • GAlib [4] was used for GA x Rover 0 computation Fig. 5. Positions of the rovers (unit: mm). 23


 Experiment 1: estimated positions by GA (to be used as the initial values in Newton’s method) Table 1. The result of estimation by the genetic algorithm with the setting of 10 × 10. Estimated position Actual position (The initial value for Newton’s method) Relative error (8) i 1 (20000 . 0 , 30000 . 0) (18411 . 987305 , 33178 . 985596) 0 . 052413 2 (10000 . 0 , 10000 . 0) (10122 . 98584 , 10196 . 990967) 0 . 016006 3 (10000 . 0 , 20000 . 0) (11677 . 993774 , 20789 . 993286) 0 . 066396 4 (30000 . 0 , 40000 . 0) (26173 . 995972 , 43239 . 990234) 0 . 010896 5 (80000 . 0 , 50000 . 0) (77340 . 995789 , 55555 . 999756) 0 . 009400 6 (90000 . 0 , 60000 . 0) (89379 . 997253 , 64320 . 999146) 0 . 018041 7 (50000 . 0 , 70000 . 0) (50185 . 989380 , 75285 . 995483) 0 . 051808 8 (0 , 60000 . 0) (2766 . 006470 , 64238 . 998413) 0 . 071642 — Average — 0 . 037075 Relative error: | d i − d ′ � i | / d i : the given distance of the rover from the origin, : the estimated distance of the rover from the origin d i d ′ � i 24