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Kevyn Young December 3, 2014 Introduction Previous Work Implementation/Algorithms Testing Results Image from http://modelaircraft.com/1758/first-test-flight-blade-350qx-quadcopter/ Use Structure from Motion to find AboveGround Level Altitude ◦ Use for terrain-following, station-keeping, landing OpenCV 2.4.9 (C++) Use Shi-Tomasi to find features in images Use Optical Flow to track features across images RANSAC to find ground plane points and camera homography 𝐻 (Discrete Space) 1 𝐻𝐶 = 𝜔 + 𝑣𝑁 𝑇 𝑑 1 + 𝑇𝑁 𝑇 𝑑 𝑋2 = 𝐻𝐷 𝑋1 → 𝐻𝐷 = 𝑅 Four Possible Solutions: Solution 1: 𝑅1 = 𝑊1 𝑈1𝑇 𝑁1 = 𝑣2 × 𝑢1 Solution 3: 𝑅3 = −𝑅1 𝑁3 = −𝑁1 1 1 𝑇3 = − 𝑇1 𝑑 𝑑 𝑅2 = 𝑊2 𝑈2𝑇 𝑁2 = 𝑣2 × 𝑢2 Solution 4: 𝑅4 = −𝑅2 𝑁4 = −𝑁2 1 1 𝑇4 = − 𝑇2 𝑑 𝑑 1 𝑇 = 𝐻𝐷 − 𝑅1 𝑁1 𝑑 1 Solution 2: 1 𝑇 = 𝐻𝐷 − 𝑅2 𝑁2 𝑑 2 Use single value decomposition to find 𝑢1 , 𝑢2 , 𝑣1 , 𝑣2 ◦ 𝑣1 and 𝑣2 are not velocities! 𝑈1 = 𝑣2 , 𝑢1 , 𝑣2 𝑢1 , 𝑊1 = 𝐻𝑣2 , 𝐻𝑢1 , 𝐻𝑣2 𝐻𝑢1 𝑈2 = 𝑣2 , 𝑢2 , 𝑣2 𝑢2 , 𝑊2 = [𝐻𝑣2 , 𝐻𝑢2 , 𝐻𝑣2 𝐻𝑢2 ] Only 2 real solutions → 𝑁𝑧 < 0 so that d is positive Use most vertical normal (largest |𝑁𝑧 |) as solution 𝑅 Δ𝑡 Find angular velocity: 𝜔 = Use continuous motion constraint: → Δ𝑡 = 1 𝐹𝑃𝑆 𝑋 𝑡 =𝜔 𝑡 𝑋 𝑡 +𝑣 𝑡 In image space: Continuous Homography Constraint: 𝑥 𝐻𝐶 𝑥 = 𝑥 𝑢 Remove angular velocity from optical flow 𝑢𝑥′ 𝑢𝑦′ −𝑥𝑥 𝑥𝑦 𝑢𝑥 = 𝑢 − 𝑦 − 1 + 𝑥𝑦 2 1 + 𝑥𝑥2 −𝑥𝑦 𝑥𝑥 𝑥𝑦 𝑥𝑥 With angular velocity removed: 1 𝐻𝐶 = 𝑣𝑁 𝑇 𝑑 𝜔 4-point algorithm (use RANSAC inliers): 𝑎𝑇 𝐻 𝑆 = 𝑥 𝑢 → 𝐴𝐻 𝑆 = 𝐵, 𝐴 = 𝑎1 , … , 𝑎𝑛 𝑇 ∈ ℝ3𝑛×9 , 𝐵 = 𝑥1 𝑢1 , … , 𝑥𝑛 𝑢𝑛 𝑇 ∈ ℝ3𝑛 , 𝑎 =𝑥⊗𝑥 𝐻 𝑆 = 𝐻11 , 𝐻21 , … , 𝐻33 𝑇 ∈ ℝ9 Reconstruct 𝐻𝐶 from 𝐻 𝑆 (normalize 𝐻𝐶 ) Known: 𝐻𝐶 , 𝑣, 𝑁 Solve for 𝑑 ∈ ℝ 𝑑 = 𝑁 𝑇 𝐻 −1 𝑣 → [𝑚𝑚] Synthetic Data ◦ No Rotation ◦ Translation in y-axis only (y and z in camera frame) Known Test Data ◦ Collected Using SCORbot in W220 ◦ Data collected using Raspberry Pi Camera Module Poor results, couldn’t prove implementation due to inadequate data ◦ No instantaneous velocity data ◦ No focal length information ◦ Camera wasn’t fixed with fixed zoom Low number of features ◦ Shi-Tomasi settings were incorrect Estimated, constant velocity was inaccurate Correct Homography found (Discrete Space) ◦ Confirmed by comparing results to planeDectection program provided by Professor Hoff (uses SURF for feature matching) Data Collected from SCORbot ◦ Tagged with velocity data ◦ Collected corresponding positional data Can confirm if altitude detected is correct