"""Bölüm 14: hiperbolik TDOA konum belirleme laboratuvarı. Model yalnız iki boyutlu, sabit yayılma hızlı, eşzamanlı saatli ve gürültüsüz bir sistemi temsil eder. Çok yollu yayılım, engeller, kırılma, saat sürüklenmesi, hız değişimi, sensör belirsizliği ve üç boyutlu konum modele dahil değildir. """ from __future__ import annotations import numpy as np from scipy.optimize import least_squares from agbook import ( canonical_array_sha256, hyperbola_diagnostics, hyperbola_frame_from_foci, rotation_matrix_2d, ) def signed_distance_difference( point_m: np.ndarray, first_receiver_m: np.ndarray, second_receiver_m: np.ndarray, ) -> float: """İlk ve ikinci alıcıya uzaklıkların işaretli farkını metreyle verir.""" return float( np.linalg.norm(point_m - first_receiver_m) - np.linalg.norm(point_m - second_receiver_m) ) def tdoa_residuals_m( point_m: np.ndarray, receivers_m: np.ndarray, measured_differences_m: np.ndarray, ) -> np.ndarray: """AB ve AC çiftlerinin model-ölçüm uzaklık farkı artıklarını verir.""" receiver_a_m, receiver_b_m, receiver_c_m = receivers_m modeled_m = np.array( [ signed_distance_difference(point_m, receiver_a_m, receiver_b_m), signed_distance_difference(point_m, receiver_a_m, receiver_c_m), ] ) return modeled_m - measured_differences_m def tdoa_jacobian( point_m: np.ndarray, receivers_m: np.ndarray, measured_differences_m: np.ndarray | None = None, ) -> np.ndarray: """İki uzaklık-fark denkleminin noktaya göre boyutsuz Jacobianını verir.""" del measured_differences_m offsets = point_m - receivers_m distances_m = np.linalg.norm(offsets, axis=1) if np.any(distances_m == 0.0): raise ValueError("Kaynak konumu bir alıcıyla çakışamaz.") unit_directions = offsets / distances_m[:, None] return np.vstack( [ unit_directions[0] - unit_directions[1], unit_directions[0] - unit_directions[2], ] ) def main() -> None: wave_speed_m_s = 343.0 receivers_m = np.array( [ [-6.0, 0.0], [6.0, 0.0], [0.0, 8.0], ] ) true_source_m = np.array([2.0, 3.0]) measured_differences_m = np.array( [ signed_distance_difference(true_source_m, receivers_m[0], receivers_m[1]), signed_distance_difference(true_source_m, receivers_m[0], receivers_m[2]), ] ) measured_tdoa_s = measured_differences_m / wave_speed_m_s pair_indices = ((0, 1), (0, 2)) frames = [] branches = [] sample_diagnostics = [] sample_parameters = np.linspace(-1.5, 1.5, 13) for difference_m, (first_index, second_index) in zip( measured_differences_m, pair_indices, strict=True, ): semi_transverse_m = 0.5 * abs(float(difference_m)) frame = hyperbola_frame_from_foci( receivers_m[first_index], receivers_m[second_index], semi_transverse_m, ) branch = 1 if difference_m > 0.0 else -1 frames.append(frame) branches.append(branch) sample_diagnostics.append( hyperbola_diagnostics( frame.center, frame.transverse_direction, frame.semi_transverse, frame.semi_conjugate, sample_parameters, branch=branch, ) ) initial_guess_m = np.array([1.0, 2.0]) result = least_squares( tdoa_residuals_m, initial_guess_m, jac=tdoa_jacobian, args=(receivers_m, measured_differences_m), method="trf", xtol=1e-14, ftol=1e-14, gtol=1e-14, max_nfev=100, ) estimated_source_m = result.x final_residuals_m = tdoa_residuals_m( estimated_source_m, receivers_m, measured_differences_m, ) position_error_m = float(np.linalg.norm(estimated_source_m - true_source_m)) jacobian = tdoa_jacobian(estimated_source_m, receivers_m) singular_values = np.linalg.svd(jacobian, compute_uv=False) condition_number = float(singular_values[0] / singular_values[-1]) assert result.success np.testing.assert_allclose(estimated_source_m, true_source_m, atol=2e-13) np.testing.assert_allclose(final_residuals_m, 0.0, atol=2e-13) assert np.linalg.matrix_rank(jacobian) == 2 for difference_m, branch, diagnostic in zip( measured_differences_m, branches, sample_diagnostics, strict=True, ): np.testing.assert_allclose( diagnostic.signed_focal_differences, difference_m, atol=3e-14, ) assert diagnostic.branch == branch assert diagnostic.maximum_absolute_algebraic_residual < 2e-14 assert diagnostic.maximum_absolute_focal_difference_residual < 4e-14 rotation = rotation_matrix_2d(np.deg2rad(29.0)) translation_m = np.array([11.0, -4.0]) moved_receivers_m = receivers_m @ rotation.T + translation_m moved_source_m = true_source_m @ rotation.T + translation_m moved_differences_m = np.array( [ signed_distance_difference( moved_source_m, moved_receivers_m[first_index], moved_receivers_m[second_index], ) for first_index, second_index in pair_indices ] ) np.testing.assert_allclose(moved_differences_m, measured_differences_m, atol=3e-15) signature_values = np.concatenate( [ np.array([wave_speed_m_s]), receivers_m.ravel(), true_source_m, measured_differences_m, measured_tdoa_s, initial_guess_m, estimated_source_m, final_residuals_m, jacobian.ravel(), singular_values, np.array(branches, dtype=float), sample_parameters, sample_diagnostics[0].points.ravel(), sample_diagnostics[1].points.ravel(), rotation.ravel(), translation_m, moved_receivers_m.ravel(), moved_source_m, ] ) print("uzunluk_birimi: m") print("zaman_birimi: s") print("model: sentetik_iki_boyutlu_gurultusuz_tdoa") print("gercek_konumlandirma_sistemi_mi: False") print("yayilma_hizi_m_s:", wave_speed_m_s) print("alicilar_m:", receivers_m.tolist()) print("gercek_kaynak_m:", true_source_m) print("olculen_uzaklik_farklari_m:", measured_differences_m) print("olculen_tdoa_s:", measured_tdoa_s) print("hiperbol_kollari:", branches) print("tahmin_edilen_kaynak_m:", estimated_source_m) print("konum_hatasi_m:", f"{position_error_m:.3e}") print("son_uzaklik_fark_artiklari_m:", final_residuals_m) print("jacobian_ranki:", int(np.linalg.matrix_rank(jacobian))) print("jacobian_kosul_sayisi:", f"{condition_number:.6f}") print( "en_buyuk_ornekleme_cebirsel_artigi:", f"{max(d.maximum_absolute_algebraic_residual for d in sample_diagnostics):.3e}", ) print( "en_buyuk_ornekleme_odak_fark_artigi_m:", f"{max(d.maximum_absolute_focal_difference_residual for d in sample_diagnostics):.3e}", ) print("rijit_harekette_olculer_korundu_mu:", bool(np.allclose(moved_differences_m, measured_differences_m))) print("bilimsel_imza:", canonical_array_sha256(signature_values)) if __name__ == "__main__": main()