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Управление траекториями воздушных судов в условиях оптимизации навигационного сеанса при автоматическом зависимом наблюдении..pdf

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2015
ɇȺɍɑɇɕɃ ȼȿɋɌɇɂɄ ɆȽɌɍ ȽȺ
ʋ 221
ɍȾɄ 621.396
ɍɉɊȺȼɅȿɇɂȿ ɌɊȺȿɄɌɈɊɂəɆɂ ȼɈɁȾɍɒɇɕɏ ɋɍȾɈȼ
ȼ ɍɋɅɈȼɂəɏ ɈɉɌɂɆɂɁȺɐɂɂ ɇȺȼɂȽȺɐɂɈɇɇɈȽɈ ɋȿȺɇɋȺ
ɉɊɂ ȺȼɌɈɆȺɌɂɑȿɋɄɈɆ ɁȺȼɂɋɂɆɈɆ ɇȺȻɅɘȾȿɇɂɂ
ȼ.ȼ. ȿɊɈɏɂɇ
Ɋɚɫɫɦɨɬɪɟɧɵ ɚɥɝɨɪɢɬɦɵ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɜɨɡɞɭɲɧɨɝɨ ɫɭɞɧɚ ɜ ɢɧɬɟɝɪɢɪɨɜɚɧɧɨɣ ɫɢɫɬɟɦɟ ɧɚɜɢɝɚɰɢɢ
ɢ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɬɪɚɟɤɬɨɪɢɟɣ. Ɋɟɡɭɥɶɬɚɬɵ ɢɫɫɥɟɞɨɜɚɧɢɣ ɩɚɪɚɦɟɬɪɨɜ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɫɟɚɧɫɚ ɢ
ɬɨɱɧɨɫɬɧɵɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɨɰɟɧɤɢ ɦɟɫɬɨɩɨɥɨɠɟɧɢɹ ɩɨɤɚɡɚɥɢ, ɱɬɨ ɩɪɢɦɟɧɟɧɢɟ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ
ɬɪɚɟɤɬɨɪɢɟɣ ɩɨɡɜɨɥɹɟɬ ɩɨɜɵɫɢɬɶ ɬɨɱɧɨɫɬɶ ɧɚɜɢɝɚɰɢɨɧɧɵɯ ɨɩɪɟɞɟɥɟɧɢɣ ɩɪɢ ɧɟɩɨɥɧɨɦ ɫɨɡɜɟɡɞɢɢ ɧɚɜɢɝɚɰɢɨɧɧɵɯ
ɫɩɭɬɧɢɤɨɜ.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɬɪɚɟɤɬɨɪɢɹ, ɨɩɬɢɦɚɥɶɧɨɟ ɭɩɪɚɜɥɟɧɢɟ, ɧɚɜɢɝɚɰɢɹ.
ȼȼȿȾȿɇɂȿ
Ɉɞɧɢɦ ɢɡ ɧɚɩɪɚɜɥɟɧɢɣ ɩɨɜɵɲɟɧɢɹ ɷɮɮɟɤɬɢɜɧɨɫɬɢ ɚɜɢɚɰɢɨɧɧɨɣ ɬɪɚɧɫɩɨɪɬɧɨɣ ɫɢɫɬɟɦɵ
(ȺɌɋ) ɹɜɥɹɟɬɫɹ ɫɨɜɟɪɲɟɧɫɬɜɨɜɚɧɢɟ ɫɢɫɬɟɦɵ ɨɪɝɚɧɢɡɚɰɢɢ ɢ ɭɩɪɚɜɥɟɧɢɹ ɜɨɡɞɭɲɧɵɦ ɞɜɢɠɟɧɢɟɦ
(ɍȼȾ), ɛɚɡɢɪɭɸɳɟɟɫɹ ɧɚ ɜɧɟɞɪɟɧɢɢ ɪɚɡɪɚɛɨɬɚɧɧɨɣ Ɇɟɠɞɭɧɚɪɨɞɧɨɣ ɨɪɝɚɧɢɡɚɰɢɟɣ ɝɪɚɠɞɚɧɫɤɨɣ
ɚɜɢɚɰɢɢ (ɂɄȺɈ) ɤɨɧɰɟɩɰɢɢ – ɫɜɹɡɶ, ɧɚɜɢɝɚɰɢɹ, ɧɚɛɥɸɞɟɧɢɟ ɢ ɨɪɝɚɧɢɡɚɰɢɹ ɜɨɡɞɭɲɧɨɝɨ
ɞɜɢɠɟɧɢɹ (CNS/ȺɌɆ), ɨɫɧɨɜɚɧɧɨɣ ɧɚ ɩɪɢɧɰɢɩɚɯ ɚɜɬɨɦɚɬɢɱɟɫɤɨɝɨ ɡɚɜɢɫɢɦɨɝɨ ɧɚɛɥɸɞɟɧɢɹ
(ȺɁɇ) ɢ ɟɝɨ ɞɚɥɶɧɟɣɲɟɝɨ ɪɚɡɜɢɬɢɹ Free Flight - "ɫɜɨɛɨɞɧɵɣ ɩɨɥɺɬ" [1].
Ʉɨɧɰɟɩɰɢɹ «ɫɜɨɛɨɞɧɵɣ ɩɨɥɟɬ» ɩɨɡɜɨɥɹɟɬ ɜɨɡɞɭɲɧɨɦɭ ɫɭɞɧɭ (ȼɋ) ɜɵɩɨɥɧɹɬɶ ɩɨɥɟɬ ɩɨ
ɜɵɛɢɪɚɟɦɨɣ ɷɤɢɩɚɠɟɦ ɨɩɬɢɦɚɥɶɧɨɣ ɞɥɹ ɞɚɧɧɵɯ ɭɫɥɨɜɢɣ ɜɨɡɞɭɲɧɨɣ ɨɛɫɬɚɧɨɜɤɢ ɬɪɚɟɤɬɨɪɢɢ
ɩɪɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɢ ɫ ɞɪɭɝɢɦɢ ȼɋ ɢ ɨɛɴɟɤɬɚɦɢ ɫɢɫɬɟɦɵ ɍȼȾ – ɧɚɜɢɝɚɰɢɨɧɧɵɦɢ ɨɩɨɪɧɵɦɢ
ɬɨɱɤɚɦɢ (ɇɈɌ). ɉɪɢ ɷɬɨɦ ɩɨɹɜɥɹɟɬɫɹ ɜɨɡɦɨɠɧɨɫɬɶ ɛɨɥɟɟ ɷɮɮɟɤɬɢɜɧɨɝɨ ɢɫɩɨɥɶɡɨɜɚɧɢɹ
ɫɬɪɭɤɬɭɪɵ ɜɨɡɞɭɲɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɩɨɜɵɲɟɧɢɹ ɷɤɨɧɨɦɢɱɧɨɫɬɢ ɢ ɧɚɞɟɠɧɨɫɬɢ
ɮɭɧɤɰɢɨɧɢɪɨɜɚɧɢɹ ȺɌɋ, ɱɬɨ ɨɩɪɟɞɟɥɹɟɬ ɜɵɫɨɤɢɟ ɬɪɟɛɨɜɚɧɢɹ, ɩɪɟɞɴɹɜɥɹɟɦɵɟ ɤ ɤɚɱɟɫɬɜɭ
ɧɚɜɢɝɚɰɢɨɧɧɨ-ɜɪɟɦɟɧɧɨɝɨ ɨɛɟɫɩɟɱɟɧɢɹ ȼɋ. ȼɵɫɨɤɭɸ ɬɨɱɧɨɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ ɧɚɜɢɝɚɰɢɨɧɧɨɜɪɟɦɟɧɧɵɯ ɩɚɪɚɦɟɬɪɨɜ (ɇȼɉ) ȼɋ ɩɪɢ ɡɨɧɚɥɶɧɨɣ ɧɚɜɢɝɚɰɢɢ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɨɛɟɫɩɟɱɢɬɶ ɩɭɬɟɦ
ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɫɩɭɬɧɢɤɨɜɵɯ ɫɢɫɬɟɦ ɧɚɜɢɝɚɰɢɢ (ɋɋɇ) (ɪɢɫ. 1,ɚ).
ɋɭɳɟɫɬɜɭɸɳɢɟ ɋɋɇ ɜɬɨɪɨɝɨ ɩɨɤɨɥɟɧɢɹ (ȽɅɈɇȺɋɋ, GPS) ɫɨɡɞɚɸɬ ɞɨɫɬɚɬɨɱɧɨ
ɬɨɱɧɨɟ ɝɥɨɛɚɥɶɧɨɟ ɧɚɜɢɝɚɰɢɨɧɧɨ-ɜɪɟɦɟɧɧɨɟ ɩɨɥɟ, ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɪɟɲɚɬɶ ɨɫɧɨɜɧɵɟ
ɡɚɞɚɱɢ ɫɚɦɨɥɟɬɨɜɨɠɞɟɧɢɹ ɧɚ ɜɨɡɞɭɲɧɵɯ ɬɪɚɫɫɚɯ ɢ ɜ ɡɨɧɟ ɚɷɪɨɞɪɨɦɨɜ ɫ ɬɪɟɛɭɟɦɵɦ
ɭɪɨɜɧɟɦ ɛɟɡɨɩɚɫɧɨɫɬɢ ɩɨɥɟɬɨɜ. Ɇɟɠɞɭ ɬɟɦ, ɋɋɇ ɢɦɟɸɬ ɪɹɞ ɫɭɳɟɫɬɜɟɧɧɵɯ ɧɟɞɨɫɬɚɬɤɨɜ –
ɧɢɡɤɚɹ ɩɨɦɟɯɨɡɚɳɢɳɟɧɧɨɫɬɶ, ɧɚɪɭɲɟɧɢɹ ɰɟɥɨɫɬɧɨɫɬɢ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɨɛɟɫɩɟɱɟɧɢɹ ɢ
ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɨɛɫɥɭɠɢɜɚɧɢɹ ɢ ɞɪ., ɤɨɬɨɪɵɟ ɧɟ ɩɨɡɜɨɥɹɸɬ ɢɫɩɨɥɶɡɨɜɚɬɶ ɢɯ ɜ ɤɚɱɟɫɬɜɟ
ɨɫɧɨɜɧɵɯ ɧɚɜɢɝɚɰɢɨɧɧɵɯ ɫɢɫɬɟɦ. ɉɪɢ ɷɬɨɦ ɬɨɱɧɨɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ȼɋ
ɛɭɞɟɬ ɜ ɡɧɚɱɢɬɟɥɶɧɨɣ ɫɬɟɩɟɧɢ ɡɚɜɢɫɟɬɶ ɨɬ ɭɫɥɨɜɢɣ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɫɟɚɧɫɚ: ɜɡɚɢɦɧɨɝɨ
ɩɨɥɨɠɟɧɢɹ ɢ ɩɨɝɪɟɲɧɨɫɬɟɣ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ɇɈɌ, ɮɭɧɤɰɢɢ ɤɨɬɨɪɵɯ ɦɨɝɭɬ ɜɵɩɨɥɧɹɬɶ
ɜ ɬɨɦ ɱɢɫɥɟ ȼɋ [2].
ɉɨɜɵɲɟɧɢɟ ɬɨɱɧɨɫɬɢ ɢ ɧɚɞɟɠɧɨɫɬɢ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɨɛɟɫɩɟɱɟɧɢɹ ɜɨɡɦɨɠɧɨ
ɤɨɦɩɥɟɤɫɢɪɨɜɚɧɢɟɦ ɜ ɫɨɫɬɚɜɟ ɢɧɬɟɝɪɢɪɨɜɚɧɧɨɣ ɫɢɫɬɟɦɵ ɧɚɜɢɝɚɰɢɢ (ɂɋɇ) ɩɪɢɟɦɨɢɧɞɢɤɚɬɨɪɚ
(ɉɂ) ɋɋɇ, ɛɨɪɬɨɜɨɝɨ ɬɟɪɦɢɧɚɥɚ ɫɢɫɬɟɦɵ ȺɁɇ ɢ ɢɧɟɪɰɢɚɥɶɧɨɣ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɫɢɫɬɟɦɵ (ɂɇɋ).
ɇɚɥɢɱɢɟ ɜ ɫɢɫɬɟɦɟ ɨɛɴɟɤɬɨɜ, ɢɦɟɸɳɢɯ ɫɬɟɩɟɧɢ ɫɜɨɛɨɞɵ ɫɜɨɟɝɨ ɩɟɪɟɞɜɢɠɟɧɢɹ, ɨɩɪɟɞɟɥɹɟɬ
ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɜɨɡɦɨɠɧɨɫɬɶ ɨɩɬɢɦɢɡɚɰɢɢ ɭɫɥɨɜɢɣ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɫɟɚɧɫɚ ɞɥɹ ɩɨɜɵɲɟɧɢɹ
ɬɨɱɧɨɫɬɢ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɨɛɟɫɩɟɱɟɧɢɹ ȼɋ (ɪɢɫ. 1,ɛ). ȼ ɷɬɨɣ ɫɜɹɡɢ ɚɤɬɭɚɥɶɧɨɣ ɡɚɞɚɱɟɣ ɹɜɥɹɟɬɫɹ
ɫɢɧɬɟɡ ɚɥɝɨɪɢɬɦɨɜ ɤɨɦɩɥɟɤɫɧɨɣ ɨɛɪɚɛɨɬɤɢ ɢɧɮɨɪɦɚɰɢɢ (ɄɈɂ) ɢ ɭɩɪɚɜɥɟɧɢɹ ɬɪɚɟɤɬɨɪɢɹɦɢ
ɩɨɥɟɬɚ ȼɋ.
ɍɩɪɚɜɥɟɧɢɟ ɬɪɚɟɤɬɨɪɢɹɦɢ ɜɨɡɞɭɲɧɵɯ ɫɭɞɨɜ ɜ ɭɫɥɨɜɢɹɯ…
19
ɚ
ɛ
Ɋɢɫ. 1
ɚ) ɨɛɨɛɳɟɧɧɚɹ ɫɬɪɭɤɬɭɪɚ ɧɚɜɢɝɚɰɢɢ, ɧɚɛɥɸɞɟɧɢɹ ɢ ɨɪɝɚɧɢɡɚɰɢɢ ɜɨɡɞɭɲɧɨɝɨ ɞɜɢɠɟɧɢɹ;
ɛ) ɧɟɭɩɪɚɜɥɹɟɦɚɹ ɢ ɭɩɪɚɜɥɹɟɦɚɹ ɬɪɚɟɤɬɨɪɢɢ ɩɨɥɟɬɚ ȼɋ
ɉɈɋɌȺɇɈȼɄȺ ɁȺȾȺɑɂ ɎɂɅɖɌɊȺɐɂɂ ɌɊȺȿɄɌɈɊɂɃ
Ⱦɥɹ ɩɫɟɜɞɨɞɚɥɶɧɨɦɟɪɧɨɝɨ ɦɟɬɨɞɚ ɭɪɚɜɧɟɧɢɟ ɧɚɛɥɸɞɟɧɢɹ ɨɬ i-ɝɨ ɇɋ (ɇɈɌ) ɞɥɹ k-ɝɨ
ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:
ξ kij = s(tk , xki , xkj ) + nkij = cTkij + nkij = Rkij + cΔtki + nkij ,
(1)
ɝɞɟ Tkij - ɜɪɟɦɹ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɫɢɝɧɚɥɚ; xki , xkj - ɜɟɤɬɨɪɚ ɤɨɨɪɞɢɧɚɬ ɨɛɴɟɤɬɨɜ;
Rkij = [( xi − x j ) 2 + ( yi − y j ) 2 + ( z i − z j ) 2 ]1 / 2 - ɞɚɥɶɧɨɫɬɶ ɞɨ j-ɝɨ ɢɫɬɨɱɧɢɤɚ ɢɧɮɨɪɦɚɰɢɢ (ɂɂ); cΔt ki ɫɦɟɳɟɧɢɟ ɲɤɚɥɵ ɜɪɟɦɟɧɢ ȼɋ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɧɨɝɨ; nkij - ɲɭɦ ɧɚɛɥɸɞɟɧɢɹ, ɩɪɢɧɢɦɚɟɦɵɣ
ɞɢɫɤɪɟɬɧɵɦ ȻȽɒ (ȾȻȽɒ) ɫɨ ɫɬɚɬɢɫɬɢɱɟɫɤɢɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ: Ɇ{ nkij }=0 ɢ Ɇ{ nkij 2}= σ ξ ij .
2
ɉɪɢ ɫɢɧɬɟɡɟ ɚɥɝɨɪɢɬɦɚ ɄɈɂ ɜ ɂɋɇ ɫɱɢɬɚɟɦ, ɱɬɨ ɨɰɟɧɤɚ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ ɩɪɨɢɡɜɨɞɢɬɫɹ
ɧɚ ɨɫɧɨɜɟ ɫɨɜɦɟɫɬɧɨɣ ɨɛɪɚɛɨɬɤɢ ɧɚɛɥɸɞɟɧɢɣ ɂɇɋ, ɋɋɇ ɢ ȺɁɇ, ɢɫɩɨɥɶɡɭɟɬɫɹ ɟɞɢɧɚɹ ɲɤɚɥɚ
ɜɪɟɦɟɧɢ, ɨɛɴɟɤɬɵ ȺɁɇ ɜɵɩɨɥɧɹɸɬ ɫɢɧɯɪɨɧɢɡɚɰɢɸ ɫɜɨɢɯ ɱɚɫɨɜ ɩɨ ɫɢɝɧɚɥɚɦ ɇɋ ɋɋɇ. Ⱦɥɹ
ɩɨɥɭɱɟɧɢɹ ɄɈɂ ɩɪɟɞɥɚɝɚɟɬɫɹ ɢɫɩɨɥɶɡɨɜɚɬɶ ɦɨɞɟɪɧɢɡɢɪɨɜɚɧɧɵɣ ɜɚɪɢɚɧɬ ɤɨɦɩɥɟɤɫɢɪɨɜɚɧɢɹ.
ɉɪɟɞɫɬɚɜɢɦ ɫɢɝɧɚɥ ɂɋɇ ɜ ɜɢɞɟ:
η k = xk + λk ,
(2)
ɝɞɟ x = x , Vx , y , V y , z , Vz , τ c , f - ɜɟɤɬɨɪ ɇȼɉ; λ = δ x , δ Vx , δ y , δ V y , δ z , δ Vz , δτ c , δ f ɜɟɤɬɨɪ ɩɨɝɪɟɲɧɨɫɬɟɣ ɨɩɪɟɞɟɥɟɧɢɹ ɇȼɉ.
Ⱦɢɧɚɦɢɤɚ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɜɡɚɢɦɨɫɜɹɡɢ ɦɟɠɞɭ ɤɨɦɩɨɧɟɧɬɚɦɢ ɜɟɤɬɨɪɨɜ x k ɢ λ k ɜ
ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ tk-1 ɢ tk ɡɚɞɚɧɚ ɫɬɨɯɚɫɬɢɱɟɫɤɢɦɢ ɪɚɡɧɨɫɬɧɵɦɢ ɭɪɚɜɧɟɧɢɹɦɢ:
T
T
xk = Φ X k xk −1 + nxk , λk = Φ λk λk −1 + nλk ,
ɜ ɤɨɬɨɪɵɯ Φ X ɢ
Φ λ - ɦɚɬɪɢɰɵ, ɨɩɢɫɵɜɚɸɳɢɟ ɞɢɧɚɦɢɤɭ ɩɨɥɟɬɚ ȼɋ ɢ ɨɲɢɛɨɤ ɂɇɋ.
20
ȼ.ȼ. ȿɪɨɯɢɧ
ɇɟɫɦɨɬɪɹ ɧɚ ɬɨ, ɱɬɨ ɜɯɨɞɹɳɢɟ ɜ ɧɚɛɥɸɞɟɧɢɟ (2) ɜɟɥɢɱɢɧɵ x ɢ λ ɧɟɪɚɡɞɟɥɢɦɵ,
ɩɨɫɬɭɩɚɟɦ ɮɨɪɦɚɥɶɧɨ, ɩɨɞɫɬɚɜɢɜ x k = η k − λ k ɜ (1), ɬɨɝɞɚ: ξ k = s (t k ,η k − λ k ) + n k .
ɨɛɪɚɡɨɦ, ɩɚɪɚɦɟɬɪɨɦ, ɩɨɞɥɟɠɚɳɢɦ ɨɰɟɧɤɟ ɩɨ ɫɨɜɨɤɭɩɧɨɫɬɢ ɧɚɛɥɸɞɟɧɢɣ, ɹɜɥɹɟɬɫɹ
ɨɲɢɛɨɤ ɂɋɇ λ k . ɂɫɩɨɥɶɡɭɹ ɦɟɬɨɞɢɤɭ ɥɨɤɚɥɶɧɨɣ ɝɚɭɫɫɨɜɨɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɧɚ
ɫɨɜɨɤɭɩɧɨɫɬɢ ɧɚɛɥɸɞɟɧɢɣ ɧɚ ɜɯɨɞɟ ɂɋɇ, ɫɢɧɬɟɡɢɪɭɟɦ ɚɥɝɨɪɢɬɦ ɨɰɟɧɤɢ ɤɨɨɪɞɢɧɚɬ
ɤɪɢɬɟɪɢɸ ɦɢɧɢɦɭɦɚ ɫɪɟɞɧɟɝɨ ɤɜɚɞɪɚɬɚ ɨɲɢɛɤɢ (ɋɄɈ):
{
Ɍɚɤɢɦ
ɜɟɤɬɨɪ
ɨɫɧɨɜɟ
ȼɋ ɩɨ
}
ε k2min = M ( λk − λˆ ) 2k = ³ ( λk − λˆk ) 2 p ( λk ξ 0 k , η 0 k ) d λk .
ȺɅȽɈɊɂɌɆ ɎɂɅɖɌɊȺɐɂɂ ɌɊȺȿɄɌɈɊɂɃ
ɉɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɞɚɧɧɨɦɭ ɫɥɭɱɚɸ ɦɧɨɝɨɦɟɪɧɨɣ ɞɢɫɤɪɟɬɧɨɣ ɮɢɥɶɬɪɚɰɢɢ ɭɪɚɜɧɟɧɢɹ
ɧɚɛɥɸɞɟɧɢɹ ɢ ɫɨɨɛɳɟɧɢɹ ɢɦɟɸɬ ɫɥɟɞɭɸɳɭɸ ɮɨɪɦɭ ɡɚɩɢɫɢ:
ξ k = H k λk + nξ k , λk = ĭk λk −1 + nλk ,
T
ª ∂R ɇɋ ∂RkɇɈɌ º
ɝɞɟ H k = « k
» - ɦɚɬɪɢɰɚ ɧɚɩɪɚɜɥɹɸɳɢɯ ɤɨɫɢɧɭɫɨɜ (ɧɚɛɥɸɞɟɧɢɣ) ɧɚ i-ɣ ɂɂ (ɇɋ,
∂λk ¼
¬ ∂λk
ɇɈɌ); nξk , nλk - ɜɟɤɬɨɪɵ ȾȻȽɒ ɫ ɧɭɥɟɜɵɦɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦɢ ɨɠɢɞɚɧɢɹɦɢ ɢ ɤɨɪɪɟɥɹɰɢɨɧɧɵɦɢ
ɦɚɬɪɢɰɚɦɢ Vk ɢ Q k . Ⱦɢɚɝɨɧɚɥɶɧɵɦɢ ɷɥɟɦɟɧɬɚɦɢ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɦɚɬɪɢɰɵ Vk ɹɜɥɹɸɬɫɹ
ɞɢɫɩɟɪɫɢɢ ɨɩɪɟɞɟɥɟɧɢɹ ɩɫɟɜɞɨɞɚɥɶɧɨɫɬɟɣ ɞɨ ɇɋ ɢ ɇɈɌ ( σ Di , i=1..N – ɤɨɥɢɱɟɫɬɜɨ ɇɋ ɢ ɇɈɌ).
Ɋɟɤɭɪɪɟɧɬɧɵɣ ɚɥɝɨɪɢɬɦ ɨɩɬɢɦɚɥɶɧɨɝɨ ɨɰɟɧɢɜɚɧɢɹ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ ɧɚ ɨɫɧɨɜɟ
ɪɚɫɲɢɪɟɧɧɨɝɨ ɮɢɥɶɬɪɚ Ʉɚɥɦɚɧɚ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ:
~
1) λˆ = ĭ λˆ + K (ξ − ξ ) - ɨɰɟɧɤɚ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ;
2
k
k
k −1
k
k
k
~
2) R k = (I − K k H k )R k - ɤɨɪɪɟɥɹɰɢɨɧɧɚɹ ɦɚɬɪɢɰɚ ɨɲɢɛɨɤ ɨɰɟɧɢɜɚɧɢɹ;
~
3) R k = ĭ k R k −1ĭ kT + Q k - ɷɤɫɬɪɚɩɨɥɢɪɨɜɚɧɧɚɹ ɤɨɪɪɟɥɹɰɢɨɧɧɚɹ ɦɚɬɪɢɰɚ ɨɲɢɛɨɤ ɨɰɟɧɢɜɚɧɢɹ;
~
4) K k = R k H kT (H k R k H kT + Vk ) −1 - ɤɨɷɮɮɢɰɢɟɧɬ ɭɫɢɥɟɧɢɹ ɮɢɥɶɬɪɚ,
~
ɝɞɟ ξ k - ɜɟɤɬɨɪ ɷɤɫɬɪɚɩɨɥɢɪɨɜɚɧɧɵɯ ɧɚɛɥɸɞɟɧɢɣ, ɜɵɱɢɫɥɟɧɧɵɯ ɧɚ ɨɫɧɨɜɟ ɨɩɪɟɞɟɥɟɧɢɹ
ɤɨɨɪɞɢɧɚɬ ɫɢɫɬɟɦɨɣ ɫɱɢɫɥɟɧɢɹ ɢ ɩɟɪɟɞɚɧɧɵɯ ɡɧɚɱɟɧɢɹɯ ɤɨɨɪɞɢɧɚɬ ɂɂ.
Ɍɨɱɧɨɫɬɶ ɨɰɟɧɤɢ ɇȼɉ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɦɚɬɪɢɰɟɣ ɨɲɢɛɨɤ ɮɢɥɶɬɪɚɰɢɢ
ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ R. Ⱦɢɚɝɨɧɚɥɶɧɵɦɢ ɷɥɟɦɟɧɬɚɦɢ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɦɚɬɪɢɰɵ R ɹɜɥɹɸɬɫɹ
ɞɢɫɩɟɪɫɢɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ȼɋ: σ x2 , σ Vx2 , σ y2 , σ Vy2 , σ z2 , σ Vz2 , σ Δ2t , σ 2f .
ɉɈɋɌȺɇɈȼɄȺ ɁȺȾȺɑɂ ɍɉɊȺȼɅȿɇɂə ɌɊȺȿɄɌɈɊɂəɆɂ
ɉɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɡɚɞɚɱɟ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɬɪɚɟɤɬɨɪɢɹɦɢ ɭɪɚɜɧɟɧɢɹ ɧɚɛɥɸɞɟɧɢɹ ɢ
ɞɢɧɚɦɢɤɢ ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ ɜɡɚɢɦɨɫɜɹɡɢ ɦɟɠɞɭ ɤɨɦɩɨɧɟɧɬɚɦɢ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ:
ȟ k = sk (Ȝk ,uk ) + nȟ , Ȝk = g k (Ȝk −1 ,uk ) + nȜk ,
k
ɝɞɟ s k (Ȝk ,u k ) – m-ɦɟɪɧɚɹ ɮɭɧɤɰɢɹ, ɡɚɜɢɫɹɳɚɹ ɨɬ Ȝk ɢ u k ;
ɮɭɧɤɰɢɹ, ɡɚɜɢɫɹɳɚɹ ɨɬ Ȝk ɢ ɩɪɢɥɨɠɟɧɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ u k .
(3)
g k (Ȝk −1 ,u k ) – n-ɦɟɪɧɚɹ
ɍɩɪɚɜɥɟɧɢɟ ɬɪɚɟɤɬɨɪɢɹɦɢ ɜɨɡɞɭɲɧɵɯ ɫɭɞɨɜ ɜ ɭɫɥɨɜɢɹɯ…
21
ɇɚ ɨɫɧɨɜɚɧɢɢ ɭɪɚɜɧɟɧɢɣ (3) ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɬɚɤɨɟ ɨɩɬɢɦɚɥɶɧɨɟ ɭɩɪɚɜɥɟɧɢɟ
u v = u (t Ȟ ) ɬɪɚɟɤɬɨɪɢɟɣ ɞɜɢɠɟɧɢɹ ɧɚ ɨɫɧɨɜɟ ɜɫɟɣ ɪɚɫɩɨɥɚɝɚɟɦɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ ɇɋ ɢ ɇɈɌ, ɩɪɢ
ɤɨɬɨɪɨɦ ɞɨɫɬɢɝɚɟɬɫɹ ɷɤɫɬɪɟɦɭɦ ɫɪɟɞɧɟɝɨ ɡɧɚɱɟɧɢɹ ɧɟɤɨɬɨɪɨɝɨ ɮɭɧɤɰɢɨɧɚɥɚ ɬɟɤɭɳɢɯ ɩɨɬɟɪɶ
ɩɪɢ ɧɚɯɨɠɞɟɧɢɢ ɭɩɪɚɜɥɟɧɢɹ ɜ ɬɟɤɭɳɢɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ (ɥɨɤɚɥɶɧɵɣ ɤɪɢɬɟɪɢɣ) ɢɥɢ ɜ ɤɨɧɟɱɧɵɣ
ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɫ ɭɱɟɬɨɦ ɜɫɟɝɨ ɩɟɪɟɯɨɞɧɨɝɨ ɩɪɨɰɟɫɫɚ (ɢɧɬɟɝɪɚɥɶɧɵɣ ɤɪɢɬɟɪɢɣ) [3].
ɍɪɚɜɧɟɧɢɟ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɟ ɬɪɚɟɤɬɨɪɢɸ ɞɜɢɠɟɧɢɹ ȼɋ, ɢɦɟɟɬ ɜɢɞ:
xɜɫ k = ĭ k xɜɫ k −1 + Ǻ k uɜɫ k + nɜɫ k
ɝɞɟ ĭk – ɦɚɬɪɢɰɚ ɜɡɚɢɦɨɫɜɹɡɢ ɤɨɨɪɞɢɧɚɬ ȼɋ ɜ ɫɨɫɟɞɧɢɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ tk ɢ tk-1;
u ɜɫT = u1 ,u 2 ,....,u r – ɜɟɤɬɨɪ ɬɪɚɟɤɬɨɪɢɟɣ ȼɋ ɪɚɡɦɟɪɧɨɫɬɢ r; B k – ɦɚɬɪɢɰɚ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɜɟɤɬɨɪɚ ɭɩɪɚɜɥɟɧɢɹ ɪɚɡɦɟɪɧɨɫɬɢ n×r; r – ɱɢɫɥɨ ɪɟɚɥɢɡɭɟɦɵɯ ɤɚɧɚɥɨɜ ɭɩɪɚɜɥɟɧɢɹ
ɨɩɪɟɞɟɥɹɸɳɢɦɫɹ ȼɋ.
ȼɵɛɨɪ ɜɢɞɚ ɮɭɧɤɰɢɨɧɚɥɚ ɜɨ ɦɧɨɝɨɦ ɨɩɪɟɞɟɥɹɟɬ ɯɚɪɚɤɬɟɪ ɪɟɲɟɧɢɹ ɢ ɫɬɪɭɤɬɭɪɭ
ɨɩɬɢɦɚɥɶɧɨɝɨ ɡɚɤɨɧɚ ɭɩɪɚɜɥɟɧɢɹ u ɜɫT = u1 ,u 2 ,....,u r . ȼ ɢɧɠɟɧɟɪɧɨɣ ɩɪɚɤɬɢɤɟ ɢ ɫɨɜɪɟɦɟɧɧɨɣ
ɬɟɨɪɢɢ ɨɩɬɢɦɚɥɶɧɵɯ ɫɢɫɬɟɦ ɲɢɪɨɤɨɟ ɩɪɢɦɟɧɟɧɢɟ ɧɚɯɨɞɢɬ ɨɛɨɛɳɟɧɧɵɣ ɤɜɚɞɪɚɬɢɱɟɫɤɢɣ
ɮɭɧɤɰɢɨɧɚɥ ɨɲɢɛɤɢ ɭɩɪɚɜɥɟɧɢɹ. Ʉɜɚɞɪɚɬɢɱɧɚɹ ɮɭɧɤɰɢɹ ɩɨɬɟɪɶ ɢɦɟɟɬ ɜɢɞ:
Ck ( Ȝ k , u k ) = (Ȝ k − Ȝ kɡɚɞ )T Ĭ k (Ȝ k − Ȝ kɡɚɞ ) + u kT K k u k ,
(4)
ɝɞɟ Ȝ ɡɚɞ - ɡɚɞɚɧɧɚɹ ɬɪɚɟɤɬɨɪɢɹ; Ĭ k ɢ Ʉ k - ɧɟɨɬɪɢɰɚɬɟɥɶɧɨ ɨɩɪɟɞɟɥɟɧɧɵɟ ɜɟɫɨɜɵɟ
ɦɚɬɪɢɰɵ ɲɬɪɚɮɨɜ.
ɉɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɩɨɬɟɪɢ ɡɚ ɨɲɢɛɤɭ (ɨɬɤɥɨɧɟɧɢɟ ɨɬ ɡɚɞɚɧɧɨɣ ɬɪɚɟɤɬɨɪɢɢ),
ɜɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ – ɡɚɬɪɚɬɵ ɧɚ ɭɩɪɚɜɥɟɧɢɟ. Ɏɭɧɤɰɢɨɧɚɥɵ ɩɨɬɟɪɶ ɞɥɹ ɢɧɬɟɝɪɚɥɶɧɨɝɨ ɢ
ɥɨɤɚɥɶɧɨɝɨ ɤɪɢɬɟɪɢɟɜ ɨɩɬɢɦɢɡɚɰɢɢ ɢɦɟɸɬ ɜɢɞ:
{ (
N
J k = M ­® C k §¨ Ȝk , Ȝˆk , u k ·¸ ½¾ , J = M ¦ Ck Ȝk , Ȝˆk ,uk
¹¿
¯ ©
v =1
ɝɞɟ
Jk
ɨɩɪɟɞɟɥɟɧɢɣ
)},
(5)
– ɮɭɧɤɰɢɨɧɚɥ ɬɟɤɭɳɢɯ ɩɨɬɟɪɶ, ɨɬɪɚɠɚɸɳɢɣ ɤɚɱɟɫɬɜɨ ɧɚɜɢɝɚɰɢɨɧɧɵɯ
ɜ
ɬɟɤɭɳɢɣ
ɦɨɦɟɧɬ
ɜɪɟɦɟɧɢ;
J –
ɫɭɦɦɚɪɧɵɣ
ɮɭɧɤɰɢɨɧɚɥ
ɩɨɬɟɪɶ,
ˆ
ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɣ ɭɫɥɨɜɢɹ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɫɟɚɧɫɚ ɧɚ ɜɫɟɣ ɬɪɚɟɤɬɨɪɢɢ; Ck §¨ Ȝk , Ȝk , u k ·¸ –
©
¹
ɡɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɭɫɥɨɜɢɹ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɫɟɚɧɫɚ [4].
Ʉɪɢɬɟɪɢɣ ɨɩɬɢɦɢɡɚɰɢɢ (5) ɞɨɥɠɟɧ ɨɛɟɫɩɟɱɢɜɚɬɶ ɦɚɤɫɢɦɚɥɶɧɭɸ ɬɨɱɧɨɫɬɶ ɨɰɟɧɢɜɚɧɢɹ
ɩɚɪɚɦɟɬɪɨɜ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ ȼɋ. ɉɨɷɬɨɦɭ ɮɭɧɤɰɢɸ ɫɬɨɢɦɨɫɬɢ ɜ (5) ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ
ɤɜɚɞɪɚɬɢɱɧɨɝɨ ɤɪɢɬɟɪɢɹ ɨɩɬɢɦɢɡɚɰɢɢ (4) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ:
T
Ck §¨ Ȝk , Ȝˆk , u k ·¸ = §¨ Ȝk − Ȝˆk ·¸ Ĭ §¨ Ȝk − Ȝˆk ·¸ ,
©
¹ ©
¹
©
¹
(6)
ɝɞɟ Ȝk – ɢɫɬɢɧɧɨɟ ɡɧɚɱɟɧɢɟ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ ȼɋ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ tv, Ȝˆ k - ɨɰɟɧɤɚ
ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ, ɮɨɪɦɢɪɭɟɦɚɹ ɜ ɪɟɡɭɥɶɬɚɬɟ ɤɨɦɩɥɟɤɫɧɨɣ ɨɩɬɢɦɚɥɶɧɨɣ ɨɛɪɚɛɨɬɤɢ ɢɧɮɨɪɦɚɰɢɢ.
22
ȼ.ȼ. ȿɪɨɯɢɧ
Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ ɩɨ Ȝv ɪɚɜɧɨ:
­
½
M ®C k §¨ Ȝk , Ȝˆk ,u k ·¸ ¾ = tr {Ĭ R k } ,
¯
©
(7)
¹¿
ɝɞɟ Rk - ɡɧɚɱɟɧɢɟ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɦɚɬɪɢɰɵ ɨɲɢɛɨɤ ɮɢɥɶɬɪɚɰɢɢ; tr - ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ
ɨɩɟɪɚɰɢɹ ɧɚɯɨɠɞɟɧɢɹ ɫɥɟɞɚ ɦɚɬɪɢɰɵ, ɬ.ɟ. ɫɭɦɦɵ ɞɢɚɝɨɧɚɥɶɧɵɯ ɷɥɟɦɟɧɬɨɜ ɦɚɬɪɢɰɵ.
Ɇɢɧɢɦɢɡɚɰɢɹ ɮɭɧɤɰɢɨɧɚɥɚ ɬɟɤɭɳɢɯ ɩɨɬɟɪɶ, ɢɫɯɨɞɹ ɢɡ ɜɵɪɚɠɟɧɢɹ (6), ɞɨɫɬɢɝɚɟɬɫɹ ɩɪɢ
ɦɢɧɢɦɭɦɟ ɡɧɚɱɟɧɢɹ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɦɚɬɪɢɰɵ ɨɲɢɛɨɤ ɮɢɥɶɬɪɚɰɢɢ ɜ ɬɟɤɭɳɢɣ ɦɨɦɟɧɬ Rk,
ɤɨɬɨɪɚɹ ɞɥɹ ɢɫɫɥɟɞɭɟɦɨɣ ɫɢɫɬɟɦɵ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɪɟɤɭɪɪɟɧɬɧɨɦɭ ɭɪɚɜɧɟɧɢɸ:
R −k 1 = (Ɏ k R k −1Ɏ k T + Ȍ k )−1 + Ǿ k (u k )T Vk−1Ǿ k (u k ).
(8)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɪɟɲɟɧɢɹ ɩɨɫɬɚɜɥɟɧɧɨɣ ɡɚɞɚɱɢ ɧɟɨɛɯɨɞɢɦɨ ɨɩɪɟɞɟɥɢɬɶ ɨɩɬɢɦɚɥɶɧɨɟ
ɭɩɪɚɜɥɟɧɢɟ ɞɥɹ ɫɢɫɬɟɦɵ (3), ɤɨɬɨɪɨɟ ɦɢɧɢɦɢɡɢɪɭɟɬ ɡɧɚɱɟɧɢɟ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɦɚɬɪɢɰɵ ɨɲɢɛɨɤ
ɮɢɥɶɬɪɚɰɢɢ (8). Ɉɫɨɛɟɧɧɨɫɬɶ ɞɚɧɧɨɣ ɡɚɞɚɱɢ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɩɨɢɫɤ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ
ɜɵɩɨɥɧɹɟɬɫɹ ɧɚ ɨɫɧɨɜɟ ɫɬɨɯɚɫɬɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɭɩɪɚɜɥɟɧɢɹ.
ȼɟɤɬɨɪ ɭɩɪɚɜɥɟɧɢɹ u k ɜ ɜɵɪɚɠɟɧɢɢ (8) ɜɥɢɹɟɬ ɬɨɥɶɤɨ ɧɚ ɡɧɚɱɟɧɢɟ ɜɬɨɪɨɝɨ ɫɥɚɝɚɟɦɨɝɨ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɢɧɢɦɢɡɚɰɢɹ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɦɚɬɪɢɰɵ ɨɲɢɛɨɤ ɮɢɥɶɬɪɚɰɢɢ ɡɚ ɫɱɟɬ
ɨɪɝɚɧɢɡɚɰɢɢ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɬɪɚɟɤɬɨɪɢɟɣ ȼɋ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ tk ɞɨɫɬɢɝɚɟɬɫɹ ɩɪɢ
ɦɚɤɫɢɦɭɦɟ ɜɵɪɚɠɟɧɢɹ H (u k )T Vk−1H (u k ) , ɤɨɬɨɪɨɟ ɜɵɛɢɪɚɟɬɫɹ ɜ ɤɚɱɟɫɬɜɟ ɮɭɧɤɰɢɢ ɫɬɨɢɦɨɫɬɢ.
ȼ ɡɚɞɚɱɚɯ ɨɰɟɧɢɜɚɧɢɹ ɞɢɧɚɦɢɱɟɫɤɢɯ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɫɢɫɬɟɦ ɬɨɱɧɨɫɬɶ ɱɢɫɥɟɧɧɨɝɨ
ɪɟɲɟɧɢɹ ɡɚɜɢɫɢɬ ɨɬ ɫɨɫɬɚɜɚ, ɤɨɥɢɱɟɫɬɜɚ ɧɚɛɥɸɞɚɟɦɵɯ ɫɢɝɧɚɥɨɜ ɢ ɩɪɨɞɨɥɠɢɬɟɥɶɧɨɫɬɢ ɜɪɟɦɟɧɢ
ɧɚɛɥɸɞɟɧɢɹ. ɏɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɭɤɚɡɚɧɧɨɣ ɬɨɱɧɨɫɬɢ ɹɜɥɹɟɬɫɹ «ɦɟɪɚ ɧɚɛɥɸɞɚɟɦɨɫɬɢ»
Ƚ = H(t )T H(t )Δt [4]. Ⱦɥɹ ɩɪɚɤɬɢɱɟɫɤɨɝɨ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɜɜɨɞɹɬ ɩɨɧɹɬɢɟ ɞɟɬɟɪɦɢɧɚɧɬɚ
ɦɚɬɪɢɰɵ Ƚɪɚɦɚ («ɦɟɪɚ ɧɚɛɥɸɞɚɟɦɨɫɬɢ»), ɤɨɬɨɪɵɣ ɱɢɫɥɟɧɧɨ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɩɥɨɳɚɞɶ,
«ɨɯɜɚɬɵɜɚɟɦɭɸ» ɮɚɡɨɜɨɣ ɬɪɚɟɤɬɨɪɢɟɣ ɜɟɤɬɨɪɚ ɧɚɛɥɸɞɟɧɢɣ ɧɚ ɢɧɬɟɪɜɚɥɟ ɨɰɟɧɢɜɚɧɢɹ (t0, tk) ɢ
ɨɩɪɟɞɟɥɹɟɦɭɸ ɜɵɪɚɠɟɧɢɟɦ:
tk
det Ƚ = det ³ H(t )T H(t )dt = H(t )T H(t )Δt.
t0
ɑɟɦ ɛɨɥɶɲɟ ɜɟɥɢɱɢɧɚ det Ƚ, ɬ.ɟ. ɜɟɥɢɱɢɧɚ «ɦɟɪɵ ɧɚɛɥɸɞɚɟɦɨɫɬɢ», ɬɟɦ ɥɭɱɲɟ
ɧɚɛɥɸɞɚɟɦɨɫɬɶ ɢ ɭɩɪɚɜɥɹɟɦɨɫɬɶ ɫɢɫɬɟɦɵ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɬɟɦ ɜɵɲɟ ɬɨɱɧɨɫɬɶ ɨɰɟɧɤɢ
ɩɟɪɟɦɟɧɧɵɯ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ, ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ – ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ ȼɋ.
ɉɪɢ ɡɚɞɚɧɧɵɯ ɫɬɨɯɚɫɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɯ ɲɭɦɨɜ ɧɚɛɥɸɞɟɧɢɹ ɪɟɲɚɸɳɟɟ ɩɪɚɜɢɥɨ
ɨɩɬɢɦɢɡɚɰɢɢ ɩɪɢɦɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
uˆk = arg max(det(H(uk )T H(uk ))).
(9)
uk
Ɏɭɧɤɰɢɹ det(H (uk ) H (uk )) ɜ ɜɵɪɚɠɟɧɢɢ (8) ɨɩɪɟɞɟɥɹɟɬ ɡɚɜɢɫɢɦɨɫɬɶ ɬɨɱɧɨɫɬɢ
ɧɚɜɢɝɚɰɢɨɧɧɵɯ ɨɩɪɟɞɟɥɟɧɢɣ ɨɬ ɜɡɚɢɦɧɨɝɨ ɪɚɫɩɨɥɨɠɟɧɢɹ ɩɨɬɪɟɛɢɬɟɥɟɣ ɢ ɢɫɬɨɱɧɢɤɨɜ
T
T
ɢɧɮɨɪɦɚɰɢɢ ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ «ɦɟɪɚ ɧɚɛɥɸɞɚɟɦɨɫɬɢ» Ƚ = H HΔt . ɉɪɢ ǻt=1 ɦɟɪɚ
ɧɚɛɥɸɞɚɟɦɨɫɬɢ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɜɯɨɞɢɬ ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ
ɍɩɪɚɜɥɟɧɢɟ ɬɪɚɟɤɬɨɪɢɹɦɢ ɜɨɡɞɭɲɧɵɯ ɫɭɞɨɜ ɜ ɭɫɥɨɜɢɹɯ…
([
23
])
−1 1/ 2
ɮɚɤɬɨɪɚ (ȽɎ) Ʉ Ƚ = tr (H T H )
. ɉɨɷɬɨɦɭ ɜ ɤɚɱɟɫɬɜɟ ɤɪɢɬɟɪɢɟɜ ɨɩɬɢɦɢɡɚɰɢɢ ɬɪɚɟɤɬɨɪɢɢ ȼɋ
ɩɪɢɧɢɦɚɟɬɫɹ ɦɢɧɢɦɭɦ ɤɨɷɮɮɢɰɢɟɧɬɚ ɝɟɨɦɟɬɪɢɢ ɢ ɦɚɤɫɢɦɭɦ «ɦɟɪɵ ɧɚɛɥɸɞɚɟɦɨɫɬɢ».
Ɉɩɬɢɦɚɥɶɧɨɟ ɭɩɪɚɜɥɟɧɢɟ û k ɩɪɢ ɥɨɤɚɥɶɧɨɣ ɨɩɬɢɦɢɡɚɰɢɢ ɧɚɯɨɞɢɬɫɹ ɢɡ ɭɫɥɨɜɢɹ
T
ɦɚɤɫɢɦɭɦɚ det(H (uk ) H (uk )) ɧɚ ɤɚɠɞɨɦ ɲɚɝɟ, ɬ.ɟ. ɦɚɤɫɢɦɭɦɚ ɦɟɪɵ ɧɚɛɥɸɞɚɟɦɨɫɬɢ, ɜ
ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɜɵɪɚɠɟɧɢɟɦ (8) [4].
Ⱦɥɹ ɩɪɚɤɬɢɱɟɫɤɨɣ ɪɟɚɥɢɡɚɰɢɢ ɨɩɬɢɦɚɥɶɧɵɟ ɬɪɚɟɤɬɨɪɢɢ ɞɨɥɠɧɵ ɚɩɩɪɨɤɫɢɦɢɪɨɜɚɬɶɫɹ
ɬɪɚɟɤɬɨɪɢɹɦɢ ɫ ɨɝɪɚɧɢɱɟɧɢɹɦɢ. Ɍɚɤ ɤɚɤ ɫɭɳɟɫɬɜɭɸɬ ɨɝɪɚɧɢɱɟɧɢɹ ɧɚ ɭɩɪɚɜɥɟɧɢɟ ɜ ɜɢɞɟ ɧɟɪɚɜɟɧɫɬɜ
u k ≤ U m , ɬɨ ɜ ɤɚɱɟɫɬɜɟ ɭɩɪɚɜɥɟɧɢɹ ɧɚ ɨɛɴɟɤɬ ɩɨɞɚɸɬɫɹ ɡɧɚɱɟɧɢɹ, ɹɜɥɹɸɳɢɟɫɹ ɩɪɨɟɤɰɢɟɣ ɡɚɤɨɧɚ
ɭɩɪɚɜɥɟɧɢɹ ɨɩɬɢɦɢɡɢɪɭɸɳɢɣ ɜɵɛɪɚɧɧɵɣ ɤɪɢɬɟɪɢɣ uk ɧɚ ɦɧɨɠɟɫɬɜɨ ɞɨɩɭɫɬɢɦɵɯ ɡɧɚɱɟɧɢɣ:
­uk ,
°
uk = ®
°U m sign (uk ),
¯
ɟɫɥɢ
uk ≤U m
ɟɫɥɢ
u
k
>U .
m
ɈȽɊȺɇɂɑȿɇɂə ɇȺ ɉȺɊȺɆȿɌɊɕ ɍɉɊȺȼɅȿɇɂə
ɍɩɪɚɜɥɟɧɢɟ ɬɪɚɟɤɬɨɪɢɟɣ ɞɜɢɠɟɧɢɹ ȼɋ ɦɨɠɟɬ ɛɵɬɶ ɪɟɚɥɢɡɨɜɚɧɨ ɩɨ ɤɚɧɚɥɚɦ ɭɝɥɚ ɚɬɚɤɢ,
ɫɤɨɪɨɫɬɧɨɝɨ ɭɝɥɚ ɤɪɟɧɚ ɢ ɬɹɝɢ ɞɜɢɝɚɬɟɥɹ (ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɢ ɩɨ ɫɤɨɪɨɫɬɢ). ɋɨɜɨɤɭɩɧɨɫɬɶ
ɭɩɪɚɜɥɟɧɢɣ u ɞɨɥɠɧɚ ɭɱɢɬɵɜɚɬɶ ɨɝɪɚɧɢɱɟɧɢɹ ɧɚ ɭɩɪɚɜɥɟɧɢɟ, ɩɚɪɚɦɟɬɪɵ ɬɪɚɟɤɬɨɪɢɢ, ɮɚɡɨɜɵɟ
ɤɨɨɪɞɢɧɚɬɵ, ɜɪɟɦɹ ɦɚɧɟɜɪɚ ɢ ɨɩɬɢɦɢɡɢɪɨɜɚɬɶ ɜɵɛɪɚɧɧɵɣ ɤɪɢɬɟɪɢɣ ɤɚɱɟɫɬɜɚ ɭɩɪɚɜɥɟɧɢɹ.
Ɉɝɪɚɧɢɱɟɧɢɹ ɧɚ ɩɚɪɚɦɟɬɪɵ ɭɩɪɚɜɥɟɧɢɹ (ɉɍ) ɫɜɟɞɟɧɵ ɜ ɫɢɫɬɟɦɭ ɧɟɪɚɜɟɧɫɬɜ ɫɥɟɞɭɸɳɟɝɨ ɜɢɞɚ:
U mɉɍ
­α min ( p ) ≤ α ≤ α min ( p ), α ≤ α max ( p ), γ ≤ γ max ( p ), γ ≤ γmax ( p ),
°
°Vmin ≤ V ≤ Vmax ,0 ≤ a ≤ amax ,
=®
°qmax − qɞɨɩ ≤ 0, nx max − nx ɞɨɩ ≤ 0, n y max − n y ɞɨɩ ≤ 0,
°m − m ≤ 0,
T ɞɨɩ
¯ T
ɝɞɟ ɪ - ɜɟɤɬɨɪ ɩɚɪɚɦɟɬɪɨɜ ɬɪɚɟɤɬɨɪɢɢ, ɨɬ ɤɨɬɨɪɵɯ ɡɚɜɢɫɹɬ ɤɨɧɤɪɟɬɧɵɟ ɡɧɚɱɟɧɢɹ
ɨɝɪɚɧɢɱɟɧɢɣ ɧɚ ɭɩɪɚɜɥɹɸɳɢɟ ɜɨɡɞɟɣɫɬɜɢɹ; amax - ɦɚɤɫɢɦɚɥɶɧɨ ɞɨɩɭɫɬɢɦɨɟ ɭɫɤɨɪɟɧɢɟ ȼɋ; Vmin ɢ
Vmax - ɦɢɧɢɦɚɥɶɧɨ ɢ ɦɚɤɫɢɦɚɥɶɧɨ ɜɨɡɦɨɠɧɵɟ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ȼɋ; q - ɫɤɨɪɨɫɬɧɨɣ ɧɚɩɨɪ; nx ɢ
ny - ɩɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ ɩɟɪɟɝɪɭɡɤɢ ɧɚ ɩɪɨɞɨɥɶɧɭɸ ɢ ɧɨɪɦɚɥɶɧɭɸ ɨɫɢ ɫɜɹɡɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ.
ɉɨɦɢɦɨ ɨɝɪɚɧɢɱɟɧɢɣ, ɫɜɹɡɚɧɧɵɯ ɫ ɜɨɡɦɨɠɧɨɫɬɹɦɢ ȼɋ ɢ ɷɤɢɩɚɠɚ ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ
ɨɩɬɢɦɢɡɚɰɢɢ, ɧɚɤɥɚɞɵɜɚɸɬɫɹ ɬɪɚɟɤɬɨɪɧɵɟ ɨɝɪɚɧɢɱɟɧɢɹ (ɌɈ), ɨɩɪɟɞɟɥɹɟɦɵɟ ɰɟɥɟɜɨɣ ɡɚɞɚɱɟɣ
ɩɨɥɺɬɚ, ɫɜɹɡɚɧɧɵɟ ɫ ɨɝɪɚɧɢɱɟɧɢɹɦɢ ɧɚ ɬɟɤɭɳɢɟ ɢ ɤɨɧɟɱɧɵɟ ɡɧɚɱɟɧɢɹ ɮɚɡɨɜɵɯ ɤɨɨɪɞɢɧɚɬ
ɰɟɧɬɪɚ ɦɚɫɫ ȼɋ. ɗɬɢ ɨɝɪɚɧɢɱɟɧɢɹ ɢɦɟɸɬ ɜɢɞ ɧɟɪɚɜɟɧɫɬɜ:
U mɌɈ
­V (t ) − Vɞɨɩ ≤ (≥)0, ϕ (t ) − ϕ ɞɨɩ ≤ (≥)0, γ ≤ γ max ( p ), H (t ) − H ɞɨɩ ≤ (≥)0,
°
= ®ΔV (t ) − ΔVɞɨɩ ≤ (≥)0, Δϕ (t ) − Δϕ ɞɨɩ ≤ (≥)0, ΔH (t ) − ΔH ɞɨɩ ≤ (≥)0,
°T − T ≤ (≥)0, ΔT − ΔT ≤ (≥)0,
ɞɨɩ
ɞɨɩ
¯
ɝɞɟ Ɍ - ɩɪɨɞɨɥɠɢɬɟɥɶɧɨɫɬɶ ɦɚɧɺɜɪɚ; n - ɤɨɥɢɱɟɫɬɜɨ ɬɪɚɟɤɬɨɪɧɵɯ ɨɝɪɚɧɢɱɟɧɢɣ;
V - ɫɤɨɪɨɫɬɶ ȼɋ ɨɬɧɨɫɢɬɟɥɶɧɨ Ɂɟɦɥɢ; ij - ɭɝɨɥ ɤɭɪɫɚ; ɇ - ɜɵɫɨɬɚ ɧɚɞ ɩɨɜɟɪɯɧɨɫɬɶɸ Ɂɟɦɥɢ;
ΔT = T − Tɬɪɟɛ .
24
ȼ.ȼ. ȿɪɨɯɢɧ
ɆȺɌȿɆȺɌɂɑȿɋɄɂȿ ɆɈȾȿɅɖ ɉɈȾȼɂɀɇɕɏ ɈȻɔȿɄɌɈȼ
Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɦɨɞɟɥɶ ɬɪɚɟɤɬɨɪɧɨɝɨ ɞɜɢɠɟɧɢɹ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɭɩɪɚɜɥɹɟɦɨɝɨ ɨɛɴɟɤɬɚ
(ȼɋ) ɜ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ ɫɬɚɧɞɚɪɬɧɨɣ ɧɨɪɦɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɨɩɢɫɵɜɚɟɬɫɹ
ɫɥɟɞɭɸɳɟɣ ɫɢɫɬɟɦɨɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ (ɪɢɫ. 2):
­ x ' = V cosψ ,
° z ' = V sin ψ ,
°
®
°ψ ' = ω max u ,
°¯ω max = k / V ,
(10)
ɝɞɟ x – ɨɪɞɢɧɚɬɚ ɩɨɥɨɠɟɧɢɹ ȼɋ ɩɨ ɨɫɢ OX ɜ ɫɬɚɧɞɚɪɬɧɨɣ ɧɨɪɦɚɥɶɧɨɣ ɫɢɫɬɟɦɟ
ɤɨɨɪɞɢɧɚɬ; z – ɚɛɫɰɢɫɫɚ ɩɨɥɨɠɟɧɢɹ ȼɋ ɩɨ ɨɫɢ OZ ɜ ɫɬɚɧɞɚɪɬɧɨɣ ɧɨɪɦɚɥɶɧɨɣ ɫɢɫɬɟɦɟ
ɤɨɨɪɞɢɧɚɬ; ȥ – ɧɚɩɪɚɜɥɟɧɢɟ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ ɢɥɢ ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɬɪɚɟɤɬɨɪɢɢ
(ɭɝɨɥ ɨɬɫɱɢɬɵɜɚɟɬɫɹ ɨɬ ɨɫɢ OX ɩɨ ɱɚɫɨɜɨɣ ɫɬɪɟɥɤɟ); Ȧmax – ɦɚɤɫɢɦɚɥɶɧɚɹ
ɜɟɥɢɱɢɧɚ ɞɨɩɭɫɬɢɦɨɣ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɢ; k > 0 – ɦɚɤɫɢɦɚɥɶɧɚɹ ɜɟɥɢɱɢɧɚ
ɞɨɩɭɫɬɢɦɨɝɨ ɛɨɤɨɜɨɝɨ ɭɫɤɨɪɟɧɢɹ ȼɋ; V = const > 0 – ɩɪɢɛɨɪɧɚɹ ɫɤɨɪɨɫɬɶ ɫɚɦɨɥɟɬɚ
ɜ
ɨɬɫɭɬɫɬɜɢɟ
ɜɨɡɦɭɳɟɧɢɣ;
u
–
ɭɩɪɚɜɥɟɧɢɟ,
ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɟ
ɨɝɪɚɧɢɱɟɧɢɸ
|u1| ” 1.
ɋɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ (10) ɫ ɞɨɫɬɚɬɨɱɧɨɣ ɬɨɱɧɨɫɬɶɸ ɨɩɢɫɵɜɚɟɬ ɩɪɨɰɟɫɫ ɪɚɡɜɨɪɨɬɚ ɢ
ɞɜɢɠɟɧɢɹ ɧɚ ɩɪɹɦɨɥɢɧɟɣɧɵɯ ɭɱɚɫɬɤɚɯ ɢ ɢɫɩɨɥɶɡɭɟɬɫɹ ɞɥɹ ɫɬɚɧɞɚɪɬɧɵɯ ɧɚɜɢɝɚɰɢɨɧɧɵɯ
ɪɚɫɱɟɬɨɜ.
ɚ
ɛ
ɜ
Ɋɢɫ. 2
ɚ, ɛ) ɩɚɪɚɦɟɬɪɵ ɬɪɚɟɤɬɨɪɢɢ ɩɨɥɟɬɚ ȼɋ; ɜ) ɫɬɚɧɞɚɪɬɧɚɹ ɧɨɪɦɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ
ɆɈȾȿɅɂɊɈȼȺɇɂȿ, ɂɋɋɅȿȾɈȼȺɇɂȿ ɏȺɊȺɄɌȿɊɂɋɌɂɄ,
ɈȻɋɍɀȾȿɇɂȿ ɊȿɁɍɅɖɌȺɌɈȼ
ɂɫɫɥɟɞɨɜɚɧɢɹ ɩɚɪɚɦɟɬɪɨɜ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɫɟɚɧɫɚ ɢ ɬɨɱɧɨɫɬɧɵɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɇȼɈ
ɩɪɨɜɨɞɢɥɢɫɶ ɧɚ ɨɫɧɨɜɟ ɪɚɡɪɚɛɨɬɚɧɧɨɣ ɦɨɞɟɥɢ ɞɜɢɠɟɧɢɹ ɇɋ ɨɪɛɢɬɚɥɶɧɨɣ ɝɪɭɩɩɢɪɨɜɤɢ. ɇɚ
ɪɢɫ. 3 ɩɪɟɞɫɬɚɜɥɟɧɵ ɩɨɝɪɟɲɧɨɫɬɢ ɨɰɟɧɤɢ ɤɨɨɪɞɢɧɚɬ ɢ ɦɟɫɬɨɩɨɥɨɠɟɧɢɹ (Ɇɉ) ȼɋ,
ɨɛɟɫɩɟɱɢɜɚɟɦɵɟ ɂɋɇ ɩɪɢ 2 ɇɋ ɫ ɢɧɟɪɰɢɚɥɶɧɨɣ ɩɨɞɞɟɪɠɤɨɣ (ɚ), ɜ ɤɚɱɟɫɬɜɟ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ
ɢɫɬɨɱɧɢɤɨɜ ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɢɫɩɨɥɶɡɨɜɚɥɢɫɶ 2 ɇɈɌ ȺɁɇ (ɧɚɡɟɦɧɚɹ ɢ ɜɨɡɞɭɲɧɚɹ)
(ɛ). ɉɪɢ ɷɬɨɦ ɪɚɞɢɚɥɶɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɧɚ ɜɵɯɨɞɟ ɂCɇ ɫɧɢɠɚɟɬɫɹ ɫ 36 ɦ (ɨɲɢɛɤɚ ɫɢɫɬɟɦɵ
ɫɱɢɫɥɟɧɢɹ) ɞɨ 20 ɦ.
ɍɩɪɚɜɥɟɧɢɟ ɬɪɚɟɤɬɨɪɢɹɦɢ ɜɨɡɞɭɲɧɵɯ ɫɭɞɨɜ ɜ ɭɫɥɨɜɢɹɯ…
25
Ɋɢɫ. 3. ɉɨɝɪɟɲɧɨɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ȼɋ ɜ ɂɋɇ ɩɪɢ ɪɚɛɨɬɟ ɩɨ ɧɟɩɨɥɧɨɦɭ ɫɨɡɜɟɡɞɢɸ ɇɋ:
ɚ) ɩɨɝɪɟɲɧɨɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɨɪɞɢɧɚɬɵ X; ɛ) ɪɚɞɢɚɥɶɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ Ɇɉ ȼɋ
ɇɚ ɪɢɫ. 4,ɚ ɩɨɤɚɡɚɧɚ ɞɢɧɚɦɢɤɚ ɢɡɦɟɧɟɧɢɹ «ɦɟɪɵ ɧɚɛɥɸɞɚɟɦɨɫɬɢ» ɢ ȽɎ ɜ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ
ɩɥɨɫɤɨɫɬɢ: ɤɪɢɜɵɟ 1, 2 – ɦɟɪɚ ɧɚɛɥɸɞɚɟɦɨɫɬɢ ɢ ɝɨɪɢɡɨɧɬɚɥɶɧɵɣ ȽɎ ɛɟɡ ɭɩɪɚɜɥɟɧɢɹ, ɤɪɢɜɵɟ 3,
4 – ɩɪɢ ɨɩɬɢɦɚɥɶɧɨɦ ɭɩɪɚɜɥɟɧɢɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɇɚ ɪɢɫ. 4,ɛ ɩɪɟɞɫɬɚɜɥɟɧɵ ɪɟɡɭɥɶɬɚɬɵ
ɢɫɫɥɟɞɨɜɚɧɢɹ ɬɨɱɧɨɫɬɧɵɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɂɋɇ ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɤɨɧɮɢɝɭɪɚɰɢɹɯ ɂɂ ɫ
ɩɪɢɦɟɧɟɧɢɟɦ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɬɪɚɟɤɬɨɪɢɣ ȼɋ: ɤɪɢɜɚɹ 1 - ɞɢɧɚɦɢɤɚ ɪɚɞɢɚɥɶɧɨɣ
ɩɨɝɪɟɲɧɨɫɬɢ ɩɪɢ ɪɚɛɨɬɟ ɂɋɇ ɩɨ 2 ɇɋ ɫ ɢɧɟɪɰɢɚɥɶɧɨɣ ɩɨɞɞɟɪɠɤɨɣ; ɤɪɢɜɚɹ 2 - ɪɚɞɢɚɥɶɧɚɹ
ɩɨɝɪɟɲɧɨɫɬɶ ɩɪɢ ɪɚɛɨɬɟ ɂɋɇ ɩɨ 2 ɇɋ ɢ 2 ɇɈɌ ȺɁɇ (ɧɚɡɟɦɧɚɹ ɢ ɜɨɡɞɭɲɧɚɹ); ɤɪɢɜɚɹ 3 ɪɚɞɢɚɥɶɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ Ɇɉ ɞɥɹ 2 ɇɋ ɢ 2 ɇɈɌ ȺɁɇ (ɧɚɡɟɦɧɚɹ ɢ ɜɨɡɞɭɲɧɚɹ) ɩɪɢ ɨɩɬɢɦɚɥɶɧɨɦ
ɭɩɪɚɜɥɟɧɢɢ ɬɪɚɟɤɬɨɪɢɟɣ ȼɋ; ɤɪɢɜɚɹ 4 – ɪɚɞɢɚɥɶɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ Ɇɉ ɩɪɢ ɪɚɛɨɬɟ ɩɨ 4-ɦ ɇɋ ɫ
ɢɧɟɪɰɢɚɥɶɧɨɣ ɩɨɞɞɟɪɠɤɨɣ.
ɋɪɚɜɧɢɬɟɥɶɧɵɣ ɚɧɚɥɢɡ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɩɪɢɦɟɧɟɧɢɟ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ
ɬɪɚɟɤɬɨɪɢɟɣ ȼɋ ɩɨɡɜɨɥɹɟɬ ɩɨɜɵɫɢɬɶ ɬɨɱɧɨɫɬɶ ɧɚɜɢɝɚɰɢɨɧɧɵɯ ɨɩɪɟɞɟɥɟɧɢɣ ɩɨ ɞɚɧɧɵɦ
ɂɋɇ ɩɪɢ ɧɟɩɨɥɧɨɦ ɫɨɡɜɟɡɞɢɢ ɇɋ ɋɇɋ ɞɨ 2σ r ≈ 10 ɦ (ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ
ɮɚɤɬɨɪɚ) ɢ ɩɪɢɛɥɢɡɢɬɶ ɟɟ ɤ ɬɨɱɧɨɫɬɧɵɦ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦ ɫɢɫɬɟɦɵ ɩɪɢ ɪɚɛɨɬɟ
ɩɨ 4-ɦ ɇɋ.
Ɋɢɫ. 4
ɚ) ɝɟɨɦɟɬɪɢɱɟɫɤɢɟ ɭɫɥɨɜɢɹ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɫɟɚɧɫɚ:
ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɮɚɤɬɨɪ Ʉxy (ɚ);
ɛ) ɪɚɞɢɚɥɶɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ Ɇɉ
ɦɟɪɚ
ɧɚɛɥɸɞɚɟɦɨɫɬɢ
ȽXY,
ɝɨɪɢɡɨɧɬɚɥɶɧɵɣ
ȼɕȼɈȾɕ
ɂɡ ɚɧɚɥɢɡɚ ɩɨɥɭɱɟɧɧɵɯ ɪɟɡɭɥɶɬɚɬɨɜ ɫɥɟɞɭɟɬ, ɱɬɨ ɨɪɝɚɧɢɡɚɰɢɹ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ
ɬɪɚɟɤɬɨɪɢɟɣ ȼɋ ɩɨ ɜɵɛɪɚɧɧɨɦɭ ɤɪɢɬɟɪɢɸ ɩɪɢɜɟɥɚ ɤ ɭɥɭɱɲɟɧɢɸ ɭɫɥɨɜɢɣ ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ
ɫɟɚɧɫɚ ɢ ɩɨɜɵɲɟɧɢɸ ɬɨɱɧɨɫɬɢ ɨɩɪɟɞɟɥɟɧɢɹ Ɇɉ ȼɋ. ɉɪɢɦɟɧɟɧɢɟ ɫɢɧɬɟɡɢɪɨɜɚɧɧɵɯ ɚɥɝɨɪɢɬɦɨɜ
ɮɢɥɶɬɪɚɰɢɢ ɢ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɬɪɚɟɤɬɨɪɢɹɦɢ ɩɨɡɜɨɥɹɟɬ ɩɨɜɵɫɢɬɶ ɬɨɱɧɨɫɬɢ
26
ȼ.ȼ. ȿɪɨɯɢɧ
ɦɟɫɬɨɨɩɪɟɞɟɥɟɧɢɹ ȼɋ ɜ ɫɪɟɞɧɟɦ ɧɚ 28%. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɨɜɦɟɫɬɧɨɟ ɩɪɢɦɟɧɟɧɢɟ ɋɋɇ, ɂɇɋ ɢ
ɫɢɫɬɟɦɵ ȺɁɇ ɜ ɫɨɫɬɚɜɟ ɢɧɬɟɝɪɢɪɨɜɚɧɧɨɣ ɫɢɫɬɟɦɵ ɧɚɜɢɝɚɰɢɢ ɩɨɡɜɨɥɢɬ ɩɨɜɵɫɢɬɶ ɬɨɱɧɨɫɬɶ
ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɨɪɞɢɧɚɬ ȼɋ ɧɚ ɪɚɡɥɢɱɧɵɯ ɷɬɚɩɚɯ ɩɨɥɟɬɚ.
ɅɂɌȿɊȺɌɍɊȺ
1. Ʉɪɵɠɚɧɨɜɫɤɢɣ Ƚ.Ⱥ. Ʉɨɧɰɟɩɰɢɹ ɢ ɫɢɫɬɟɦɵ CNS/ATM ɜ ɝɪɚɠɞɚɧɫɤɨɣ ɚɜɢɚɰɢɢ. - Ɇ.:
ɂɄɐ Ⱥɤɚɞɟɦɤɧɢɝɚ, 2003.
2. Ɋɭɤɨɜɨɞɫɬɜɨ ɩɨ ɬɪɟɛɭɟɦɵɦ ɧɚɜɢɝɚɰɢɨɧɧɵɦ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦ (RNP). Doc 9613-AN/937,
ICAO, 1994.
3. ɋɟɣɞɠ ɗ.ɉ., ɍɚɣɬ ɑ.ɋ. Ɉɩɬɢɦɚɥɶɧɨɟ ɭɩɪɚɜɥɟɧɢɟ ɫɢɫɬɟɦɚɦɢ. - Ɇ.: Ɋɚɞɢɨ
ɢ ɫɜɹɡɶ, 1982.
4. ɋɤɪɵɩɧɢɤ Ɉ.ɇ., ȿɪɨɯɢɧ ȼ.ȼ., ɋɥɟɩɱɟɧɤɨ Ⱥ.ɉ. Ɉɩɬɢɦɢɡɚɰɢɹ ɭɫɥɨɜɢɣ
ɧɚɜɢɝɚɰɢɨɧɧɨɝɨ ɫɟɚɧɫɚ ɞɥɹ ɩɨɜɵɲɟɧɢɹ ɬɨɱɧɨɫɬɢ ɧɚɜɢɝɚɰɢɨɧɧɨ-ɜɪɟɦɟɧɧɵɯ ɨɩɪɟɞɟɥɟɧɢɣ ɜ
ɥɨɤɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ // ɇɚɭɱɧɵɣ ɜɟɫɬɧɢɤ ɆȽɌɍ ȽȺ. 2010. ʋ 159. ɋ. 55-62.
CONTROL OF AIRCRAFT TRAJECTORIES IN THE CONDITIONS
OF THE NAVIGATION SESSION OPTIMIZATION
AT AUTOMATIC DEPENDENT SURVEILLANCE
Erokhin V.V.
Algorithms of determination of coordinates of the aircraft in the integrated system of navigation and optimum
control of a trajectory are considered. Results of researches of parameters of a navigation session and precision
characteristics of an assessment of location showed that application of optimum control of a trajectory allowɭɜ to increase
the accuracy of navigation definitions in case of incomplete constellation of navigation satellites.
Key words: trajectory, optimal control, navigation.
REFERENCES
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and CNS/ATM systems in civil aviation). - M.: IKTs of Akademkniga, 2003. (In Russian).
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navigation session for increase of accuracy of navigation and time definitions in local system of
coordinates // Nauchniy vestnik MGTU GA. 2010. No. 159. Pp. 55-62. (In Russian).
ɋȼȿȾȿɇɂə ɈȻ ȺȼɌɈɊȿ
ȿɪɨɯɢɧ ȼɹɱɟɫɥɚɜ ȼɥɚɞɢɦɢɪɨɜɢɱ, 1975 ɝ.ɪ., ɨɤɨɧɱɢɥ ɂɪɤɭɬɫɤɨɟ ȼȼȺɂɍ (1998), ɞɨɰɟɧɬ,
ɤɚɧɞɢɞɚɬ ɬɟɯɧɢɱɟɫɤɢɯ ɧɚɭɤ, ɞɨɰɟɧɬ ɤɚɮɟɞɪɵ ȺɊɗɈ ɂɎ ɆȽɌɍ ȽȺ, ɚɜɬɨɪ 39 ɧɚɭɱɧɵɯ ɪɚɛɨɬ,
ɨɛɥɚɫɬɶ ɧɚɭɱɧɵɯ ɢɧɬɟɪɟɫɨɜ – ɫɩɭɬɧɢɤɨɜɵɟ ɫɢɫɬɟɦɵ ɧɚɜɢɝɚɰɢɢ, ɤɨɦɩɥɟɤɫɧɚɹ ɨɛɪɚɛɨɬɤɚ
ɧɚɜɢɝɚɰɢɨɧɧɨɣ ɢɧɮɨɪɦɚɰɢɢ.
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