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Page 1: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University
Page 2: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

• 𝑥𝑓 𝑥 < 𝑓 𝑡∗ 𝑡∗

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Page 3: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

• 𝐼𝐺 ℎ

• 𝑥

• 𝑓 𝑥 < 𝑓 𝑡∗ = 𝐶∗

• 𝐵𝑥

3

𝑥

𝑠

Page 4: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

• 𝐺′ 𝑡(𝑥, 𝑡) ℎ(𝑥)

• ℎ′ ℎ ℎ′ 𝑡 = 0

• ℎ′

o

𝑡

o (𝑦, 𝑡)ℎ′ 𝑦 = ℎ 𝑦 ≤ 𝑐 𝑦, 𝑥 + ℎ 𝑥= 𝑐 𝑦, 𝑡 = 𝑐 𝑦, 𝑡 + ℎ′ 𝑡

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𝑠

𝑡

ℎ(𝑥)

𝑥

𝐺

𝐺′

Page 5: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

• 𝐼′ 𝐺′ℎ′ 𝑡

𝑔 𝑡 + ℎ′ 𝑡 = 𝑔 𝑥 + ℎ 𝑥 < 𝐶∗

• 𝐵 𝑥 𝐼′𝐼 𝐵

𝐶∗

𝐵∎

5

𝑡

ℎ(𝑥)

𝐺

𝐺′

𝑠

𝑥

Page 6: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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Page 7: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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𝑎𝑎

𝑏 𝑐ℎ = 1 ℎ = 20 ℎ = 6

3 1 1

1

ℎ = 1 ℎ = 1

𝑎

11 1

𝑠

𝑡

Page 8: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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Page 9: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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Page 10: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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Page 11: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

o

o

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Page 12: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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Page 13: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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Page 14: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

o 𝑥1, … , 𝑥𝑛

o 𝑥𝑘 𝑥𝑗

o 𝑥1, … , 𝑥𝑗 , 𝑥𝑘 , … , 𝑥𝑛

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Page 15: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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Page 16: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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Page 17: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

o 𝑔(𝑦) = 𝑔(𝑥) + 𝑐(𝑥, 𝑦)

o 𝑦𝑓 𝑦 = 𝑔(𝑥) + 𝑐(𝑥, 𝑦) + ℎ(𝑦)

o (𝑥) 𝑦 𝑦𝑓(𝑦) = 𝑔( (𝑥)) + 𝑐( (𝑥), 𝑦) + ℎ(𝑦)

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Page 18: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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Page 19: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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Page 20: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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Page 21: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

• ⇒ 𝑝

𝑝

• 𝑝

21

𝑝

Page 22: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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𝑝

𝑝

Page 23: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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Page 24: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

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= 𝑠, 𝑡= (𝑥, 𝑦) 𝑦 𝑥

Page 25: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

• 𝑛

1. Θ 𝑛

2. Θ 𝑛

3. Θ 𝑛2

4. Θ 𝑛3

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Page 26: PowerPoint Presentationarielpro/15780s18/slides/780s18-3.pdf · JORGE 2009 "HIS PATH-PLANNING MAY BE SUB-OPTIMAL, BUT IT's GOT FLAIR." Carnegie Mellon University

o

o

o

o

o

o

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