Adaptive stochastic path planning for robots – real-time optimization by means of neural networks
K. Marti, Shihong Qu
Abstract
The problem of trajectory planning for robots can be described [1-4] mathematically by the following variational problem: (1a) http://www.w3.org/1998/Math/MathML"> min β ( . ) , q ( . ) ∫ s 0 s e f 0 s , q , q ′ , q ″ , β , β ′ , p d s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203737422/5666775f-c50d-4241-afa4-822ed4f1cd9a/content/eq2761.tif" xmlns:xlink="http://www.w3.org/1999/xlink"/> (1b) http://www.w3.org/1998/Math/MathML"> q s 0 = q 0 , q s e = q e , β s 0 = β s e = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203737422/5666775f-c50d-4241-afa4-822ed4f1cd9a/content/eq2762.tif" xmlns:xlink="http://www.w3.org/1999/xlink"/> (1c) http://www.w3.org/1998/Math/MathML"> q min ≤ q ≤ q max , q ˙ min ≤ q ′ β ≤ q ˙ max , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203737422/5666775f-c50d-4241-afa4-822ed4f1cd9a/content/eq2763.tif" xmlns:xlink="http://www.w3.org/1999/xlink"/> (1d) http://www.w3.org/1998/Math/MathML"> τ min , i ≤ a i q , q ′ , p β ′ + b i q , q ′ , q ″ , p β + c i ( q , p ) ≤ τ max , i , i = 1 , 2 , … n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203737422/5666775f-c50d-4241-afa4-822ed4f1cd9a/content/eq2764.tif" xmlns:xlink="http://www.w3.org/1999/xlink"/> (1e) http://www.w3.org/1998/Math/MathML"> β ( s ) ≥ 0 , s 0 ≤ s ≤ s f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203737422/5666775f-c50d-4241-afa4-822ed4f1cd9a/content/eq2765.tif" xmlns:xlink="http://www.w3.org/1999/xlink"/>