American Institute of Mathematical Sciences

Advanced
2011, 2011(Special): 1413-1422. doi: 10.3934/proc.2011.2011.1413

Generalizations of Naismith's problem: Minimal transit time between two points in a heterogenous terrian

Erik I. Verriest 1,

1. 

School of ECE, Georgia Institute of Technology, Atlanta, GA 30332-0250, United States

Received  July 2010 Revised  March 2011 Published  October 2011

Naismith obtained a set of empirical rules for the time required to move through a terrain. In this paper we solve the problem of determining the path which minimizes the transit time between two points on a given terrain. We give an interpretation of Naismith’s rule which leads to an elegant geometric construction of the optimal solution. This problem is a paradigm for the navigation of an autonomous vehicle in a heterogenous terrain.
Keywords: Optimal control, path planning..
Mathematics Subject Classification: Primary: 49K05, 49K15; Secondary: 65K1.
Citation: Erik I. Verriest. Generalizations of Naismith's problem: Minimal transit time between two points in a heterogenous terrian. Conference Publications, 2011, 2011 (Special) : 1413-1422. doi: 10.3934/proc.2011.2011.1413
[1]

Yi Gao, Rui Li, Yingjing Shi, Li Xiao. Design of path planning and tracking control of quadrotor. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2221-2235. doi: 10.3934/jimo.2021063
[2]

Jinlong Guo, Bin Li, Yuandong Ji. A control parametrization based path planning method for the quad-rotor uavs. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1079-1100. doi: 10.3934/jimo.2021009
[3]

Matthias Gerdts, René Henrion, Dietmar Hömberg, Chantal Landry. Path planning and collision avoidance for robots. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 437-463. doi: 10.3934/naco.2012.2.437
[4]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[5]

Sheng-I Chen, Yen-Che Tseng. A partitioning column approach for solving LED sorter manipulator path planning problems. Journal of Industrial and Management Optimization, 2022, 18 (3) : 2033-2047. doi: 10.3934/jimo.2021055
[6]

Yongkun Wang, Fengshou He, Xiaobo Deng. Multi-aircraft cooperative path planning for maneuvering target detection. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1935-1948. doi: 10.3934/jimo.2021050
[7]

Louis Caccetta, Ian Loosen, Volker Rehbock. Computational aspects of the optimal transit path problem. Journal of Industrial and Management Optimization, 2008, 4 (1) : 95-105. doi: 10.3934/jimo.2008.4.95
[8]

Ta-Wei Hung, Ping-Ting Chen. On the optimal replenishment in a finite planning horizon with learning effect of setup costs. Journal of Industrial and Management Optimization, 2010, 6 (2) : 425-433. doi: 10.3934/jimo.2010.6.425
[9]

Thalya Burden, Jon Ernstberger, K. Renee Fister. Optimal control applied to immunotherapy. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 135-146. doi: 10.3934/dcdsb.2004.4.135
[10]

Ellina Grigorieva, Evgenii Khailov. Optimal control of pollution stock. Conference Publications, 2011, 2011 (Special) : 578-588. doi: 10.3934/proc.2011.2011.578
[11]

Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial and Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967
[12]

Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial and Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275
[13]

Cristiana J. Silva, Helmut Maurer, Delfim F. M. Torres. Optimal control of a Tuberculosis model with state and control delays. Mathematical Biosciences & Engineering, 2017, 14 (1) : 321-337. doi: 10.3934/mbe.2017021
[14]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019
[15]

Changjun Yu, Shuxuan Su, Yanqin Bai. On the optimal control problems with characteristic time control constraints. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1305-1320. doi: 10.3934/jimo.2021021
[16]

Jorge San Martín, Takéo Takahashi, Marius Tucsnak. An optimal control approach to ciliary locomotion. Mathematical Control and Related Fields, 2016, 6 (2) : 293-334. doi: 10.3934/mcrf.2016005
[17]

C.Z. Wu, K. L. Teo. Global impulsive optimal control computation. Journal of Industrial and Management Optimization, 2006, 2 (4) : 435-450. doi: 10.3934/jimo.2006.2.435
[18]

Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455
[19]

Filipe Rodrigues, Cristiana J. Silva, Delfim F. M. Torres, Helmut Maurer. Optimal control of a delayed HIV model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 443-458. doi: 10.3934/dcdsb.2018030
[20]

Antonio Fernández, Pedro L. García. Regular discretizations in optimal control theory. Journal of Geometric Mechanics, 2013, 5 (4) : 415-432. doi: 10.3934/jgm.2013.5.415

 Impact Factor: 

Tools

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy