Journal of Intelligent & Robotic Systems

, Volume 89, Issue 1–2, pp 51–67 | Cite as

Abnormal Optimal Trajectory Planning of Multi-Body Systems in the Presence of Holonomic and Nonholonomic Constraints

  • Andrea L’AfflittoEmail author
  • Wassim M. Haddad


In optimal control problems, the Hamiltonian function is given by the weighted sum of the integrand of the cost function and the dynamic equation. The coefficient multiplying the integrand of the cost function is either zero or one; and if this coefficient is zero, then the optimal control problem is known as abnormal; otherwise it is normal. This paper provides a characterization of the abnormal optimal control problem for multi-body mechanical systems, subject to external forces and moments, and holonomic and nonholonomic constraints. This study does not only account for first-order necessary conditions, such as Pontryagin’s principle, but also for higher-order conditions, which allow the analysis of singular optimal controls.


Optimal trajectory planning Singular controls Pontryagin’s principle Normal and abnormal optimal control 


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This work was supported in part by NOAA/Office of Oceanic and Atmospheric Research under NOAA-University of Oklahoma Cooperative Agreement #NA16OAR4320115, the U.S. Department of Commerce, and the Air Force Office of Scientific Research under Grant FA9550-16-1-0100.


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© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.School of Aerospace and Mechanical EngineeringThe University of OklahomaNormanUSA
  2. 2.School of Aerospace EngineeringGeorgia Institute of TechnologyAtlantaUSA

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