Feedback synthesis for underactuated systems using sequential second-order needle variations

Giorgos Mamakoukas*, Malcolm A. MacIver, Todd D. Murphey

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


This paper derives nonlinear feedback control synthesis for general control affine systems using second-order actions, the second-order needle variations of optimal control, as the basis for choosing each control response to the current state. A second result of this paper is that the method provably exploits the nonlinear controllability of a system by virtue of an explicit dependence of the second-order needle variation on the Lie bracket between vector fields. As a result, each control decision necessarily decreases the objective when the system is nonlinearly controllable using first-order Lie brackets. Simulation results using a differential drive cart, an underactuated kinematic vehicle in three dimensions, and an underactuated dynamic model of an underwater vehicle demonstrate that the method finds control solutions when the first-order analysis is singular. Finally, the underactuated dynamic underwater vehicle model demonstrates convergence even in the presence of a velocity field.

Original languageEnglish (US)
Pages (from-to)1826-1853
Number of pages28
JournalInternational Journal of Robotics Research
Issue number13-14
StatePublished - Dec 1 2018


  • dynamics
  • kinematics
  • motion control
  • underactuated robots

ASJC Scopus subject areas

  • Software
  • Modeling and Simulation
  • Mechanical Engineering
  • Electrical and Electronic Engineering
  • Artificial Intelligence
  • Applied Mathematics


Dive into the research topics of 'Feedback synthesis for underactuated systems using sequential second-order needle variations'. Together they form a unique fingerprint.

Cite this