Journal of Dynamical and Control Systems

, Volume 22, Issue 1, pp 129–168 | Cite as

A Unified Approach to Input-output Linearization and Concurrent Control of Underactuated Open-chain Multi-body Systems with Holonomic and Nonholonomic Constraints

  • Robin Chhabra
  • M. Reza EmamiEmail author


This paper presents a unified geometric framework to input-output linearization of open-chain multi-body systems with symmetry in their reduced phase space. This leads us to an output tracking controller for a class of underactuated open-chain multi-body systems with holonomic and nonholonomic constraints. We consider the systems with multi-degree-of-freedom joints and possibly with non-zero constant total momentum (in the holonomic case). The examples of these systems are free-base space manipulators and mobile manipulators. We first formalize the control problem, and rigorously state an output tracking problem for such systems. Then, we introduce a geometrical definition of the end-effector pose and velocity error. The main contribution of this paper is reported in Section 5, where we solve for the input-output linearization of the highly nonlinear problem of coupled manipulator and base dynamics subject to holonomic and nonholonomic constraints. This enables us to design a coordinate-independent controller, similar to a proportional-derivative with feed-forward, for concurrently controlling a free-base, multi-body system. Finally, by defining a Lyapunov function, we prove in Theorem 3 that the closed-loop system is exponentially stable. A detailed case study concludes this paper.


Open-chain multi-body system Dynamical reduction Nonholonomic constraints Input-output linearization Nonlinear control 

Mathematical Subject Classifications (2010)

70E60 70H33 70Q05 



We would like to thank Professor Yael Karshon from the Department of Mathematics, University of Toronto, for her useful insights throughout the work.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.MacDonald, Dettwiler and Associates Ltd.BramptonCanada
  2. 2.Institute for Aerospace StudiesUniversity of TorontoTorontoCanada
  3. 3.Space Technology Division, Department of Computer Science, Electrical and Space EngineeringLuleå University of TechnologyKirunaSweden

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