Partial Linearization of Mechanical Systems with Application to Observer Design

We consider general mechanical systems and establish a necessary and sufficient condition for the existence of a suitable change in the generalized momentum coordinates such that the new dynamics become linear in the transformed momenta. The class of systems which can be (partially) linearized by the proposed approach is characterized by (the solvability of) a set of partial differential equations and is shown to be larger than the class reported in all the previous works on linearization. We employ this linearization procedure to design an observer for mechanical systems where, we first (partially) linearize the system to make it affine in the new momenta and then construct a globally exponentially stable reduced order observer (which estimates the new momenta) by using the Immersion and Invariance approach.

[1]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[2]  Abhinandan Jain,et al.  Diagonalized Lagrangian robot dynamics , 1995, IEEE Trans. Robotics Autom..

[3]  J. Tsinias.  Further results on the observer design problem , 1990 .

[4]  John M. Lee.  Introduction to Smooth Manifolds , 2002 .

[5]  Arjan van der Schaft,et al.  Non-linear dynamical control systems , 1990 .

[6]  Alessandro Astolfi,et al.  Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems , 2001, IEEE Trans. Autom. Control..

[7]  Mark W. Spong.  Remarks on robot dynamics: canonical transformations and Riemannian geometry , 1992, Proceedings 1992 IEEE International Conference on Robotics and Automation.

[8]  D. Koditschek.  Robot kinematics and coordinate transformations , 1985, 1985 24th IEEE Conference on Decision and Control.

[9]  Arjan van der Schaft,et al.  Control of underactuated mechanical systems: Observer design and position feedback stabilization , 2008, 2008 47th IEEE Conference on Decision and Control.

[10]  Nazareth Bedrossian,et al.  Nonlinear control using linearizing transformations , 1991 .

[11]  Alessandro Astolfi,et al.  Nonlinear and adaptive control with applications , 2008 .

[12]  J. W. Humberston.  Classical mechanics , 1980, Nature.

[13]  V. Sundarapandian,et al.  Reduced order observer design for nonlinear systems , 2006, Appl. Math. Lett..

[14]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[15]  Gildas Besancon,et al.  Global output feedback tracking control for a class of Lagrangian systems , 2000, Autom..

[16]  Rob Dekkers,et al.  Control of Robot Manipulators in Joint Space , 2005 .