An Exponential Observer for Systems on SE.3/ with Implicit Outputs

This paper considers the state estimation problem of a class of systems described by implicit outputs and whose state lives in the special Euclidean group SE(3). This type of systems are motivated by applications in dynamic vision such as the estimation of the motion of a camera from a sequence of images. We propose an observer in the group of motion SE(3) and discuss conditions under which the linearized state estimation error converges exponentially fast. We also analyze the problem when the system is subject to disturbances and noises. We show that the estimate converges to a neighborhood of the real solution. The size of the neighborhood increases/decreases gracefully with the bound of the disturbance and noise.

[1]  João Pedro Hespanha,et al.  Robust filtering for deterministic systems with implicit outputs , 2009, Syst. Control. Lett..

[2]  N. Harris McClamroch,et al.  Tracking and maneuver regulation control for nonlinear nonminimum phase systems: application to flight control , 2002, IEEE Trans. Control. Syst. Technol..

[3]  R. Mahony,et al.  Observer design for invariant systems with homogeneous observations , 2008, 0810.0748.

[4]  P. Rouchon,et al.  A non-linear symmetry-preserving observer for velocity-aided inertial navigation , 2006, 2006 American Control Conference.

[5]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[6]  Philippe Martin,et al.  Symmetry-Preserving Observers , 2006, IEEE Transactions on Automatic Control.

[7]  D. Sattinger,et al.  Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics , 1986 .

[8]  Francesco Bullo,et al.  Stabilization of relative equilibria for underactuated systems on Riemannian manifolds , 2000, Autom..

[9]  Robert E. Mahony,et al.  Observers for systems with invariant outputs , 2009, 2009 European Control Conference (ECC).

[10]  B. Kolman,et al.  A survey of Lie groups and Lie algebras with applications and computational methods , 1972 .

[11]  Karim M. Abadir,et al.  Matrix Algebra: Notation , 2005 .

[12]  Roger Skjetne,et al.  Robust output maneuvering for a class of nonlinear systems , 2004, Autom..

[13]  Antonio Loría,et al.  Relaxed persistency of excitation for uniform asymptotic stability , 2001, IEEE Trans. Autom. Control..

[14]  S. Shankar Sastry,et al.  An Invitation to 3-D Vision , 2004 .

[15]  D. Luenberger.  Observing the State of a Linear System , 1964, IEEE Transactions on Military Electronics.

[16]  Robert E. Mahony,et al.  Gradient-Like Observers on Semidirect Products , 2009, 2009 European Control Conference (ECC).

[17]  João P. Hespanha,et al.  Trajectory-Tracking and Path-Following of Underactuated Autonomous Vehicles With Parametric Modeling Uncertainty , 2007, IEEE Transactions on Automatic Control.

[18]  M. Morari,et al.  Optimal control of piecewise affine systems: A dynamic programming approach , 2005 .

[19]  Marshall C. Pease,et al.  Methods of Matrix Algebra. , 1967 .

[20]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[21]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[22]  M. C. Pease,et al.  Methods of Matrix Algebra , 1965 .

[23]  B. Ghosh,et al.  Motion and shape parameters identification with vision and range , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[24]  B. Ghosh,et al.  A Perspective Theory for Motion and Shape Estimation in Machine Vision , 1995 .

[25]  R. H. Thurston,et al.  Scientific Books: Transactions of the American Society of Mechanical Engineers , 1900 .

[26]  João Pedro Hespanha,et al.  Minimum-energy state estimation for systems with perspective outputs , 2006, IEEE Transactions on Automatic Control.

[27]  Lorenzo Marconi,et al.  Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems , 2005 .

[28]  F. Leite,et al.  A Variational Approach for High Order Least Squares Problems on Riemannian Manifolds , 2008 .

[29]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[30]  Robert E. Mahony,et al.  Gradient-Like Observers for Invariant Dynamics on a Lie Group , 2008, IEEE Transactions on Automatic Control.

[31]  S. Shankar Sastry,et al.  An Invitation to 3-D Vision: From Images to Geometric Models , 2003 .

[32]  Christian Lageman,et al.  State observers for invariant dynamics on a Lie group , 2008 .

[33]  John R. Hauser,et al.  Aggressive flight maneuvers , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[34]  P. Rouchon,et al.  On invariant asymptotic observers , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[35]  Philippe Martin,et al.  Non-linear observer on Lie Groups for left-invariant dynamics with right-left equivariant output , 2008 .

[36]  Philippe Martin,et al.  Non-Linear Symmetry-Preserving Observers on Lie Groups , 2007, IEEE Transactions on Automatic Control.