Dynamic interpolation for obstacle avoidance on Riemannian manifolds

This work is devoted to studying dynamic interpolation for obstacle avoidance. This is a problem that consists of minimising a suitable energy functional among a set of admissible curves subject to...

[1]  R. Giambò,et al.  An analytical theory for Riemannian cubic polynomials , 2002 .

[2]  Leonardo Colombo,et al.  Higher-order variational problems on lie groups and optimal control applications , 2014 .

[3]  P. Crouch,et al.  The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces , 1995 .

[4]  Timothy Bretl,et al.  Quasi-static manipulation of a Kirchhoff elastic rod based on a geometric analysis of equilibrium configurations , 2014, Int. J. Robotics Res..

[5]  Xiaoming Hu,et al.  A geodesic feedback law to decouple the full and reduced attitude , 2017, Syst. Control. Lett..

[6]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[7]  Domenico D'Alessandro,et al.  The optimal control problem on SO(4) and its applications to quantum control , 2002, IEEE Trans. Autom. Control..

[8]  James D. Biggs,et al.  Optimal under-actuated kinematic motion planning on the ϵ-group , 2018, Autom..

[9]  L. Machado,et al.  Higher-order smoothing splines versus least squares problems on Riemannian manifolds , 2010 .

[10]  Jacob R. Goodman.  Local minimizers for variational obstacle avoidance on Riemannian manifolds , 2022, Journal of Geometric Mechanics.

[11]  A. D. Lewis,et al.  Geometric Control of Mechanical Systems , 2004, IEEE Transactions on Automatic Control.

[12]  A. Vershik,et al.  Nonholonomic problems and the theory of distributions , 1988 .

[13]  P. Alam.  ‘Z’ , 2021, Composites Engineering: An A–Z Guide.

[14]  Taeyoung Lee,et al.  Geometric tracking control of a quadrotor UAV on SE(3) , 2010, 49th IEEE Conference on Decision and Control (CDC).

[15]  Fabio Giannoni,et al.  Optimal Control on Riemannian Manifolds by Interpolation , 2004, Math. Control. Signals Syst..

[16]  K. Lynch.  Nonholonomic Mechanics and Control , 2004, IEEE Transactions on Automatic Control.

[17]  A. Bloch,et al.  Dynamic Coverage Optimal Control for Multiple Spacecraft Interferometric Imaging , 2007 .

[18]  Darryl D. Holm,et al.  The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories , 1998, chao-dyn/9801015.

[19]  James Biggs,et al.  Trajectory generation using sub-Riemannian curves for quadrotor UAVs , 2015, 2015 European Control Conference (ECC).

[20]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[21]  Daniel E. Koditschek,et al.  Robot planning and control via potential functions , 1989 .

[22]  F. Silva Leite,et al.  Geometry and the Dynamic Interpolation Problem , 1991, 1991 American Control Conference.

[23]  James Biggs,et al.  Geometric Attitude Motion Planning for Spacecraft with Pointing and Actuator Constraints , 2016 .

[24]  Peter E. Crouch,et al.  On the equivalence of higher order variational problems and optimal control problems , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[25]  Jean-Marie Souriau,et al.  On Geometric Mechanics , 2007 .

[26]  Leonardo Colombo,et al.  Second-order constrained variational problems on Lie algebroids: applications to optimal control , 2017, 1701.04772.

[27]  Darryl D. Holm.  Book Review: Geometric Mechanics, Part II: Rotating, Translating and Rolling , 2008 .

[28]  Lyle Noakes,et al.  Cubic Splines on Curved Spaces , 1989 .

[29]  Naomi Ehrich Leonard,et al.  Motion Control of Drift-Free, , 1995 .

[30]  Yuliang Bai,et al.  Attitude guidance and tracking for spacecraft with two reaction wheels , 2018, Int. J. Control.

[31]  Peter E. Crouch,et al.  A Non-Holonomic Dynamic Interpolation Problem , 1991 .

[32]  Vijay Kumar,et al.  On the generation of smooth three-dimensional rigid body motions , 1998, IEEE Trans. Robotics Autom..

[33]  Naomi Ehrich Leonard,et al.  Virtual leaders, artificial potentials and coordinated control of groups , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[34]  A. Bloch,et al.  Nonholonomic and Vakonomic Control Systems on Riemannian Manifolds , 1993 .

[35]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[36]  S. Smale.  On Gradient Dynamical Systems , 1961 .

[37]  Peter E. Crouch,et al.  Elastic Curves as Solutions of Riemannian and Sub-Riemannian Control Problems , 2000, Math. Control. Signals Syst..

[38]  Zhiyong Geng,et al.  Finite-time optimal formation control of multi-agent systems on the Lie group SE(3) , 2013, Int. J. Control.

[39]  M. Rumpf,et al.  Variational time discretization of Riemannian splines , 2017, 1711.06069.

[40]  Naomi Ehrich Leonard,et al.  Motion control of drift-free, left-invariant systems on Lie groups , 1995, IEEE Trans. Autom. Control..

[41]  Leonardo Colombo,et al.  Variational obstacle avoidance problem on riemannian manifolds , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[42]  Oussama Khatib,et al.  Real-Time Obstacle Avoidance for Manipulators and Mobile Robots , 1986 .

[43]  Leonardo Colombo,et al.  Variational collision avoidance problems on Riemannian manifolds , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[44]  N. Perkins,et al.  Nonlinear dynamics and loop formation in Kirchhoff rods with implications to the mechanics of DNA and cables , 2005 .

[45]  Jesse Freeman,et al.  in Morse theory, , 1999 .

[46]  Leonardo Colombo,et al.  Optimal Control Problems with Symmetry Breaking Cost Functions , 2017, SIAM J. Appl. Algebra Geom..

[47]  Darryl D. Holm,et al.  Invariant Higher-Order Variational Problems , 2010, Communications in Mathematical Physics.

[48]  Lyle Noakes,et al.  Elastic helices in simple lie groups , 2014 .

[49]  Claudio Altafini,et al.  REDUCTION BY GROUP SYMMETRY OF SECOND ORDER VARIATIONAL PROBLEMS ON A SEMIDIRECT PRODUCT OF LIE GROUPS WITH POSITIVE DEFINITE RIEMANNIAN METRIC , 2004 .

[50]  A. Bloch,et al.  Nonholonomic Control Systems on Riemannian Manifolds , 1995 .