Constrained motion planning for multiple vehicles on SE(3)

Abstract

Abstract-This paper proposes a computational method to solve constrained cooperative motion planning problems for multiple vehicles undergoing translational and rotational motions. The problem is solved by means of the Lie group projection operator approach, a recently developed optimization strategy for solving continuous-time optimal control problems on Lie groups. State constraints (for collision avoidance) are handled by means of a set of barrier functions, turning the optimization approach into an interior point method. A sample computation is shown to demonstrate the effectiveness of the method.

Cooperative Motion Planning for Multiple Autonomous Marine Vehicles

“…that behaves as the standard barrier functionβ δ for small and negative z, but goes to zero for z → ∞ (Saccon et al, 2012).…”

Covariant differentiation of a map in the context of geometric optimal control

“…Modifications of the strategy for handling a terminal condition and mixed input-state constraints (through a barrier function approach) are discussed, for control problems on R n , in Hauser (2003) and Hauser and Saccon (2006). A constrained optimal control problem on Lie groups is discussed in Saccon et al (2012).…”

Fourier Transform on the Homogeneous Space of 3D Positions and Orientations for Exact Solutions to PDEs

“…Let U be in the domain of the generator Q α given by 7, of our evolution (6). For a formal definition of this domain we refer to [77,Eq.…”

“…Indeed separation of variables (also known as 'the Fourier method') directly provides a Sturm-Liouville problem [1] and an orthonormal basis of eigenfunctions for L 2 (Ω), which is complete due to compactness of the associated self-adjoint kernel operator. When dilating the subset Ω to the full space R d , the discrete set of eigenvalues start to fill R Nowadays, in fields such as mechanics/robotics [4][5][6][7], mathematical physics/harmonic analysis [8], machine learning [9][10][11][12][13] and image analysis [14][15][16][17][18][19] it is worthwhile to extend the spatial domain of functions on M = R d (or M = Z d ) to groups G = M T that are the semi-direct product of an Abelian group M and another matrix group T. This requires a generalization of the Fourier transforms on the Lie group (R d , +) towards the groups G = R d T. Then the Fourier transform gives rise to an invertible decomposition of a square integrable function into irreducible representations. This is a powerful mechanism in view of the Schur's lemma [20,21] and spectral decompositions [22,23].…”