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# Communications in Mathematical Sciences

## Volume 14 (2016)

### Number 1

### Bi-Jacobi fields and Riemannian cubics for left-invariant $SO(3)$

Pages: 55 – 68

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n1.a3

#### Authors

#### Abstract

Bi-Jacobi fields are generalized Jacobi fields, and are used to efficiently compute approximations to Riemannian cubic splines in a Riemannian manifold $M$. Calculating bi-Jacobi fields is straightforward when $M$ is a symmetric space such as bi-invariant $SO(3)$, but not for Lie groups whose Riemannian metric is only left-invariant. Because left-invariant Riemannian metrics occur naturally in applications, there is also a need to calculate bi-Jacobi fields in such cases. The present paper investigates bi-Jacobi fields for left-invariant Riemannian metrics on $SO(3)$, reducing calculations to quadratures of Jacobi fields. Then left-Lie reductions are used to give an easily implemented numerical method for calculating bi-Jacobi fields along geodesics in $SO(3)$, and an example is given of a nearly geodesic approximate Riemannian cubic.

#### Keywords

Lie group, Riemannian manifold, Jacobi field, trajectory planning, mechanical system, rigid body, nonlinear optimal control, Riemannian cubic

#### 2010 Mathematics Subject Classification

Primary 34E05, 49K99, 53A99, 70E17. Secondary 34H05, 49S05, 70E60.

Published 16 September 2015