Previous |  Up |  Next


Darboux polynomials; drag power; Euler–Lagrange equations; grading; integrability; vector fields
This paper deals with integrability issues of the Euler-Lagrange equations associated to a variational problem, where the energy function depends on acceleration and drag. Although the motivation came from applications to path planning of underwater robot manipulators, the approach is rather theoretical and the main difficulties result from the fact that the power needed to push an object through a fluid increases as the cube of its speed.
[1] Crouch, P., Leite, F. Silva: The dynamic interpolation problem: on Riemannian manifolds, Lie groups and symmetric spaces. J. Dynam. Control Systems 1 (1995), 2, 177-202. DOI 10.1007/bf02254638 | MR 1333770
[2] Camarinha, M., Leite, F. Silva, Crouch, P.: On the geometry of Riemannian cubic polynomials. Differential Geom. Appl. 15 (2001), 15, 107-135. DOI 10.1016/s0926-2245(01)00054-7 | MR 1857558
[3] Goriely, A.: Integrability and Nonintegrability of Dynamical Systems. Advanced Series in Nonlinear Dynamics 19, World Scientific Publishing Co., Inc., River Edge, NJ 2001. DOI 10.1142/9789812811943 | MR 1857742 | Zbl 1002.34001
[4] Hartman, P.: Ordinary Differential Equations. John Wiley and Sons Inc., New York 1964. MR 0171038 | Zbl 1009.34001
[5] Häusler, A. J., Saccon, A., Aguiar, A. P., Hauser, J., Pascoal, A.: Cooperative motion planning for multiple autonomous marine vehicles. In: Proc. 9th IFAC Conference on Manoeuvring and Control of Marine Craft 2012 (G. Bruzzone, ed.), International Federation of Automatic Control, 2012, pp. 244-249. DOI 10.3182/20120919-3-it-2046.00042
[6] Kruger, D., Stolkin, R., Blum, A., Briganti, J.: Optimal AUV path planning for extended missions in complex, fast-flowing estuarine environments. In: ICRA'07, IEEE 2007, pp. 4265-4270. DOI 10.1109/robot.2007.364135
[7] Noakes, L., Heinzinger, G., Paden, B.: Cubic splines on curved spaces. IMA J. Math. Control Inform. 6 (1989), 4, 465-473. DOI 10.1093/imamci/6.4.465 | MR 1036158 | Zbl 0698.58018
Partner of
EuDML logo