Keenan Crane
Lie Group Integrators for Animation and Control of Vehicles
ACM Transactions on Graphics 2009
Marin Kobilarov Keenan Crane Mathieu Desbrun
This paper is concerned with the animation and control of vehicles with complex dynamics such as helicopters, boats, and cars. Motivated by recent developments in discrete geometric mechanics we develop a general framework for integrating the dynamics of holonomic and nonholonomic vehicles by preserving their state-space geometry and motion invariants. We demonstrate that the resulting integration schemes are superior to standard methods in numerical robustness and efficiency, and can be applied to many types of vehicles. In addition, we show how to use this framework in an optimal control setting to automatically compute accurate and realistic motions for arbitrary user-specified constraints.
Video - Snakeboard
The authors wish to thank Jerrold E. Marsden for inspiration and guidance, Gaurav Sukhatme for his assistance and support, and the reviewers for their feedback. This research project was partially funded by the NSF (ITR DMS-043145, CCF-0811373, and CMMI-0757106), DOE (DE-FG02-04ER25657), the Caltech IST Center for the Mathematics of Information, and Pixar Animation Studios.
Video - Helicopter
Video - Optimal Control
The following books provide useful introductions / definitions / background material.
Manifolds, Tensor Analysis, and Applications, Ralph Abraham, Jerrold Marsden and Tudor Ratiu
Introduction to Mechanics and Symmetry, Jerrold Marsden and Tudor Ratiu
Video - Boat
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@article{Kobilarov:2009:LGI, author = {Kobilarov, Marin and Crane, Keenan and Desbrun, Mathieu}, title = {Lie group integrators for animation and control of vehicles}, journal = {ACM Trans. Graph.}, volume = {28}, issue = {2}, month = {May}, year = {2009}, }
In this animation generated with our integrator for holonomic systems, a helicopter is manipulated through a joystick.
Our vehicle integrator can easily be coupled with standard simulation methods for other physical phenomena. Here a holonomic boat with rear thrusters interacts with a turbulent free-surface flow.
Optimized trajectories of a helicopter flying through in a complex canyon (left), and automatically landing on the mark (right).