Geometric control for autonomous underwater vehicles: overcoming a thruster failure Michael Andonian, Dario Cazzaro, Luca Invernizzi, Monique Chyba, Sergio Grammatico Abstract— The goal of this paper is to show how geometric control theory can be used to design efficient trajectories for an autonomous underwater vehicle descending into a basin, as well as performing its recovery after experiencing an actuator failure. The underwater vehicle is modeled as a forced affine connection control system, and the control strategies are developed through the use of integral curves of rank one and kinematic reductions. Such a method is particularly efficient in case of actuator failure and it provides a constructive way to design trajectories for the new under-actuated system. A typical scenario of basin descent is presented, control signals are computed to realize the desired trajectories and some simulations are provided. I NTRODUCTION In recent years, Autonomous Underwater Vehicles (AUVs) have allowed researchers to explore underwater environments too hostile and dangerous for man or manned vehicles. As technology exponentially advances, the sophistication of AUVs grows as well. Such projects as the NASA funded Deep Phreatic THermal eXplorer (DEPTHX) and its second generation, the Environmentally Non-Disturbing Under-ice Robotic ANtarctiC Explorer (ENDURANCE) are examples of such state-of-the-art AUVs. Both projects involved de- ploying an AUV to survey, exploiting Simultaneous Local- ization And Mapping techniques (SLAM), the underwater environment of Lake Bonney in Antarctica (ENDURANCE) and a group of five sinkholes of Sistema Zacatn (DEPTHX), in preparation and anticipation for the opportunity to explore Europa, a moon of Jupiter, searching for life in its icy oceans [1]. Other approaches to explore and study hostile underwater environments include strategies involving teams of AUVs to survey hydrothermal vents [2], by using sonar sensors to create maps, eventually to photograph hydrothermal vent sites [3], and by predetermining trajectories for the AUV to follow while sampling the water to choose a viable site to study [4]. For a long-duration mission, during which the AUV does not have the possibility to recharge its batteries, it is absolutely critical to take the energy demands of the vehicle into consideration [5]. Moreover, the fact remains that the environments AUVs have to explore are hazardous to the vehicles as well. Precautionary techniques are therefore M. Andonian, M. Chyba are with the Mathematics Department, College of Natural Sciences, University of Hawai’i, Honolulu, HI 96822, USA. {andonian, mchyba}@math.hawaii.edu D. Cazzaro, L. Invernizzi are with the College of Engineering, Sant’Anna School of Advanced Studies, Pisa, PI, 56127, Italy. {d.cazzaro, l.invernizzi}@sssup.it S. Grammatico is with the Department of Electrical Systems and Au- tomation, College of Engineering, University of Pisa, PI, 56127, Italy. s.grammatico@dsea.unipi.it implemented to help the vehicle to return to its starting point safely and intact, in case of unexpected damages during the mission. Even so, some works have placed emphasis on fault-accommodating allocation of thruster forces on AUVs, for instance exploiting the excess number of thrusters to tolerate some faults during operation, see [6], [7] as survey references. Our work differs from the previously cited ones in several aspect. First it uses differential geometric techniques to exploit the fact that autonomous underwater vehicles can be modeled as forced affine connection control system. Second, our work incorporates under-actuated scenarios for which the actuator failure implies an actual reduction of the available degrees of freedom (DOF). In those cases more sophisticated techniques than thruster’s reallocation is required. Finally our method do not rely heavily on the sensors of the vehicle as it is often the case. More precisely, the designed mission for the AUV is to descent into a basin, map as accurately as possible the walls of the cave and find its way back after an actuator failure occurred. Since the AUV has to map the walls of an underwater cave, performing efficient trajectories that trade off between the magnitude of the mapped area and the minimization of the energy consumption is a priority. This is a complex optimal control problem since the cost function cannot be expressed in terms of the control function or the trajectory exclusively. We here design trajectories using the geometric framework to produce simple motions that can be implemented on a real vehicle and that are efficient with respect to the goals of the mission. In addition, due to the fact that dead reckoning using vehicle state estimation generally tends to be very inaccurate, this geometric approach provides accurate state estimations with no instruments. This strategy is also particularly useful when mapping a new environment. With satisfactory knowledge of the environment, the vehicle can implement this geometric control strategy to be more efficient and effective when exploring its environment. Therefore, a coupling of this control strategy with other localization and mapping methods would result in a more robust overall strategy. The presented geometric control is an open-loop control strategy, therefore its validity is mainly theoretical, because of the unavoidable presence of unmodeled dynamics and external disturbances. A practical implementation of the proposed control law is indeed limited to the case of adequate knowledge of the environment. The layout of the paper is as follows. The equations of motion of a submerged rigid body are developed from a differential geometry perspective in Section I. The resulting dynamic equations are then used to compute the control 49th IEEE Conference on Decision and Control December 15-17, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA 978-1-4244-7746-3/10/$26.00 ©2010 IEEE 7051