Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993 THRUSTER MODELLING AND ENERGY-OPTIMIZATION FOR AN AUTONOMOUS UNDERWATER VEHICLE I. Spangelo and O. Egeland Department 0/ Engineering Cybernetics, The Norwegian institUle o/Technology, University o/Trondheim, N-7034 Trondheim, Norway Abstract: Energy-optimal trajectories for an autonomous underwater vehicle are investi- gated using control vector parameterization and single shooting. The vehicle has 6 propellers driven by DC-motors. Optimal trajectories are calculated using three different thruster mod- els. The results show that in order to achieve valid trajectories the thrust coefficient's linear dependence on the vehicle velocity has to be modelled. Optimal performance index as a function of time consumption is shown. Key Words: Underseas vehicles; Thruster models; Optimal Control; Nonlinear Program- ming; Numerical methods. 1 INTRODUCTION The use of energy-optimization can extend the period of operation for an autonomous battery powered underwater vehicle significantly. In or- der for the resulting trajectories to be valid the model used in the computations has to be ade- quate. The thruster characteristics for the screw pro- peller are highly nonlinear and depends on the vehicle velocity as well as the velocity of the pro- peller (Todd and Taylor, 1967). However, for pos- itive advance ratio the thrust coefficient decreases approximately linearly with increasing advance ratio. Blanke (1982) explains this theoretically for a propeller in an ideal fluid. Experimental results for the thrusters of the Nor- wegian Experimental Remotely Operated Vehi- cle (NEROV) (Sagatun and Fossen, 1990) devel- oped at the Norwegian Institute of Technology confirmed this linear behaviour for positive ad- vance ratio. For negative advance ratio, however the highly nonlinear effects seemed to dominate. In Spangelo and Egeland (1992a,b) optimal tra- jectories for the NEROV vehicle were calculated using a constant thrust coefficient, that is the thrust force was proportional to the square of the propeller velocity. In this work energy-optimization using the sim- ple thruster model is compared to two more com- plex thruster models involving the vehicle veloc- ity. Numerical calculations were performed using the trajectory-optimization program PROMIS (Jansch and Schnepper, 1991) developed at the German Aerospace Research Establishment, us- ing control vector parameterization and nonlinear programming with direct shooting. 1005 2 MATHEMATICAL MODELS 2.1 Vehicle Model The equations of motion for a 6 DOF underwater vehicle can be written (Fossen, 1991): Mv + C{v)v + D{v)v + 1'(e) = B(v)T (1) Here M is the 6 X 6 inertia matrix contain- ing vehicle inertia and hydrodynamic added inertia. The six dimensional vector v = (u v w p q T) T is the velocity in vehi- cle coordinates where u, v, and ware the linear velocities in the vehicle x, y and z directions, and p, q and T are the angular velocities about the vehicle x, y and z axes. C is a 6 x 6 matrix so that Cv is the vector of Coriolis and centripetal forces due to vehicle and hydrodynamic added in- ertia. D is a 6 x 6 matrix containing coefficients describing dissipative hydrodynamic terms. 1'{e) is the 6 dimensional vector of restoring forces and moments caused by gravity and buoyancy. B is the 6 x 6 input matrix, and T is a 6 dimensional vector of thruster forces. The vector e = (x y z </> 8 1/1) T denotes the global position where x, y and z are the posi- tion coordinates and </>, 8 and 1/1 are the roll, pitch and yaw angles. Accordingly (2) The Jacobian J{e) can be found in e.g. Spong and Vidyasagar (1989, p. 46). It is assumed that 181 < i so that J is bounded. The system can be described in state space form by defining the 12 dimensional state vector (3)