This article presents the development, implementation, and comparison of two trajectory tracking nonlinear controllers for underactuated surface vessels. A control approach capable of stabilizing all the states of any planar vehicle is specifically adapted to surface vessels. The method relies on transformation of the six position and velocity state dynamics into a four-state error dynamics. The backstepping and sliding mode control laws are then derived for stabilization of the error dynamics and proven to stabilize all system states. Simulations are presented in the absence and presence of modeling uncertainties and unknown disturbances. An experimental setup is then described, followed by successful experimental implementation and comparison of the two controllers.

Summary This paper presents the derivation of robust trajectory‐tracking nonlinear control laws for general three‐dimensional vehicle models with one degree of underactuation where all of the state tracking errors are stabilized. The method is based on a novel transformation of the trajectory tracking problem into a reduced‐order error dynamics. Two traditional nonlinear controllers based on sliding mode and backstepping approaches are developed and shown to stabilize the trajectory tracking errors in presence of modeling uncertainties and bounded disturbances. The performance of the two controllers are compared in absence and presence of disturbances.

A common approach to control of underactuated vehicles is a prelude to cooperative control of heterogeneous networks. This article presents a novel nonlinear trajectory tracking control framework for general three-dimensional models of underactuated vehicles with twelve states and four control inputs, including underwater, air, and space vehicles. Given a desired reference trajectory for four of the six degrees of freedom, feasible state trajectories are generated using the second-order nonholonomic constraints in the vehicle equations of motion. A novel transformation is then introduced to formulate the error dynamics in a simplified form. The error dynamics is stabilized using a traditional nonlinear control approach. The control law is shown to uniformly asymptotically stabilize all six pose states assuming bounded unknown uncertainties and disturbances. Adaptation of the method to underwater vehicles is presented along with simulation under highly nonlinear conditions. The approach is also applied to quadrotors and both simulation and experimental results are presented. K E Y W O R D Sbackstepping control, trajectory tracking control, underactuated vehicles INTRODUCTIONAutonomous vehicles have become more practical and thus more prevalent as a consequence of sensors, computers, and power electronics becoming better, cheaper, and smaller. However, controlling the motion of such vehicles is complex, as they tend to be underactuated, that is, fewer actuators than degrees of freedom. These include cars, surface vessels, underwater vessels, and air vehicles. In one of our previous works 1 we considered developing a control algorithm that can be applied to any planar underactuated vehicle. We then extended that work to Spatial vehicle models but with only one degree of underactuation. 2 In this work, we attempt to develop a control algorithm applicable to a variety of spatial underactuated vehicles with two degrees of underactuation including underwater vessels and air vehicles.Considering underwater vehicles, a submarine's propeller produces a surge force along the axis of motion; this is the translational actuator. There are also rotational actuators. A rudder induces yaw motion and diving planes induce pitching motion. 3 Roll motion, however, is a bit more complicated. Many manned submarines are designed to be passively self-righting and therefore do not have active roll control, with some exceptions where roll motion is desired, such as the DeepFlight Super Falcon. 4 There are several works that ignore the roll motion or at least its coupling with the other degrees of freedom and use five degree-of-freedom (DOF) models. [5][6][7] These methods concentrate on uncertainty and disturbance rejection as well as other important topics such as actuation limits and velocity estimation. 8

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