This paper addresses the problem of designing a trajectory tracking control law for a quadrotor UAV, subsequent to complete failure of a single rotor. The control design problem considers the reduced state space which excludes the angular velocity and orientation about the vertical body axis. The proposed controller enables the quadrotor to track the orientation of this axis, and consequently any prescribed position trajectory using only three rotors. The control design is carried out in two stages. First, in order to track the reduced attitude dynamics, a geometric controller with two input torques is designed on the Lie-Group @math . This is then extended to @math by designing a saturation based feedback law, in order to track the center of mass position with bounded thrust. The control law for the complete dynamics achieves exponential tracking for all initial conditions lying in an open-dense subset. The novelty of the geometric control design is in its ability to effectively execute aggressive, global maneuvers despite complete loss of a rotor. Numerical simulations on models of a variable pitch and a conventional quadrotor have been presented to demonstrate the practical applicability of the control design.
In recent times, globally stabilizing geometric controllers which exploit the intrinsic structure of the underlying manifold, have been developed. Here, singularities due to attitude parameterizations or input-output decoupling are avoided. Approaches such as that in @cite_29 and @cite_11 stabilize mechanical systems on Lie Groups using nonlinear proportional-derivative (PD) control. One such pioneering control design for quadrotors on the Lie Group @math has been presented in @cite_9 and @cite_19 . Reduced attitude stabilization to a fixed point on @math with two control torques is presented in @cite_6 . In @cite_12 , the primary axis of a rigid body on @math as well as the angular velocity about this axis, are tracked using three independent torques. In @cite_3 , global reduced attitude tracking with three torques is achieved by constructing a synergistic family of potential functions on @math . To the best of our knowledge, none of the above mentioned control laws are suitable for trajectory tracking with three functioning rotors (i.e. two torque inputs and a net thrust).
@cite_9 @cite_29 @cite_6 @cite_3 @cite_19 @cite_12 @cite_11
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This paper provides new results for the tracking control of a quadrotor unmanned aerial vehicle (UAV). The UAV has four input degrees of freedom, namely the magnitudes of the four rotor thrusts, that are used to control the six translational and rotational degrees of freedom, and to achieve asymptotic tracking of four outputs, namely, three position variables for the vehicle center of mass and the direction of one vehicle body-fixed axis. A globally defined model of the quadrotor UAV rigid body dynamics is introduced as a basis for the analysis. A nonlinear tracking controller is developed on the special Euclidean group SE(3) and it is shown to have desirable closed loop properties that are almost global. Several numerical examples, including an example in which the quadrotor recovers from being initially upside down, illustrate the versatility of the controller. We present a general intrinsic tracking controller design for fully-actuated simple mechanical systems, when the configuration space is one of a general class of Lie groups. We first express a state-feedback controller in terms of a function-the "error function"-satisfying certain regularity conditions. If an error function can be found, then a general smooth and bounded reference trajectory may be tracked asymptotically from almost every initial condition, with locally exponential convergence. Asymptotic convergence from almost every initial condition is referred to as "almost-global" asymptotic stability. Error functions may be shown to exist on any compact Lie group, or any Lie group diffeomorphic to the product of a compact Lie group and R sup n . This covers many cases of practical interest, such as SO(n), SE(n), their subgroups, and direct products. We show here that for compact Lie groups the dynamic configuration-feedback controller obtained by composing the full state-feedback law with an exponentially convergent velocity observer is also almost-globally asymptotically stable with respect to the tracking error. We emphasize that no invariance is needed for these results. However, for the special case where the kinetic energy is left-invariant, we show that the explicit expression of these controllers does not require coordinates on the Lie group. The controller constructions are demonstrated on SO(3), and simulated for the axi-symmetric top. Results show excellent performance. This paper focuses on a new geometric approach to (fully actuated) control systems on the sphere. Our control laws exploit the basic and intuitive notions of geodesic direction and of distance between points, and generalize the classical proportional plus derivative feedback (PD) without the need of arbitrary local coordinate charts. The stability analysis relies on an appropriate Lyapunov function, where the notion of distance and its properties are exploited. This methodology then applies to spin-axis stabilization of a spacecraft actuated by only two control torques: discarding the rotation about the unactuated axis, a reduced system is considered, whose state is in fact defined on the sphere. For this reduced stabilization problem our approach allows us not only to deal optimally with the inevitable singularity, but also to achieve simplicity, versatility and (coordinate independent) adaptive capabilities. Abstract In this paper, we propose a hybrid feedback based on a “synergistic” potential function that achieves global asymptotic stabilization of a desired orientation on the n -sphere with a nominal robustness to measurement disturbances, a task that is not possible by classical feedback–be it smooth, nonsmooth, periodic, or any combination thereof–due to the topological structure of the sphere. We extend this basic result to a tracking controller for the reduced attitude–or pointing direction–of a rigid body and provide a method to remove jumps in the controlled torque by backstepping. The proposed hybrid feedback is compared with a similar smooth feedback in simulation, where it is illustrated that the hybrid feedback overcomes performance limitations inherent to the smooth feedback. We provide two examples of a synergistic potential function–one defined on a general sphere and the other on the unit circle. The control task of tracking a reference pointing direction (the attitude about the pointing direction is irrelevant) while obtaining a desired angular velocity (PDAV) around the pointing direction using geometric techniques is addressed here. Existing geometric controllers developed on the two-sphere only address the tracking of a reference pointing direction while driving the angular velocity about the pointing direction to zero. In this paper a tracking controller on the two-sphere, able to address the PDAV control task, is developed globally in a geometric frame work, to avoid problems related to other attitude representations such as unwinding (quaternions) or singularities (Euler angles). An attitude error function is constructed resulting in a control system with desired tracking performance for rotational maneuvers with large initial attitude angular velocity errors and the ability to negotiate bounded modeling inaccuracies. The tracking ability of the developed control system is evaluated by comparing its performance with an existing geometric controller on the two-sphere and by numerical simulations, showing improved performance for large initial attitude errors, smooth transitions between desired angular velocities and the ability to negotiate bounded modeling inaccuracies. This talk will outline a comprehensive set of modeling, analysis and design techniques for a class of mechanical systems. We concern ourselves with simple mechanical control systems, that is, systems whose Lagrangian is kinetic energy minus potential energy. Example devices include robotic manipulators, aerospace and underwater vehicles, and mechanisms that locomote exploiting nonholonomic constraints. Borrowing techniques from nonlinear control and geometric mechanics, we propose a coordinateinvariant control theory for this class of systems. First, we take a Riemannian geometric approach to modeling systems dened on smooth manifolds, subject to nonholonomic constraints, external forces and control forces. We also model mechanical systems on groups and symmetries. Second, we analyze some control-theoretic properties of this class of systems, including controllability, averaged response to oscillatory controls, and kinematic reductions. Finally, we exploit the modeling and analysis results to tackle control design problems. Starting from controllability and kinematic reduction assumptions we propose some algorithms for generating and tracking trajectories.