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A hybrid feedback controller for robust global trajectory tracking of quadrotor-like vehicles with minimized attitude error
Abstract
In this paper, we tackle the problem of trajectory tracking for a particular class of underactuated vehicles with full torque actuation and a single force direction (thrust) that is fixed relative to a body attached frame. Additionally, we consider that thrust reversal is not available. We present the design of a hybrid controller that, under some given assumptions, is able to globally asymptotically stabilize the vehicle to a reference position trajectory while minimizing the angle to the desired attitude trajectory. This objective is achieved robustly and globally, in the sense that small perturbations do not lead to instability and it is achieved regardless of the initial state of the vehicle. The algorithm is tested in a experimental setup, using a small scale quadrotor vehicle and the VICON motion capture system.
... Recent progress in the development of stability and control theory for hybrid dynamical systems has led to a new framework [8]. In [6], a hybrid feedback control was proposed to trajectory tracking of quadrotor-like vehicles. In addition, the fault detection problem of discretetime hybrid systems was addressed in [30]. ...
In this paper, a hybrid control strategy using appropriate switching between a set of linear controllers is presented for stabilization of a class of hybrid systems with input constraints. The input constraints under consideration are novel reverse polytopic constraints avoiding a set of specified input values. The design procedure of the switching controllers is performed by solving a set of linear matrix inequalities for the full state feedback case and by solving a set of bilinear matrix inequalities for the output feedback case. Under the designed controllers, the closed-loop system is shown to be globally uniformly pre-asymptotically stable if a specific set of matrix inequalities ensuring the existence of a common Lyapunov function are satisfied. To avoid the Zeno behavior, a tunable parameter related to the controller gains is introduced and assessed. It is shown that the proposed switching control is superior to continuous state feedback control with input constraints. In addition, besides continuous linear systems, the proposed control strategy is applicable to hybrid or impulsive systems. Numerical examples are included to illustrate the proposed results.
The paper surveys recent advances in underwater vehicle navigation and identifies future research challenges. Improvements in underwater navigation sensor technology and underwater navigation algorithms are enabling novel underwater vehicles and novel underwater vehicle missions. This paper first reviews advances in underwater navigation sensor technology. Second, advances in deterministic and stochastic underwater navigation methodologies and algorithms are reviewed. Finally, future challenges in underwater vehicle navigation are articulated, including near-bottom navigation, vehicle state estimation, optimal survey, environmental estimation, multiple-vehicle navigation, and mid-water navigation. Advances in vehicle navigation will enable new missions for underwater vehicle (commercial, scientific, and military) which were previously considered impractical or infeasible.
A control approach is proposed for a class of underactuated vehicles in order to stabilize reference trajectories either in thrust direction, velocity, or position. The basic modeling assumption is that the vehicle is pro-pulsed via a thrust force along a single body-fixed direction and that it has full torque actuation for attitude control (i.e., a typical actuation structure for aircrafts, vertical take-off and landing (VTOL) vehicles, submarines, etc.). Additional assumptions on the external forces applied to the vehicle are also introduced for the sake of control design and stability analyses. They are best satisfied for vehicles which are subjected to an external force field (e.g., gravity) and whose shape induces lift forces with limited amplitude, unlike airplanes but as in the case of many VTOL drones. The interactions of the vehicle with the surrounding fluid are often difficult to model precisely whereas they may significantly influence and perturb its motion. By using a standard Lyapunov-based approach, novel nonlinear feedback control laws are proposed to compensate for modeling errors and perform robustly against such perturbations. Simulation results illustrating these properties on a realistic model of a VTOL drone subjected to wind gusts are reported.
In this paper, a two step path-planning algorithm for UAVs is
proposed. The algorithm generates a stealthy path through a set of enemy
radar sites of known location, and provides an intuitive way to
trade-off stealth versus path length. In the first step, a suboptimal
rough-cut path is generated through the radar sites by constructing and
searching a graph based on Voronoi polygons. In the second step, a set
of nonlinear ordinary differential equations are simulated, using the
graph solution as an initial condition. The ODEs describe the dynamics
of a set of virtual masses located in a virtual force field. The virtual
forces push the masses away from the radar and toward one another. The
ODEs are simulated to find a locally, exponentially stable equilibrium
solution, which is interpreted as the optimal path. A simulation is
provided to illustrate the idea
This paper proposes a new methodology for integrated design of
guidance and control systems for autonomous underwater vehicles (AUVs).
The methodology developed leads to a systematic procedure for the design
of controllers for AUVs to accurately track reference trajectories
defined in an inertial reference frame. The paper illustrates the
application of this procedure to the design of a tracking controller for
the AUV MARIUS. The design phase is summarized, and the performance of
the resulting controller is assessed in simulation using dynamic models
of the vehicle and its sensor suite
Nonlinear tracking control of unmanned underwater vehicles (UUVs)
in 6 degrees of freedom (DOF) is discussed. The 4-parameter unit
quaternion is used in a singularity-free representation of attitude.
Several control laws based on a generalized Lyapunov function for the
attitude dynamics are derived, including feedback from the vector
quaternion, the Euler rotation vector and the Rodrigues parameter
vector. A new feedback gain matrix for translational motion is also
proposed. Lyapunov analysis is used in the tracking error convergence
analysis. Extensions to adaptive control are made. The control laws are
tested in a simulation study
We address the problem of position trajectory-tracking and path-following control design for underactuated autonomous vehicles in the presence of possibly large modeling parametric uncertainty. For a general class of vehicles moving in either 2- or 3-D space, we demonstrate how adaptive switching supervisory control can be combined with a nonlinear Lyapunov-based tracking control law to solve the problem of global boundedness and convergence of the position tracking error to a neighborhood of the origin that can be made arbitrarily small. The desired trajectory does not need to be of a particular type (e.g., trimming trajectories) and can be any sufficiently smooth bounded curve parameterized by time. We also show how these results can be applied to solve the path-following problem, in which the vehicle is required to converge to and follow a path, without a specific temporal specification. We illustrate our design procedures through two vehicle control applications: a hovercraft (moving on a planar surface) and an underwater vehicle (moving in 3-D space). Simulations results are presented and discussed.
Hybrid dynamical systems exhibit continuous and instantaneous changes, having features of continuous-time and discrete-time dynamical systems. Filled with a wealth of examples to illustrate concepts, this book presents a complete theory of robust asymptotic stability for hybrid dynamical systems that is applicable to the design of hybrid control algorithms--algorithms that feature logic, timers, or combinations of digital and analog components. With the tools of modern mathematical analysis,Hybrid Dynamical Systemsunifies and generalizes earlier developments in continuous-time and discrete-time nonlinear systems. It presents hybrid system versions of the necessary and sufficient Lyapunov conditions for asymptotic stability, invariance principles, and approximation techniques, and examines the robustness of asymptotic stability, motivated by the goal of designing robust hybrid control algorithms. This self-contained and classroom-tested book requires standard background in mathematical analysis and differential equations or nonlinear systems. It will interest graduate students in engineering as well as students and researchers in control, computer science, and mathematics.
The unit quaternion is a pervasive representation of rigid-body attitude used for the design and analysis of feedback control laws. Because the space of unit quaternions constitutes a double cover of the rigid-body-attitude space, quaternion-based control laws are often—by design—inconsistent, i.e., they do not have a unique value for each rigid-body attitude. Inconsistent quaternion-based control laws require an additional mechanism that uniquely converts an attitude estimate into its quaternion representation; however, conversion mechanisms that are memoryless—e.g., selecting the quaternion having positive scalar component—have a limited domain where they remain injective and, when used globally, introduce discontinuities into the closed-loop system. We show—through an explicit construction and Lyapunov analysis—that such discontinuities can be hijacked by arbitrarily small measurement disturbances to stabilize attitudes far from the desired attitude. To remedy this limitation, we propose a hybrid-dynamic algorithm for smoothly lifting an attitude path to the unit-quaternion space. We show that this hybrid-dynamic mechanism allows us to directly translate quaternion-based controllers and their asymptotic stability properties (obtained in the unit-quaternion space) to the actual rigid-body-attitude space. We also show that when quaternion-based controllers are not designed to account for the double covering of the rigid-body-attitude space by a unit-quaternion parameterization, they can give rise to the unwinding phenomenon, which we characterize in terms of the projection of asymptotically stable sets. Finally, we employ the main results to show that certain hybrid feedbacks can globally asymptotically stabilize the attitude of a rigid body.
This paper addresses the design and experimental evaluation of a globally stabilizing controller to steer a quadrotor vehicle along a predefined path. The proposed solution consists of a nonlinear adaptive state feedback controller for thrust and angular velocity actuation that i) guarantees convergence of the closed-loop path following error to zero in the presence of constant force disturbances and ii) ensures that the actuation does not grow unbounded as a function of the position error. A prototyping and testing architecture, developed to streamline the implementation and the tuning of the controller, is also described. Experimental results, where the quadrotor is subject to external wind disturbances, are presented to demonstrate the performance and robustness of the proposed controller.
In this paper, we address the problem of trajectory tracking for a class of underactuated vehicles with full torque actuation and only one dimensional force actuation (thrust). For this class of vehicles, the desired thrust is defined by a saturated control law that achieves global asymptotic stabilization of the position tracking error. The proposed control law also assures that the third component of the angular velocity is regulated to zero. To accomplish this task we propose a hybrid controller that is designed using backstepping techniques and recent developments on synergistic Lyapunov functions. Simulations validating the results are also provided.
During the course of the last year, a number of very exciting developments have arisen concerning the use of unmanned aerial vehicles (UAVs). As a strengthening link in the overall architecture of reconnaissance assets, or sometimes-called ‘system of systems’, UAVs have arrived and promise to expand well beyond their predominately tactical military use of today.
This classic book is the first comprehensive presentation of data, theory, and practice in attitude analysis. It was written by 33 senior technical staff members in the Spacecraft Attitude Department of the Computer Sciences Corporation and incorporates their experience in supporting more than 30 space missions. Because of the extensive cross-references, complete index, and 13 technical appendices, this book can be either a self teaching text, or a reference handbook. Among its unique features are orthographic globe projections to eliminate confusion in vector drawings; discussions of common data anomalies, data validation, attitude hardware, and associated mathematical models; and a presentation of new geometrical procedures for mission analysis and attitude accuracy studies which can eliminate many complex simulations.
This paper addresses the problem of steering a quadrotor vehicle along a prescribed trajectory. The problem is formulated so as to enforce bounds on the thrust actuation and present robustness against possible disturbances resulting from errors in the estimation of the aircraft state. The proposed solution consists of a nonlinear state feedback controller for thrust and torque actuation that i) guarantees asymptotic stability of the closed-loop system in the presence of exponentially decaying observer errors, ii) ensures global regulation of the trajectory tracking errors, and iii) defines the thrust actuation as a saturated function of the position and linear velocity errors. Simulation results are presented to assess the performance and robustness of the proposed controller.
In this paper, we present a new design method for the flight control of an autonomous quadrotor helicopter based on sliding mode control. Due to the under-actuated property of a quadrotor helicopter, the controller can make the helicopter move three positions (x, y, z) and the yaw angle to their desired values and stabilize the pitch and roll angles. A sliding mode control is proposed to stabilize a class of cascaded under-actuated systems. The global stability analysis of the closed-loop system is presented. The advantage of sliding mode control is its insensitivity to the model errors, parametric uncertainties and other disturbances. Simulations show that the control law robustly stabilizes a quadrotor
This paper treats the practical and challenging control problem of tracking a prescribed continuous trajectory for an autonomous underwater vehicle immersed in the presence of gravity, buoyancy, hydrodynamic and other uncertain forces and moments. These uncertain forces and moments are bounded and may be difficult to model accurately, but they act persistently or over long periods of time. The trajectory is specified in terms of desired attitude and translational motion for a rigid body model of the vehicle. For an autonomous underwater vehicle (AUV) in a dynamic environment like the ocean, the presence of changing ocean currents create hydrodynamic forces and moments that are not well-known or predictable, even though they are bounded. In the absence of such forces and moments, it is possible to model the AUV dynamics very accurately and use the model for trajectory generation and tracking, as has been shown in prior research. However, the presence of uncertainties makes it difficult to accurately model the dynamics. This in turn makes the control task of tracking a desired or prescribed trajectory very challenging as an accurate model of the system dynamics is not available. We develop a robust feedback tracking scheme for autonomous underwater vehicles that can track a prescribed trajectory, while rejecting the effects of disturbances due to poorly known inputs with a simple yet general internal model.
Robust stability and control for systems that combine continuous-time and discrete-time dynamics. This article is a tutorial on modeling the dynamics of hybrid systems, on the elements of stability theory for hybrid systems, and on the basics of hybrid control. The presentation and selection of material is oriented toward the analysis of asymptotic stability in hybrid systems and the design of stabilizing hybrid controllers. Our emphasis on the robustness of asymptotic stability to data perturbation, external disturbances, and measurement error distinguishes the approach taken here from other approaches to hybrid systems. While we make some connections to alternative approaches, this article does not aspire to be a survey of the hybrid system literature, which is vast and multifaceted.
We address the controller design and the trajectory generation for a quadrotor maneuvering in three dimensions in a tightly constrained setting typical of indoor environments. In such settings, it is necessary to allow for significant excursions of the attitude from the hover state and small angle approximations cannot be justified for the roll and pitch. We develop an algorithm that enables the real-time generation of optimal trajectories through a sequence of 3-D positions and yaw angles, while ensuring safe passage through specified corridors and satisfying constraints on velocities, accelerations and inputs. A nonlinear controller ensures the faithful tracking of these trajectories. Experimental results illustrate the application of the method to fast motion (5-10 body lengths/second) in three-dimensional slalom courses.
It is well known that controlling the attitude of a rigid body is subject to topological constraints. We illustrate, with examples, the problems that arise when using continuous and (memoryless) discontinuous quaternion-based state-feedback control laws for global attitude stabilization. We propose a quaternion-based hybrid feedback scheme that solves the global attitude tracking problem in three scenarios: full state measurements, only measurements of attitude, and measurements of attitude with angular velocity measurements corrupted by a constant bias. In each case, the hybrid feedback is dynamic and incorporates hysteresis-based switching using a single binary logic variable for each quaternion error state. When only attitude measurements are available or the angular rate is corrupted by a constant bias, the proposed controller is observer-based and incorporates an additional quaternion filter and bias observer. The hysteresis mechanism enables the proposed scheme to simultaneously avoid the “unwinding phenomenon” and sensitivity to arbitrarily small measurement noise that is present in discontinuous feedbacks. These properties are shown using a general framework for hybrid systems, and the results are demonstrated by simulation.
The problem of robust, global asymptotic stabilization of a rigid body is hampered by major topological obstructions. These obstructions prevent a continuous state feedback from solving the problem and also lead to robustness issues when (non-hybrid) discontinuous feedback is applied. In this paper, we extend a hybrid control strategy proposed in a companion paper for robust, global asymptotic stabilization of rigid body attitude to the case where translation is also considered. Through Lyapunov analysis, we develop quaternion-based hysteretic hybrid control laws in the kinematic and dynamic settings. In the dynamic setting, two control laws are derived: one from an energy-based Lyapunov function and one by backstepping. Robustness to measurement noise is asserted by employing recently developed stability theory for hybrid systems. A comparison between discontinuous and hysteretic feedback under measurement noise is shown in simulation.
This paper addresses the problem of stabilizing a fully-actuated rigid body. The problem is formulated by considering the natural configuration space for rigid bodies, the Special Euclidean group SE(3). The proposed solution consists of a landmark-based controller for force and torque actuation that guarantees almost global asymptotic stability of the desired equilibrium point. As such the equilibrium point is asymptotically stable and only a nowhere dense set of measure zero lies outside its region of attraction. The controller uses velocity measurements and the position coordinates of a collection of landmarks fixed in the environment. As an additional feature, the control law is designed so as to verify prescribed bounds on the actuation.
We show that a continuous dynamical system on a state space that has the structure of a vector bundle on a compact manifold possesses no globally asymptotically stable equilibrium. This result is directly applicable to mechanical systems having rotational degrees of freedom. In particular, the result applies to the attitude motion of a rigid body. In light of this result, we explain how attitude stabilizing controllers obtained using local coordinates lead to unwinding instead of global asymptotic stability.
Several definitions are in use for the derivative of an m × p matrix function F(X) with respect to its n × q matrix argument X. We argue that only one of these definitions is a viable one, and that to study smooth maps from the space of n × q matrices to the space of m × p matrices it is often more convenient to study the map from nq-space to mp-space. Also, several procedures exist for a calculus of functions of matrices. It is argued that the procedure based on differentials is superior to other methods of differentiation, and leads inter alia to a satisfactory chain rule for matrix functions.
We show that when a compact set is globally asymptotically stable under the action of a differential inclusion satisfying certain regularity properties, there exists a smooth differential equation rendering the same compact set globally asymptotically stable. The regularity properties assumed in this work stem from the consideration of Krasovskii/Filippov solutions to discontinuous differential equations and the robustness of asymptotic stability under perturbation. In particular, the results in this work show that when a compact set cannot be globally asymptotically stabilized by continuous feedback due to topological obstructions, it cannot be robustly globally asymptotically stabilized by discontinuous feedback either. The results follow from converse Lyapunov theory and parallel what is known for the local stabilization problem.
This paper addresses the problem of cooperative path planning for a fleet of unmanned aerial vehicles (UAVs). The paths are optimized to account for uncertainty/adversaries in the environment by modeling the probability of UAV loss. The approach extends prior work by coupling the failure probabilities for each UAV to the selected missions for all other UAVs. In order to maximize the expected mission score, this stochastic formulation designs coordination plans that optimally exploit the coupling effects of cooperation between UAVs to improve survival probabilities. This allocation is shown to recover real-world air operations planning strategies, and to provide significant improvements over approaches that do not correctly account for UAV attrition. The algorithm is implemented in an approximate decomposition approach that uses straight-line paths to estimate the time-of-flight and risk for each mission. The task allocation for the UAVs is then posed as a mixed-integer linear program that can be solved using CPLEX.
In this paper we present a tracking controller for a class of
underactuated mechanical systems, based on a backstepping procedure.
This class includes an approximation of small helicopter dynamics. The
need to avoid artificial singularities due to the attitude
representation is the main driver behind the control design presented in
this paper: to achieve this goal, we will operate directly in the
configuration manifold of the vehicle. The control design provides
asymptotic tracking for an approximate model of small helicopters, and
bounded tracking when more complete models are considered. Simulation
examples, including both point stabilization and aggressive maneuver
tracking, are presented and discussed
We present a general framework for the control of Lagrangian
systems with as many inputs as degrees of freedom. Relying on the geometry
of mechanical systems on manifolds, we propose a design algorithm for
the tracking problem. The notion of error function and transport map
lead to a proper definition of configuration and velocity error. These
are the crucial ingredients in designing a proportional derivative
feedback and feedforward controller. The proposed approach includes as
special cases a variety of results on control of manipulators, pointing
devices and autonomous vehicles. Our design provides particular insight
into both aerospace and underwater applications where the configuration
manifold is a Lie group.
Theory of Applied Robotics: Kinematics, Dynamics, And Control