Trim Trajectories Research Articles

Observability analysis for a cooperative range-based navigation system that uses a rotating single beacon

This work addresses the observability analysis for a cooperative range-based navigation system that uses a beacon with circular motion installed on-board the support platform to find the location of an underwater vehicle. First, the observability analysis is carried out without taking into account the ocean current that affects the vehicle. To that end, an augmented state-space realization is obtained in order to convert the nonlinear system into a linear time varying (LTV) system. Then, the observability analysis is performed considering constraints on the initial conditions due to the state transformation. Based on these results, a second observability analysis is carried out taking into account ocean current in the model. We show that, by knowing the initial beacon’s position, the system is observable when the vehicle is moving in a trim trajectory with and without ocean currents. The obtained results include information about the initial conditions of the beacon and the ocean currents in the observability analysis. This result is important from a theoretical point of view, and represents a new way of implementing range-based navigation systems, without relying on deviation of the vehicle’s mission or the need of another vehicle to ensure observability.

Hovering and forward flight of the hawkmoth Manduca sexta: trim search and 6-DOF dynamic stability characterization

We show that the forward flight speed affects the stability characteristics of the longitudinal and lateral dynamics of a flying hawkmoth; dynamic modal structures of both the planes of motion are altered due to variations in the stability derivatives. The forward flight speed ue is changed from 0.00 to 1.00 m s−1 with an increment of 0.25 m s−1. (The equivalent advance ratio is 0.00 to 0.38; the advance ratio is the ratio of the forward flight speed to the average wing tip speed.) As the flight speed increases, for the longitudinal dynamics, an unstable oscillatory mode becomes more unstable. Also, we show that the up/down (wb) dynamics become more significant at a faster flight speed due to the prominent increase in the stability derivative Zu (up/down force due to the forward/backward velocity). For the lateral dynamics, the decrease in the stability derivative Lv (roll moment due to side slip velocity) at a faster flight speed affects a slightly damped stable oscillatory mode, causing it to become more stable; however, the thalf (the time taken to reach half the amplitude) of this slightly damped stable oscillatory mode remains relatively long (∼12T at ue = 1 m s−1; T is wingbeat period) compared to the other modes of motion, meaning that this mode represents the most vulnerable dynamics among the lateral dynamics at all flight speeds. To obtain the stability derivatives, trim conditions for linearization are numerically searched to find the exact trim trajectory and wing kinematics using an algorithm that uses the gradient information of a control effectiveness matrix and fully coupled six-degrees of freedom nonlinear multibody equations of motion. With this algorithm, trim conditions that consider the coupling between the dynamics and aerodynamics can be obtained. The body and wing morphology, and the wing kinematics used in this study are based on actual measurement data from the relevant literature. The aerodynamic model of the flapping wings of a hawkmoth is based on the blade element theory, and the necessary aerodynamic coefficients, including the lift, drag and wing pitching moment, are experimentally obtained from the results of previous work by the authors.

Optimal control of a small fixed-wing UAV about concatenated trajectories

The paper focuses on the control of a small fixed-wing unmanned aerial vehicle (UAV) along trajectories. The trajectories are generated in real-time from a library of pre-specified motion primitives. Two concatenated primitive trajectories are considered: one formed from seven primitives and exhibiting a figure-8 geometric path and another composed of a Split-S maneuver that settles into a level-turn trim trajectory. Switched control systems stemming from ℓ2-induced norm synthesis approaches are designed for discrete-time linearized models of the nonlinear UAV system. These controllers are analyzed based on simulations in a realistic operational environment and are further implemented on the physical UAV.

Observability analysis of 3D AUV trimming trajectories in the presence of ocean currents using range and depth measurements

We analyze the observability properties of the kinematic model of an autonomous underwater vehicle (AUV) moving in 3D, under the influence of ocean currents, using range and depth measurements. The results obtained shed light into the types of trajectories that an AUV may be requested to undergo in order to ensure observability, which is a crucial step in the design of single or multiple beacon positioning systems. We assume that the AUV is equipped with two sensor suites: the first computes the distance (range) of the AUV to single or multiple fixed transponders, while the second measures the vehicle’s depth. In both situations, the vehicle has access to its heading angle. We further assume that the AUV undergoes maneuvers commonly known as trimming trajectories, that are naturally obtained when the inputs (thruster rpms and control surface deflections) are held constant. This is done for two main reasons: (i) the class of trajectories thus generated is sufficiently rich for a vast number of applications and (ii) from an observability-analysis standpoint they lead to mathematical tractability and allow for an intuitive physical interpretation. These facts stand in sharp contrast to common approaches adopted in the literature, where the characterization of trajectories that yield observability is only implicit and defies a simple interpretation.In the set-up adopted, the trimming trajectories are completely characterized by three variables: (a) linear body speed ‖v‖; (b) flight-path angle γ; and (c) yaw rate ψ˙. We assume that ‖v‖ > 0, γ, and ψ˙ are constant but otherwise arbitrary (within the constraints of the vehicle capabilities) and examine the observability of the resulting system with the two above mentioned sensor suites. We adopt definitions of observability and weak observability that seek inspiration from those proposed by Herman and Krener (1977) but reflect the fact that we consider specific kinds of maneuvers in 3D.We start with the single transponder case. For range measurements only, we show that in the absence of ocean currents the 3D kinematic model of an AUV undergoing trimming trajectories with nonzero flight-path angle and yaw rate is observable. In the case of non-zero but known ocean currents, identical results apply subject to the condition that the flight-path angle satisfies a current-related constraint. However, if the current is non-zero and unknown, the model is only weakly observable. The situation changes completely when both range and depth measurements are available. In this case, under the assumption that the yaw rate is different from zero, observability is obtained even when the flight-path angle is zero (vehicle moving in a horizontal plane) and there are non-zero unknown currents. These obvious advantages are lost if yaw rate is equal to zero, for in this case the model is only weakly observable. In all situations where the model is weakly observable we give a complete characterization of the sets of states that are indistinguishable from a given initial state. Finally, we show that the extended model that is obtained by considering multiple (at least two) transponders is observable in all situations if the yaw rate is different from zero.

Observability Analysis of 3D AUV Trimming Trajectories in the Presence of Ocean Currents using Single Beacon Navigation

This paper addresses the observability properties of a 3D autonomous underwater vehicle (AUV) model in the presence of ocean currents, under the assumption that the vehicle can only measure its distance to a fixed transponder using an acoustic ranging device. In the set-up adopted, the AUV may undergo a wide range of maneuvers that are usually described as trimming trajectories. The latter are of paramount importance in flight dynamics and can be completely parametrized by three variables: i) linear body speed v, ii) flight-path angle γ, and iii) yaw rate ψ. We assume that v > 0, γ, and ψ are constant but otherwise arbitrary (within the constraints of the vehicle capabilities) and examine the observability of the resulting system with the output (measured) variable described above. We adopt weaker definitions of observability that are akin to those proposed by Herman and Krener (Hermann and Krener, 1977) but reflect the fact that we consider specific kinds of maneuvers in 3D. We show that in the presence of known constant ocean currents the 3D kinematic model of the AUV that corresponds to trimming trajectories with nonzero flight path angle and yaw rate is observable. In case the latter conditions fail, we give a complete characterization of the sets of states that are indistinguishable from a given initial state. We further show that in the case of unknown constant ocean currents the model is locally weakly observable for yaw rate different from zero but fails to be locally weakly observable for zero yaw rate. In both the cases we give a complete characterization of the sets of states that are indistinguishable from a given initial state.

Observability Analysis for AUV Range-only Localization and Mapping Measures of Unobservability and Experimental Results

We investigate the observability problem of the Simultaneous Localization and Mapping (SLAM) process of an Autonomous Underwater Vehicle (AUV) equipped with inertial sensors, a depth sensor, and an acoustic ranging device that provides relative range measurements to stationary beacons. For trimming trajectories (that is, when the motion of the AUV is in steady-state with constant linear and angular velocities expressed in the body-frame), we provide conditions under which it is possible to reconstruct the initial state of the resulting SLAM system (and in particular the position of the AUV). We show that the unobservable subspace UO restricted to the assumption that the position of one of the beacons or the initial position of the AUV is known, contains only the zero vector with exception of a distinct case where the UO is composed by a finite set of isolated points. Another problem that we also address in this paper is to understand (independently of the observer scheme) how difficult it is to accurately estimate the state of the system from the pair input/output measured signals. To this end, we compute explicitly the unobservability index and the estimation condition number that provide measures of the degree of unobservability. Simulation and experimental results with the Medusa robotic vehicle are presented and discussed.

Modeling and Dynamic Exploration of a Tilt-Rotor VTOL Aircraft

In this paper we introduce and study a novel model of a Tilt-Rotor VTOL aircraft. The aircraft is structured as a blended wing body equipped with a tilting rotor (the propulsion unit) that gives the vehicle the capability of vertical take off and transition to a forward flight configuration. This model captures the main features of novel tilt-rotor aircraft architectures with more challenging control and maneuvering capabilities. We introduce a complex nine degrees of freedom model of the aircraft to explore the dynamics and the maneuvering capabilities of the vehicle. We perform an analysis of the equilibrium manifold of the aircraft (namely a parametrized family of trimming trajectories) and, as main contribution of the paper, we introduce a set of optimal control based strategies to explore the trajectory manifold of the vehicle in order to generate non-stationary and highly aggressive trajectories. In particular, we provide numerical computations showing how to generate a trajectory for transitions from near hover to forward flight.

Trajectory-Tracking and Path-Following of Underactuated Autonomous Vehicles With Parametric Modeling Uncertainty

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.

Tracking control of trim trajectories of a blimp for ascent and descent flight manoeuvres

A blimp is a small airship that has no metal framework and collapses when deflated. It belongs to family of unmanned aerial vehicles (UAVs). In this paper we address the problem of designing tracking feedback control of an underactuated autonomous UAV. The ascent and descent flight conditions as one in which the rate of change (of magnitude) of the airship's state vector is zero and the resultant of the applied forces and moments is constant lead to trimmed flight trajectories. The subject of the tracking control is to stabilize the engine around the planned flight. Using a combined integrator backstepping approach and Lyapunov theory, the stability results are local and overcome the minimum number of actuators (inputs) with respect to the blimp's six degrees of freedom. Considering physic limits in UAVs, other trimmed flights are investigated and compared.

Human-Inspired Control Logic for Automated Maneuvering of Miniature Helicopter

The maneuvering control logic that was developed to implement aerobatic maneuvers fully automatically on a miniature helicopter is described. The key component of this system is a state-machine maneuver execution logic that was inspired by an analysis of human pilot strategies. Conventional multivariable trim trajectory controllers were used before and on the exit from the maneuvers; bumpless transfer between these control modes was achieved through re-initialization of controller integrator states. Flight tests with this control logic demonstrated smooth maneuver entry, automatic recovery to a steady-state trim trajectory, and an ability to sequence maneuvers in a fully autonomous airshow-like sequence. This approach was flight tested with split-S, hammerhead, and 360-deg axial roll maneuvers, as well as a split-S-hammerhead maneuver sequence. The maneuvering control logic can be used to automate a variety of other maneuvers.

Trajectory Tracking Controllers for AUVS: An Integrated Approach to Guidance and Control System Design

This paper describes a solution to the problem of 3-D trajectory tracking for Autonomous Underwater Vehicles (AUVs). The trajectories considered consist of segments of trimming helices, which are parameterized by the vehicle's linear speed, yaw rate and flight path angle. Motivated by recent developments in robotics, the linear position of an AUV is given in terms of its location with respect to the closest point on a desired trajectory, together with the length of an imaginary curve traced along that trajectory. Tracking of a trimming trajectory by the vehicle at a fixed speed is then converted into the problem of driving a generalized error vector - that implicitly includes the distance to the trajectory - to zero. By showing that the linearization of the generalized error dynamics about trimming trajectories is time-invariant, the problem of trajectory tracking is posed and solved in the framework of gain scheduled control theory. This leads to a new method for the integrated design of guidance and control systems for AUVs.