Andrew Kurdila - Academia.edu (original) (raw)

Papers by Andrew Kurdila

Research paper thumbnail of A Switched Reduced-Order Dynamical System for Fluid Flows under Time-Varying Flow Conditions

Research paper thumbnail of Switched Dynamic Systems for Reduced-Order Flow Modeling

Aiaa Journal, Mar 1, 2008

Research paper thumbnail of Nonrecursive formulations of multibody dynamics and concurrent multiprocessing

Since the late 1980's, research in recursive formulations of multibody dynamics has flourished. H... more Since the late 1980's, research in recursive formulations of multibody dynamics has flourished. Historically, much of this research can be traced to applications of low dimensionality in mechanism and vehicle dynamics. Indeed, there is little doubt that recursive order N methods are the method of choice for this class of systems. This approach has the advantage that a minimal number of coordinates are utilized, parallelism can be induced for certain system topologies, and the method is of order N computational cost for systems of N rigid bodies. Despite the fact that many authors have dismissed redundant coordinate formulations as being of order N(exp 3), and hence less attractive than recursive formulations, we present recent research that demonstrates that at least three distinct classes of redundant, nonrecursive multibody formulations consistently achieve order N computational cost for systems of rigid and/or flexible bodies. These formulations are as follows: (1) the preconditioned range space formulation; (2) penalty methods; and (3) augmented Lagrangian methods for nonlinear multibody dynamics. The first method can be traced to its foundation in equality constrained quadratic optimization, while the last two methods have been studied extensively in the context of coercive variational boundary value problems in computational mechanics. Until recently, however, they have not been investigated in the context of multibody simulation, and present theoretical questions unique to nonlinear dynamics. All of these nonrecursive methods have additional advantages with respect to recursive order N methods: (1) the formalisms retain the highly desirable order N computational cost; (2) the techniques are amenable to concurrent simulation strategies; (3) the approaches do not depend upon system topology to induce concurrency; and (4) the methods can be derived to balance the computational load automatically on concurrent multiprocessors. In addition to the presentation of the fundamental formulations, this paper presents new theoretical results regarding the rate of convergence of order N constraint stabilization schemes associated with the newly introduced class of methods.

Research paper thumbnail of Nonlinear Dynamic Simulation and Control of Military Ground Vehicles

... None (c) Papers Presented at Meetings (1) D. McDaniel, N. Fitz-Coy, A. Kurdila (Florida, Univ... more ... None (c) Papers Presented at Meetings (1) D. McDaniel, N. Fitz-Coy, A. Kurdila (Florida, Univ., Gainesville), and M. Hale (US Army ... Submitted to ARO None (7) Participating Scientific Personnel PI: Andrew J. Kurdila Co-PI: Norman Fitz-Coy Students: Dwayne McDaniel (PhD ...

Research paper thumbnail of A receding horizon controller for motion planning in the presence of moving obstacles

Proceedings Ieee International Conference on Robotics and Automation, May 3, 2010

Research paper thumbnail of Maneuvering and Tracking for a Micro Air Vehicle Using Vision-Based Feedback

SAE Technical Paper Series, 2004

Missions envisioned for micro air vehicles may require a high degree of autonomy to operate in un... more Missions envisioned for micro air vehicles may require a high degree of autonomy to operate in unknown environments. As such, vision is a critical technology for mission capability. This paper discusses an autopilot that uses vision coupled with GPS and altitude sensors. One use of vision processing analyzes a horizon to estimate roll and pitch information. Another use tracks a feature point to estimate position relative to a target. This paper presents examples of waypoint navigation and hom- ing using vision-based feedback. The examples indicate the vi- sion provides sufcient information to achieve the missions.

Research paper thumbnail of A Hardware-in-the-Loop Experiment and Simulation Facility for Vision-Based Control of Micro-Air-Vehicles

Research paper thumbnail of Extension of the Rubin-Ungar theorem and its implications for multibody simulation

Sufficient conditions for the convergence of a class of penalty approximation methods are derived... more Sufficient conditions for the convergence of a class of penalty approximation methods are derived by extending the Rubin-Ungar theorem. The approach includes penalty terms that are quadratic in the derivatives of the constrain coordinates. The approach not only simplifies the original Rubin-Ungar derivation, but makes it possible to control the prohibitively large frequency of oscillation that is associated with the constraint degrees of freedom. The importance of retaining a full-rank constraint Jacobian matrix throughout the simulation when using the class of penalty formulations is emphasized.

Research paper thumbnail of On the Presence of Internal Resonances in Aeroelastic Systems

Herein, the authors examine the presence of internal resonance in aeroelastic structures, and ill... more Herein, the authors examine the presence of internal resonance in aeroelastic structures, and illustrate the pathologies that internal resonance may have on these systems. These studies consider an aeroelastic model which possesses nonlinear aerodynamic loads such as those that arise from aerodynamic stall. Analyses of the equations of motion representing a physical aeroelastic structure are presented. The aerodynamics are in

Research paper thumbnail of Multiscale Wavelet Methods for Partial Differential Equations

This latest volume in the Wavelets Analysis and Its Applications Series provides significant and ... more This latest volume in the Wavelets Analysis and Its Applications Series provides significant and up-to-date insights into recent developments in the field of wavelet constructions in connection with partial differential equations. Specialists in numerical applications and ...

Research paper thumbnail of Shaping of a system's frequency response using an array of subordinate oscillators

The Journal of the Acoustical Society of America, Jul 1, 2009

The frequency response of an oscillating structure can be tailored by attaching one or more subor... more The frequency response of an oscillating structure can be tailored by attaching one or more subordinate oscillators. This paper shows how the magnitude and phase of the frequency response can be deliberately shaped by prescribing the distributions of the dynamic properties in an array of such subordinate oscillators. Exact analytic governing equations of motion are derived for the coupled system composed of the primary system and the subordinate array. For a relatively small number (<100) of attached oscillators whose total mass is small (<1%) relative to the primary structure, it is possible to engineer frequency-response functions of the primary oscillator to have, for example, nearly linear phase or constant amplitude over a frequency band of interest. The frequency range over which response shaping is achieved is determined by the band of the attached oscillators. It is shown that the common analytic methodology for designing a dynamic vibration absorber represents the limiting case of a single oscillator in the subordinate set. Moreover, increasing the number of subordinate oscillators (without increasing the total added mass) offers a number of advantages in reshaping the dominant system's frequency response.

Research paper thumbnail of Model Classes, Approximation, and Metrics for Dynamic Processing of Urban Terrain Data

Research paper thumbnail of Nonlinear flutter of composite plates with damage evolution

The investigators present a study of dynamic and aeroelastic response of structures which evolve ... more The investigators present a study of dynamic and aeroelastic response of structures which evolve due to damage. Aeroelastic response is shown to be dependent upon the distribution and accumulation of damage. In turn, the damage is dependent upon the presence of the aerodynamic loads. Dynamic characteristics are unique to the coupled damage/aeroelastic system and are developed as part of the solution methodology. In this study, the damage is due to the natural progression of microcracking of the composite structure; yet, the control model presented is appropriate for distributed actuation systems. The stability boundary for aeroelastic flutter and divergence evolves due to damage. Control design based upon the min-max control theory is presented which addresses model uncertainties.

Research paper thumbnail of Control Design for Evolutionary Structural Systems

Research paper thumbnail of Efficient computation of level sets for path planning

2009 Ieee Rsj International Conference on Intelligent Robots and Systems, Oct 1, 2009

Research paper thumbnail of Low Reynolds number flow control via active material actuators

An experimental testbed is described that is used to study the feasibility of control of a class ... more An experimental testbed is described that is used to study the feasibility of control of a class of flows that have low Reynolds numbers. The experimental testbed is comprised of a thin airfoil with a backward facing step machined into the upper surface. A thin PZT composite flap is mounted at the edge of the backward facing step to enable modification of the flow. Output measurement sensors consist of MEMs-based shear stress sensors, and conventional pressure taps, located on the surface of the airfoil. This paper derives a control framework for the synthesis of control methodologies for the testbed. A reduced order control model is obtained by employing reduced basis approximations of the two dimensional Navier-Stokes equations. Preliminary open loop experimental results are reported that illustrate the existence of convected large scale structures in the flow.

Research paper thumbnail of An Investigation of Internal Resonance in Aeroelastic Systems

Nonlinear Dynamics, 2003

Although the study of internal resonance in mechanical systems has been given significant conside... more Although the study of internal resonance in mechanical systems has been given significant consideration, minimal attention has been given to internal resonance for systems which consider the presence of aerodynamic forces. Herein, the investigators examine the possible existence of internal resonances, and the related nonlinear pathologies that such responses may have, for an aeroelastic system which possesses nonlinear aerodynamic loads.

Research paper thumbnail of Volterra Kernel Extrapolation for Modeling Nonlinear Aeroelastic Systems at Novel Flight Conditions

Journal of Aircraft, May 22, 2012

ABSTRACT

Research paper thumbnail of Vision-Based Kalman Filtering for Aircraft State Estimation and Structure from Motion

In this paper, we provide a critical, albeit brief, overview of feature point tracking, SFM algor... more In this paper, we provide a critical, albeit brief, overview of feature point tracking, SFM algorithms and Kalman Filtering as they apply specifically to the state estimation problem for aircraft. We naturally focus on those attributesof the methods that are well-suitedto aircraft estimation problems. We also note features of the algorithms that require further work to develop a stable, consistent vision-based state estimation algorithm for aircraft. We specifically discuss the direct, and quite natural, extension of the work of9 and1 that incorporates aircraft dynamical system models in the state propagation step of the filter. Withthis framework, we effectively augment the baseline vision-based motion estimation with the benefits of a dynamical model. He structure from motion (SFM) problem, in which the objective is to estimate camera motion and the three- dimensional structure of the scene using features extracted from two-dimensional images, has been the subject of an enormous body of literature in the computer vision community. It has been widely recognized for some time that the ability to accurately and robustly solve the structure from motion problem has important implications for the guidance, navigation, and control of autonomous vehicles. One such application is the autonomous control of unmanned and micro air vehicles in complex urban environments using vision as the primary sensor. Clearly, accurate estimation of aircraft states is valuable from a controls standpoint while knowledge of the scene is critical for obstacle avoidance. It should be remembered that much of the research in SFM algorithms has been Carried out for slow moving ground vehicles or robotic systems. In some Instances, SFM algorithms have been derived for post-processing large collections of video data that can be processed simultaneously, as opposed to causally or incrementally in time. This point is discussed more completely shortly below. The implications of these observations is, however, that the SFM problemcan be especially challengingfor aircraft applicationssince the images taken fromaircraft can be subject to noise, occlusions, frame dropout due to telemetry problems, and loss of feature tracking resulting from fast vehicle dynamics. Therefore, any SFM algorithm that is to be applied to aircraft, and particularly micro air vehicles, must be robust with respect to noise in the two-dimensional image plane. Structure from motion is by nature a very general problem and many specific different problem statements can be defined based on the nature of the two-dimensional image data used and the form of the three-dimensional structure of the scene. For example, SFM algorithms have been employed that make use of tracked feature points, lines, and line segments. In addition, homography-basedapproaches can be used in cases where there are known planar features in the images. Likewise, the form of the three-dimensional scene that is computed from SFM can vary in complexity from the locations of discrete feature points in inertial space to complete geometrical descriptions of the environment. Another important distinction in SFM techniques is the choice between batch implementations that make use of all the images obtained from a video sequence and causal methods that only use information from images up to the present time. This choice is usually application dependent since, as noted in Ref. 1, given the luxury of being able to compute SFM estimates in an off-line batch procedure, one can almost certainly obtain better results than from a causal, on- line estimation process. It is clear that the application of SFM to the real-time guidance, navigation, and control of

Research paper thumbnail of Identification for a class of nonlinear models for shape memory alloy (SMA) embedded elastomeric rods

This paper presents a formulation of the transient response of elastomeric rods with embedded sha... more This paper presents a formulation of the transient response of elastomeric rods with embedded shape memory alloy actuators that incorporate the inherent coupling between the dynamics of deformation at the structural level, the thermal response and the constitutive law describing the shape memory alloy. Previous work by the authors has shown that the incorporation of shape memory alloy actuation in distributed parameter systems can induce a large class of nonlinearities including hysteresis effects in the SMA constitutive law, nonlinear kinematics of large deformation, and, in some cases, local plasticity effects. To derive a control methodology for the class of SMA embedded elastomeric rods considered in this paper, it is essential that the system characteristics of the nonlinear distributed parameter system be identified accurately. This paper presents an identification theory applicable to the coupled system of partial differential equations. The results are validated using both numerical simulation and experimental results.

Research paper thumbnail of A Switched Reduced-Order Dynamical System for Fluid Flows under Time-Varying Flow Conditions

Research paper thumbnail of Switched Dynamic Systems for Reduced-Order Flow Modeling

Aiaa Journal, Mar 1, 2008

Research paper thumbnail of Nonrecursive formulations of multibody dynamics and concurrent multiprocessing

Since the late 1980's, research in recursive formulations of multibody dynamics has flourished. H... more Since the late 1980's, research in recursive formulations of multibody dynamics has flourished. Historically, much of this research can be traced to applications of low dimensionality in mechanism and vehicle dynamics. Indeed, there is little doubt that recursive order N methods are the method of choice for this class of systems. This approach has the advantage that a minimal number of coordinates are utilized, parallelism can be induced for certain system topologies, and the method is of order N computational cost for systems of N rigid bodies. Despite the fact that many authors have dismissed redundant coordinate formulations as being of order N(exp 3), and hence less attractive than recursive formulations, we present recent research that demonstrates that at least three distinct classes of redundant, nonrecursive multibody formulations consistently achieve order N computational cost for systems of rigid and/or flexible bodies. These formulations are as follows: (1) the preconditioned range space formulation; (2) penalty methods; and (3) augmented Lagrangian methods for nonlinear multibody dynamics. The first method can be traced to its foundation in equality constrained quadratic optimization, while the last two methods have been studied extensively in the context of coercive variational boundary value problems in computational mechanics. Until recently, however, they have not been investigated in the context of multibody simulation, and present theoretical questions unique to nonlinear dynamics. All of these nonrecursive methods have additional advantages with respect to recursive order N methods: (1) the formalisms retain the highly desirable order N computational cost; (2) the techniques are amenable to concurrent simulation strategies; (3) the approaches do not depend upon system topology to induce concurrency; and (4) the methods can be derived to balance the computational load automatically on concurrent multiprocessors. In addition to the presentation of the fundamental formulations, this paper presents new theoretical results regarding the rate of convergence of order N constraint stabilization schemes associated with the newly introduced class of methods.

Research paper thumbnail of Nonlinear Dynamic Simulation and Control of Military Ground Vehicles

... None (c) Papers Presented at Meetings (1) D. McDaniel, N. Fitz-Coy, A. Kurdila (Florida, Univ... more ... None (c) Papers Presented at Meetings (1) D. McDaniel, N. Fitz-Coy, A. Kurdila (Florida, Univ., Gainesville), and M. Hale (US Army ... Submitted to ARO None (7) Participating Scientific Personnel PI: Andrew J. Kurdila Co-PI: Norman Fitz-Coy Students: Dwayne McDaniel (PhD ...

Research paper thumbnail of A receding horizon controller for motion planning in the presence of moving obstacles

Proceedings Ieee International Conference on Robotics and Automation, May 3, 2010

Research paper thumbnail of Maneuvering and Tracking for a Micro Air Vehicle Using Vision-Based Feedback

SAE Technical Paper Series, 2004

Missions envisioned for micro air vehicles may require a high degree of autonomy to operate in un... more Missions envisioned for micro air vehicles may require a high degree of autonomy to operate in unknown environments. As such, vision is a critical technology for mission capability. This paper discusses an autopilot that uses vision coupled with GPS and altitude sensors. One use of vision processing analyzes a horizon to estimate roll and pitch information. Another use tracks a feature point to estimate position relative to a target. This paper presents examples of waypoint navigation and hom- ing using vision-based feedback. The examples indicate the vi- sion provides sufcient information to achieve the missions.

Research paper thumbnail of A Hardware-in-the-Loop Experiment and Simulation Facility for Vision-Based Control of Micro-Air-Vehicles

Research paper thumbnail of Extension of the Rubin-Ungar theorem and its implications for multibody simulation

Sufficient conditions for the convergence of a class of penalty approximation methods are derived... more Sufficient conditions for the convergence of a class of penalty approximation methods are derived by extending the Rubin-Ungar theorem. The approach includes penalty terms that are quadratic in the derivatives of the constrain coordinates. The approach not only simplifies the original Rubin-Ungar derivation, but makes it possible to control the prohibitively large frequency of oscillation that is associated with the constraint degrees of freedom. The importance of retaining a full-rank constraint Jacobian matrix throughout the simulation when using the class of penalty formulations is emphasized.

Research paper thumbnail of On the Presence of Internal Resonances in Aeroelastic Systems

Herein, the authors examine the presence of internal resonance in aeroelastic structures, and ill... more Herein, the authors examine the presence of internal resonance in aeroelastic structures, and illustrate the pathologies that internal resonance may have on these systems. These studies consider an aeroelastic model which possesses nonlinear aerodynamic loads such as those that arise from aerodynamic stall. Analyses of the equations of motion representing a physical aeroelastic structure are presented. The aerodynamics are in

Research paper thumbnail of Multiscale Wavelet Methods for Partial Differential Equations

This latest volume in the Wavelets Analysis and Its Applications Series provides significant and ... more This latest volume in the Wavelets Analysis and Its Applications Series provides significant and up-to-date insights into recent developments in the field of wavelet constructions in connection with partial differential equations. Specialists in numerical applications and ...

Research paper thumbnail of Shaping of a system's frequency response using an array of subordinate oscillators

The Journal of the Acoustical Society of America, Jul 1, 2009

The frequency response of an oscillating structure can be tailored by attaching one or more subor... more The frequency response of an oscillating structure can be tailored by attaching one or more subordinate oscillators. This paper shows how the magnitude and phase of the frequency response can be deliberately shaped by prescribing the distributions of the dynamic properties in an array of such subordinate oscillators. Exact analytic governing equations of motion are derived for the coupled system composed of the primary system and the subordinate array. For a relatively small number (<100) of attached oscillators whose total mass is small (<1%) relative to the primary structure, it is possible to engineer frequency-response functions of the primary oscillator to have, for example, nearly linear phase or constant amplitude over a frequency band of interest. The frequency range over which response shaping is achieved is determined by the band of the attached oscillators. It is shown that the common analytic methodology for designing a dynamic vibration absorber represents the limiting case of a single oscillator in the subordinate set. Moreover, increasing the number of subordinate oscillators (without increasing the total added mass) offers a number of advantages in reshaping the dominant system's frequency response.

Research paper thumbnail of Model Classes, Approximation, and Metrics for Dynamic Processing of Urban Terrain Data

Research paper thumbnail of Nonlinear flutter of composite plates with damage evolution

The investigators present a study of dynamic and aeroelastic response of structures which evolve ... more The investigators present a study of dynamic and aeroelastic response of structures which evolve due to damage. Aeroelastic response is shown to be dependent upon the distribution and accumulation of damage. In turn, the damage is dependent upon the presence of the aerodynamic loads. Dynamic characteristics are unique to the coupled damage/aeroelastic system and are developed as part of the solution methodology. In this study, the damage is due to the natural progression of microcracking of the composite structure; yet, the control model presented is appropriate for distributed actuation systems. The stability boundary for aeroelastic flutter and divergence evolves due to damage. Control design based upon the min-max control theory is presented which addresses model uncertainties.

Research paper thumbnail of Control Design for Evolutionary Structural Systems

Research paper thumbnail of Efficient computation of level sets for path planning

2009 Ieee Rsj International Conference on Intelligent Robots and Systems, Oct 1, 2009

Research paper thumbnail of Low Reynolds number flow control via active material actuators

An experimental testbed is described that is used to study the feasibility of control of a class ... more An experimental testbed is described that is used to study the feasibility of control of a class of flows that have low Reynolds numbers. The experimental testbed is comprised of a thin airfoil with a backward facing step machined into the upper surface. A thin PZT composite flap is mounted at the edge of the backward facing step to enable modification of the flow. Output measurement sensors consist of MEMs-based shear stress sensors, and conventional pressure taps, located on the surface of the airfoil. This paper derives a control framework for the synthesis of control methodologies for the testbed. A reduced order control model is obtained by employing reduced basis approximations of the two dimensional Navier-Stokes equations. Preliminary open loop experimental results are reported that illustrate the existence of convected large scale structures in the flow.

Research paper thumbnail of An Investigation of Internal Resonance in Aeroelastic Systems

Nonlinear Dynamics, 2003

Although the study of internal resonance in mechanical systems has been given significant conside... more Although the study of internal resonance in mechanical systems has been given significant consideration, minimal attention has been given to internal resonance for systems which consider the presence of aerodynamic forces. Herein, the investigators examine the possible existence of internal resonances, and the related nonlinear pathologies that such responses may have, for an aeroelastic system which possesses nonlinear aerodynamic loads.

Research paper thumbnail of Volterra Kernel Extrapolation for Modeling Nonlinear Aeroelastic Systems at Novel Flight Conditions

Journal of Aircraft, May 22, 2012

ABSTRACT

Research paper thumbnail of Vision-Based Kalman Filtering for Aircraft State Estimation and Structure from Motion

In this paper, we provide a critical, albeit brief, overview of feature point tracking, SFM algor... more In this paper, we provide a critical, albeit brief, overview of feature point tracking, SFM algorithms and Kalman Filtering as they apply specifically to the state estimation problem for aircraft. We naturally focus on those attributesof the methods that are well-suitedto aircraft estimation problems. We also note features of the algorithms that require further work to develop a stable, consistent vision-based state estimation algorithm for aircraft. We specifically discuss the direct, and quite natural, extension of the work of9 and1 that incorporates aircraft dynamical system models in the state propagation step of the filter. Withthis framework, we effectively augment the baseline vision-based motion estimation with the benefits of a dynamical model. He structure from motion (SFM) problem, in which the objective is to estimate camera motion and the three- dimensional structure of the scene using features extracted from two-dimensional images, has been the subject of an enormous body of literature in the computer vision community. It has been widely recognized for some time that the ability to accurately and robustly solve the structure from motion problem has important implications for the guidance, navigation, and control of autonomous vehicles. One such application is the autonomous control of unmanned and micro air vehicles in complex urban environments using vision as the primary sensor. Clearly, accurate estimation of aircraft states is valuable from a controls standpoint while knowledge of the scene is critical for obstacle avoidance. It should be remembered that much of the research in SFM algorithms has been Carried out for slow moving ground vehicles or robotic systems. In some Instances, SFM algorithms have been derived for post-processing large collections of video data that can be processed simultaneously, as opposed to causally or incrementally in time. This point is discussed more completely shortly below. The implications of these observations is, however, that the SFM problemcan be especially challengingfor aircraft applicationssince the images taken fromaircraft can be subject to noise, occlusions, frame dropout due to telemetry problems, and loss of feature tracking resulting from fast vehicle dynamics. Therefore, any SFM algorithm that is to be applied to aircraft, and particularly micro air vehicles, must be robust with respect to noise in the two-dimensional image plane. Structure from motion is by nature a very general problem and many specific different problem statements can be defined based on the nature of the two-dimensional image data used and the form of the three-dimensional structure of the scene. For example, SFM algorithms have been employed that make use of tracked feature points, lines, and line segments. In addition, homography-basedapproaches can be used in cases where there are known planar features in the images. Likewise, the form of the three-dimensional scene that is computed from SFM can vary in complexity from the locations of discrete feature points in inertial space to complete geometrical descriptions of the environment. Another important distinction in SFM techniques is the choice between batch implementations that make use of all the images obtained from a video sequence and causal methods that only use information from images up to the present time. This choice is usually application dependent since, as noted in Ref. 1, given the luxury of being able to compute SFM estimates in an off-line batch procedure, one can almost certainly obtain better results than from a causal, on- line estimation process. It is clear that the application of SFM to the real-time guidance, navigation, and control of

Research paper thumbnail of Identification for a class of nonlinear models for shape memory alloy (SMA) embedded elastomeric rods

This paper presents a formulation of the transient response of elastomeric rods with embedded sha... more This paper presents a formulation of the transient response of elastomeric rods with embedded shape memory alloy actuators that incorporate the inherent coupling between the dynamics of deformation at the structural level, the thermal response and the constitutive law describing the shape memory alloy. Previous work by the authors has shown that the incorporation of shape memory alloy actuation in distributed parameter systems can induce a large class of nonlinearities including hysteresis effects in the SMA constitutive law, nonlinear kinematics of large deformation, and, in some cases, local plasticity effects. To derive a control methodology for the class of SMA embedded elastomeric rods considered in this paper, it is essential that the system characteristics of the nonlinear distributed parameter system be identified accurately. This paper presents an identification theory applicable to the coupled system of partial differential equations. The results are validated using both numerical simulation and experimental results.