A Comparison of Two Methods Used to Deal with Saturation of Multiple, Redundant Aircraft Control Effectors

TL;DR: In this article, the authors present a list of acknowledgements and acknowledgements for the work of the authors of this article. But they do not mention the authors' work in this paper.

Abstract: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of

Summary

List of Figures

• Star represents the starting vertex of the AMS.
• The method of control vector scaling is implemented.
• The method of pitch prioritization is implemented.
• 42 5.7 Desired moment time histories plotted with attained moments for the offset left approach using the method of scaling the moment direction.
• . . . 46 5.10 Longitudinal and lateral stick time histories for the offset high left approach using both methods.

1.1 Background

• Classically, aircraft flight controls have been designed to use a single control effector to produce a desired moment about a specific rotational degree of freedom.
• These additional arise from freeing opposing moment generators and controlling them independently.
• Left and right horizontal tails can be used individually to produce roll, pitch or yaw moments.
• With this arrangement the number of control effectors on an aircraft could total close to 20.
• Most of these solutions are not realistic for an actual aircraft due to control limits created by the physical geometry of the effector and aerodynamic constraints on the surfaces.

1.2 Allocation Problem

• The distribution of controls to obtain a certain objective is the general allocation problem.
• These equations come from Newton’s Second Law of Motion and are frequently linearized about some reference condition and written ẋ = Ax+Bu (1.1).
• The problem is solved by finding a control vector to produce desired aircraft dynamics.
• This research examines such a method in dealing with these unattainable moments.
• Note that other difficulties arise in control allocation such as the fact that the linearized equation md =.

1.3 Allocation Methods

• A schematic of the general arrangement of a control system, with control law and control allocator, is presented in Figure 1.1.
• Dynamic inversion allows one to tailor the shape of the responses by calculating the forces and moments required of the controls.
• Some of these methods of control allocation are presented in Bordignon [4].
• Some methods such as daisy chaining and generalized inverses are often used [4].
• All of these methods provide a simple linear approach to solving the allocation.

1.4 Attainable Moments

• The term attainable refers to the set of moments that can be generated given the physical limits of the effectors.
• Control vector solutions that are within the physical limits are admissible solutions.
• The calculation of the attainable moment subset (AMS) is simple and defines individual facets of the polytope [5].
• One facet is defined by two controls free to vary, with all other controls being saturated.
• An additional constraint to the control allocation problem could involve the rate limits for each effector.

1.5 Research Objectives

• It was the purpose of this research to provide an insight into the problem of unattainable moments.
• A method of pitch prioritization was investigated to allocate controls in the case of an unattainable moment and compared with the method of scaling the moment direction.
• Pitch prioritization allocates controls while preserving the stability augmentation in the longitudinal axis of an unstable airframe.
• An F-18-like airframe was used as a testbed to perform the research and evaluate the performance of prioritization.
• Prioritization of a single objective during situations that ask for unattainable moments was.

2.1 Introduction

• Before the days of redundant control effectors and complex control laws the control allocation problem was simply dealt with by the use of mechanical linkages between the pilot’s stick and pedals, and the control surfaces.
• Control systems since then have increased in complexity such as using an aileron/rudder interconnect to offset the adverse yaw effect during a turn.
• Even more complexity comes about when additional controls are developed and added to the airframe in an attempt to provide more maneuverability.
• Redundancy in aircraft controls has significantly altered the complexity of the control allocation problem.
• Several algorithms have been developed over the years to solve this problem.

2.2 Background

• Less complex algorithms such as generalized inverses have been used successfully in the past.
• The most popular of these control allocation algorithms is the Minimum Norm solution.
• This solution arranges the controls to produce the desired objectives while minimizing control energy.
• This method simply uses the pseudo-inverse solution to minimize the 2-norm of the control vector solution.
• There are several well-known ways to derive the pseudo-inverse solution that may be found in any text dealing with the underdetermined systems of linear 9.

2.3 Cascading Generalized Inverses

• One method called Cascading Generalized Inverse (CGI) was developed by Bordignon [4].
• This method uses the idea that if a generalized inverse commands a solution in which one or more controls exceed a limit, so long as not all are saturated, the controls are set at the limits, and the rest of the controls are redistributed to achieve the desired objectives.
• In other words, if a control is commanded past its limit, it is set at the limit, and the effects of the control at saturation are subtracted from the desired objectives.
• The method is repeated every time a control position is exceeded.
• If all controls become saturated, the moment is unattainable using this method.

2.4 Facet Search

• The allocation method used in this research is based on the original research by Durham [1,2] and later refined by Bordignon and Durham [3, 4].
• This method of control allocation takes the control effectiveness matrix B, the set of admissible controls, and some desired objective.

2.5 Bisecting Edge Searching Algorithm

• The method described above is more or less a brute force method of determining the intersection of md with the surface of the AMS.
• The method involved looking at a single facet defined by a pair of controls and determining whether the desired moment direction pointed toward this facet.
• There is no guarantee that the intersection would be found before the last facet was tested.
• The potential of searching the entire AMS meant that the number of floating point operations and computation time could be very high with the addition of control effectors.
• The reader is referred to Durham and Scalera [6, 7] for the theory behind the algorithm and a comparison of the performance with other allocators.

2.5.1 Two-Dimensional Problem

• The two dimensional problem is introduced as part of the solution to the three dimensional problem.
• That is, the three dimensional problem can be solved by systematically repeated solutions of the two dimensional projections of the three dimensional problem.
• B will then consist of two rows, with each element describing the effect each control has on the objectives.
• Figure 2.1 depicts an attainable moment subset, AMS, created from a four-control two-objective problem.
• The vertex with the maximum x-component is identified on the AMS.

2.5.2 Three-Dimensional Problem

• For the three-dimensional problem, consider md to consist of three objectives, causing the control effectiveness matrix to have three rows.
• From this view point the problem resembles the two-dimensional problem and is solved in the same way.
• There is an angle where the AMS can be rotated about the +x-axis such that the intersection of md, which is aligned with the +x-axis, with the AMS will lie exactly on a limb of the AMS as viewed from the +z-axis.
• Small perturbations in this angle will identify two edges that define the facet that contains the intersection.
• The rotation angle can not be calculated analytically, however, the angle is not needed, just.

3.2 F/A-18 Testbed

• The testbed airframe used in this research was based off of an F/A-18A airframe simulation.
• The greatest area in which the test-bed simulation differs from the original F/A-18A airframe is in the treatment of control effectors.
• These effectors were retained to make the simulation compatible with existing F/A-18 aircraft simulations.
• These controls are only used for initial trim and subsequent scheduling.
• The F/A18 in PA mode schedules leading-edge flaps based on angle-of-attack.

3.3 Airframe Simulation

• There are six files that were used in the simulation of the airframe: AERO.F, AEROPA.F, CONTROL.F, CONSTANTS.F, ENGINE.F, AND ALLOC.F.
• This section explains the purpose of each routine.

3.3.1 Aero.f

• AERO.F is an executive subroutine that calls subroutines based on the flight phase of the aircraft.
• In this specific case AERO.F calls AEROPA.
• The code then combines the aerodynamics from the non-linear scheduled/trimmed flight condition with the aerodynamics from the UGLOBAL array to produce the total aerodynamic forces and moments.

3.3.2 Aeropa.f

• AEROPA.F is taken from the F/A-18 simulation and modified slightly to mesh with AERO.F and to include proper control scheduling.
• This is the only code that gives the airframe F/A-18 like characteristics.

3.3.3 Control.f

• Stick and rudder pedal commands are taken as inputs and converted into an angle-of-attack command αcmd, sideslip command βcmd, and a roll rate command pcmd.
• These commands are input to a simple dynamic inversion control law that generates desired moments for the control allocation subroutine.

3.3.6 Alloc.f

• This code is the BESA control allocator that produces required control deflections for desired moments.
• The BESA method was explained in Chapter 2.

3.4 Simulation Environment

• Naval Air Systems Command provided the original F/A-18 simulation code in FORTRAN that was implemented on a Silicon Graphics Origin 2000TM Deskside System with two CPU’s running at 180 MHz with 256 MB of RAM and a 4 GB disk.
• CASTLE is a 6-degree-of-freedom non-linear aircraft simulation architecture developed by Naval Air Warfare Center’s Manned Flight Simulator branch (MFS) [11].
• The visual scene is calligraphic depicting a dusk or night environment.
• The visual database includes scenes for a carrier landing approach, and a naval air station.
• The stick feel is produced using an electronic control loader and can be modified through software to produce a realistic feel of any actual aircraft stick.

3.5 Airframe Validity

• The airframe in this research was modeled after the F-18A airframe for ease of implementation on other users computers that already had the original F-18 airframe.
• The modifications that were made were simply added subroutines to transform the airframe as needed.

4.1 Introduction

• The method of prioritization has rarely been used for problems such as unattainable moments that lead to inadmissible control solutions.
• The use of pitch prioritization provides one method of dealing with unattainable moment commands.
• As previously stated, pitch axis prioritization was chosen because of the popularity of relaxing the static margin of an aircraft to reduce trim drag.
• To stabilize the airframe the control law requires a certain amount of pitching moment.
• Prioritization the pitching-moment requirement will ensure that the maximum amount of pitch can be attained during any maneuver.

4.2 Sizing the Attainable Moment Subset

• Control effectiveness was determined by the linearization of the F/A-18 aerodynamic database described in Chapter 3.
• The controls that were added to the testbed airframe were given the 25.

4.3 Method of Pitch Prioritization and Moment Direc-

• Tion Preservation Scaling the control solution vector preserves the moment direction while decreasing the magnitude.
• Preserving the moment direction will simply decrease the magnitude of the moment vector, using the solution on the boundary of the AMS.
• The light green arrow indicates the desired moment for a specific time during the maneuver.
• Prioritizing the pitch axis, the control effectors are able to provide the full amount of desired pitching moment.

5.1 Background

• Maneuvers that cause the control law to require unattainable moments and require the majority of the controls to be saturated are used to investigate the comparison between the two methods of dealing with unattainable moments.
• An important aspect of the results in this research was how the airplane felt to the pilot in addition to the hard data that was plotted.
• One evaluation pilot was used to fly all the data represented in this paper.
• Several flights were flown by other pilots, however those results are not included here.
• The test pilot has a background in Navy fighters, carrier suitability, and flight testing.

5.2 Offset Carrier Approach maneuver

• Aerodynamic control effectiveness reduces with a decrease in the dynamic pressure.
• As a result, an aircraft requires larger control deflections to produce the desired amount of force and moment.
• An offset powered approach to an aircraft carrier was chosen for evaluation in this research since it provided the low dynamic pressure flight condition as well as large stick inputs and a high pilot workload.

5.2.1 Offset Carrier Approach maneuver: Description

• For this maneuver the static margin of the airframe was relaxed so that the airframe was unstable in the longitudinal axis.
• The control law kept the airframe stable during normal flight with small stick inputs.
• Offset Left - Airframe was offset to the left of centerline eight degrees and required more lateral maneuvering than pitch to correct the offset.
• The approach was offset eight degrees left of centerline and three degrees above the desired glide-slope.
• The pilot began the maneuver at the same point.

5.2.2 Offset Left Carrier Approach maneuver: Results

• The following data represents two different offset left approaches flown from the same initial conditions.
• One approach implemented the method of scaling the moment direction while the other used the method of pitch prioritization.
• Figure 5.2 plots the saturation of each method for an offset left maneuver.
• The yellow hatching indicates when the controls are saturated.
• The area of saturation for the lateral-directional requirement occured when the pitch requirement was near saturation indicating that most of the controls had been used for pitching moment.

5.2.3 Offset High Left Carrier Approach maneuver: Results

• An additional offset approach was used to further evaluate the two methods of dealing with unattainable moments.
• The prioritization mode was able to achieve all of the pitching moment without saturating any of the controls on the airframe.
• The saturation plot for the scaling method ends early because the airframe diverged in pitch as soon as the corrective maneuver was applied.
• These percentages are calculated as previously explained in section 5.2.2.
• Figure 5.10 plots the stick movement throughout the entire flight using both methods.

5.2.4 Offset Carrier Approach maneuvers: Pilot Interpretation

• The main trend seen in all the test flights was the pilot adaptation to how the aircraft handled.
• This learning curve made it difficult to investigate all of the differences between the two different methods.
• The one disconcerting tendency of the airframe while in pitch prioritization mode was the continuously changing amount of lateral control power during flight.
• The method of scaling the desired moment direction gave the aircraft an unstable feel to the pilot during any kind of aggressive maneuver.
• Overall, the airframe with pitch prioritization was “more smooth” compared to the flight with the method of scaling the moment direction implemented.

Summary and Conclusions

• A comparison of two methods to deal with unattainable moments in an aircraft was performed using a testbed airframe that resembled an F/A-18 with a highly redundant control effector suite.
• Control allocation utilizing the Bisecting, Edge Searching Algorithm was implemented on this airframe and a dynamic inversion control law was used to produce the desired aircraft handling qualities.
• In extreme unstable cases prioritizing the pitch axis produced favorable characteristics with respect to the alternative of preserving the moment direction by scaling the control solution vector that the allocator produces.
• Representative maneuvers that offered real-time evaluation of the two methods were flown with a pilot-in-the-loop.
• The validity of these tasks were subject to pilot compensation, to the point where the pilot was required to remove any learning of how the aircraft feels from the desired task.

Figures

Figure 5.10: Longitudinal and lateral stick time histories for the offset high left approach using both methods.

Figure 2.3: Three-dimensional AMS rotated such that the view point is looking down the z-axis of the problem.

Figure 4.4: AMS described by the original Control Effectivness Matrix B with Scaled Final B used in the Research.

Figure 4.3: AMS described by the scaled Control Effectivness Matrix B plotted with the desired moments of the offset approach task.

Figure 5.7: Desired moment time histories plotted with attained moments for the offset left approach using the method of scaling the moment direction.

Figure 1.2: General Attainable Moment Subset for a 4 Control Problem

Figure 5.2: Comparison of the saturation of control surfaces between scaling the moment direction and prioritizing the pitch axis for an offset left maneuver.

Figure 5.11: Angle of attack time histories for the offset high left approach using both methods.

Figure 5.15: Desired moment time histories plotted with attained moments for the offset high left approach using the method of pitch prioritization.

Figure 5.14: Desired moment time histories plotted with attained moments for the offset high left approach using the method of preserving the moment direction.

Table 3.1: Control Surfaces on the testbed airframe.

Figure 4.5: AMS as viewed down the +x-axis with the desired moments of the offset approach maneuver superimposed.

Figure 2.1: Two-dimensional AMS. Star represents the starting vertex of the AMS. x and y axes correspond to arbitrary moment directions.

Figure 5.8: Desired moment time histories plotted with attained moments for the offset left approach using the method of pitch prioritization.

Figure 5.13: Desired moment time histories for the offset high left approach using both methods.

Figure 3.1: Block Diagram of the Testbed Airframe Simulation Structure

Figure 1.1: Control Loop Block Diagram

Figure 4.6: AMS as viewed down the +x-axis. The method of control vector scaling is implemented.

Figure 4.7: AMS as viewed down the +x-axis. The method of pitch prioritization is implemented.

Figure 4.1: AMS described by the original Control Effectivness Matrix B.

Figure 5.4: Angle of attack time histories for the offset left approach using both methods.

Full text

A Comparison of Two Methods Used to Deal with Saturation of
Multiple, Redundant Aircraft Control Effectors
Mark D. Nelson
Thesis submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Master of Science
in
Aerospace Engineering
Dr. Wayne Durham - chair
Dr. Frederick Lutze
Dr. Chris Hall
August 2001
Blacksburg, Virginia
Keywords: Control Allocation, Moment Direction Preservation, Moment Prioritization,
Control Saturation, Redundant Aircraft Controls
Copyright 2001, Mark D. Nelson

A Comparison of Two Methods Used to Deal with Saturation of
Multiple, Redundant Aircraft Control Effectors
Mark D. Nelson
(ABSTRACT)
A comparison of two methods to deal with allocating controls for unattainable moments
in an aircraft was performed using a testbed airframe that resembled an F/A-18 with a
large control effector suite. The method of preserving the desired moment direction to deal
with unattainable moments is currently used in a specific control allocator. A new method
of prioritizing the pitch axis is compared to the moment-direction preservation. Realtime
piloted simulations are completed to evaluate the characteristics and performance of these
methods.
A direct comparison between the method of preserving the moment direction by scaling the
control solution vector and prioritizing the pitching moment axis is performed for a specific
case. Representative maneuvers are flown with a highly unstable airframe to evaluate the
ability to achieve the specific task. Flight performance and pilot interpretation are used to
evaluate the two methods.
Pilot comments and performance results favored the method of pitch-axis prioritization.
This method provided favorable flight characteristics compared to the alternative method of
preserving the moment direction for the specific tasks detailed in this paper.

Acknowledgments
I would like to thank first and foremost my parents for their support and unending advice
that have helped me to make the most of myself and live my life to the fullest. To my sister
Julie and my brother Daniel with whom I have the fortune of knowing. Julie’s pursuit to
make the most of life in the face of constant hurdles, and Dan’s quest to never be in the
shadow of older siblings has made our relationships both exhaustive and rewarding.
I would like to thank my advisor Dr. Wayne Durham for his persistence and patience with me
and my academic career. He has given me the perfect work environment and has constantly
pushed me, with much resistance, to achieve all that I could. To the remaining members of
my commitee, Dr. Fred Lutze and Dr. Chris Hall, thank you for making my education at
Virginia Tech a wonderful experience and for sharing everyday life experiences with me.
I must thank my friends that have made my life at Virginia Tech so enjoyable. To Mike
Henry, Kevin Waclawicz, Dan Hart, Todd Norell, Trevor Wallace, Roger Beck, Josh Durham,
Bill Oetjens, and others that mean so much to me. To the friend that I lost, and the ones
that I have recently gained, I will always remem ber you. Finally, I would like to thank the
few friends back home who have offered me support from a long ways away and have never
lost touch.
iii

Contents
Abstract......................................... ii
Acknowledgments.................................... iii
TableofContents.................................... iv
ListofFigures...................................... v
ListofTables ...................................... vi
Nomenclature...................................... 1
1 Introduction 1
1.1 Background .................................... 1
1.2 AllocationProblem................................ 2
1.3 AllocationMethods................................ 3
1.4 AttainableMoments ............................... 5
1.5 ResearchObjectives................................ 7
2 Control Allocation 9
2.1 Introduction.................................... 9
2.2 Background .................................... 9
2.3 CascadingGeneralizedInverses ......................... 10
2.4 FacetSearch.................................... 10
iv

2.5 BisectingEdgeSearchingAlgorithm....................... 11
2.5.1 Two-DimensionalProblem........................ 12
2.5.2 Three-DimensionalProblem ....................... 13
3 Airframe Simulation and Implementation 16
3.1 Introduction.................................... 16
3.2 F/A-18Testbed.................................. 16
3.3 AirframeSimulation ............................... 20
3.3.1 Aero.f ................................... 20
3.3.2 Aeropa.f.................................. 20
3.3.3 Control.f.................................. 20
3.3.4 Constants.f ................................ 22
3.3.5 Engine.f .................................. 23
3.3.6 Alloc.f ................................... 23
3.4 SimulationEnvironment ............................. 23
3.5 AirframeValidity ................................. 23
4 Pitch Prioritization 25
4.1 Introduction.................................... 25
4.2 Sizing the Attainable Moment Subset . ..................... 25
4.3 MethodofPitchPrioritizationandMomentDirectionPreservation ..... 29
5 Represen tative maneuver 32
5.1 Background .................................... 32
5.2 Offset Carrier Approach maneuver . . ..................... 32
5.2.1 Offset Carrier Approach maneuver: Description . . .......... 33
v