Flight control system for a reusable rocket booster on the return flight through the atmosphere
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1 Flight control system for a reusable rocket booster on the return flight through the atmosphere Aaron Buysse 1, Willem Herman Steyn (M2) 1, Adriaan Schutte 2 1 Stellenbosch University Banghoek Rd, Stellenbosch 7600, South Africa Phone: , Mail: aaron.j.buysse@gmail.com, whsteyn@sun.ac.za 2 Heliaq Advanced Engineering United Kingdom Mail: adriaan@heliaq.com Abstract: This article presents a flight control system for the subsonic return flight of a reusable first stage rocket booster of a launch vehicle dedicated to launching small satellites. Each first stage booster has a deployable wing and an aero engine with deployable propellers. The wing rotates about a single pivot point, resulting in an oblique wing aircraft while partially deployed. Control algorithms are presented for the rocket and aircraft configurations. Proportional-integral feedback controllers are designed to follow a pitch-up trajectory and stabilize lateral motion in the rocket configuration. A multiple-loop control architecture using linear-quadratic servo controllers is designed for the aircraft configuration. This controller consists of a fast inner-loop that controls the angular velocities, and a slow outer-loop that controls the attitude. Guidance controllers are designed to track airspeed and heading angle references by providing references to the aircraft attitude controller. Control allocation and gain scheduling is performed to handle the significant change in booster configuration and the large flight envelope. 1. INTRODUCTION The rapid pace of technology advancements has significantly reduced the cost and size of satellites in recent years, allowing universities and small companies to design and manufacture highly-capable, small satellites. Currently, launch costs limit the trend of cost reduction in part due to the practice of utilizing expendable rocket boosters for the launch vehicle. Furthermore, small satellites are often secondary payloads on launch vehicles, restricting their orbits and launch date to that of the largest payload. The Austral Launch Vehicle (ALV) is a three-stage launch vehicle dedicated to launching small satellites with the goal of significantly reducing launch costs [1]. The ALV consists of several reusable first stage rocket boosters each with a deployable wing and deployable propellers, as well as second and third stage rocket boosters. The first stage boosters deliver the latter stages to the edge of the atmosphere, then separate and the latter stages place the satellite into orbit. After separation, the first stage boosters enter an unpowered, ballistic flight to re-enter the atmosphere. A high angle of attack is maintained during hypersonic and supersonic re-entry before entering a dive across the transonic region. During subsonic flight, the boosters pull out of the dive, deploy the wing and slow down to deploy the propellers. The boosters follow pre-programmed waypoints to return to the launch pad, where a human pilot remotely lands the boosters. This article presents the flight control algorithms for a first stage booster, shown in Figure 1, during the subsonic phase of the return flight prior to starting the aero engine. The wing is initially stowed with the span along the body -axis and rotates about a single pivot point, resulting in an oblique wing aircraft until fully deployed. The wing sweep angle,, is measured from the body y-axis as shown. Each booster has two ailerons on
2 the wing, and two ailerators and two ruddervators on the empennage. The propellers are stowed in the booster nose and deploy by centrifugal force once the aero engine starts. The nose landing gear deploys at the same time as the propellers, therefore the engine is started at the design cruise speed of to reduce drag. The control objective is to stabilize the booster s attitude during the descent and the wing deployment, and to obtain the cruise speed. After wing deployment, the controllers must maximize the time available to start the aero engine and begin flying towards the launch pad to reduce fuel requirements. These objectives are met with classical controllers in the rocket configuration, a multiple-loop linear-quadratic servo controller in the aircraft configuration, and aircraft guidance controllers. Control allocation facilitates a smooth transition between the configurations, and gain scheduling manages the controllers throughout the descent. 2. DESCENT TRAJECTORY The airfoil s critical Mach number,, restricts the velocity at which the wing is deployed, but is larger for swept oblique wings [2]. Table 1 shows the results of computational fluid dynamics (CFD) analysis using the SU2 software [3] to determine. Wing Sweep Angle ( ) Figure 1. ALV First stage booster in the oblique wing configuration Critical Mach Number ( ) Table 1. Wing critical Mach number as a function of the wing sweep angle. Flight tested oblique wing aircraft have found a significant reduction in stability and aileron control authority for [4]. Therefore, the wing is deployed to at Mach 0.65, allowing for a margin of error in the CFD analysis. Once the wing reaches, an airspeed controller obtains a minimum sink rate glide by commanding a pitch angle, causing the booster to pull up and slow down by exchanging kinetic energy for potential energy. The rate of the pull up maneuver is limited by the maximum load factor,, the booster structure can sustain, which is expressed as [5] (1) where is the gravitational acceleration and is the velocity. A value of is used to ensure the load factor during the descent remains below 3.8, which is the minimum load factor the structure must be able to handle for FAR airworthiness requirements [6]. After obtaining the minimum sink rate glide, a heading angle controller is activated to begin following pre-programmed waypoints by commanding a roll angle. 3. FLIGHT CONTROL SYSTEM Separate sets of flight controllers are designed for the rocket configuration and the aircraft configuration to follow reference attitude commands.
3 3.1 Flight Dynamics Modeling The rigid body booster dynamics are modeled in the North-East-Down coordinate system fixed to a flat, non-rotating Earth, and the nonlinear equations of motion are linearized about an equilibrium condition as in [7]. The booster dynamics are described as (2) where [ ] is the perturbation state with velocity, sideslip angle, angle of attack, body angular velocity vector [ ], roll angle, and pitch angle. The control vector is [ ] where the right and left ailerons are and, the right and left ailerators are and, and the right and left ruddervators are and, respectively. The unique aerodynamic properties of the oblique wing aircraft are described in [4] and [8], and are modeled in this work using the potential flow panel method software PANUKL [9]. 3.2 Control Allocation Since the effectiveness and availability of the physical control surfaces vary significantly from the rocket configuration to the aircraft configuration, control allocation is used to establish pseudo control effectors for use in the control algorithms. The pseudo control effectors used are the typical aileron,, elevator, and rudder,, effectors, and the control allocation problem selects a control allocation matrix,, such that [ ] (3) Explicit ganging [10] is used to determine. The ailerators are used for in the rocket configuration and the physical ailerons are used in the aircraft configuration. Both the ailerators and ruddervators are used for, and the ruddervators are used for. Thus, the control allocation matrices are [ ] [ ] A positive pseudo control deflection causes a negative moment, and a positive physical control deflection is towards the positive body -direction. The four elements of mapping to the physical control effectors are linearly varied between and over five seconds to achieve a smooth transition. 3.3 Rocket Flight Controller Classical proportional-integral controllers are designed for the rocket case by assuming small roll angles so that the dynamics may be decoupled into longitudinal and lateral dynamics [7]. A pitch angle controller is designed using to dampen pitch rate dis- (4)
4 , with zero steady state error. The con- turbances and to track a pitch angle reference, trol law is (5) Similarly, a roll angle controller is designed using to dampen roll rate disturbances and to track a roll angle reference,, with zero steady state error. The control law is (6) The booster empennage is sized to provide weathercocking under sideslip conditions in the rocket configuration, so must only dampens yaw motions. The control law is (7) All of the gains are positive to achieve negative feedback, since a positive pseudo control deflection causes a negative moment. 3.4 Aircraft Flight Controller A multiple loop control architecture is designed for the aircraft configuration using linear quadratic servo controllers for each loop. Figure 2 shows the controller block diagram. A slow outer-loop controller tracks a reference attitude, [ ], by providing a reference angular velocity, [ ], to a fast inner-loop controller. Figure 2 Oblique Wing Controller block diagram Angular Velocity Inner-loop Controller The angular velocity states,, are given by [ ] (8) The control input is chosen to be the body moments,, and integrator states,, are added to track with zero steady state errors. Thus, the state space model of the angular velocity dynamics is [ ] [ ] [ ] [ ] [ ] (9)
5 where and are the zero matrix and identity matrix, respectively, is the inertia tensor and extracts the angular velocity dynamics from the system matrix in Equation 2. The cost function is selected as ([ ] [ ] [ ] ) (10) where, and are diagonal weighting matrices. The resulting algebraic Riccati equation is then solved [11] resulting in the control law [ ] [ ] (11) where is the pseudo control effectiveness matrix, and and are proportional and integral gain matrices, respectively, acting on the angular velocity error Attitude Outer-loop Controller The attitude states,, are given by [ ] (12) Integrator states,, are added to track with zero steady-state errors. The state space model of the attitude dynamics is [ ] [ ] [ ] [ ] [ ] (13) where and extract the attitude dynamics from the system matrix in Equation 2. The cost function is selected as ([ ] [ ] [ ] ) (14) where, and are diagonal weighting matrices. The resulting algebraic Riccati equation is then solved [11] resulting in the controller [ ] [ ] (15) where and are proportional and integral gain matrices, respectively, acting on the attitude error. 3.5 Aircraft Guidance Controllers The aircraft guidance controllers provide to the aircraft controller, using and to track a reference airspeed and heading angle, respectively, while is constantly zero.
6 3.5.1 Airspeed Controller The airspeed controller obtains a minimum sink-rate velocity,, at each wing sweep to allow for maximum time to deploy the propellers. The minimum sink rate velocity is [5] ( ) (16) where is the booster s weight, is the drag-due-to-lift factor, is the parasitic drag coefficient, is the wing planform area and is the air density. Equation 16 is calculated for each value of. A proportional-integral controller is designed to achieve zero steady state error. The control law is (17) where and are both negative gains. Equation 1 limits the rate of pull up to remain within the booster s load factor limits Heading Angle Controller The heading angle controller tracks a heading angle reference,. The dynamics are [5] (18) The natural integration from to makes the system type 1, therefore a proportional controller will achieve zero steady state error. The heading control law is thus { (19) where is positive and the conditional statement ensures that the shortest turning angle is taken to achieve. A bank angle increases the sink rate by a factor of [5], so the value of is saturated to to limit the sink rate increase to less than 10%. Also, is limited to reduce the overshoot of the attitude controller. A rate limit of is used to comply with the FAR for the minimum roll rate on approach of in four seconds [6]. 3.6 Gain Scheduling The booster s physical configuration and control objectives vary significantly throughout the descent trajectory, therefore gain scheduling is required. Table 1 summarizes the reference commands and the schedule switching criteria. The rocket controllers maintains zero roll angle and track a pitch trajectory as a function of altitude,, to remain near an equilibrium state. The aircraft flight controller is activated once the wing begins to deploy and initially maintains a constant attitude, thus is set to the last pitch angle prior to deploying the wing,. During this initial deployment, is used
7 until, where the transition to begins. The airspeed controller is activated at. Once the booster airspeed is near the minimum sink rate airspeed (approximately ), the wing is deployed to and the heading angle controller is activated. After, the wing is deployed at increments every of altitude loss, which allows the aircraft flight controller to settle for each configuration. Schedule Number #1 #2 #3 #4 #5 #6 #7 Wing Sweep Angle Pitch angle reference Roll angle reference Table 2. Gain scheduling reference commands and scheduling criteria. Control Allocation Matrix Scheduling Criteria 4. RESULTS The flight controllers are implemented on a six degree-of-freedom simulation using the full nonlinear rigid body dynamics. Figure 3 shows the altitude, Mach number and wing sweep angle throughout the descent trajectory, where the tick marks on the -axis signify the different phases of the gain scheduling as described in Table 2. Figure 4 shows the booster attitude, where the dashed red lines are the reference values. The initial heading angle is, with a heading angle reference of initiated at gain schedule #4. Figure 5 shows the airspeed, sideslip angle and angle of attack, with the airspeed controller initiated at gain schedule #3. Figure 6 shows the physical and pseudo control effector deflections for the initial wing deployment, and are plotted together, i.e. the plot for shows, and. The five second transition from to can be seen in Figure 6, as physical aileron deflections begin one second before switching to gain schedule #3, and the physical elevator deflections converge to the pseudo elevator deflection four seconds after switching to gain schedule #3. As the wing approaches, the roll moment produced by the ailerators becomes insufficient to stabilize roll alone. This motivated the premature transition to using the wing ailerons. The wing is fully deployed at an altitude of approximately and a velocity of. Figure 3. Altitude, Mach number and wing sweep angle for descent trajectory with the gain schedule number. Figure 4. Booster attitude for the descent trajectory.
8 Figure 5. Velocity, sideslip angle and angle of attack for the descent trajectory. Figure 6. Physical and pseudo control effector deflections for the initial wing deployment. 5. CONCLUSION Reusable launch vehicles can significantly reduce launch costs for small satellites. Flight controllers for the subsonic return flight of a reusable rocket booster with a deployable wing are presented. These controllers stabilize the booster attitude throughout the deployment of the wing, and track a minimum sink rate airspeed and heading angle in the aircraft configuration. Gain scheduling and control allocation are employed to smoothly transition between the two configurations, resulting in the wings fully deployed at the design cruise speed and with sufficient altitude to start the aero engine. 6. REFERENCES [1]A. Schutte, The Austral Launch Vehicle: 2014 Progress in Reducing Space Transportation Cost through Reusability, Modularity and Simplicity, 12th Reinventing Space Conf., BIS-RS , [2]R. T. Jones, The oblique wing-aircraft design for transonic and low supersonic speeds, Acta Astronautica, 4, , [3]F. Palacios, et al., Stanford University Unstructured (SU2): An open-source integrated computational environment for multi-physics simulation and design, AIAA Paper , [4]A. G. Sim and R. E. Curry, Flight Characteristics of the AD-1 Oblique-Wing Research Aircraft, NASA Technical Report, NASA-TP-2223, [5]D. P. Raymer, Aircraft Design: A Conceptual Approach, AIAA Education Series, Virginia, [6]Anon., Part 23, Airworthiness Standards, Normal Utility, Acrobatic and Commuter Category Airplanes, Federal Aviation Regulations, U.S. Government Printing Office, Washington D.C., [7]M. Drela, Flight Vehicle Aerodynamics, MIT Press, Massachusetts, [8]W. Lixin, et al., Dynamic characteristics analysis and flight control design for the oblique wing aircraft, Chinese Jour. Of Aeronautics, [9]Anon., Users manual for PANUKL, Warsaw University of Technology, 2013, URL: [10]M. W. Oppenheimer, et al., Control Allocation for Over-actuated Systems, IEEE 14 th Mediterranean Conf. on Control and Automation, [11] F. L. Lewis, V. L. Syrmos, Optimal Control, John Wiley & Sons, Inc., New York, 1995.
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