Document
A Robust Uniform Control Approach for VTOL Aircraft
Jacob Cook Irene Gregory Research Engineer Senior Technologist Dynamic Systems and Control Branch Dynamic Systems and Control Branch NASA Langley Research Center NASA Langley Research Center Hampton, VA, U.S.A. Hampton, VA, U.S.A.
ABSTRACT We present a uniform control approach for transitioning vertical take-off and landing aircraft. The approach combines several well-understood linear techniques, including robust servomechanism linear quadratic regulation, control al- location, and gain scheduling, to provide a practical control framework that can be used to unify the control design process in all flight regimes. The choice of command variables provides a pilot/operator with a uniform set of intuitive control inputs through all phases of flight while also being easily integrated into autonomous trajectory tracking op- erations. The control method is applied to the NASA LA-8 aircraft, a tandem tiltwing distributed electric propulsion research vehicle designed at NASA Langley Research Center. The trim envelope of the aircraft is explored and the aircraft control authority analyzed throughout the transition corridor. The uniform control approach is then used to develop reference command tracking controllers for both the longitudinal and lateral-directional dynamics. An exam- ple trajectory is simulated to demonstrate how the controller effectively transitions the aircraft from hover to forward flight and vice versa while tracking a desired trajectory.
Compounding these complexities is the plethora of configu- INTRODUCTION rations being proposed for the UAM market, each with their own unique aerodynamic properties and available control ef- Urban Air Mobility (UAM) operations are based on vertical fectors. From the outset, it is easy to see that an automatic take-off and landing (VTOL) aircraft, some of which, tran- control system is needed to improve vehicle flyability and pro- sition from a rotor or thrust- borne-flight to a more efficient vide foundational capabilities to build on as we transition to wing- or lift-borne flight. Transitioning UAM vehicles tend autonomous operations.
to fall into one of the three general classes of VTOL con- figurations: tiltwing, where the wing and propulsion rotate Traditional fly-by-wire systems made use of robust linear as a unit from a vertical to horizontal position; a tiltrotor, methods to develop control and stabilization algorithms for where the wing is fixed and the propulsion units rotate be- conventional aircraft (Refs. 1–4). Because of the broad range tween a vertical and horizontal position; or lift+cruise, where of flight conditions at which a VTOL aircraft are expected both the wings and propulsion units are fixed and there are to operate, various different approaches have been proposed separate propulsors that provide lift in hover and thrust in for- and implemented. In (Refs. 5, 6), a velocity control con- ward flight. Common across all these platforms is the use of cept is proposed that makes use of a cascaded loop archi- propulsion units and aerodynamic surfaces as control actua- tecture. Redundant controls are abstracted through the use tors. Their combination in different flight regimes provides of virtual control effectors and feed forward commands are redundancy in control force and moment generation; however provided by a predefined trim envelope. The F-35 control their individual effectiveness and energy requirements vary as law (Refs. 7, 8) uses an on board model of the aircraft aerody- the flight envelope is traversed.
namic and propulsion characteristics and a control allocation methodology (cascaded generalized inverse algorithm) to pro- Transitioning VTOL aircraft must operate in three general vide rate control of the aircraft in all phases of flight. A non- flight modes: hover, forward flight, and transition. Hover and linear dynamic inversion approach is taken in (Refs. 9–11) forward flight are well understood flight regimes but opera- where the changing relative degree of the control inputs are tionally present two very distinct modes of flight. Transition, handled by introducing the pitch and roll angle as virtual con- a less well understood regime, must seamlessly stitch the two trols. Additionally, the heading frame velocities are used to flight modes together, taking into account the changing aero- provide a common set of control commands throughout the dynamics of the aircraft as well as the dynamic nature of con- flight envelope.
trol force and moment production from available actuators.
This paper presents a Robust Uniform Control Architec- The combination of the transitioning aerodynamics and con- ture for VTOL aircraft. The main contribution being a trol actuation make VTOL aircraft, in general, very complex.
configuration-independent framework which unifies the con- trol design across all flight regimes and thus provides a uni- Presented at the VFS Autonomous VTOL Technical Meeting and Electric VTOL Symposium, January 26–28, 2021. form set of control commands throughout the entire flight en- velope. angles (Ref. 13), effectively mapping out an expected flight envelope. The subsequent test runs used Design of Experi- The control architecture builds upon the foundational princi- ments (DOE) methods to build a high-fidelity 23-factor model ples of Robust Servomechanism Linear Quadratic Regulator that captures the many interactive effects of this complex (RSLQR) control theory, but formulates the problem using aircraft (Ref. 14). Additional isolated propeller testing was general accelerations as the commanded input. The desired performed to determine the propeller performance through a control accelerations are mapped to the set of physical and broad range of incidence angles and capture the deviations in virtual control inputs by way of a weighted pseudo-inverse.
thrust and torque generation, as well as the off axis forces and The formulation enables a consistent control strategy over the moments generated when subject to flow at a high incidence entire operating envelope of a VTOL aircraft.
angle (Ref. 15).
The effectiveness of the Robust Uniform Controller is demon- strated in simulation using a model of NASA Langley’s LA- 8 tandem tiltwing research aircraft (Ref. 12). A Trajectory tracking maneuver, that transitions the aircraft from hover to forward flight and back to hover, is demonstrated.
AIRCRAFT DESCRIPTION AND AERODYNAMIC MODEL The Langley Aerodrome No. 8 (LA-8) is a distributed electric propulsion, VTOL aircraft designed and built at the NASA Langley Research Center (Ref. 12). The aircraft, shown in figure 1, is a tandem tiltwing with four propellers mounted along the leading edge of each wing. The wings rotate, from the zero degree fixed-wing position, a full 90 degrees such that the wing and propeller thrust axis is vertical. The wing tilting mechanisms operate independently of one another such Figure 2. Control surface and rotor diagram of the LA-8 that the wings may be at different tilt angles simultaneously.
Each wing has a set of flaps located on the inner portion of The resulting six-degree-of-freedom model provides the aero- the wing and a set of elevons on the outer portion. These are propulsive forces and moments over a range of flight con- shown in figure 2. An inverted V-tail at the rear of the aircraft ditions from hover to forward flight centered around a body includes a set of ruddervator surfaces. Each surface may be level flight condition. Of note, the model contains asymme- operated independently. The propellers of the LA-8 are speed tries discovered while testing, among them, is a difference in controlled and are also commanded independently. The di- thrust production of the clockwise and counterclockwise pro- rection of rotation of the propellers alternates as indicated in pellers, attributed to the different manufacturing processes by figure 2.
which they were obtained. Since the wind tunnel model is also the flight vehicle, there was no attempt to symmetrize the model; the asymmetries, are therefore present in this pre- sentation. Additionally, while testing at higher dynamic pres- sures (5 PSF) the elevon servos burned out causing the model to reflect diminished control authority of these actuators in forward flight. Due to extenuating circumstance (COVID-19 pandemic), additional wind tunnel testing could not be per- formed to rectify the elevon measurements. Therefore, the ar- tificial control reduction remains in the model and provide an effective example of how the control architecture allows the designer to easily distribute control actuation to other avail- able effectors.
REFERENCE FRAMES Three reference frames are used throughout this study: the in- Figure 1. LA-8 aircraft undergoing wind tunnel testing ertial North-East-Down (NED) frame, the aircraft body frame, and the heading frame. The aircraft body frame is defined The aerodynamic model of the LA-8 was developed over mul- with the origin at the nominal center of mass with the x-axis tiple wind tunnel experimental runs. The initial wind tunnel pointing out the nose, y-axis out the right wing, and z-axis test used a one-factor-at-a-time approach to explore the tran- completes the right hand rule. The rotation from body frame sition corridor by trimming the aircraft at different wing tilt to the inertial frame is parameterized by the Euler angles, T complicate the feedback control design and degrade the per- η = [ φ , θ , ψ ] , (roll, pitch, and yaw respectively). The ro- formance of the system.
tation matrix R ∈ SO ( 3 ) is defined as A purely lateral maneuver, such as sideways flight in hover, is T R = [ R ( φ ) R ( θ ) R ( ψ )] (1) x y z commanded using the lateral component of the velocity in the heading frame, ¯ v . Turning flight and stationary yaw maneu- d where R ( · ) , R ( · ) , and R ( · ) are the intermediate single-axis x y z vers are incorporated into the guidance commands by includ- rotation matrices, explicitly defined as ing the desired turn rate ˙ ψ such that the complete guidance d command is comprised of the desired heading frame velocity 1 0 0 T T and turn rate, r ( t ) = [ ¯ v ( t ) , ˙ ψ ( t )] .
R ( x ) = 0 cos x sin x (2) x d d 0 − sin x cos x The formulation of the guidance commands in the heading frame provides seamless integration into existing trajectory cos x 0 − sin x generation tools, in which smooth, four-dimensional trajecto- R ( x ) = 0 1 0 (3) y ries may be decomposed into the four guidance components.
sin x 0 cos x Additionally, the heading frame commands provide an intu- cos x sin x 0 itive set of uniform control inputs for a pilot or operator to − sin x cos x 0 R ( x ) = . (4) z direct the aircraft in all phases of flight.
0 0 1 AIRCRAFT DYNAMICS The heading frame is defined as the inertial frame rotated about the z-axis by the heading angle ψ , with the origin co- The angular kinematics of the vehicle are expressed as the incident with the aircraft center of mass. The rotation from time derivatives of the Euler angles body frame to the heading frame is defined as the abbreviated T rotation ¯ R = [ R ( φ ) R ( θ )] . A vector expressed in the body x y ˙ η = S ω , (5) frame is mapped to the heading frame through the transforma- ¯ tion ¯ v = R v , where the subscript b denotes a vector expressed where ω is the angular rate of the aircraft expressed in the b in the body frame, whereas the over-bar indicates the heading body frame with components ( p , q , r ) and S is the non- frame. orthogonal transformation from the body angular rates to the time derivatives of the Euler angles GUIDANCE COMMANDS 1 sin φ tan θ cos φ tan θ 0 cos φ − sin φ S = . (6) Flight maneuvers of a transitioning VTOL aircraft are ex- 0 sin φ / cos θ cos φ / cos θ pected to encompass those of rotor-craft in addition to conven- tional flight. This includes stationary hover, and pure lateral ¯ Because we seek to track heading frame velocities, v = and vertical maneuvers. For this reason, typical aircraft guid- T [ ¯ u , ¯ v , ¯ w ] , it makes sense to formulate the translational dy- ance commands, such as airspeed, flight path angle, and head- namics in the heading frame as ing angle, are inadequate to capture the full operating range, particularly in hover where the flight path angle is undefined ˙ ¯ ¯ ˙ ¯ v = − ψ ˆ e × ¯ v + g + F ( ¯ v , ω , R , u ) , (7) at zero total velocity. 3 m Additionally, UAM aircraft are expected to operate in highly where ¯ F ( · ) is the aero-propulsive forces acting on the aircraft congested airspace. To ensure vehicle separation, both spatial T expressed in the heading frame, g = [ 0 , 0 , a ] is the gravita- g and temporal adherence to a scheduled trajectory is necessary.
tional acceleration vector, u is the vector of available control Therefore, inertially referenced speeds, such as ground speed, effectors, and ˆ e is the unit vector in the body z-direction. Ex- are used to describe a desired trajectory (Ref. 16).
panding (7) gives The use of heading frame coordinates to describe the de- ¯ ¯ ˙ sired trajectory provides a set of guidance commands that ˙ ψ ¯ v + X ( ¯ v , ω , R , u ) ¯ u m can be used to describe the full range of maneuvers of a ˙ ¯ ¯ ˙ ¯ v = − ψ ¯ u + Y ( ¯ v , ω , R , u ) . (8) m VTOL aircraft. The desired heading frame velocity vector ˙ ¯ ¯ ¯ w a + Z ( ¯ v , ω , R , u ) g m T ¯ v = [ ¯ u , ¯ v , ¯ w ] , is comprised of two horizontal compo- d d d d The rotational dynamics are given by nents, ¯ u and ¯ v , and the vertical component ¯ w . The forward d d d and vertical components, ¯ u and ¯ w , are analogous to the total d d ¯ J ˙ ω = − ω × J ω + τ ( ¯ v , ω , R , u ) (9) velocity and flight path angle, but do not suffer from the same ill conditioning when the velocity goes to zero. Additionally, where J is the inertia matrix, and τ is the aero-propulsive mo- as the forward velocity increases the vertical velocity com- ments expressed in the body frame. Expanding (9) gives the mand may remain constant when performing, for instance, a following set of equations for the rotational dynamics: transition with a constant climb rate. In contrast, commands ¯ ˙ p J rq − J qp − J qr L ( ¯ v , ω , R , u ) y xz z constructed of total speed and flight path angle would neces- − 1 2 2 ¯ ¯ ˙ q = J J p − J r + M ( v , ω , R , u ) . (10) x z sitate both commands changing in concert, which may further ¯ ˙ r J qp + J qr − J pq N ( ¯ v , ω , R , u ) x xz y EQUILIBRIUM MANIFOLD EXPLORATION min J ( φ , θ , u ) φ ∈ Φ , θ ∈ Θ , u ∈ U ¯ subject to X ( ¯ u , ¯ w , φ , θ , u ) = 0 0 0 The equilibrium manifold is the set of states at which the air- ¯ Y ( ¯ u , ¯ w , φ , θ , u ) = 0 0 0 craft is in a constant operating (trim) condition. The trajec- , (18) ¯ Z ( ¯ u , ¯ w , φ , θ , u ) = − ma 0 0 g tories described by the equilibrium manifold are the set of L ( ¯ u , ¯ w , φ , θ , u ) = 0 0 0 constant radius helical trajectories which includes level flight M ( ¯ u , ¯ w , φ , θ , u ) = 0 0 0 and steady level turns. For transitioning aircraft, exploring the N ( ¯ u , ¯ w , φ , θ , u ) = 0 0 0 equilibrium manifold provides valuable insight into how the where the cost function J ( φ , θ , u ) may be used to produce aircraft might fly including: how the effectiveness of the of a trim condition that, for instance, minimizes the energy us- the control surfaces varies throughout the envelope, how the age or maximizes the control authority at that particular flight lifting force is distributed between the lifting rotors and the condition. The optimization problem can be solved with a wing, and at what speeds the aircraft is fully transitioned to commercial-off-the-shelf nonlinear programming solver, but wing-borne flight. The trim points, at select conditions in the it should be noted that the solutions are not unique and depend flight envelope, provide the basis for control design, as well on where the algorithm is initialized. In this study, the non- as feed forward actuation and pitch and roll commands as a turning level flight equilibrium points are presented, but the function of the aircraft’s desired speed, climb rate and turn method can be extended to turning flight with the additional rate.
complexity of adding the body angular rates as free variables.
The aircraft is said to be in equilibrium (or trimmed) when The LA-8 aircraft can be flown in many different ways, so the translational and rotational dynamics as well as the deriva- some decisions were made up front for this study. We con- ˙ tives of the pitch and roll angles are equal to zero ( ˙ ¯ v = ω = sider level flight at speeds within in the range of 0-60 ft/s to T ˙ ˙ [ 0 , 0 , 0 ] and φ = θ = 0). When considering non-turning stay within the bounds of the aerodynamic model. To sim- steady flight ( ˙ ψ = p = q = r = 0), the trim condition con- plify the optimization problem, the body pitch angle was held straints simplify to the following set of equations.
◦ at 0 , and the elevons and ruddervators — reserved for control ◦ actuation — were set to their neutral positions, δ = δ = 0 .
e r ¯ ¯ X ( ¯ v , ω , R , u ) = 0 (11) The free variables used to trim the aircraft were the roll angle, ¯ ¯ Y ( ¯ v , ω , R , u ) = 0 (12) wing tilt angles, the propeller speeds, and the flaps on both ¯ ¯ Z ( ¯ v , ω , R , u ) = − ma (13) sets of wings. The control effectors were allowed to move in- g dependently of one another, i.e., no effectors were ganged to- ¯ L ( ¯ v , ω , R , u ) = 0 (14) gether. The cost function used to produce the trim conditions ¯ M ( ¯ v , ω , R , u ) = 0 (15) presented in figures 3-5 used a combination of quadratic cost ¯ N ( ¯ v , ω , R , u ) = 0 (16) on the usage of the propeller and flaps as well as quadratic cost on the difference between wing tilt angles, propeller speeds, For symmetric aircraft models, and control inputs mirrored and flap deflections on each wing.
about the XZ plane, the constraints may be further reduced to T T 2 J = w ω ω + w δ δ + w ( δ − δ ) the three longitudinal equations. Since the LA-8 model is not 1 p 2 f 3 i i p f 2 1 symmetric, all six constraints must be met. 2 2 + w ( δ − δ ) + w ( δ − δ ) (19) 4 f f 5 f f 2 1 4 3 Equilibrium conditions are found by specifying the desired + w ( ω − ω ) .
6 ∑ p p
i + 1 i forward and vertical speed, ¯ u and ¯ w , then solving for the 0 0 i = 1 free variable values that satisfy the nonlinear constraints. In this case, the free variables are the pitch angle, roll angle, and The development of the cost function was the result of an iter- control inputs u . The input vector u , is comprised of the pro- ative process, of which many combinations of weighting func- T peller speeds ω = [ ω , . . . , ω ] , the tilt angle of each wing p p p tions and weights were considered. The resulting trim curves 1 8 T T δ = [ δ , δ ] , the elevon deflections δ = [ δ , . . . , δ ] , the i i i e e e were evaluated for relative smoothness and available control 1 2 1 4 T flap defections δ = [ δ , . . . , δ ] , and the ruddervator de- f f f authority of the chosen free-variables. The level-flight trim 1 4 T flections δ = [ δ , δ ] , such that r r r curve presented in figures 3, 4, and 5, show the wing tilt an- 1 2 gles, propeller speeds, and flap angles at each trim point for [ ] T T T T T T a range of air speeds between 0 and 60 ft/s. The trim curve ω δ δ δ δ u = . (17) p i e f r represents one of many possible solutions.
Due to the complexity of the nonlinear equations, solving for The wing tilt angles presented in figure 3 show the wing an- the free variables directly requires use of a nonlinear solver. gles decreasing from a trimmed hover condition of approx- ◦ ◦ On simpler systems, the Newton-Raphson method can be em- imately 84 and 83 for the forward and rear wing, respec- ◦ ◦ ployed to great effect, but due to the multitude of redundant tively, down to 11 and 10 at a forward flight speed of 60 control effectors of the LA-8, and UAM aircraft in general, the ft/s. The wing offset from vertical in hover is expected and is trimming problem is setup as an optimization problem with attributed to the need to balance the rear directional lift force nonlinear equality constraints generated by the wing in the slipstream of the propeller.
The propeller and flap curves shown in figures 4 and 5 show the need to balance significant asymmetries in the aircraft model. These manifest themselves as both yaw and roll mo- ments in hover where it can be noted that significant differ- ences in flap and propeller speeds are needed when compar- ing actuation on the left and right side of the aircraft. Flaps 1 and 3 are used to balance out a positive yaw moment while the propellers are used to balance out a negative roll moment.
0 10 20 30 40 50 60 As the speed is increased, and the wing tilt angle decreased, the asymmetries become less severe between 30 and 45 ft/s ( ≈ Figure 3. Trim wing tilt angles ◦ ◦ 50 and ≈ 25 wing tilt angles), as noted by the decrease in the difference between the left and right control effectors. As the speed is increase beyond 45 ft/s, significant flap differential is needed to balance the roll moment.
The set of trim points form the basis of the transition con- troller. The dynamics are linearized about a subset of these points, and linear control techniques are employed (described 4500 in more detail in the next section) to produce a robust reference-tracking controller. The control gains are scheduled based on the flight speed, and the trim solutions used as feed- 0 10 20 30 40 50 60 forward control inputs as the aircraft transitions between flight regimes.
AIRCRAFT CONTROL AUTHORITY As with many different classes of VTOL aircraft, the LA-8 transitions from thrust-borne flight to lift-borne flight, and in doing so changes how it affects control. Understanding the changing effectiveness of the control actuators, and how much 0 10 20 30 40 50 60 control authority is available as the aircraft traverses the flight envelope is important, since the LA-8 is predominantly open- Figure 4. Trim propeller speeds loop unstable throughout the flight envelope.
To understand how control authority changes, or is transferred between effectors, we look at the linearization of the dynam- ics at each trim point. From the input Jacobian matrix, insight 20 as to how the effectiveness changes as the aircraft transitions between regimes may be obtained. Additionally, approxima- tions of the total control authority of an individual effector may be produced.
This type of study, following wind tunnel testing and mod- eling of the aircraft dynamics, can provide early feedback as 0 10 20 30 40 50 60 to how the vehicle is expected to perform and whether or not it can meet the design requirements of the proposed vehicle mission.
The effectiveness and the control authority of the effectors are presented in two plots for each moment direction, i.e., roll, pitch, and yaw. The control effectiveness is presented as a nor- malized value between -1 and 1, and was obtained by divid- ing the derivative value in the Jacobian by the absolute value of the maximum derivative value (for that actuator) over all 0 10 20 30 40 50 60 trim points. The control authority of an actuator is approx- imated by taking the value of the derivative and multiplying it by a nominal change in either direction, taking into account Figure 5. Trim flap deflections the limits of the actuator by truncating the deflection at the ab- solute effector limit. The estimated nominal change for each Table 1. Nominal control deflection ∆ ω ∆ δ ∆ δ ∆ δ ∆ δ p i e f r ◦ ◦ ◦ ◦ 200 RPM 2 10 5 10 actuator, including propellers, wing-tilt, and surface deflec- tions is listed in Table 1.
Looking first at the pitch moment generation, the actuator effectiveness and control authority can be seen in figures 6 and 7. The top two plots in figure 6 show that the two banks of propellers on the two wings provide opposing pitch moments throughout the transition. In addition, although the effective- 0.8 0.6 0 10 20 30 40 50 60 70 -0.5 -1 Figure 7. Pitch rate control authority 0 10 20 30 40 50 60 70 active feedback control, it is worth noting how the individual tilt of each wing may be used to produce a pitch moment.
-1 0 10 20 30 40 50 60 70 Figure 8 shows that in hover and lower transition speeds, tilt actuation is not as effective a means with which to produce a pitch moment compared to the propellers, but as the speed -1 ◦ is increased and the wing-tilt angles drop below 30 it can be 0 10 20 30 40 50 60 70 used to great effect.
-0.5 -1 0 10 20 30 40 50 60 70 Figure 6. Pitch rate control effectiveness ness of the propellers is reduced when transitioning to forward flight, the propellers remain a viable means for providing pitch control, due to the vertical offset of the wings relative to the center of mass. The deflecting surfaces (elevons, flaps, and ruddervators), shown in the bottom three plots, do not become effective at generating a pitch moment until there is sufficient dynamic pressure and the wing tilt angle is reduced below ◦ 45 . The reduction in the effectiveness of the elevons at high speeds is due to the elevon servo malfunction during wind tun- nel testing that was mentioned earlier. The control authority plots in figure 7 reiterate the consistent pitch moment author- ity of the propeller but show that the deflecting surfaces may provide much more control authority at higher flight speeds.
Indeed, if the general quadratic trend that is present in the flaps and ruddervators was reflected in the elevons, it can be Figure 8. Wing tilt pitch control effectiveness and author- expected that the elevons would provide two to three times ity the pitch moment provided by the propellers in forward flight, possibly at a much lower energy cost. The asymmetric nature of the control authority of flaps 2 and 4 is due to the physical Moving on to the lateral direction, the roll moment effec- limits of the flap deflections.
tiveness and control authority plots are presented in figures Although, in this study, the wing tilt actuator is not used in 9 and 10 respectively. The effectiveness and control authority plots show that as the aircraft transitions from hover to for- ward flight, roll authority is transferred from the propellers to the deflecting surfaces. In both cases, left-right differential ac- tuation may be used to produce a roll moment, and adequate -1 blending of the propellers and surfaces can provide consistent 0 10 20 30 40 50 60 70 roll authority throughout the transition. The ruddervator sur- faces provide little in terms of roll moment production.
-1 0 10 20 30 40 50 60 70 -1 0 10 20 30 40 50 60 70 -1 0 10 20 30 40 50 60 70 -1 0 10 20 30 40 50 60 70 -1 0 10 20 30 40 50 60 70 -1 0 10 20 30 40 50 60 70 -1 0 10 20 30 40 50 60 70 Figure 11. Yaw rate control effectiveness -1 0 10 20 30 40 50 60 70 -1 0 10 20 30 40 50 60 70 Figure 9. Roll rate control effectiveness Figure 12. Yaw rate control authority blown-wing effect in hover, where the local increase in dy- namic pressure at the wing due to the propeller slip stream produces a lifting force acting primarily in the direction to- ward the rear of the aircraft. Figure 12 shows that the outside propellers produce a slightly larger yaw moment in hover. As the wing tilt angle decreases, the yaw moment sign of the pro- pellers switches as the thrust force of the propeller becomes Figure 10. Roll rate control authority the primary contributor to the yaw moment production.
In hover, the elevons provide the majority of the control au- The yaw moment production is particularly interesting, be- thority by directing the slipstream of the propeller and chang- cause the propeller yaw moment sign flips during the transi- ing the lift production of the wing. As the aircraft transitions, tion, as seen in figure 11. This phenomenon is due to the the outside propellers and the ruddervator surfaces provide the n m best means of yaw moment production, while the elevons and where x ∈ R is the system state, u ∈ R is the input, and p flaps produce a yaw moment due to the increase/decrease in y ∈ R is a set of outputs. We wish the output, y ( t ) , to track drag. Use of aerodynamic torque by differentially actuating a constant reference signal with zero steady-state error. The the clockwise and counter-clockwise propellers, like that of a error signal is defined by the difference between the reference quad rotor, does not appear to be an effective method of pro- and the output signal, e = r − y . To drive the steady state error ducing yaw as the torque is overpowered by the aerodynamic to zero, we raise the system type by adding integral action to ∫ effects of the blown wing. the error signal, x = − e , and augment the system with this i new state: As demonstrated by the preceding analysis, the transition of [ ] [ ] [ ] [ ] [ ] a tiltwing VTOL aircraft is a complex maneuver, in which ˙ x 0 C x D − I i i = + u + r . (21) the changing aerodynamics are compounded by the changing ˙ x 0 A x B 0 control effectiveness and authority of the control effectors.
The saving grace, for the LA-8, lies in the multitude of ac- Taking the derivative of (21) and noting that ˙ r = 0, we arrive tuators, where redundancy and overlap provide sufficient con- at the servo design model trol authority throughout the transition. The question remains, ˜ ˜ how to design a controller that takes into account the dynamic ˙ z = Az + Bv , (22) nature of the aircraft and optimizes its usage of the available n + p where the design model state vector z ∈ R is comprised of control effectors at every stage of the transition, while provid- the tracking error e and the time derivative of x , and the input ing safe and reliable trajectory tracking performance.
m v ∈ R is the time derivative of u , resulting in the state and The next section describes a uniform control architecture that T input vectors z = [ ˙ x , ˙ x ] and v = ˙ u . The servo design system i provides a straightforward approach to design a trajectory- matrices are defined as tracking controller for transitioning VTOL aircraft.
[ ] [ ] 0 C D ˜ ˜ A = , B = , (23) UNIFORM CONTROL FRAMEWORK 0 A B The control structure provides a unified approach to control The tracking controller is obtained by applying the LQR algo- design throughout the entire flight envelope. This approach rithm to (22) using the quadratic cost brings together several design strategies that are described ∫ below to build a method that is applicable across all flight ∞ T T J = z Qz + v Rv dt , (24) regimes (Ref. 17).
T T Robust Servo-mechanism Linear Quadratic Regulator ˜ ˜ where Q = Q > 0, R = R ≥ 0, ( A , B ) is stabilizable, and ˜ ( A , Q ) is detectable. The optimal control is v = − Kz where The robust servo-mechanism linear quadratic (RSLQR) struc- − 1 T ˜ K = R B P , and P is the unique positive definite solution to ture (Ref. 18) builds on the well-known linear quadratic opti- the algebraic Riccati equation mal control theory. When applied to the linearized dynamics of the aircraft at an equilibrium point, the linear quadratic reg- T − 1 T ˜ ˜ ˜ ˜ A P + P A − P BR B P = 0 , (25) ulator (LQR) forces the system to the origin, forming a type 0 system. When used to track a desired state of the system The resulting controller drives the tracking error and the state or reject a constant disturbance, the closed loop response will derivatives to zero but allows the system state to settle in at have a constant steady-state offset, and therefore integral er- nonzero values. The design model control, v = − Kz , is then ror control action is needed. RSLQR makes use of the internal integrated to obtain the control input model principal (Ref. 19), augmenting the state space repre- [ ] sentation by embedding a model of the class of signals (e.g., [ ] x i u = − K K . (26) step, sinusoidal) to be tracked, then applying optimal control i x x theory. The resulting control structure has two parts: a servo tracking controller for command following and state feedback where the feedback gain has both integral and proportional for stabilization. Together, they provide accurate command components.
tracking and predictable robust performance. In practice, if The cost function’s parameters Q and R are chosen in order a constant reference command is sufficient for obtaining the to elicit the desired response. A good rule of thumb is to desired performance, then the system type is raised to one, start with weights on just the integrator states. The resultant providing zero steady-state error command tracking. This ap- closed-loop system is shown in Figure 13.
proach has been successfully deployed in many aerospace sys- tems, both autonomous and piloted, and is quickly reviewed Virtual Controls and Relative Degree here for completeness.
Consider the linear time-invariant system The relative degree of an output to an input is the number of times that output must be differentiated with respect to time ˙ x = Ax + Bu , (20) such that the derivative is affected directly by that input. For y = Cx + Du , The virtual control is used as a reference for x feedback, and the difference is added to the x reference signal. The de- sign relies on proper time separation of the x and x closed- 1 2 loop dynamics to ensure good performance, which can be ac- complished by weighting the cost matrices Q and R appropri- ately. The resulting closed loop system is depicted in figure Figure 13. RSLQR control block diagram 14, which shows the general structure of the RSLQR with vir- example, the vertical velocity of an aircraft in hover has a rel- tual control feedback. The controller dynamics are defined ative degree of one to the thrust of the rotors, since the thrust as of the rotors directly affects the vertical acceleration. On the ν ν other hand, in cruise, the relative degree of the vertical veloc- − ˙ x = r − K K x − [ K K + K C + C ] x , i ν i ν ν ν x i (33) u u ity to the elevator is two. The elevator input causes a pitch u = − K x − K x , i x i moment which in turn changes the angle of attack increasing T the lift production and thus the vertical acceleration.
where, for this particular example, K = [ 0 , 1 ] and C = ν ν [ 0 , 0 , 1 ] . For proper dimensions, zeros are appended to the Virtual controls are introduced to handle the changing rela- ν 1 × 3 u right side of the K matrices such that K ∈ R and K ∈ tive degree of the aircraft in flight (Ref. 20). The use of vir- x x x 1 × 3 R .
tual controls is analogous to successive loop closure where the feedback loop is closed first on the faster dynamics and time- scale separation of inner and outer loops ensures good overall performance of the system. Here, the use of virtual controls is demonstrated within the RSLQR frame-work using a simpli- fied system representative of a pitch-for-speed type dynamics.
Consider the linear system ˙ x = ax , 1 3 ˙ x = bu , (27) Figure 14. RLSQR with virtual controls ˙ x = x .
3 2 Expanding into state space gives Performance Design and Control Allocation ˙ x 0 0 a x 0 1 1 ˙ x = 0 0 0 x + b u . (28) VTOL aircraft that transition from thrust-borne flight to the 2 2 more efficient wing-borne flight must traverse a large flight ˙ x 0 1 0 x 0 3 3 envelope where the effectiveness of a control input may vary Suppose that we would like to use the RSLQR algorithm to greatly, and there are inevitably regions of cross-over where track the first two states such that the tracking variables are there are multiple effectors capable of producing similar ac- defined as celerations. To simplify the design process it is helpful to [ ] x 1 0 0 break up the design into two parts: the performance, and the y = x . (29) 0 1 0 actuation. The performance being the desired response to a x command input in any part of the flight envelope, and the actu- Using system matrices defined in (28) and (29) and putting ation pertaining to how the control effort is distributed among them into the servo design model (22), we see that the result- the available effectors. This may be accomplished by apply- ing system is not stabilizable. This is remedied by including ing theory presented in (Ref. 21), which is briefly described the kinematic state x as a virtual control, ν = x , and defining 3 3 here for the linear quadratic case.
T a new state vector ¯ x = [ x , x ] . The resulting design model is 1 2 Consider the linear system then [ ] [ ] [ ] [ ] [ ] ˙ x 0 0 x 0 a u 1 1 ˙ x = Ax + B u , (34) u = + , ˙ x 0 0 x b 0 ν 2 2 (30) n × n n × m n [ ] [ ] with A ∈ R , B ∈ R , x ( t ) ∈ R is the state, and u ( t ) ∈ u 1 0 x 1 m y = . R is the control input. Assume B does not have full column u 0 1 x rank, implying that it can be factorized as Applying the RSLQR formulation of Eq. (21) results in the B = B B , (35) feedback law [ ] [ ] u μ u x i = − K , (31) n × k k × m ν ¯ x where B ∈ R and B ∈ R both have rank k for some μ k < m . This gives way to an alternative description where the control gain matrix has the form [ ] u u K K ˙ x = Ax + B μ , μ i x K = . (32) (36) ν ν K K μ = Bu , i x k arise when using different types of effectors and helps target where μ ( t ) ∈ R may be interpreted as the total control effort specific dynamics that need to be more responsive than others.
of the effectors, most naturally thought of in our application Once the desired performance is achieved, the second step al- as the total accelerations. Since k < m , B and B have a non- u lows control allocation solutions to be evaluated and tuned trivial null space in which u may be perturbed without produc- using the weighting matrix W . Because the response is the ing an acceleration μ . Simply put, there are multiple ways to same, provided no effectors are saturated, an apples-to-apples actuate the control input u that produce the same commanded comparison can be made on a per W basis.
total acceleration μ , and therefore there are built-in redundan- cies in the control actuation.
LONGITUDINAL CONTROL DESIGN When designing optimal feedback control, we explore two ap- proaches. The first poses the optimal control policy in terms The uniform control approach described above is applied here of the input u , and the second, poses the optimal control pol- to the longitudinal dynamics at a non-turning trim condition.
icy in terms of μ , where the solution is then mapped onto u For non-turning flight ( ˙ ψ = 0), the longitudinal dynamics sim- by solving a quadratic optimization problem. We describe the plify to ¯ two methods below.
˙ X ( x , u ) ¯ u lon m ¯ ˙ ¯ w a + Z ( x , u ) g lon Method 1. Consider the linear system description (34). De- m = , (41) ˙ q M ( x , u ) termine u ( t ) by solving lon J y ˙ θ q ∫ ∞ T T min [ x Qx + u R u ] dt , (37) T u where the state vector is given by x = [ ¯ u , ¯ w , q , θ ] , and the lon u ( t ) input u is the set of effectors listed in eq. (17). Given the trim T condition ( x , u ) the Jacobian matrices are computed such 0 0 where Q ≥ 0, and R = R > 0. This is the standard linear u u that quadratic regulator (LQR) problem. If the pair ( A , B ) is stabi- u 1 1 1 1 lizable and the pair ( A , Q ) is detectable, the optimal control ¯ ¯ ¯ ¯ X X X X ¯ u ¯ w q θ m m m m − 1 T 1 1 1 1 is u = − R B Px , where P is the unique positive definite so- ¯ ¯ ¯ ¯ Z Z Z Z u u ¯ u ¯ w q θ m m m m A = , (42) lon 1 1 1 1 lution to the algebraic Riccati equation M M M M ¯ u ¯ w q θ J J J J y y y y 0 0 1 0 T − 1 T A P + PA − PB R B P = 0 . (38) u u 1 1 ¯ ¯ X X ω r δ m m s 1 1 ¯ ¯ Z Z ω δ m r m s B = , (43) lon 1 1 Method 2. Consider the linear system description (36). De- M M ω r δ J J s y y termine μ ( t ) by solving 0 0 ∫ ∞ where the subscript denotes the variable that the derivative of T T min [ x Qx + μ R μ ] dt , (39) μ the force ( ¯ X , ¯ Z ), or moments ( M ) is taken with respect to.
v ( t ) As stated previously, the horizontal and vertical velocities T where Q ≥ 0, and R = R > 0. Again, solving the LQR μ μ ( ¯ u , ¯ w ) are the variables to be tracked and the pitch rate, q , is 2 to be regulated, which leaves the pitch angle, θ , to be treated problem with ( A , B ) stabilizable and ( A , Q ) detectable, the μ − 1 T as a virtual control input.
solution to (39) is μ = − R B Px . Then u ( t ) is determined μ μ by solving The performance design model therefore has states ¯ x = lon T T min u Wu , [ ¯ u , ¯ w , q ] , and the linear design system is u ( t ) (40) subject to μ = Bu , ¯ ˙ ¯ x = A ¯ x + μ , lon lon lon lon (44) y = ¯ x , T lon lon where W = W > 0. The solution to this quadratic problem is − 1 T − 1 T − 1 the weighted generalized inverse u = W B ( BW B ) μ .
¯ where A is the upper left 3 × 3 submatrix of A cor- lon lon responding to the performance states, and μ is the gen- The main result of (Ref. 21) states that, assuming the matrices lon − 1 T − 1 ∗ eral longitudinal control accelerations such that μ = R and R are related such that BR B = R , then u and lon u μ u μ ∗ T [ ¯ a , ¯ a , α ] . Applying the RSLQR structure to the system μ are the optimal controls associated with design 1 and 2, x z q ∗ ∗ (44), the control design matrices are respectively, and μ = Bu and the corresponding trajectories [ ] [ ] are the same. Further, if for a given R and W the matrix R μ u 0 I 0 T − 1 T − ˜ ˜ is chosen as R = W + B [ R − ( BW B ) 1 ] B , the control A = , B = . (45) u μ lon lon ¯ 0 A I lon laws for both design methods will be the same.
The components of the Q and R matrices are selected to obtain Thus, by splitting up the control design process, as in method the desired tracking performance. The desired control accel- two, the control engineer designs the aircraft response to a erations are then command, before determining how that response would be [ ] [ ] physically implemented via the available effectors. Using x i lon lon lon μ = − K K . (46) lon i x general acceleration as inputs alleviates scaling issues that ¯ x lon The control inputs are determined using the weighted pseudo- LATERAL CONTROL DESIGN inverse The lateral dynamics in non-turning flight simplify to the fol- [ ] u lowing set of equations − 1 T − 1 T − 1 ¯ ¯ ¯ = W B ( B W B ) μ , (47) lon lon lon lon θ des 1 ¯ Y ( x , u ) ˙ ¯ v m T ( J N ( x , u ) − J L ( x , u )) xz z lat ˙ p 2 where W = W is a positive definite weighting matrix, and J − J J x z xz = , (51) − 1 ¯ ˙ r B is the input Jacobian augmented with the pitch angle ( J N ( x , u ) − J L ( x , u )) lon 2 x xz J − J J x z xz ˙ φ derivatives p + q sin φ tan θ + r cos φ tan θ 1 1 1 ¯ ¯ ¯ X X X ω δ θ r T m m s m where the lateral state vector is given by x = [ ¯ v , p , r , φ ] , lat 1 1 1 ¯ ¯ ¯ ¯ Z Z Z B = ω θ . (48) lon r δ m m s m and the input u is the set of inputs defined in eq (17). Given 1 1 1 M M M ω δ θ r s J J m y y the trim condition ( x , u ) the state space matrices can be com- 0 0 puted and are listed in the appendix.
The control inputs are executed while the desired pitch angle Of the lateral dynamics, we wish to track the lateral velocity ¯ v is fed back, differenced with the measured pitch angle, and as well as the body roll and yaw rates, p and r . The roll angle, used to build the pitch rate reference signal.
φ , is used as virtual control input.
Denoting the pseudo inverse matrix M = lon The uniform control design process is then repeated, with − 1 T − 1 T − 1 ¯ ¯ ¯ W B ( B W B ) , it can be decomposed into lon lon lon the lateral performance model state vector defined as ¯ x = lat the submatrices corresponding to the input and pitch angle, T [ ¯ v , p , r ] and the general control accelerations as μ = lat T [ ] [ ¯ a , α , α ] .
y p r lon M u M = . (49) lon Applying the RSLQR algorithm, the lateral integral and state M θ lat lat feedback gains, K and K , are computed after selecting the i x appropriate Q and R matrices.
Combining the feedback gains, virtual control, and control al- location, the longitudinal control law is given by the following It follows that the control inputs are determined by applying linear system, (the lon subscript is dropped in the interest of the weighted pseudo-inverse using the roll angle augmented space), matrix, ¯ B , defined in the appendix.
lat The lateral pseudo-inverse matrix is M = lat − ˙ x = − K M K x − ( K M K + K C + C ) x − 1 T − 1 T − 1 i θ θ i i θ θ x θ θ ¯ ¯ ¯ W B ( B W B ) , and can be decomposed into lat lat lat + Fr , (50) submatrices corresponding to the input and roll angle u = − M ( K x + K x ) , u i i x [ ] lat M u M = .
lat M φ T where r ( t ) = [ ¯ u , ¯ w ] , is the horizontal and vertical veloc- d d ity commands; F maps the command inputs to the integrated [ ] The lateral-directional reference commands include the de- T 1 0 0 sired lateral velocity in the heading frame, ¯ v ( t ) and the turn states, and is given by F = ; C pulls the pitch d θ 0 1 0 rate ˙ ψ ( t ) , such that the lateral-directional reference is d angle from the state vector, and is defined as C = [ 0 , 0 , 0 , 1 ] ; θ T and C selects the tracking states from the state vector, and is r ( t ) = [ ¯ v ( t ) , ˙ ψ ( t )] . (52) lat d d defined as The turn rate command is fed forward, via G in figure 16, 1 0 0 0 to directly command a roll acceleration proportional to the C = 0 1 0 0 .
trimmed forward velocity ¯ u , and also used to construct the 0 0 1 0 body r reference command. In figure 16 the F and G matrices are defined as The corresponding block diagram of the controller is shown in figure 15. 1 0 0 0 F = 0 0 , G = 0 ¯ u 0 1 0 0 The lateral controller is described mathematically by the fol- lowing dynamic system in which the lat subscript is omitted in the interest of space.
− ˙ x = − K M K x − ( K M K + K C + C ) x i φ φ i i φ φ x φ φ Figure 15. Longitudinal control block diagram +( F + K M G ) r , (53) φ φ u = − M ( K x + K x − Gr ) .
u i i x going into the longitudinal and lateral controller is [ ] ¯ v ( t ) + K ¯ e ( t ) d ¯ e r ( t ) = , (54) ψ + K e ( t ) d ψ ψ where ¯ e ( t ) is the inertial position error expressed in the head- ing frame, K is a tunable feedback gain, and e ( t ) is the head- ¯ e ψ ing angle error with associated gain K .
ψ Figure 16. Lateral control block diagram The results of the trajectory tracking simulation are presented in figures 17–22, which show good tracking capabilities in all TRANSITION TRAJECTORY TRACKING flight regimes while performing the transition from hover to forward flight and back to hover while climbing, turning and The control architecture is demonstrated by simulating a flight descending. The position error in figure 18 shows a maxi- in which the aircraft tracks a desired trajectory. The trajectory, mum path deviation of 14 ft over the course of the flight, with pictured in figure 17, starts in a stationary hover state, begins the largest deviations occurring at the inflection points of the a vertical ascent, which is followed by a constant radius as- commands.
cending transition. The aircraft is accelerated to a forward flight speed of 55 ft/s then levels out for a short period at the cruise velocity. The cruise portion of the trajectory is followed by a constant radius helical descent and decelerate to hover.
0 20 40 60 80 100 120 140 160 180 200 Figure 18. Position error The heading frame velocity and turn rate commands are pre- sented in figure 19, where it is shown that the scheduled con- troller sufficiently tracks the command variables as it per- Figure 17. Flight path and ground track of the trajectory forms the transition. The largest deviations occur at the in- flection points of the commands, most notably, 110 seconds into the simulation where the aircraft begins to descend and The transition controller is implemented by applying the uni- decelerate, a known troublesome part of the transition, as any form control architecture at 13 equilibrium points at 5 ft/s in- change in wing tilt angle results in large changes in the lifting crements between 0 and 60 ft/s. The points were taken from force generated.
the steady level flight trim map described in a previous sec- tion.
The control actuation throughout the flight is shown in fig- ures 20–22. The speeds of the propellers, which provide lift The control effector weighting matrices were set such that the in hover and thrust in forward flight while also serving as roll, primary control effectors are the propellers, elevons, and rud- pitch, and yaw effectors in all flight regimes, are shown in fig- dervators, as well as the roll and pitch angle virtual controls. A ure 20. In figure 21, the control surface actuation is shown, large weight was placed on the flaps in order to use them as lit- including the wing tilt angles. The wing tilt and flap deflec- tle as possible as control effectors. The weightings were tuned tion plots show that they are primarily driven by the trim table to avoid effector saturation in all flight regimes, which neces- schedule, with only slight actuation of the flaps to assist in roll sitated increasing the weight of the elevons at higher flight torques in forward flight. This was by design, since the wing speeds due to decrease in the elevon effectiveness represented tilt was omitted from the control effectors when designing the in the LA-8 aerodynamic model.
controller and a large weighting was used to limit the usage of The desired forward flight speed was used to schedule the con- the flaps.
troller gains as well as the feed forward propellers, flaps, and As the aircraft transitions to forward flight, elevons and rud- wing tilt angle settings.
dervators are increasingly used as control effectors. The The heading frame velocity and turn rate commands are aug- elevons are used as both roll and pitch effectors, while the rud- mented with the position error, expressed in the heading dervators are used for yaw moment generation. The roll and frame, and heading angle error, to more accurately track the pitch angles, shown in figure 22, are driven by the controller inertially referenced trajectory. The complete reference signal as virtual controls. The plot shows the roll angle increasing 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 -2 -10 -4 -20 -6 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 -2 -4 0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 0 0 -10 -20 -5 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200 Figure 19. Heading frame command tracking Figure 21. LA-8 control surface deflections as the aircraft speed is increased while maintaining the con- stant radius turn. It then levels off in the cruise portion before increasing in the opposite direction as the aircraft begins its descending transition. The pitch angle is used to decrease the wing angle of attack and point the thrust vector of the pro- 0 -10 pellers forward while accelerating. In the descent, the aircraft -20 is pitched nose up to decelerate.
-30 -40 0 20 40 60 80 100 120 140 160 180 200 Figure 22. Roll and pitch angles off and landing aircraft that must transition from thrust-borne flight to lift-borne flight, and, in doing so, must transition be- tween different control effectors. The formulation of the dy- 0 20 40 60 80 100 120 140 160 180 200 namics in the heading frame allows for the use of uniform commands to be used as reference inputs throughout all flight regimes. The uniform framework augments the robust ser- vomechanism linear quadratic control with virtual controls and splits the control design process into the two distinct steps of performance and control allocation. The effectiveness of the control process was demonstrated on a tandem tiltwing 3500 distributed electric propulsion VTOL aircraft. The trajectory- 0 20 40 60 80 100 120 140 160 180 200 tracking simulation results demonstrated that the controller, designed using the robust uniform method, provided accu- Figure 20. LA-8 propellers speeds rate trajectory tracking performance while transitioning the aircraft from hover to forward flight and vice versa.
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APPENDIX Lateral Directional State Space Matrices 1 1 1 1 ¯ ¯ ¯ ¯ Y Y Y Y ¯ v p r φ m m m m 1 1 1 1 ( J N − J L ) ( J N − J L ) ( J N − J L ) ( J N − J L ) xz ¯ v z ¯ v xz p z p xz r z r xz φ z φ ¯ ¯ ¯ ¯ J J J J A = lat − 1 − 1 − 1 − 1 ( J N − J L ) ( J N − J L ) ( J N − J L ) ( J N − J L ) x ¯ v xz ¯ v x p xz p x r xz r x φ xz φ ¯ ¯ ¯ ¯ J J J J 0 1 cos φ tan θ 0 1 1 ¯ ¯ Y Y ω δ p s m m 1 1 ( J N − J L ) ( J N − J L xz ω z ω xz z ¯ p p ¯ δ δ ) s s J J B = lat − 1 − 1 ( J N − J L ) ( J N − J L ) x ω xz ω x xz ¯ p p ¯ δ δ s s J J 0 0 Lateral Directional Control Allocation Matrix 1 1 1 ¯ ¯ ¯ Y Y Y ω δ φ p s m m m 1 1 1 ¯ B = ( J N − J L ) ( J N − J L ( J N − J L ) xz ω z ω xz z xz φ z φ lat ¯ p p ¯ δ δ ) ¯ s s J J J − 1 − 1 − 1 ( J N − J L ) ( J N − J L ) ( J N − J L ) x ω xz ω x xz x φ xz φ ¯ p p ¯ δ δ ¯ s s J J J