SDRE CONTROL WITH NONLINEAR FEEDFORWARD COMPENSATION FOR A SMALL UNMANNED HELICOPTER
Alexander Bogdanov, Eric Wan OGI School of Science and Engineering, OHSU 20000 NW Walker Rd, Beaverton, Oregon 97006 In this paper we report on the state-dependent Riccati equation (SDRE) control of a small unmanned helicopter for autonomous agile maneuvering. SDRE control requires reformulation of the vehicle dynamics into a pseudo-linear form. For a helicopter application, however, this results in a number of terms not accounted in the SDRE design. To overcome this problem, we employ a nonlinear feedforward compensator that is designed to match the vehicle response to the model used in the SDRE design. This paper provides new control results and additonal details based on work described previously by Bogdanov, et al. 1
NOMENCLATURE
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vehicle velocities in longitudinal, lateral and vertical directions vehicle angular (roll, pitch and yaw) velocities Euler angles (roll, pitch and yaw) vehicle position in inertial frame longitudinal and lateral cyclic controls main rotor collective and tail rotor collective control inputs wind velocities in longitudinal, lateral and vertical directions helicopter mass moments of inertia around rolling, pitching and yawing axes main rotor thrust, tail rotor thrust rotor thrust coefficient rotor torque coefficient inflow ratio advance ratio normal airflow component speed of the rotor blade tip rotor blade drag coefficient air density main rotor disk area main rotor radius main rotor blade lift curve slope rotor solidity ratio coeff. of non-ideal wake contraction commanded and actual main rotor speed tail rotor speed tail rotor gear ratio
P Senior Research Associate, AIAA member,
[email protected] Q Associate Professor, AIAA member,
[email protected]
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main rotor hub height above center of gravity (c.g.) tail rotor hub height above c.g. tail rotor hub location behind c.g. vertical tail blockage factor fuselage drag and main rotor induced force in longitudinal direction fuselage drag, main rotor induced force, tail rotor induced force and vertical tail force in lateral direction fuselage drag, main rotor induced force, horizontal tail force in vertical direction main and tail rotors induced rolling moment rolling moment from the vertical tail main and tail rotor induced pitching moment rolling moment from the horizontal tail yawing moment from the tail rotor and vertical tail engine torque engine power and max. engine power total moment of inertia as measured on the main rotor shaft proportional gain of the engine governor integral gain of the engine governor longitudinal and lateral flapping angles scaling factor of flap response to speed variation main rotor hub torsional stiffness effective time constant of a rotor with a stabilizer bar longitudinal and lateral cyclic to flap gains
INTRODUCTION We are developing nonlinear controllers to provide automatic vehicle control for a helicopter capable of per-
1 A MERICAN I NSTITUTE OF A ERONAUTICS AND A STRONAUTICS
forming a broad spectrum of maneuvers and in-flight situations. Previous work has been reported in Wan, et. al and Bogdanov, et. al. 1, 6 One technique that has shown considerable promise is called State-Dependent Riccati Equation (SDRE) control. This involves reformulating the system dynamics into a pseudo-linear form and then iteratively solving a Riccati equation on-line, providing state-feedback optimized around the system state at each time step. Details of this approach will be provided later in this paper. As pointed by Bogdanov, et al,1 direct application of the SDRE technique to a helicopter model results in the necessity to further simplify the vehicle model due to a number of terms that cannot be effectively presented in state-dependent form. Thus, the actual response of the vehicle may significantly differ from the one expected from SDRE. To overcome this problem, a fixed or scheduled “trim” control can be added (usually to provide altitude and heading hold in hover or level flight). However it is more appropriate to design a feedforward control that compensates for the differences between the actual vehicle model and the model used in the SDRE design. This approximately matches the vehicle response to the one used by the SDRE. In this paper, we present further details on the design of the feedforward compensator for a small helicopter. For simulation purposes and control design, we have used a full analytic nonlinear dynamic model of the helicopter,3 consisting of a six-degree-of-freedom, quaternion model augmented with simplified analytic models for the rotor forces, torque, and thrust, flapping dynamics, horizontal stabilizer and vertical tail forces and moments, fuselage drag, and actuator states. For actual flight tests, we use a instrumented X-Cell60 acrobatic helicopter, which is a popular platform among competition R/C pilots for its capability to perform aerobatic maneuvers. The research vehicle (Figure 1) is a clone of a vehicle developed by MIT.4 The custom avionics package includes an inertial measurement unit (IMU) with three gyroscopes and three accelerometers, a GPS receiver, a barometric altimeter and a triaxial magnetoresistive compass. Wireless communications and an on-board microprocessor with compact flash memory is included.
SDRE AND FEEDFORWARD DESIGN In brief, the SDRE approach2 involves manipulating the vehicle dynamic equations
q rts jvuxwzy q r {Gr)|
(1)
into a pseudo-linear form (SD-parameterization), in which system matrices are explicit functions of the current state:
q rts j u~}y q r |$q rv y q r |{ r
(2)
Fig. 1 X-Cell 60 helicopter with sensors and avionics box
A standard Riccati Equation can then be solved at each time step to design the state feedback control law on-line (a 50 Hz sampling rate is used in the flight experiments). The SDRE regulator is specified as
9 {Gr uv
j y q r|$ y q r| y q r)|( vy q r)|$q r
(3)
where y q r | is a steady state solution of the difference Riccati equation, obtained by solving the discrete-time algebraic Riccati equation using state-dependent matrifX ces }y q r | and y q r | , which are treated as being constant. For tracking problems with the desired state qr , the SDRE control can be implemented as
fX { r uvy q r | y q r q r | vy q r | r
(4)
For helicopter control, we define the observable states 9 to correspond to the standard 12 states of a 6-DoF rigid
q r g
Y
| u y 9, body model: and vector of controls {r uy !#$ &!# | corresponding to rotor blade pitch angles. Unfortunately, reformulation of the nonlinear dynamic equations specific to the helicopter into a state-dependent form does not yield an exact parameterization due to various approximations, disturbances and terms which can not be effectively presented in the pseudo-linear form. In this case, one may expect the dynamic response of the actual and approximated systems to differ. To overcome this problem, we split the control design into two parts. First, we design the SDRE control for the approximate stateX dependent system. Let us denote the computed SDRE control as { r . Next, we design a feedforward compen r , which provides sator {
X X wzy q r { r {G r)| }y q r)|#q r y q r|{ r
(5)
The control law is then described as a combination of the SDRE X control and the feedforward compensator, { r u { r { r , as is graphically illustrated in Figure 2. The purpose of the feedforward compensator can be viewed as matching the vehicle’s response (Eqn. (1)) to the response of the dynamic system (Eqn. (2)) to correct for inherent inaccuracies in a state-dependent reformulation
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of the vehicle dynamics. The feedforward control provides system performance close to the potential of the ideal SDRE design.
state for the purpose of deriving a control law as (see Gavrilets, et al,3 Bogdanov, et al1 )
kj ' i % k3j u¬ o U\p ® ] @ (8) ? BADC Kj ' Kj +' i 2 @ Kj u o U\p @ ? ? BA¯C BADC (9) p where Kj and k_j are the longitudinal and lateral flap 3± ± iVl yB¶ Y· "!#
