Abstract
This paper treats numerical methods for an efficient prediction of the rotorcraft emergency procedures after engine failures. The analytical means of compliance for the Category-A requirements can be used for the type certification of the transport-category rotorcraft when their fidelities are approved by the civil airworthiness authority. However, the most promising trajectory optimization approaches to the Category-A maneuver analyses typically suffer from a dimensionality problem when a high-fidelity math model is adopted. To cope with such difficulties, the paper proposes new techniques, where the system states except the initial ones and all dynamic constraints are removed from the resultant nonlinear programming problem. For these proposes, the controls are parameterized using the Hermit splines with the local support and efficient recursive formulas to predict the constraint-function Jacobians are derived. The efficiency of the proposed techniques is compared with that using the pseudo-spectral collocation method. In addition to an autorotational descent maneuver, four Category-A procedures for the continued takeoff, rejected takeoff, continued landing, and balked landing maneuvers are analyzed with varying the engine-failure conditions and with a suitable consideration on the pilot-delay time to validate the usefulness of the proposed methods.
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Abbreviations
- Cat-A:
-
Category-A
- FAA:
-
Federal aviation administration
- AC:
-
Advisory circular
- NOCP:
-
Nonlinear optimal control problem
- NLP:
-
Nonlinear programming problem
- NAE:
-
Nonlinear algebraic equation
- SNOPT:
-
Sparse nonlinear optimizer
- OEI:
-
One-engine-inoperative
- KKT:
-
Karush–Kuhn–Tucker
- DDSA:
-
Direct dynamic-simulation approach
- SQP:
-
Sequential quadratic programming
- LGL:
-
Legendre–Gauss–Lobatto
- CTO:
-
Continued takeoff
- RTO:
-
Rejected takeoff
- CL:
-
Continued landing
- BL:
-
Balked landing
- TDP:
-
Takeoff decision point
- LDP:
-
Landing decision point
- \(J\) :
-
Objective function
- \(\phi (*)\) :
-
Boundary objective function
- \(f_{obj}\) :
-
Integral objective function
- \({\mathbf{f}}\) :
-
Forcing function in the system dynamics
- \({{\boldsymbol{\psi}}}(*)\) :
-
Equality constraints
- \({\mathbf{g}}(*)\) :
-
Inequality constraints
- \({\mathbf{x}}\) :
-
State vector of system dynamics
- \({\mathbf{u}}\) :
-
Control vector of system dynamics
- \(m\) :
-
Number of system controls
- \(n\) :
-
Number of system states
- \(L_{e}\) :
-
Number of equality constraints
- \(L_{i}\) :
-
Number of inequality constraints
- \(t\,\,(t_{0} ,\,\,\,t_{f} )\) :
-
Time variables (initial and final)
- \({\mathbf{c}}(*)\) :
-
Continuity condition
- \(J_{N,M}\) :
-
Integral part of objective function
- \(M\) :
-
Number of horizons
- \(N\) :
-
Number of collocation nodes
- \({\overline{\mathbf{y}}}\) :
-
Design variables for nonlinear programming
- \({\overline{\mathbf{v}}}\) :
-
Control vector of all horizons
- \({\mathbf{v}}\) :
-
Control and control derivatives vector of one horizon
- \(\overline{{\mathbf{h}}}\) :
-
Equality constraints for nonlinear programming
- \({\overline{\mathbf{g}}}\) :
-
Inequality constraints for nonlinear programming
- \({{\boldsymbol{\mu}}},\,\,{{\boldsymbol{\gamma}}}\) :
-
Lagrange multipliers of equality and inequality constraints
- \(L(*)\) :
-
Lagrangian function
- \(\tau\) :
-
Nondimensionalized time variable
- \(K\) :
-
Order of control derivatives
- \(\frac{\partial *}{{\partial {\mathbf{x}}_{0} }},\frac{\partial *}{{\partial {\mathbf{v}}_{k} }},\frac{\partial *}{{\partial t_{f} }}\) :
-
Jacobian matrices with respect to states, control parameters, and final time
- \(\delta\) :
-
Kronecker-delta function
- \({\mathbf{u}}^{(l)}\) :
-
\(l\)-Th time derivative of control \(( = d^{l} {\mathbf{u}}/dt^{l} )\)
- \(I_{jk}\) :
-
Integral approximation matrix
- \(\varepsilon\) :
-
Tolerance for pseudo-spectral integrator
- \(u,\,\,w\) :
-
Longitudinal and vertical velocity
- \(x,\,\,h\) :
-
Horizontal distance and height
- \(P_{*}\) :
-
Power
- \(\Omega\) :
-
Main-rotor RPM
- \(C_{T}\) :
-
Thrust coefficient
- \(C_{P}\) :
-
Power coefficient
- \(\alpha\) :
-
Main-rotor tip-path plane angle
- \(m_{h}\) :
-
Helicopter mass
- \(R\) :
-
Main-rotor radius
- \(\sigma\) :
-
Solidity of main rotor
- \(f_{e}\) :
-
Equivalent flat plate area
- \(C_{d0}\) :
-
Rotor drag coefficient
- \(a\) :
-
Lift curve slope
- \(I_{R}\) :
-
Main-rotor moment of inertia
- \(H_{R}\) :
-
Main-rotor height
- \(g\) :
-
Gravity coefficient
- \(\tau_{P}\) :
-
Time constant
- \(V_{toss}\) :
-
Takeoff safety speed
- \(\tilde{u}\) :
-
Nondimensionalized velocity
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Acknowledgements
This work is supported by the Korea Agency for Infrastructure Technology Advancement (KAIA) grant funded by the Ministry of Land, Infrastructure and Transport (Grant 20CHTR-C139566-04).
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Nam, Y.H., Kim, CJ., Lee, S.H. et al. Direct Dynamic-Simulation Approach to Trajectory Optimization for Rotorcraft Category-A Maneuver Procedures. Int. J. Aeronaut. Space Sci. 22, 648–662 (2021). https://doi.org/10.1007/s42405-020-00322-2
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DOI: https://doi.org/10.1007/s42405-020-00322-2