# A Novel Inverse Simulation Method of Helicopter Maneuvering Flight
## 1 Introduction  
The conventional simulation solves the helicopter flight state response to selected control inputs, to generate a time history of the flight path. The inverse simulation conducts the reverse of that process, where the time history of a flight path is defined first, and the time histories of the flight states and control inputs are then solved using a series of numerical algorithms. It is an effective method for studying the maneuverability of aircraft [1]. This is because by using the inverse simulation to obtain the control inputs, we can check if the control inputs exceed the limits of the aircraft, and can also check if some key variables such as power, speed, angular velocity, etc., exceed the usage limits of the aircraft. If they exceed, it indicates that the aircraft cannot fly this maneuvering action, and needs to change the definition of the trajectory. If a certain limit is reached, it indicates that the aircraft's ultimate maneuvering ability has been reached. The inverse simulation can also be used in the conceptual design of aircraft, to identify deficiencies in the design proposals during preliminary assessment [2]. Some other aspects of research, such as the flying quality evaluation [3], flying trajectory optimization [4], maneuvering strategies optimization [5], and flight dynamics characteristics [6], etc., can be conducted using the inverse simulation. This is a very helpful technique for aircraft design, and application in the aviation field. The first inverse simulation was conducted in the 1930s, when a simple linearized aircraft model to study the effect of gusts was inverted [7]. Later, [PERSON] [8] discussed simple inverse simulation methods applied to problems such as achieving a prescribed response. Subsequently, an energy-based technique [9] that investigated the ability of helicopters to maneuver was proposed. However, the techniques required to achieve inverse simulation only came into early use in practical engineering in the mid-1980s, when a practical method of inverse simulation for studying helicopter agility was developed by [PERSON] and [PERSON] [10]. Their method was capable of simulating a wide range of maneuvers, and a wide range of practical problems [2, 11]. To date, conventional inverse simulation methods can be mainly classified as differentiation methods, integration methods, and two time-scale methods [12]. The differentiationmethod discretizes a maneuver based on the numerical differentiation method, and solves the model at each time step using an iterative scheme. It has been adopted by many researchers [13; 14; 15], and has been updated and improved [16; 17; 18]. The integration method integrates the helicopter model through a time interval that is considerably greater than the integration time step while these control inputs are fixed [19; 20; 21]. It has been adopted to assess the reliability of simplified rotorcraft models in the evaluation of maneuvering potential [22]. Helicopters undergoing unsteady maneuvers can also be analyzed using this method [23]. The two time-scale methods attempt to reduce the helicopter system by considering that the translational equations of motion are solved, with the assumption that the rotational equations have attained a dynamic balance, with the angular velocity of zero [24; 25]. This article presents a novel method of inverse simulation that names a degeneration method, by constructing a transitional vector, to prescribe the time histories of an output, and, meanwhile, to express the state vector of the dynamics system. Thus, the first-order differential equations can be degenerated into nonlinear equations in the process of inverse simulation. This degeneration method will be applied to a helicopter system. The new idea behind the method is that the quantitative relationships between the flight state and control input, as well as the prescribed output (i.e., the flight path), can be analyzed (or deduced) almost directly, without any complex numerical algorithm.