二自由度动力学Simulink模型(包括详细模型分析)

Miniature helicopters are naturally more agile than their full-scale counterparts [15]. This trait results from physical

scaling effects and specific design features found in helicopters of that size. Following physical scaling rules, as

the vehicle size decreases, the moments of inertia decrease with the fifth power of the scale factor, while the rotor

thrust nominally decreases proportionally to the vehicle mass, i.e., with the third power. Miniature helicopters can

easily have a high thrust-to-weight ratio (it can easily reach values as high as 2 to 3), which tends to accentuate

the scaling effects. Moreover, the rotor heads of miniature rotorcraft are also relatively m ore rigid than those in

full-scale helicopters, allowing for large rotor control moments. Also, many such vehicles can pro duce negative

thrust, allowing sustained inverted flight. Combined, these effects and design features allow for large and fast

angular excursions (up to 200 deg/sec), and large bi-directional load factors, which are the main ingredients of

aerobatic flight. When flown by expert pilots, miniature helicopters can execute maneuvers that outperform the

automatically execute aggressive maneuvers enables using the broadest range of behaviors when performing tasks

such as the ones described above. Pre-programmed maneuvers can be incorporated into online motion planning

techniques such as the “maneuver automaton” [6]. In a fully autonomous setting, such capabilities are key to a

timely and effe ctive reaction to unforeseen obstacles.

There exists an extensive body of literature on the dynamics of full-scale helicopters. Step-by-step procedures for

developing first-principles dynamic models have been devised and published [22, 2, 25]. The models used in full-

scale helicopter simulators are high-order and contain a large number of parameters that often cannot be measured

directly. Moreover, once developed, models require extensive validation and refinement until they can predict the

vehicle dynamic behavior accurately. Applying detailed first-principle modeling techniques, is thus not a trivial

task, and may not be warranted given the differences between full-scale and miniature helicopters. Helicopters

evolve in a wide range of aerodynamic conditions. Complex interactions take place between the rotor wake and the

fuselage or tail. In miniature helicopters, these subtle effects tend to be “overpowered” by the large rotor forces

and moments that follow quasi-instantaneously the rotor control inputs. To the authors’ knowledge, no examples

of models covering aerobatic conditions exist in the literature. In fact, few, or maybe even none of the full-scale

helicopters have the control authority of most agile small-scale helicopters.

Modeling techniques based on system identification have been used to derive linear models for control design,

study the vehicle flying qualities, and for the validation and refinement of detailed non-linear first-principle models.

Frequency domain identification methods such as CIFER [21, 26], were used to derive relatively simple linear

parameterized models that accurately capture the vehicle dynamics around specific operating points.

namics were necessary. The coupled rotor and stabilizer bar equations can be lumped into one first-order effective

rotor equation of motion (for both the lateral and longitudinal tip-path-plane flapping). The linear models accu-

rately capture the vehicle dynamics for a relatively large region around the nominal operating point. The model

accurately predicts the vehicle angular response for aggressive control inputs for the full range of angular motion.

Subsequently, comparing results obtained for the larger Yamaha R-50 (150lb, 5ft rotor radius), and smaller MIT’s

X-Cell .60 (17lb, 2.5ft rotor radius), he showed that the former is dynamically similar to a full-scale helicopter; its

characteristics relate to those of a full-scale vehicle through Froude scaling rules. The latter, on the other hand,

belongs to an entirely different dynamic class; it relates to the larger vehicle via Mach scaling rules, which predict

a dramatic increase in agility with reduction of the vehicle size.

Following the work by Mettler, LaCivita et al. [11] developed a technique that makes use of the frequency responses

speeds [10].

The goal of this paper is to describe the analytical model used in our simulation model; providing the rationale for

the derivation of the equations of motion, as well as documenting the experimental methods used for the estimation

of the model’s parameters.

An essential aspect of our effort was the availability of a fully instrumented aerobatic vehicle, which allowed us

to collect accurate flight-test data. The helicopter selected for our effort is an X-Cell .60 SE manufactured by

Miniature Aircraft [20]. This helicopter is popular among expert hobby pilots for its extreme maneuverability. Its

characteristics result from a stiff rotor head, allowing the transmission of large control moments from the rotor to

the fuselage, a large thrust-to-weight ratio, and a fast rotor speed. It is equipped with a powerful .90 size engine

and an electronic governor for maintaining a near constant rotor speed. Like most miniature helicopters it is also

the helicopter avionics: an inertial measurement unit, consisting of three gyros and three accelerometers; a GPS

receiver; and a pressure altimeter. Without an air data system, we had to limit our flights to calm days. Full

estimated vehicle state and servo command data are available at 50 Hz rate.

The model has typical rigid body states with the quaternion attitude representation used in order to enable simu-

lation of extreme attitudes [23], two states for the lateral and longitudinal flapping angles, one for the rotorsp e ed,

and one for the integral of the rotorspeed tracking error. Governor action is modeled with a proportional-integral

Figure 1: Miniature helicopter undergoing a Split-S maneuver. The same maneuver was performed by MIT

helicopter under computer control. Drawing courtesy of Popular Mechanics magazine.

feedback from the rotorspeed tracking error to the throttle command. The model covers a large portion of the

X-Cell’s natural flight envelope: from hover flight to ab out 20 m/sec forward flight. The maximum forward speed

verified using comparison between model predicted responses and responses collected during flight test data. The

model was also “flown” by an expert RC pilot to determine how well it reproduces the piloted flying qualities.

Analytical linearization of the model with respect to forward speed was used to derive simple linear models. These

were subsequently used for a model-based design of the controllers used for the automatic execution of aerobatic

maneuvers. Before their implementation, the controllers were put to test in the hardware-in-the-loop simulator.

One of the aerobatic maneuvers performed entirely under computer control is shown in Figure 1. The actual

aggressive trajectories flown by the helicopter were adequately predicted by the simulation based on the developed

nonlinear mo del.

In the remainder of the paper we provide a full list of model parameters with the numerical values, dynamics equa-

tions of motion, and e xpressions for forces and moments exerted on the helicopter by its components. Throughout

lift curve slopes of these surfaces were estimated according to their resp ec tive aspect ratios [9]. The effective

torsional stiffness in the hub was estimated from angular rate responses to step commands in cyclic, as described

in Section 4.1.3.

Note that in the table “m.r.” stands for the main rotor, “t.r.” stands for the tail rotor.

3 Equations of motion

The rigid body equations of motion for a helicopter are given by the Newton-Euler equations shown below. Here

the cross products of inertia are neglected.

The set of forces and moments acting on the helicopter are organized by components: ()

()

()