International Journal of Aerospace Engineering

Volume 2019, Article ID 9396352, 10 pages

https://doi.org/10.1155/2019/9396352

## A Multibody Model of Tilt-Rotor Aircraft Based on Kane’s Method

^{1}School of Physics and Optoelectronic Engineering, Guangdong University of Technology, No. 100, West Ring Road, University Town, Guangzhou, 510006, China^{2}School of Data and Computer Science, Sun Yat-sen University, No. 132, Outer Ring Road East, University Town, Guangzhou, 510006, China

Correspondence should be addressed to Chengyue Su; nc.ude.tudg@usyc

Received 6 September 2018; Revised 29 December 2018; Accepted 9 January 2019; Published 16 April 2019

Academic Editor: Paul Williams

Copyright © 2019 Jianmin Su et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A tilt-rotor aircraft can switch between two flight configurations (the helicopter configuration and the fixed-wing plane configuration) by tilting its rotors. In the process of rotor tilting, the nacelles which drive the rotors tilt together with the rotors. Because the mass of the nacelles cannot be ignored compared to the mass of the whole aircraft, the tilting of the nacelles is a coupling motion of the body and the nacelles. In order to better character the aircraft dynamics during the nacelle tilting, a multibody model is established in this paper. In this multibody model, Kane’s method is used to build a dynamic model of a tilt-rotor aircraft. The generalized rates are used to describe the movement of the body and the nacelles (with rotors). The generalized active forces and generalized inertial forces of both the body and the nacelles (with rotors) are obtained, respectively, and the first-order differential equations of the generalized rates are obtained. The longitudinal trim of the XV-15 aircraft is calculated according to the single-body model and our multibody model, in this paper, and the results verify the correctness of the multibody model. In the process of nacelle inclination angle command tracking, the multibody model can provide more information about the disturbance torque of the nacelle than the single-body model, and model inversion control based on the proposed multibody model can obtain a better tracking result than a PID control method only using nacelle angle feedback information.

#### 1. Introduction

A tilt-rotor aircraft makes its flight configuration switch between the helicopter configuration and the fixed-wing airplane configuration by changing the direction of the rotor thrust. The tilting of the nacelles (with rotors) distinguishes the tilt-rotor aircraft from a helicopter or fixed-wing airplane, so tilt-rotor aircraft modeling should consider the process of the tilting of the nacelles. Because the mass of the nacelles (with rotors) cannot be ignored compared to the mass of the tilt-rotor aircraft, tilting of the nacelles (with rotors) will lead to the following features: (1) significant changes of both the center of gravity (c.g.) position and the moment of inertia of the tilt-rotor aircraft and (2) an interaction between the nacelles (with rotors) and the body of the tilt-rotor aircraft. These features make the dynamic model of the tilt-rotor aircraft more complicated.

Rosenstein et al. [1] established a mathematical model for a real-time simulation of a tilt-rotor aircraft (Boeing Vertol Model 222). In their work, the angular momentums of the body and the nacelles were added together to get the angular momentum of the tilt-rotor aircraft with respect to the tilt-rotor aircraft c.g. in the body axis, and the angular momentum theorem was used to obtain the dynamic model of the tilt-rotor aircraft. Ferguson [2] established a generic tilt-rotor aircraft model (GTRS), and the differential equations of motion variables were the same as that in Ref. [1]. Miller et al. used d’Alembert’s principle to establish the dynamic equations of a tilt-rotor aircraft [3]. Li et al. reckoned the body, the nacelles, and the rotors to be independent entities and established a realistic model in the form of multibody motion equations [4].

A numerical simulation model including the rotor, the wing, and the nacelle was built using a general purpose multibody simulation tool to study active control technology for tilt-rotor aircraft whirl flutter stability augmentation [5]. A simplified aeroelastic tilt-rotor model was built to analyze the vertical bounce phenomenon [6]. Multibody models of a tilt-rotor aircraft were implemented using the free multibody analysis software MBDyn in Politecnico di Milano. Aerodynamic loads were modeled for the wing and the rotor. The detailed control system kinematics and compliance were modeled. The overall multibody model consisted of more than 800 equations [7, 8].

Neural network-augmented model inversion control was used to provide a tilt-rotor aircraft with consistent response characteristics throughout its operating envelope [9]. Multiple model inversion controllers were designed at different flight conditions, and mode switching technology was adopted to guarantee smoothly the transition between the controllers [10]. Model predictive control was used to design a tilt-rotor aircraft flight control system and was implemented on a real-time simulator [11]. These control algorithms were verified on a GTRS model. Kim et al. designed the trajectory tracking controller for a tilt-rotor unmanned aerial vehicle based on a neural network-augmented model inversion control method [12]. In the above research on the flight control of the tilt-rotor aircraft, active control for the nacelle tilting is not found. An active control method has been developed for the control of the aeroelasticity and structural dynamics of a tilt-rotor [5]. Bernardini et al. [13] proposed an active control strategy to the alleviation of tonal noise inside the hosting passenger area of a midrange tilt-rotor. Singh et al. [14] examined the effectiveness of active control through wing flaperon and swash plate actuation for the alleviation of whirl-flutter instability of the full-scale XV-15 proprotor on a semispan wing.

In Refs. [1, 2], a tilt-rotor aircraft was treated as a single rigid body, and Euler equations were used to describe the attitude motion. A Euler pitch equation was amended according to the nacelle tilting, and the inclination angle and angular velocity of the nacelles were considered known quantities. Tilting of the nacelles is driven by a motor torque from the body, and it is the sum of the nacelle rotation and the body pitch motion. In order to describe the interaction between the body and the nacelles, a multibody model is established in this paper. In this multibody model, the body and the nacelles can be treated as rigid bodies that are hinged together. Kane’s method is a multirigid body modeling method suitable for computer programming. It uses generalized rates instead of generalized coordinates as independent variables to describe the motion of the system, avoiding the cumbersome process of dynamic function derivation [15]. Therefore, in this paper, Kane’s method is used to establish the multibody model of a tilt-rotor aircraft. The characteristics of this model are as follows: (1) the final form of the model is not a correction to the rigid body six-degree-of-freedom model, but a dynamic model of multiple rigid bodies; (2) the interaction motor torques between the body and the nacelles are introduced into the model, and the first-order differential equation of the nacelle tilting angular velocity is given; and (3) the derivation process of the model is simple and straightforward, and it is suitable for computer programming.

Using the proposed multibody model, a tilt-rotor aircraft attitude controller based on the model inverse method is designed to make Euler angles and the nacelle inclination angle of the tilt-rotor track commands. A linearized multibody model of a tilt-rotor aircraft at an equilibrium point is used in a model inverse control method. In the tracking of the nacelle inclination angle command near the equilibrium point, the model inverse control is better than a PID control, which does not use any model information.

The following is the arrangement of this paper: In Section 2, the basic assumptions and the definition of coordinate systems are given. Section 3 describes our multibody model of a tilt-rotor aircraft based on Kane’s method. In Section 4, we trim the XV-15 tilt-rotor aircraft longitudinally at various speeds based on the multibody model and single-body model. In Section 5, model inversion control is used to make Euler angles and the nacelle inclination angle of a tilt-rotor aircraft track the commands near an equilibrium point. The conclusions of this paper are included in the last section.

#### 2. Frames and Hypothesis

A tilt-rotor aircraft consists of the fuselage, wings, rudders, left nacelle, right nacelle, left rotor, and right rotor. Each component is treated as a rigid body. To simplify the problem, we make the following assumptions: (1)During the transition of the flight configuration of the tilt-rotor aircraft, the left and right nacelles are synchronously tilted(2)The tilt-rotor aircraft is composed of two rigid bodies, which are the body (including fuselage, wings, and rudders) and the nacelle (including left nacelle, right nacelle, left rotor, and right rotor)(3)Due to the synchronous tilting of the left and right nacelles, the gyro torques of the left and right rotors offset each other

The coordinates used in this paper are as follows:

(1) Earth-axis system : the origin of the Earth axis system is fixed at a point on the ground. The -axis points to the north in the horizontal plane. The -axis is perpendicular to the horizontal plane, and the -axis points the direction of gravity. The Earth axis system is considered the inertial coordinate system

(2) Body axis system : the origin of the body axis system is located at the center of gravity of the body. The -axis points forward along the central axis of the body, and the -axis is in the longitudinal plane of symmetry of the body. The -axis follows the right-hand rule

(3) Nacelle axis system : the origin of the nacelle axis system is located at the center of gravity of the nacelle. The direction of the -axis is the same as the thrust of the rotors. The direction of the -axis is the same as the shaft of the nacelle, and the -axis follows the right-hand rule

The body axis system and nacelle axis system are shown in Figure 1. Point is the joint point of the body and the nacelle, and is the inclination angle of the nacelle.