# Modeling and Control of an Articulated Multibody Aircraft
## 1 Introduction
### Scope and Contributions  
In this paper, an increased understanding of the various abdominal motions for control in insect flight is pursued through mathematical modeling and attitude control of a dragonfly-inspired straight-wing aircraft (DISWA), augmented to include an articulated abdomen (see Figure 2b). The flight of some insects consists of both flapping- and fixed-wing episodes. Since this is an isolated study of the effects of abdominal motions, the approach used will provide insight into the use of appendage motion for control. Technological systems have included dedicated active mass-distribution mechanisms for control, and in most cases, the slowly moving elements have been enclosed within a craft, creating space constraints. In the case of animals, all cases mentioned had comparatively fast, multifunctional, rotational or telescoping, external structures with physiological as well as control functions. This is indicative of the high level of optimization of biological systems, presenting inspiration for the design of technology.  
The multibody nonlinear equations of motion were developed using the Newton-Euler method. Flight dynamics involves the use of many reference frames and respective coordinate systems. Changing the reference frame, for example, from inertial to body frame, is sometimes advantageous. The change of frame is governed by [PERSON]'s generalized transformation. Classical Euler transformation factors the change in reference frame using _ordinary  
Figure 1: Comparison of a mosquito to a dragonfly: (**a**) Outline of a Culex mosquito—_Culex quinquetafscitatus_. (**b**) Outline of an Australian Emerald dragonfly—_Hemicordulia australiae_.  
Figure 2: Description of the DISWA: (**a**) Photograph of an Australian emerald dragonfly, _Hemicordulia australiae_. (**b**) Reference points and coordinate systems of the dragonfly-inspired straight-wing aircraft (DISWA).  
time derivatives_, where additional terms appear in the equations of motion (EoMs) to account for the time-dependent coordinate transformations. Using the classical formulation, the resulting expression is not a tensor due to the extra terms from time-dependent changes in coordinates, resulting in equations that are often described as complex and cumbersome [42; 50]. The formulation used in this study preserves the tensor formulation using the _rotational time derivative_ operator, offering a few major advantages. It is tensor-based and compact. It is in an invariant tensor form, which is independent of any coordinate system and therefore allows us to derive the mathematical model first without consideration of coordinate systems. Points and frames are used instead of classical radius vectors and coordinate systems.  
Since the fore and hind wings are combined to form single wings with elevons, the equations of motion are simplified to a two-body problem; however, the method can be extended to include more than two bodies. In addition, the model of the articulated abdomen for control was derived for a specific regime, which allowed the linearization of the multibody model for stable equilibrium (level cruising flight). A linear quadratic integral controller (LQI) was designed and implemented to optimally maintain attitude (pitch and yaw) of the DISWA. The LQI controller offers the advantage of simplicity in design, and application of multiple-input-multiple-output (MIMO) systems in terms of control effort allocation. The developed model will enable the study of the flight dynamics of insect-like flapping aerial vehicles and a determination of the relative importance of abdominal actuation to insect flight.  
The model we will derive treats appendages as mass elements that do not create lift or drag. Depending on the appendage, this might be a reasonable approximation. Future work could consider moments from drag on appendages. In this first formulation, it would replace one approximation (dragless slender bodies) with another (drag on slender bodies).