2. Related works#
2.1. Flight simulation#
This paper develops paraglider flight dynamics models that can be used for flight simulation, which means that this paper is built on the foundations of flight simulation. Flight simulation is simply the specific name of a dynamic simulation that involves a flight dynamics model, and developing a flight dynamics model follows the structure outlined in the Overview: understand the system, model the inertia and forces, develop the equations of motion, and integrate them over time.
The first step to creating a model of an aircraft is a familiarity with the physical system and how it behaves. Key concepts in the context of this paper include characteristics of wing geometry; conventions for axes and relative motion; flow angles (angle of attack and sideslip); aerodynamic coefficients; and control inputs, actuators, and surfaces. An approachable starting point is [3], which provides a thorough discussion of the terminology and significance of the major wing design characteristics. Another ubiquitous resource is [4], which may be more suitable to in-depth study.
Next, to model a behavior you must be able to explain the behavior. The unique characteristic of aircraft dynamics is that they experience aerodynamic forces due to their motion relative to the air. The aerodynamic forces on the surfaces of an aircraft are the results of the geometry, relative motion, and characteristics of the fluid. Key concepts include the characteristics of the flow (inviscid versus viscous, laminar versus turbulent, compressibility, etc) and the modeling intuition of Prandtl’s seminal work on boundary layers [5] (both 2D and 3D, which are vital to understanding some of the aerodynamic difficulties in simulating flow around a paraglider canopy). When selecting and working with aerodynamics models, it is highly beneficial to have a general awareness of the complexity of Navier-Stokes, and how the variety of aerodynamics models are the result of attempts to produce tractable systems of equations by applying different simplifying assumptions. An excellent introduction to these topics is [6], which provides an approachable introduction to the underlying physics, overviews of the core aerodynamic models, and how they’re derived. Another prevalent work is [7] (or any of Anderson’s works). For more targeted discussions, [8] provides clear insight into the theoretical details of common aerodynamics models, and [9] provides a guide to their computational aspects. For a less conventional approach, [10] provides a unique perspective of these aerodynamics models and the assumptions that underlie them, including an excellent discussion of some issues with the NLLT that may shed light on the difficulties that arise when using that approach.
Once the inertial properties, forces, and moments can be determined, they must be synthesized into a complete system dynamics model, which in this case are known as the equations of motion. Unlike the simple equations in the Overview, the equations describing the translational and angular accelerations of an aircraft cannot always be decoupled; the equations must be solved simultaneously. Producing the equations of motion when such relationships exist involves writing equations for the translational and angular momentum of the system and taking their derivatives with respect to time (since acceleration is the time rate of change of momentum). For a thorough explanation with a focus on aircraft dynamics see [11]; although the notation can be opaque, it provides an excellent development for conservation of momentum of multi-body systems, which is especially useful for understanding the derivations of system models that include degrees of freedom between the paraglider harness and the rest of the system.
Once the equations of motion are known, they can be used to generate simulated trajectories of the aircraft in response to different environmental and pilot inputs. Key concepts include the choice of state variables, coordinate systems and their relative advantages, encoding geometric orientation, representing the environment, and applying numerical integration to the equations of motion to produce the simulated result. For this work I found a complete reference in [12]; the opening chapters provide a masterful introduction to these key concepts, including a principled mathematical notation (adopted by this paper, see Notation and Symbols) and a thorough review of vector calculus (especially the counter-intuitive results of taking the derivative of a vector with respect to an accelerating reference frame, which is important when defining the State dynamics).
2.2. Paraglider modeling#
In addition to the general knowledge of aircraft behavior, it is necessary to understand the unique characteristics of paraglider flight. For practical knowledge, recreational pilot materials make excellent resources. One thorough introduction targeting beginner pilots [13] provides a tour of the components of a paraglider, their function, behavior, and an admirable review of their aerodynamics; if any of the paraglider-specific terminology in this paper is unclear, this book will likely clear up the confusion.
Beyond recreational sources, academic literature relevant to paraglider modeling is typically from one of two branches: parafoil-payload systems, and paragliders. Parafoil-payload systems usually (but not always) refer to large-scale ram-air gliding parachutes intended for heavy payload applications such as cargo delivery and vehicle-recovery (such as landing the X-38 experimental space plane [14], or the more recent work by SpaceX to catch rocket fairings on a boat), while the term “paraglider” usually (but not always) refers to the recreational aircraft. Although the physical characteristics of parafoil-payload systems differ significantly from paragliders due to their scale, carrying capacity, and control schemes, their similarities make much of the research informative, albeit not directly applicable. As a result this section will mix the two groups, noting their differences when significant. Also, as this project has chosen to neglect the effects of canopy deformations, research into modeling those deformations will not be discussed.
The first topic of research is on the aerodynamics of arched, inflatable wings. Their nonlinear geometry made analyses difficult, so early studies were limited to their longitudinal dynamics (fore-aft two-dimensional motion). Alternatively, simple models of their 3D dynamics divide the wing into several discrete segments that act independently (thus neglecting the 3D flow interactions of a real 3D model) [15]. Attempts to account for the full 3D aerodynamics typically involved either measuring the longitudinal and lateral aerodynamic coefficients experimentally [16], or estimating them using vortex lattice and panel methods that can account for their nonlinear geometry by neglecting viscous effects. The significance of the viscous effects led to attempts to incorporate experimental aerodynamic coefficients via extended lifting-line models; two important works regarding this approach were [17] and [18], which could estimate the 3D aerodynamics of wings with circular arcs, but were unable to account for sweep. As nonlinear lifting-line theory (NLLT) models continue to be developed, their applicability to paraglider wings has greatly improved [19]; for example, [20] successfully applied the method from [21] to a reference paraglider wing in a static flight test, confirming the merit of the of a modern NLLT to this application.
Another significant characteristic of paraglider canopies is their low density, which makes them sensitive to the effects of apparent mass [22]. Early attempts to model the apparent mass of a paraglider simplified the wing as an ellipsoid with a single center of rotation [23]. Further developments recognized the inadequacies the ellipsoid model, and adjusted the estimates to account for two separate centers of rotation for rolling and pitching motions [24]. Both models are limited by their assumption of steady flow [25] so their adequacy for simulations involving dynamic maneuvers is unclear; nevertheless, the adapted model is assumed to be adequate for the purposes of this paper.
The last major topic of research is the system model. There are many system models in literature, but their key differentiating factors in the context of this project are whether they incorporate apparent mass and how they model the attachment of the harness to the suspension lines. The inclusion of apparent mass appears to be a modeling decision driven by whether the author expected the effect to be significant; papers that exclude apparent mass do so without explicit justification. For the harness connection, models are categorized by their degrees of freedom (DoF) and the character of the connection points; a 6-DoF model does not allow the payload to move at all, a 7-DoF allows the payload to translate or rotate (relative to the suspension lines) in one dimension, an 8-DoF adds two degrees of freedom, etc. For a general understanding of the impact, [26] provides a comparative analysis of a fixed (6-DoF) model versus a 9-DoF system model. For a more thorough review of the many available system models, [27] has a seemingly exhaustive list of the models through 2005, including a discussion of those models that account for apparent mass. Two informative models that incorporate apparent mass are [15] (which used the older method in [23]) and [28] (which used the adapted apparent mass model from [24]).
In addition to topical works, there have been several more comprehensive studies. The best place to start is [29]: although it has a parafoil-payload perspective, this approachable paper is a thorough introduction to the terminology, geometric parameters, choice of airfoil, and control schemes of parafoils (which it calls a “ram-air parachute”); this paper also used geometric simplifications to study the canopy aerodynamics and drag contributions, and developed linear models of the longitudinal and lateral dynamics to study performance and stability. Next, for a paraglider perspective, [30] provides a compact survey on the sources of aerodynamic drag; it reviews the impacts of arc, flexibility, air intakes, lines, and pilot. Worth reading immediately after is [20], as it is essentially an updated revision of [30].
The most comprehensive work on paraglider flight dynamics to date is the dissertation [31] that inspired the general structure of this paper. First, it provides an overview of paraglider geometry, construction, and behavior. It then develops a foil geometry that uses the locus of quarter-chord points to position the sections, as well as intuitive parametric definitions of the underlying paraglider canopy structure. For the paraglider components, it develops a model to position the harness as a function of the accelerator control, a continuous brake deflection distribution using both brakes, and the spherical harness model used by this paper. Next, for the canopy aerodynamics it develops a pseudo-LLT (which it acknowledges is an approximation in deference to the project’s primary focus on stability and control) using constant 2D aerodynamic coefficients. From the complete aerodynamics model, it then estimates the 3D aerodynamic coefficients and stability derivatives for a linearized model that is used for the remainder of the work, which is focused on performance aspects (such as glide ratio versus equilibrium pitch angle), stability analyses (such as longitudinal stability versus riser position, and roll stability versus sideslip), and controllability (takeoff, maneuvering, and landing).
2.3. This work#
As mentioned in the previous paragraph, this project began with [31] as its starting point. While attempting to use those models to recreate commercial paraglider wings, this work identified a collection of improvements that led to newly derived models.
First, it improves the canopy geometry by developing a novel foil geometry model inspired by a suggestion in [32] that allows independent reference points for the \(x\)- and \(yz\)-positions. This increased flexibility allows accurate representations of existing wings using simple parametric equations, which this work uses to replace the parametric design curves in [31] with new parametrizations that are easier to estimate for an existing paraglider canopy. It also replaces the approximate inertia calculations for the canopy surface and volume with a mesh-based method that can account for different upper and lower surface densities, and the extra solid mass from vertical ribs.
For the canopy aerodynamics, it replaces his pseudo-LLT with a full NLLT ([21], [33]) that supports arbitrary arc, sweep, twist, specific (nonlinear model) aerodynamic coefficients for each section as a function of Reynolds number and deflection distance, and non-uniform wind vectors along the span. Also, instead of modeling trailing edge deflections as section rotations (by adding the deflection angle to the section angle of attack, effectively shifting the coefficient curves), this model uses section coefficients generated from the actual deflected geometry, and accounts for the effects of Reynolds number.
Next, it completely redesigns the suspension line model, keeping only the intuition to replace the “rigging angle” with a displacement vector in the body axes. The new model improves the representation of the brakes by first calculating the deflection distance before calculating the true change in angle of attack (which depends on the section chord), as well as improving the accuracy of the deflection distribution itself. The new model improves the representation of the accelerator by parametrizing the fore and aft connection points instead of fixing them at the leading and trailing edge of the canopy, thus allowing accurate models of commercial wings. Lastly, the new model moves the line drag away from canopy centroid and distributes it into lumped points that can model asymmetric forces between each semispan.
For the harness, the only minor change was to separate the weight shift distance from an absolute distance to a proportional one controlled by a harness parameter for the maximum displacement. Although functionality equivalent, I personally felt that this change makes simulation scenarios easier to write and understand.
For the system model, this paper derived 6-DoF and 9-DoF models (the 9-DoF is a rederivation of the model used in [34] and [35]) that may optionally incorporate the apparent mass estimates from [24]. The 9-DoF model is included for demonstration and testing purposes, and is not used in any analyses.
The implementation of all models are available as an open source library [1], including example wing models, and the simulations used in this paper are available as part of the open source materials used to produce this paper.