1. Introduction

1.1. Overview

The objective of this paper is to create a set of parametric models that can estimate the flight dynamics of commercial paraglider wings using only limited technical specifications.

In this paper, modeling refers to creating a mathematical representation of a physical characteristic or behavior. A dynamics model is a mathematical function that computes the acceleration of an object given the forces that act on it, as described by Newton’s 2nd law of motion (1.1):

(1.1)\[\begin{split}\begin{aligned} \textrm{Translational} \qquad F &= ma \\ \textrm{Angular} \qquad M &= J \alpha \end{aligned}\end{split}\]

These equations show that to compute the translational acceleration \(a\) and the rotational acceleration \(\alpha\), a dynamics model requires:

1. The mass \(m\) and mass moment of inertia \(J\)

2. The forces \(F\) and moments \(M\)

For a paraglider, the forces and moments that act on it are determined by its current velocity, the relative wind flowing past the glider, air density, gravity, and the pilot control inputs. The motion that is produced are the flight dynamics, and the equations that represent how those inputs produce the accelerations are called a flight dynamics model:

Fig. 1.1 Flight dynamics model block diagram

The purpose of these flight dynamics models is to enable dynamic simulations. A dynamic simulation is when acceleration is integrated over time to produce a record of the object’s velocity and position. The ability to simulate a system’s behavior provides opportunities such as studying that behavior, developing control models, and running statistical filtering pipelines. In fact, the inspiration for this project was a question whether statistical flight reconstruction could be used to recreate the wind fields present during a paraglider flight given only a record of its position, in much the same way as researchers attempted to locate the lost Malaysia Airlines Flight 370 [2].

The steps to producing a dynamic simulation can be summarized as follows:

1. Understand the physical system

2. Model its inertial properties and forces

3. Develop the equations of motion (Newton’s 2nd law)

4. Integrate the equations of motion over time

The majority of the work for this project is in step 2 (estimating the inertial properties and forces) because the estimation process requires accurate models of the mass distribution and aerodynamics of each component of the glider.

Attention

The remainder of this work assumes a working familiarity with fundamental aerodynamics. For the necessary background to understand this work, the section of the “Related works” covering Flight simulation provides an overview and complete list of material that I found helpful.

1.2. Modeling challenges

The existence of this project suggests that existing (and freely available) tools for aircraft simulations are inadequate for simulating paragliders. The reason is that paragliders have a variety of unique characteristics that make them difficult to model using tools built for conventional aircraft:

1. Highly curved shape

   Aerodynamics models must simplify the Navier-Stokes equations in order to produce a tractable system of equations. Those simplifications frequently make them incapable of representing the flow field around a nonlinear wing.

2. Low airspeed

   Paraglider airspeeds are typically in the range 24–72 [km/h]. They also have relatively short wing sections, with chord lengths ranging from 0.5–3 [m]. These characteristics combined with the reduced airspeed at the inside wingtip during a turn means that the canopy (and the wing tips in particular) are frequently operating at Reynolds values in the 300k range, far below the \(Re = 10^6\) range where where viscous effects start to become significant.

3. High angles of attack

   Compounding the issue of operating at low Reynolds values, paragliders frequently operate at high angles of attack, leading to flow separation and the dramatic nonlinear aerodynamic behavior that results. As they approach stall conditions, simple aircraft simulators that rely on linear aerodynamics can dramatically overestimate the true lift produced by the wing.

4. Flexible

   Paragliders are constructed from flexible nylon sheets and rely on air pressure and suspension lines to maintain their shape. Their internal cells billow and wrinkle while the canopy twists and bends in the wind. It can even collapse entirely. Systems that rely on a predetermined geometry are fundamentally incapable of modeling such behavior.

5. Air intakes

   To produce the internal pressure that forms the canopy, paragliders use air intakes at the leading edge which pressurize its volume. These air intakes violate the expected pressure gradients predicted by analyses that use the idealized airfoils used to define the section profiles. As a result, theoretical aerodynamic coefficients underestimate the section drag.

6. Lightweight

   A paraglider canopy is a large volume with a small amount of solid mass. Its low density means that a naive application of Newton’s 2nd law will overestimate acceleration because it fails to account for the momentum of the fluid surrounding the glider, an effect known as apparent mass.

In addition to these characteristics, there is another issue that is relatively unique to gliding aircraft:

7. Pilots care about the details of the wing behavior in non-uniform wind fields.

   The reason is that glider pilots rely on the ability to determine the structure of the wind field by sensing the imbalanced forces produced by differences in relative wind vectors across the wing.

Each of these characteristics introduce modeling challenges. The modeling requirements will depend on which of these characteristics the dynamics model attempts to capture.

1.3. Modeling requirements

The nuances of paraglider behavior are dominated by subtle interactions. The design philosophy for this project was to avoid simplifying assumptions whenever reasonable to avoid accidentally masking those subtle interactions. This approach was driven by a desire to answer questions such as:

• How much drag comes from each individual component?

• How important are section-specific Reynolds values?

• How important is apparent mass?

• How does a paraglider react when one side of the wing is in a stronger thermal than the other side?

The desire for accuracy must be balanced with practical limitations, choosing which characteristics to include and which to simplify away. Having considered the tradeoffs, this project chose the following set of modeling requirements, beginning with the fundamental challenges of the previous section:

1. The aerodynamics method must use the true, nonlinear geometry. It must not flatten the canopy geometry in any dimension.

2. The aerodynamics method must support variable Reynolds values.

3. The aerodynamics method must provide graceful degradation as it approaches high angles of attack. (A decrease in accuracy is acceptable, but assuming linear aerodynamics up to high alpha is not. The goal is to fly the wing into strong thermals which will rapidly increase angle of attack, so the method must at least approximate those conditions.)

4. Canopy deformations due to flexibility will be neglected. This means that glider controls that use non-brake-line manipulations will also be neglected (since they rely on canopy deformations).

5. The aerodynamics method must support empirical viscous correction factors to mitigate the issues caused by a mismatch between the theoretical and actual section profiles.

6. The system model must support apparent mass (in order to verify its significance).

7. The aerodynamics method must support non-uniform vectors along the span.

In addition to those characteristic behaviors, this project had an additional goal:

8. Computationally fast

   The fundamental goal of this project is to enable people to create models of commercial paraglider wings, and that process requires iteration, so the software should pursue simulation speed that would allow rapid iteration.

1.4. Roadmap

The majority of this work is spent producing the models that estimate the inertial properties and resultant forces for each component, but it also develops the additional models necessary to generate flight simulations. For reference, a complete flight simulation architecture is shown in Fig. 1.2. This paper will develop everything inside the “State dynamics” block.

Fig. 1.2 Flight simulation block diagram

The modeling process begins by developing a novel Foil geometry with increased flexibility compared to other open source wing modeling tools, enabling simple, parametric representations of typical paraglider canopies. It then chooses a Foil aerodynamics method that satisfies those Modeling requirements that relate to the canopy aerodynamics. Next, it develops a set of parametric Component models using parametrizations that simplify creating models of commercial paraglider systems. Finally, System dynamics models combine the components into complete flight dynamics models, and State dynamics shows how to define the derivatives of a set of state variables in terms of those system dynamics. Having completed the model derivations, the paper provides a complete demonstration of how they can be used to model a commercial paraglider wing. The penultimate chapter provides Validation data of the aerodynamics method by comparing wind tunnel measurements for a scale-model paraglider wing against simulated results, as well as comparing simulated polar curves for the demonstration model against basic flight test data. Finally, the Conclusion revisits the questions from the Modeling requirements and proposes how this material may be used in future work.