6. System dynamics#

This chapter combines the individual component models into composite system dynamics models. In this paper, a system dynamics model is a set of derivatives that define the translational and angular acceleration of groups of components that represent an aircraft, specified using a coordinate system attached to the aircraft. Developing a system model can be roughly described as a sequence of steps:

  1. Choose a set of components to represent the aircraft

  2. Characterize their connections

  3. Choose a dynamics reference point for the composite system

  4. Develop the system of equations for the accelerations

6.1. Components#

The previous chapter defined component models for the canopy, suspension lines, and harness; in the system models, these are lumped into two quasi-rigid-body groups called the body and the payload. The body of the glider is the combination of canopy and suspension lines. The payload includes the harness, pilot, and their gear (in this simplified model, the pilot and their gear are treated as additional masses that are added to the mass of the harness).

These models are quasi-rigid because the dynamics equations will only consider their instantaneous configurations when calculating their accelerations; conservation of momentum requires accounting for redistributions of mass, but doing so would require inertia derivatives as functions of time derivatives of the control input (such as weight shift, accelerator, etc), which would significantly complicate the model. Because the redistributions of mass are relatively small for typical scenarios, these models assume the affect of violating conservation of momentum is negligible.

It is important to note that the unfortunately ambiguous terminology of body is deliberate. The paraglider community typically refers to the combination of canopy and lines as a paraglider wing, but the “body” convention improves consistency with existing parafoil-payload literature (which in turn inherited the term from conventional aeronautics literature). Some texts prefer the term parafoil, but having the same prefix \(p\) for both parafoil and payload makes subscripting the variables unnecessarily difficult. Similarly, using “wing” would be preferred in this context, but subscripting with \(w\) causes confusion when discussing wind vectors. Referring to whatever group of components include the canopy as the body was a compromise chosen for consistency with existing literature.

6.2. Connections#

Next, the system model must characterize the connection between the body and payload. In literature, parafoil-payload models are commonly categorized by their degrees-of-freedom (DoF): the total number of dimensions in which the components of the system are free to move. The body has 3-DoF for translational motion and another 3-DoF for rotational motion, and if the payload is allowed to translate or rotate relative to the body, those additional DoF are added to the total DoF of the system model. For example, in a 6-DoF model, the body and payload are connected as a single rigid body, with no relative motion between them.

_images/paraglider_fbd_6dof.svg

Fig. 6.1 Diagram for a 6-DoF model.#

For typical paragliding flight maneuvers, assuming a fixed payload orientation is reasonably accurate, but with one significant failing: although the relative roll and twist are typically negligible, relative pitch about the riser connections is very common, even during static glides. Friction at the riser carabiners (and aerodynamic drag, to a lesser extent) dampen pitching oscillations, but the payload is otherwise free to pitch as necessary to maintain equilibrium. Assuming a fixed relative pitch angle introduces a fictitious pitching moment that disturbs the equilibrium conditions of the wing and artificially dampens the pitching dynamics during maneuvers. To mitigate that issue, the obvious solution is to add an additional DoF, but for demonstration purposes it is simpler to define a full 9-DoF model, where the body and payload are connected at the riser midpoint \(RM\). The connection is modeled as a spring-damper system, which produces an internal force \(\vec{F}_R\) and moment \(\vec{M}_R\):

_images/paraglider_fbd_9dof.svg

Fig. 6.2 Diagram for a 9-DoF model with internal forces.#

6.3. Reference point#

Each dynamics model must choose a reference point about which the moments and angular inertia are calculated. A common choice for conventional aircraft is the center of real mass because it decouples the translational and angular dynamics of isolated objects. For a paraglider, however, this is not possible: paragliders are sensitive to apparent mass, which depends on the direction of motion, so there is no “center” that decouples the translational and rotational terms of the apparent inertia matrix [24]. Because the system matrix cannot be diagonalized there is no advantage in choosing the center of real mass. Instead, the reference point can be chosen such that it simplifies other calculations.

In particular, the method to estimate the apparent inertia matrix requires that the reference point lies in the \(xz\)-plane of the canopy. Two natural choices in that plane are the leading edge of the central section, or the midpoint between the two risers. The riser midpoint \(RM\) has the advantage that is a fixed point in both the body and payload coordinate systems, which means it does not depend on the relative position or orientation of the payload with respect to the body. (This choice simplifies the equations for the 9-DoF model while maintaining consistency with the 6-DoF model.)

6.4. System inputs#

The inputs \(\vec{u}\) to the system model the control inputs for each component (with the exception of the trailing edge deflection distances \(\delta_d(s)\) which are computed internally using the suspension lines and foil geometry models), the wind velocity \(\vec{v}_{W/e}\), air density \(\rho_\textrm{air}\), and the gravity vector \(\vec{g}\).

(6.1)#\[\vec{u} = \left\{ \delta_a, \delta_{bl}, \delta_{br}, \delta_w, \vec{v}_{W/e}^b, \rho_\textrm{air}, \vec{g}^b, \right\}\]

Here the wind field is assumed to be uniform so the wind velocity at every control point is defined by a single, constant vector, but for non-uniform wind fields there will be a unique wind vector for each aerodynamic control point.

6.5. Equations of motion#

The equations of motion are developed by solving for the derivatives of translational momentum \({^e \dot{\vec{p}}} = \sum{\vec{F}} = m \dot{\vec{v}}\) and angular momentum \({^e \dot{\vec{h}}} = \sum \vec{M} = \mat{J} \dot{\vec{\omega}}\) for each group of components [11]. In addition to requiring the forces, moments, and inertia matrices for each component, each system model must choose a dynamics reference point and whether to account for the affects of apparent mass. The appendix includes derivations demonstrating different choices for several each model.

For the 6-DoF model, the most complete is Model 6a which accounts for the effects of apparent mass, while Model 6b and Model 6c have the advantage of simplicity (making them easier to implement and useful for validating implementations of more complex models). The derivation produces a system of equations (14) that can be solved for the two vector derivatives that describe the accelerations of the body relative to the earth frame \(\mathcal{F}_e\) taken with respect to the body frame \(\mathcal{F}_b\):

(6.2)#\[\begin{split}\begin{aligned} {^b \dot{\vec{v}}_{RM/e}} \qquad & \textrm{translational acceleration of the riser midpoint} \, RM \\ {^b \dot{\vec{\omega}}_{b/e}} \qquad & \textrm{angular acceleration of the body} \\ \end{aligned}\end{split}\]

Similarly, for the 9-DoF model, Model 9a also develops a complete system of equations (33) that account for apparent mass of the canopy, but with the addition of a separate angular acceleration for the payload with respect to the payload frame \(\mathcal{F}_p\):

(6.3)#\[\begin{split}\begin{aligned} {^b \dot{\vec{v}}_{RM/e}} \qquad &\textrm{translational acceleration of the riser midpoint} \, RM \\ {^b \dot{\vec{\omega}}_{b/e}} \qquad & \textrm{angular acceleration of the body} \\ {^p \dot{\vec{\omega}}_{p/e}} \qquad & \textrm{angular acceleration of the payload} \\ \end{aligned}\end{split}\]