5. Component models

A paraglider can be modeled as a system of three components: a canopy, a harness, and suspension lines that connect the canopy to the harness.

Fig. 5.1 Paraglider component breakdown

Diagram remixed from a Wikipedia contribution by user Mysid.

To compute the dynamics of the composite system, each component model must define three things:

1. Control inputs

2. Inertial properties

3. Resultant force

This chapter develops basic models for each component, favoring simplicity whenever possible. In particular, all models are based on a quasi-rigid body assumption; unlike a true rigid-body model where no component is allowed to move, these models (and their connections) are treated as “instantaneously rigid”, where they are allowed specific reconfigurations based on the control inputs (moving the pilot in the harness, or deflecting the trailing edges of the canopy). This may seem like a major oversimplification, but in practice it works quite well: although nearly every component of a paraglider is made from highly flexible materials, they tend to remain relatively rigid during typical flight conditions.

5.1. Canopy

A paraglider canopy (or parafoil) is a kind of ram-air parachute: inflatable lifting surfaces manufactured from nylon sheets with air intakes at the leading edge that pressurize their internal volume. The shape of an inflated parafoil is determined by a combination of surface materials, internal structure, air pressure, and suspension lines. Because the canopy is flexible, pilots can manipulate the suspension lines to change the shape of the canopy, allowing them to control its aerodynamics.

To model a parafoil, it is helpful to think of the canopy as a physical realization of some idealized foil geometry. The physical canopy is significantly more complex because it must attempt to create the foil geometry using flexible materials that deform once the canopy is pressurized (as well as meeting requirements such as weight, physical reliability, manufacturability, etc). Modeling the deformations that occur during flight (cell billowing, profile flattening, surface wrinkling, etc) are exceptionally difficult to model without resorting to complete material simulation [36], which is why this project does not consider any deformations other than deflections of the trailing edge due to brake inputs (which are calculated separately).

Instead, this model assumes that the foil geometry is an exact representation of the physical canopy, then adds small empirical corrections to account for the most significant error. It models the canopy volume with smooth upper and lower surfaces, whose extents also serve to define the section air intakes. It does not model individual cells, but it does incorporate an estimate of the additional inertia from the internal ribs between each cell. The only deformations included in the model are trailing edge deflections due to pilot control inputs, which are accounted for with precomputed section aerodynamic coefficients; it does not support manipulation via load-bearing lines (used by pilots for maneuvers such as “big ears”, C-riser control, etc) or the stabilo lines.

5.1.1. Controls

A paraglider canopy is controlled by changing its shape through manipulation of suspension lines. In theory, any of the suspension lines can be used to alter the positions, orientations, or profiles of its wing sections, but this model only supports trailing edge deflections produced by the lines connected to the left and right brake handles.

When a pilot applies the brakes, they generate a continuous deformation along the trailing edge of the canopy. In terms of the individual sections, this results in deformed variants of the undeflected section profiles. Because this canopy model does not perform material simulation, it requires that each variant has been precomputed and assigned a unique airfoil index that associates it with a given brake input. The choice of section index has a significant impact on the design of the suspension line model, and should be chosen thoughtfully.

A simplistic (but not uncommon) approach is to model the trailing edge deflection as a global rotation about some rotation point, and completely ignore profile deformations. The airfoil index in this case is the deflection angle measured between the deflected and undeflected chords. The rotation point is typically implicit; for example, lifting-line models that assume a fixed quarter-chord are implicitly rotating about the quarter-chord position.

Fig. 5.2 Deflection as a rotation of the entire profile.

By ignoring deformations of the profile geometry this model assumes the shape of the aerodynamic coefficient curves do not change with brake deflections. Instead, the deflection angle \(\delta_f\) is added directly to the angle of attack, meaning the control input produces a simple translation of the section coefficients. The appeal of this model is the fact that it only requires the section coefficient data from the undeflected profile. Unfortunately, the accuracy of the model degrades rapidly as the deflection angle is increased.

A more accurate model that is extremely common for wings built from rigid materials is to use a discrete flap which rotates about a hinge point at some fixed position along the chord:

Fig. 5.3 Deflection as a rotation of a rigid flap about a fixed hinge point.

Fixed-hinge flaps are ubiquitous due to their simplicity and acceptable accuracy for rigid wings. Unfortunately, this model is troublesome for flexible wings because there are no fixed hinge points: parafoil edge deflections develop as a variable arc, not a rigid rotation. Also, explicit deflection angles are problematic because parafoil brake inputs cannot control the deflection angles directly; they can only control the downward deflection distance \(\delta_d\) of the trailing edge:

Fig. 5.4 Deflection as a vertical displacement of the trailing edge.

Because airfoils and section coefficients are conventionally normalized to a unit chord, the natural choice of airfoil index for a parafoil is the normalized deflection distance \(\overline{\delta_d}\), a function of the deflection distance \(\delta_d\) and the chord length \(c\):

(5.1)\[\overline{\delta_d} \defas \frac{\delta_d}{c}\]

The normalized deflection distances are unusual in that, although they are control inputs to the canopy aerodynamics model, they are not direct inputs to the system model. Instead, they are computed indirectly using values provided by the suspension lines and the foil geometry so that the deflection distribution along the span is a function of section index and brake inputs:

(5.2)\[\overline{\delta_d}\left(s, \delta_{bl}, \delta_{br} \right) = \frac {\delta_d \left(s, \delta_{bl}, \delta_{br} \right)} {c \left( s \right)}\]

5.1.2. Inertia

For a parafoil canopy in-flight, the effective inertia is produced by a combination of three different masses: a solid mass, from the structural materials, an air mass, from the air enclosed in the foil, and an apparent mass, from the air surrounding the foil. (Some texts refer to the combination of the solid and enclosed air masses as the real mass [24].)

5.1.2.1. Solid mass

The solid mass is all the surface and structural materials that comprise the canopy. A rigorous model would include the upper and lower surfaces, ribs, half-ribs, v-ribs, horizontal straps, tension rods, tabs (line attachment points), stitching, etc, but for this model the calculation is restricted to the upper and lower surfaces and internal ribs. The internal ribs are assumed to be solid (non-ported), resulting in an overestimate that is somewhat mitigated by the absence of accounting for the other internal structures.

It does, however, account for the extents of the upper and lower surfaces along the section profile. This extent will be used to calculate the inertial properties of the upper and lower surface materials, as well as to calculate empirical viscous correction factors for the section drag coefficients. For this model, the extent of the upper surface and lower surface can be defined using the normalized distance along the section profile, with \(-1 \le r_\textrm{lower} \le r_\textrm{upper} \le 1\), with their symmetric spanwise extent controlled by a section index \(0 \le s_\textrm{end} \le 1\).

Fig. 5.5 Air intake parameters

Assuming the material densities are uniform, the inertial properties of the materials can be determined by first calculating the total area \(a\) and areal inertia matrix \(\mat{J}\) for each surface (using the method in Area), then scaling them by the areal densities \(\rho\) of each surface. The result is the total masses for the upper surface, lower surface, and internal ribs:

(5.3)\[\begin{split}\begin{aligned} m_{\mathrm{u}} &= \rho_{\mathrm{u}} a_{\mathrm{u}} \\ m_{\mathrm{l}} &= \rho_{\mathrm{l}} a_{\mathrm{l}} \\ m_{\mathrm{r}} &= \rho_{\mathrm{r}} a_{\mathrm{r}} \end{aligned}\end{split}\]

And their mass moments of inertia about the canopy origin \(O\):

(5.4)\[\begin{split}\begin{aligned} \mat{J}_{\mathrm{u}/\mathrm{O}} &= \rho_{\mathrm{u}} \mat{J}_{a_u/\mathrm{O}} \\ \mat{J}_{\mathrm{l}/\mathrm{O}} &= \rho_{\mathrm{l}} \mat{J}_{a_l/\mathrm{O}} \\ \mat{J}_{\mathrm{r}/\mathrm{O}} &= \rho_{\mathrm{r}} \mat{J}_{a_r/\mathrm{O}} \end{aligned}\end{split}\]

In theory the inertial properties are functions of the brake inputs since they alter the distribution of mass, but in practice the effect is negligible. For this project the centroids and moments of inertia for the solid mass are calculated once using the undeflected section profiles.

5.1.2.2. Air mass

Although the weight of the air inside the canopy is counteracted by its buoyancy, it still represents significant mass. When the canopy is accelerated the enclosed air is accelerated at the same rate, and must be included in the inertial calculations. (This model neglects surface porosity; although the canopy is porous, and thus constantly receiving an inflow of air through the intakes, in a properly functioning wing the leakage is slow enough that the volume of air can be treated as constant.)

Similar to the surface masses, the internal volume and its unscaled inertia about the canopy origin is easily computed from the Foil geometry using the method in Volume. Given the internal volume \(v\) and the current air density \(\rho_{\mathrm{air}}\), the total mass of the enclosed air \(m_{\mathrm{air}}\) is simply:

(5.5)\[m_{\mathrm{air}} = \rho_{\mathrm{air}} v\]

Similarly, for the inertia matrix of the enclosed air about the canopy origin \(O\):

(5.6)\[\mat{J}_{\mathrm{air}/O} = \rho_{\mathrm{air}} \mat{J}_{\mathrm{v}/\mathrm{O}}\]

5.1.2.3. Apparent Mass

Newton’s second law states that the acceleration of an isolated object is proportional to the net force applied to that object:

\[a = \frac{\sum{F}}{m}\]

This simple rule is sufficient and effective for determining the behavior of isolated objects, but when an object is immersed in a fluid it is longer isolated. When an object moves through a fluid there is an exchange of momentum, and so the momentum of the fluid must be taken into account as well. In fact, it is this exchange of momentum that gives rise to the aerodynamic forces on a wing. The difference is that apparent mass is an unsteady phenomena that is not accounted for by simple aerodynamic models, such as Phillips’ numerical lifting-line.

In static scenarios, where the vehicle is not changing speed or direction relative to the fluid, this exchange of momentum can be summarized with coefficients that quantify the forces and moments on the wing due to air velocity. But for unsteady flows, where the vehicle is accelerating relative to the fluid, the net force on the vehicle is no longer simply the product of the vehicle’s “real” mass and acceleration. Instead, when a net force is applied to an object in a fluid, it will accelerate more slowly than the object would have in isolation, as if the vehicle has increased its mass:

\[a = \frac{\sum{F}}{m + m_a}\]

This apparent mass \(m_a\) (or added mass [25]) tends to become more significant as the density of the vehicle approaches the density of the fluid. If the density of the vehicle is much greater than the density of the fluid then the effect is often ignored, but for lightweight aircraft the effect can be significant.

Because apparent mass effects are the result of a volume in motion relative to a fluid, its magnitude depends on the volume’s shape and the direction of the motion. Unlike the real mass, apparent mass is anisotropic, and the diagonal terms of the apparent inertia matrix are independent. Calculating the apparent mass of an arbitrary geometry is difficult. For a classic discussion of the topic, see [22]. For a more recent discussion of apparent mass in the context of parafoils, see [23], which used an ellipsoid model to establish a parametric form commonly used in parafoil-payload literature

This paper uses an updated method from [24] which added corrections to the ellipsoid model of [23]. (For a replication of the equations in that method but given in the notation of this paper, see Apparent mass of a parafoil.) The method uses several significant simplifying assumptions (the dynamics reference point must lie in the \(xz\)-plane, the foil has circular arc, uniform thickness, uniform chord lengths, etc), but the effects of deviations from the method’s assumptions are negligible for typical parafoil models.

5.1.3. Resultant force

A method for estimating the canopy aerodynamics was presented earlier. An advantage of that method is that it does not assume any particular functional form of the aerodynamic coefficients (linear, polynomial, etc), allowing their definition to use whatever form is convenient. This model uses that flexibility to compose the section coefficients as a two step process:

1. Design a set of airfoils associated with the range of trailing edge deflection, and estimate their aerodynamic coefficients.

2. Apply correction factors to each section to account for physical inaccuracies in the idealized airfoils.

The airfoils are indexed by their normalized deflection distance (5.1), which appears in Phillip’ NLLT as the control input \(\delta_i\); the indexed airfoils allow the brakes to control the canopy aerodynamics with no modifications to the NLLT. This section index allows each section to provide its own section coefficients, as well as empirical correction factors. One correction factor included in this model, \(C_{D,\textrm{surface}}\), is for “surface roughness” ([43], [30]), and the other, \(C_{D,\textrm{intakes}}\), is for the additional viscous drag due to the air intakes [30]. (See the demonstration for an example.)

Given the foil geometry and aerodynamic coefficients, the aerodynamics model estimates the aerodynamic forces \(\vec{f}_{f,\textrm{aero},n}\) (4.1) and moments \(\vec{g}_{f,\textrm{aero},n}\) (4.13) for the \(N\) foil sections.

(5.7)\[\vec{f}_{f,\textrm{weight}} = m_p \vec{g}\]

(5.8)\[\vec{f}_{f,\textrm{aero}} = \sum_{n=1}^{N} \vec{f}_{f,\textrm{aero},n}\]

(5.9)\[\vec{g}_{f/R} = \sum_{n=1}^{N} \left( \vec{r}_{CP_n/R} \times \vec{f}_{f,\textrm{aero},n} \right) + \sum_{n=1}^{N} \vec{g}_{f,\textrm{aero},n} + \vec{r}_{S/R} \times \vec{f}_{f,\textrm{weight}}\]

5.1.4. Parameter summary

In addition to the design curves that define the Foil geometry, the physical canopy model requires additional information about physical details associated with that geometry:

(5.10)\[\begin{split}\begin{aligned} r_\textrm{upper} \qquad & \textrm{Profile extent of the upper surface} \\ r_\textrm{lower} \qquad & \textrm{Profile extent of the lower surface} \\ s_\textrm{end} \qquad & \textrm{Section index where air intakes end} \\ \rho_u \qquad & \textrm{Areal density of the upper surface material} \\ \rho_r \qquad & \textrm{Areal density of the internal rib material} \\ \rho_l \qquad & \textrm{Areal density of the lower surface material} \\ N_\textrm{cells} \qquad & \textrm{Number of internal cells} \\ C_{D,\textrm{intakes}} \qquad & \textrm{Drag coefficient due to air intakes} \\ C_{D,\textrm{surface}} \qquad & \textrm{Drag coefficient due to surface characterstics} \\ \end{aligned}\end{split}\]

5.2. Suspension lines

The suspension lines connect the canopy to the harness and pilot. The lines are conventionally grouped into load-bearing sets (labeled A/B/C/D, depending on their relative positions on the section chords), brake lines (that produce the trailing edge deflections), and stabilo lines (that assist in preventing the wing tips from curling into a dangerous cravat). Starting from the canopy, the lines progressively attach together in a cascade that terminates at two risers which connect the lines to the harness. The lines are responsible for producing the arc of the canopy, suspending the harness at some position relative to the canopy, and allowing the pilot to manipulate the shape of the canopy.

For rigorous models the line geometry is a major factor in wing performance, but for this project a fully-specified suspension line model would be both tedious and redundant. It would be tedious because it would require the lengths of every segment of every line, and it would be (mostly) redundant because the canopy model is a quasi-rigid body whose arc is already defined by the \(yz\)-curve of the idealized foil geometry. As a result, the suspension lines can only affect the riser position and trailing edge deflections, so this model can reasonably use simple approximations that do not depend on an explicit line geometry.

5.2.1. Controls

The suspension lines provide two primary methods of controlling the paraglider system: through brakes, which change the canopy aerodynamics, and the accelerator, which repositions the payload underneath the canopy.

5.2.1.1. Brakes

A parafoil canopy can be manipulated by pulling on any of its many suspension lines, but two of the lines in particular are dedicated to slowing the wing or controlling its turning motion. Known as the brakes or toggles, these controls induce downward trailing edge deflections (see Fig. 5.4) along each half of the canopy, increasing drag on that side of the wing. Symmetric deflections slow the wing down, and asymmetric deflections cause the wing to turn.

Fig. 5.6 Asymmetric brake deflection.

Photograph by Frédéric Bonifas, distributed under a CC-BY-SA 3.0 license.

Fig. 5.7 Symmetric brake deflection.

Photograph by Wikimedia contributor “PiRK” under a CC-BY-SA 3.0 license.

A physically accurate model of the deflection distribution would need to model the length and angle of every line and how the angles deform during braking maneuvers. Because the line geometry was not a focus for this project, an approximation is used instead.

First, observe that as brakes are progressively applied the deflections will typically start near the middle and radiate towards the wing root and tip as the brake magnitude is increased. For small brake inputs the deflections are zero near the wing root and tip, but for large brake inputs even those sections experience deflections.

To approximate this behavior, start by assuming the deflection distances from each individual brake input are symmetric around some peak near the middle of each semispan and vary as a quartic function \(q(p)\). Define the polynomial coefficients such that the function value and slope are zero at \(p = 0\) and \(p = 1\) and a peak at \(p = 0.5\). The result is a quartic that is symmetric about \(p = 0.5\) with a peak magnitude of \(1\).

(5.11)\[\begin{split}q(p) = \begin{cases} 16p^4 - 32p^3 + 16p^2 &\mbox 0 \le p \le 1 \\ 0 & \mbox{else} \end{cases}\end{split}\]

Fig. 5.8 Truncated quartic distribution

Next define two variables for the section indices near the canopy root and tip that control the start and stop points of the deflection. Representing the start and stop positions as variables allows modeling how the deflection distribution changes with the brake inputs. For both \(s_\textrm{start}\) and \(s_\textrm{stop}\), define their values when \(\delta_{br} = 0\) and \(\delta_{br} = 1\). Then, using linear interpolation as a function of brake input:

(5.12)\[\begin{split}\begin{aligned} s_\textrm{start} &= s_\textrm{start,0} + \left( s_\textrm{start,1} - s_\textrm{start,0} \right) \delta_b\\ s_\textrm{stop} &= s_\textrm{stop,0} + \left( s_\textrm{stop,1} - s_\textrm{stop,0} \right) \delta_b \end{aligned}\end{split}\]

The start and stop points can be used to map the section indices \(s\) into the domain of the quartic \(p\), such that \(s = s_\textrm{start} \rightarrow p = 0\) and \(s = s_\textrm{stop} \rightarrow p = 1\):

(5.13)\[p(s) = \frac{s - s_\textrm{start}}{s_\textrm{stop} - s_\textrm{start}}\]

The quartic output for each brake is unit magnitude, which should be scaled by the brake input. Summing the two scaled outputs represent the fraction of maximum brake deflection distance over the entire span. The maximum brake deflection distance is a constraint set by the suspension line model parameter \(\kappa_b\), the maximum length that the model will allow the pilot to pull the brake line (although on a physical wing there isn’t a clear limit to how far the brakes can be pulled).

Finally, the total brake deflection distance is the sum of contributions from left and right brake:

(5.14)\[\delta_d(s, \delta_{bl}, \delta_{br}) = \left( \delta_{bl} \cdot q(p(-s)) + \delta_{br} \cdot q(p(s)) \right) \cdot \kappa_b\]

A feature of this design is that setting \(s_\textrm{start,1} < 0\) allows deep brake inputs to deflect the opposing semispan, and \(s_\textrm{stop,1} > 1\) allows deflections at the wing tips, as shown in Fig. 5.9.

Fig. 5.9 Quartic brake deflections, \(\delta_{bl} = 1.00\) and \(\delta_{br} = 1.0\)

Together with the Foil geometry, the absolute brake deflection distances can be used to compute each section’s airfoil index (5.1).

5.2.1.2. Accelerator

Paragliders are not powered aircraft, but pilots can increase their airspeed by adjusting how the payload is positioned relative to the canopy. The accelerator or speed bar is positioned under the pilot’s feet, and by pushing out they can shift the riser position \(RM\) forward and up. The canopy pitching angle, angle of attack, and airspeed must adjust to the new equilibrium, changing both the airspeed and the glide ratio.

The goal is to model how the riser position changes as a function of the accelerator control input \(0 \le \delta_a \le 1\).

Fig. 5.10 Paraglider wing accelerator geometry.

For notational simplicity, define \(\overline{A}\) and \(\overline{C}\) as the lengths of the lines connecting them to the riser midpoint \(RM\):

\[\begin{split}\begin{aligned} \overline{A} &\defas \norm{\vec{r}_{A/RM}} \\ \overline{C} &\defas \norm{\vec{r}_{C/RM}} \end{aligned}\end{split}\]

The default lengths of the lines are defined by two pairs of design parameters. First, the default position of the riser midpoint \(RM\) is defined with \(\kappa_x\) and \(\kappa_z\); this is the position of \(RM\) when \(\delta_a = 0\). Second, two connection points along the canopy root chord are defined with \(\kappa_A\) and \(\kappa_C\); connecting lines from these points are the physical means by which \(RM\) is positioned underneath the canopy. The \(A\) lines connect near the front of the wing, and are variable length; the pilot can use the accelerator to shorten the lengths of these lines. The \(C\) lines connect towards the rear of the canopy, and are fixed length.

Geometrically, shortening \(\overline{A}\) will move \(RM\) forward while rotating the \(C\) lines. Aerodynamically, shortening \(\overline{A}\) effectively rotates the canopy pitch down about the point \(C\), decreasing the global angle of incidence of the canopy; decreasing the angle of incidence decreases lift, and the wing must accelerate to reestablish equilibrium.

A fifth design parameter, the accelerator length \(\kappa_a\), is required to define the maximum length change produced by the accelerator; this is the maximum length that \(\overline{A}\) can be decreased. This value is limited by the physical geometry of the pulleys that give the pilot the leverage to pull the canopy into its new position. The pilot uses the accelerator control input \(\delta_a\), a value between 0 and 1, to specify the total decrease in \(\overline{A}\):

(5.15)\[\overline{A}(\delta_a) = \overline{A_0} - \delta_a \kappa_a\]

For deriving the basic geometric relations, it is convenient to normalize all the design parameters by the central chord. This avoids the extra terms in the derivation and allows a wing design to scale naturally with the canopy.

The goal is to use the physical geometry, where the risers position is determined by \(\overline{A}\) and \(\overline{C}\), to define the position of \(RM\) a function of \(\delta_a\). The first step is to determine the default line lengths by setting \(\delta_a = 0\) and applying the Pythagorean theorem:

(5.16)\[\begin{split}\begin{aligned} \overline{A_0} &= \sqrt{\kappa_z^2 + \left( \kappa_x - \kappa_A \right) ^2}\\ \\ \overline{C_0} &= \sqrt{\kappa_z^2 + \left( \kappa_C - \kappa_x \right) ^2} \end{aligned}\end{split}\]

In the general case, the line lengths are functions of \(\delta_a\):

(5.17)\[\begin{split}\begin{aligned} \overline{A}(\delta_a)^2 &= {RM}_z^2 + \left( {RM}_x - \kappa_A \right) ^2\\ \\ \overline{C}(\delta_a)^2 &= {RM}_z^2 + \left( \kappa_C - {RM}_x \right) ^2 = \overline{C_0}^2 \end{aligned}\end{split}\]

Where \(\overline{C} \equiv \overline{C_0}\) due to the physical constraint that the length of the \(C\) lines are constant.

Subtract the two equations in (5.17):

\[\overline{A}(\delta_a)^2 - \overline{C_0}^2 = \left( {RM}_x - \kappa_A \right) ^2 - \left( \kappa_C - {RM}_x \right) ^2\]

Finally, substitute (5.15) and solve for \({RM}_x\) and \({RM}_z\) as functions of \(\delta_a\):

(5.18)\[\begin{split}\begin{aligned} {RM}_x(\delta_a) &= \frac {\left( \overline{A_0} - \delta_a \kappa_a \right) ^2 - \overline{C_0}^2 - \kappa_A^2 + \kappa_C^2} {2 \left( \kappa_C - \kappa_A \right)}\\ \\ {RM}_z(\delta_a) &= \sqrt{\overline{C_0}^2 - \left( \kappa_C - {RM}_x(\delta_a) \right) ^2 }\\ \end{aligned}\end{split}\]

The final position of \(RM\) with respect to the leading edge (which is also the origin of the canopy coordinate system), scaled by the length of the central chord \(c_0\) of the wing, is then:

(5.19)\[\vec{r}_{RM/LE}^b(\delta_a) = c_0 \cdot \left\langle -{RM}_x(\delta_a), 0, {RM}_z(\delta_a) \right\rangle\]

Where \({RM}_x\) was negated since the wing \(x\)-axis is positive forward.

5.2.2. Inertia

This simplistic model assumes the inertia of the lines is negligible compared to that of the canopy; in particular, inaccuracies in the simplified canopy inertia are more significant than the line inertia, so this model simply defines the translational and rotation inertia as zero.

5.2.3. Resultant force

Although the lines are nearly invisible compared to the rest of the wing, they contribute a significant amount of aerodynamic drag. Because the total system drag of a paraglider is relatively small, even a small increase can have a large impact on sensitive characteristics such as glide ratio; in fact, paraglider suspension lines contribute upwards of 20% of the total paraglider system drag ([30], [20]), and should not be neglected.

This model does not provide an explicit line geometry, so it can’t compute the true line area distribution. Instead, it lumps the entire length of the lines into configurable control points; for example, given the total line length and average line diameter, the line area can be lumped into singularities such as the centroid of line area for each semispan. As with other similar designs [20], this model treats the drag as isotropic (because the operating ranges of alpha and beta are so small the line drag is effectively constant, and what little force exists along the \(z\)-axis is negligible compared to the lift of the canopy). Given the total area \(S_\textrm{lines}\) represented by each singularity the total aerodynamic drag at some control point \(L\) can be calculated as in [20] or [30]:

(5.20)\[S_l = \kappa_L \kappa_d\]

(5.21)\[\vec{f}_{l,\textrm{aero},n} = \frac{1}{2} \rho_\textrm{air} \norm{\vec{v}_{W/L_n}}^2 S_l C_{d,l,n} \hat{\vec{v}}_{W/L_n}\]

(5.22)\[\vec{f}_{l,\textrm{aero}} = \frac{1}{N} \sum_{n=1}^{N} \vec{f}_{l,\textrm{aero},n}\]

(5.23)\[\vec{g}_{l/R} = \frac{1}{N} \sum_{n=1}^{N} \vec{r}_{CP_n/R} \times \vec{f}_{l,\textrm{aero},n}\]

5.2.4. Parameter summary

For the harness position:

(5.24)\[\begin{split}\begin{aligned} \kappa_A \qquad & \textrm{Chord ratio to the A lines} \\ \kappa_C \qquad & \textrm{Chord ratio to the C lines} \\ \kappa_x \qquad & \textrm{Chord ratio to the } x\textrm{-coordinate of the riser midpoint} \\ \kappa_z \qquad & \textrm{Chord ratio to the } z\textrm{-coordinate of the riser midpoint} \\ \kappa_a \qquad & \textrm{Accelerator line length} \\ \end{aligned}\end{split}\]

For the brakes:

(5.25)\[\begin{split}\begin{aligned} s_{\textrm{start},0}, s_{\textrm{start},1} \qquad & \textrm{Section indices where deflections begin for } \delta_b \in \{0, 1\} \\ s_{\textrm{stop},0}, s_{\textrm{stop},1} \qquad & \textrm{Section indices where deflections end for } \delta_b \in \{0, 1\} \\ \kappa_b \qquad & \textrm{Maximum trailing edge deflection distance} \\ \end{aligned}\end{split}\]

For the aerodynamics:

(5.26)\[\begin{split}\begin{aligned} \kappa_L \qquad & \textrm{Total line length} \\ \kappa_d \qquad & \textrm{Average line diameter} \\ \vec{r}_{CP_n/R} \qquad & \textrm{Position of lumped control point} \ n \\ C_{d,l,n} \qquad & \textrm{Line drag coefficient for control point} \ n \\ \end{aligned}\end{split}\]

5.3. Harness

A paraglider harness is the seat for the pilot, which is suspended from the risers. Safety straps over the legs and chest ensure the pilot cannot fall from the harness in turbulent conditions or during unsteady maneuvers. A tensioning strap in front of the pilot’s chest controls the horizontal riser separation distance, which allows the pilot to adjust the balance between stability (sensitivity to turbulence) and wing responsiveness to weight shift control. In addition to giving the pilot a safe place to sit, the harness also provides places to store the pilot’s gear, a pouch to contain the emergency reserve parachute, and optional padding to protect the pilot in the event of a crash.

Instead of attempting to capture all the geometric irregularities of paraglider harnesses, this model calls upon a time-honored solution from physics: it considers the harness as a sphere. Moreover, the pilot, gear, and reserve parachute are accounted for by simply adding their masses to the mass of the harness. The harness, pilot, and gear are collectively referred to as the payload.

5.3.1. Controls

Paraglider harnesses allow pilots to shift their weight left and right, causing an imbalanced load on each semispan. (For a real wing this maneuver also causes a vertical shearing stress along the center of the foil, but due to the rigid body assumption of the canopy model this deformation will be neglected.) The weight imbalance causes the canopy to roll towards the shifted mass, resulting in a gentle turn in the desired direction. Although the turn rate is less than can be produced by the brakes, this maneuver causes less drag and is preferred (when suitable) for its aerodynamic efficiency.

The movement of the pilot can be arguably described as occurring inside the volume of the harness, so weight shift control can be modeled as a displacement of the payload center of mass \(P\). Given that the pilot can only shift a limited distance \(\kappa_w\) in either direction, a natural choice of control input is \(-1 \le \delta_w \le 1\). With the harness initially centered in the canopy \(xz\)-plane, the displacement due to weight shift control is \(\Delta y = \delta_w \kappa_w\). The displacement of the payload center of mass produces a moment on the risers that rolls the wing and induces the turn.

Defining the riser midpoint \(RM\) as the origin the harness-local coordinate system, the position of the displaced center of mass is then:

(5.27)\[\vec{r}_{P/RM} = \bar{\vec{r}}_{P/RM} \, + \left< 0, \delta_w \kappa_w, 0 \right>\]

5.3.2. Inertia

As in [44] (and similarly in [20]), the payload is modeled as a solid sphere of uniform density. With a total mass \(m_p\), center of mass \(P\), and projected surface area \(S_p\), the moment of inertia about the payload center of mass is simply:

\[\begin{split}\mat{J}_{p/P} = \begin{bmatrix} J_{xx} & 0 & 0 \\ 0 & J_{yy} & 0 \\ 0 & 0 & J_{zz} \end{bmatrix}\end{split}\]

where

\[J_{xx} = J_{yy} = J_{zz} = \frac{2}{5} m_p r_p^2 = \frac{2}{5} \frac{m_p S_p}{\pi}\]

5.3.3. Resultant force

Harness drag coefficients were studied experimentally in [44]. The author measured several harness models in a wind tunnel and converted the results into aerodynamic coefficients normalized by the cross-sectional area of the sphere. For a more sophisticated approach the coefficient can be adjusted to account (approximately) for angle of attack and Reynolds number [20], but this model simply treats the drag coefficient as a constant.

(5.28)\[\vec{f}_{p,\textrm{weight}} = m_p \vec{g}\]

(5.29)\[\vec{f}_{p,\textrm{aero}} = \frac{1}{2} \rho_\textrm{air} \norm{\vec{v}_{W/P}}^2 S_p C_{D,p} \hat{\vec{v}}_{W/P}\]

(5.30)\[\vec{g}_{p/R} = \vec{r}_{CP/R} \times \vec{f}_{p,\textrm{aero}} + \vec{r}_{P/R} \times \vec{f}_{p,\textrm{weight}}\]

Note that the spherical nature of the model implies isotropic drag. Although this is clearly a poor assumption for such a significantly non-spherical object, the fact that the wind is rarely more than 15 degrees off the \(x\)-axis means the such a “naive” drag coefficient will remain fairly accurate over the typical range of operation (regardless of the poor geometric accuracy). This assumption also has the downside that it will never produce an aerodynamic moment about the payload center of mass, but in the absence of experimental data on the magnitude of the missing moment, this model continues to ignore it.

5.3.4. Parameter summary

\[\begin{split}\begin{aligned} m_p \qquad & \textrm{Total payload mass} \\ \bar{\vec{r}}_{P/RM} \qquad & \textrm{Payload center of mass default position} \\ \kappa_w \qquad & \textrm{Maximum weight shift distance} \\ S_p \qquad & \textrm{Projected payload area} \\ C_{d,p} \qquad & \textrm{Payload drag coefficient} \\ \end{aligned}\end{split}\]