Derivations

Parametric design curves

The “simplified” foil geometry chose a set of variables (3.15) that describe different aspects of the shape. This section provides definitions for several of those variables using parametric functions that can approximate the structure of a typical parafoil using a small number of simple parameters.

Elliptical chord

A Foil geometry requires a chord distribution $c(s)$. For parafoils, the chords lengths are most commonly defined by a truncated elliptical function of section index, in which case the distribution is a function of two design parameters. The typical choices are either the root and wingtip chord lengths, or the root length and a taper ratio. Choosing the root and wingtip chord lengths, a truncated elliptical function over the section index $-1\le s\le 1$ is then:

(2)$$\begin{array}{r}\begin{array}{rl}a& =\frac{1}{\sqrt{1-{\left(\frac{c_{\text{tip}}}{c_{\text{root}}}\right)}^2}}\\ b& =c_{\text{root}}\\ c(s)& =b\sqrt{1-{\left(\frac{s}{a}\right)}^2}\end{array}\end{array}$$

Refer to EllipticalChord in glidersim for an implementation.

Elliptical arc

In this paper the arc of a parafoil is the vector-valued function of $⟨y,z⟩$ coordinates that position the section reference points. For parafoils, the arc is typically defined by an elliptical function.

A centered elliptical curve can be defined as a function of four parameters, but the symmetry of the wing reduces that to three free design parameters, and normalizing the arc length reduces it to just two. There are several possible parametrizations, but an intuitive choice is the mean anhedral angle ${\mathrm{\Gamma }}_{\text{tip}}$ and the section roll angle ${\varphi }_{\text{tip}}$ of the wing tips [31].

Fig. 1 Parametrized elliptical arc

Choosing those parameters to define an elliptical function that is proportional to the desired $yz$-curve produces an intermediate result:

$$\begin{array}{r}\begin{array}{rl}{k}_1& =1-\frac{\tan ({\mathrm{\Gamma }}_{\text{tip}})}{\tan ({\varphi }_{\text{tip}})}\\ k_2& =1-\frac{2\tan ({\mathrm{\Gamma }}_{\text{tip}})}{\tan ({\varphi }_{\text{tip}})}\\ A& =\frac{k_1}{\sqrt{k_2}}\\ B& =\frac{k_1}{k_2}\tan ({\mathrm{\Gamma }}_{\text{tip}})\\ \mathbf{f}(t)& =⟨A\cos (t),B\sin (t)⟩\end{array}\end{array}$$

This design requires that ${\varphi }_{\text{tip}}>2{\mathrm{\Gamma }}_{\text{tip}}$ (so the wing must be wider than it is tall and the wing tip roll cannot exceed 90°) and is valid over $t_{min}\le t\le \pi -t_{min}$, where $t_{min}=\arccos \left(\frac{1}{A}\right)$.

Next although the shape produced by this intermediate result is proportional to the desired curve, it is not directly usable by the Foil geometry. It needs two modifications:

1. Make the arc a function of the chosen section index $s$

2. Scale the arc to a total curve length of $b_{\text{flat}}$

Both can be achieved by normalizing the elliptical function to a curve length of 2. First, scale the axes to produce a new semi-ellipse with a total curve length of 1:

$$\begin{array}{r}\begin{array}{rl}L(t)& ={\int }_{\frac{\pi }{2}}^{t_{min}}\Vert \mathbf{f}(t)\Vert dt\\ k_3& =L(t_{min})\\ \overline{\mathbf{f}}(t)& =⟨\frac{A}{k_3}\cos (t),\frac{B}{k_3}\sin (t)⟩\end{array}\end{array}$$

The fact that the simplified foil geometry chose to define the section index $s$ as the linear distance along the $yz$-curve enables a convenient conversion over $\frac{\pi }{2}\le t\le t_{min}$ and $0\le s\le 1$:

$$\begin{array}{r}\begin{array}{rl}\overline{L}(t)& ={\int }_{\frac{\pi }{2}}^{t_{min}}\Vert \overline{\mathbf{f}}(t)\Vert dt=s(t)\\ t(s)& =s^{-1}(t)\end{array}\end{array}$$

Thus the complete parametric function for the $yz$-curve of the arc is thus $⟨y,z⟩(s)=\overline{\mathbf{f}}(t(\left|s\right|))$. The integrals and inverse functions are not available analytically, but are trivial to compute numerically. Refer to EllipticalArc in glidersim for an implementation.

Polynomial torsion

Like most wings, parafoils use section-relative pitch $\theta (s)$ (conventionally referred to as geometric torsion) to fine-tune wing behavior. The exact distribution of geometric torsion along a wing can be difficult to measure, but they are frequently described using simple polynomials or piecewise-linear functions. For idealized models of nonlinear geometries such as those developed here, a piecewise-polynomial function is assumed to be adequate.

Assuming a symmetric wing, define three parameters:

• $T$: the maximum torsion (in radians) at the wingtips

• $s_{start}$: the section index where the torsion begins (where $0\le s_{start}<1$)

• $\beta$: the degree of the polynomial (for example, $\beta =1$ is linear, $\beta =2$ is quadratic, etc.)

$$\begin{array}{r}\begin{array}{rl}p(s)& =\frac{|s|-s_{start}}{1-s_{start}}\\ \theta (s)& =\{\begin{array}{ll}0& |s|<s_{start}\\ T{p}^{\beta }& |s|\ge s_{start}\end{array}\end{array}\end{array}$$

Refer to PolynomialTorsion in glidersim for an implementation.

Area and Volume of a Mesh

The paraglider dynamics require the inertial properties of the canopy surface areas and volume. These include the magnitudes (total mass or volume), centroids, and inertia tensors. All of these quantities can be computed using a triangular surface mesh over the canopy surfaces.

What follows is a reproduction of the procedure developed in [51], which is a functionally equivalent to the procedure from [52] but with a more intuitive interpretation and complete equations for the inertia tensors.

Area

First, for each of the upper and lower surfaces, cover the surface with a triangulated mesh so it is represented by a set of $N$ triangles. Each triangle is defined by three points ${\{\mathrm{P}1,\mathrm{P}2,\mathrm{P}3\}}_n$ in canopy coordinates. For convenience, define position vectors for each of the three points of the nth triangle: ${\mathbf{r}}_{i,n}\stackrel{\mathrm{def}}{=}{\mathbf{r}}_{Pi/O,n}$.

The area of each triangle is easily computed using the vector cross-product of two legs of the triangle:

$$a_n=\frac{1}{2}\rho \Vert ({\mathbf{r}}_{2,n}-{\mathbf{r}}_{1,n})\times ({\mathbf{r}}_{3,n}-{\mathbf{r}}_{2,n})\Vert$$

The total area of the surface is the sum of the triangle areas:

$$a=\sum _{n=1}^N{a}_n$$

The area centroid of each triangle:

$${\bar{\mathbf{a}}}_n\stackrel{\mathrm{def}}{=}\frac{1}{3}({\mathbf{r}}_{1,n}+{\mathbf{r}}_{2,n}+{\mathbf{r}}_{3,n})$$

And the centroid $\mathrm{A}$ of the total surface area with respect to the canopy origin $\mathrm{O}$:

$${\mathbf{r}}_{\mathrm{A}/\mathrm{O}}=\frac{1}{a}\sum _{n=1}^N{a}_n{\bar{\mathbf{a}}}_n$$

The covariance matrix of the total surface area:

$${\mathbf{\Sigma }}_a=\sum _{n=1}^N{a}_n{\bar{\mathbf{a}}}_n{\bar{\mathbf{a}}}_n^T$$

The inertia tensor of the total surface area $a$ about the canopy origin $\mathrm{O}$:

$${\mathbf{J}}_{a/\mathrm{O}}=\mathrm{trace}\left({\mathbf{\Sigma }}_a\right){\mathbf{I}}_3-{\mathbf{\Sigma }}_a$$

This completes the calculation of the three relevant properties for each surface area: the total area $a$, the area centroid ${\mathbf{r}}_{\mathrm{A}/\mathrm{O}}$, and the inertia tensor ${\mathbf{J}}_{a/\mathrm{O}}$.

Volume

Now for the volume. For the purposes of computing the inertia properties of the enclosed air, it is convenient to neglect the air intakes and treat the canopy as a closed volume. Given this simplifying assumption, build another triangular mesh that covers the entire canopy surface as well as the left and right wing tip sections. For this derivation, it is essential that the points on each triangle are ordered such that a right-handed traversal produces a normal vector pointing out of the volume. It is also essential that the complete mesh does not contain any holes, or the volume may be miscounted. Given a surface triangulation over the closed canopy geometry using $N$ triangles, the volume can be computed as follows.

First, treat each triangle as the face of a tetrahedron that includes the origin. The signed volume of the tetrahedron formed by each triangle is given by:

$$v_n=\frac{1}{6}({\mathbf{r}}_{1,n}\times {\mathbf{r}}_{2,n})\cdot {\mathbf{r}}_{3,n}$$

Given that the vertices of each triangle were oriented such that they satisfy a right-hand rule, the sign of each volume will be positive if the normal vector for each triangular face points away from the origin, and negative if it points towards the origin. In essence the tetrahedrons “overcount” the volume for triangles pointing away from the origin, then the triangles facing the origin subtract away the excess volume. The final volume of the canopy is the simple sum:

$$v=\sum _{n=1}^N{v}_n$$

For the volume centroid of each tetrahedron:

$${\bar{\mathbf{v}}}_n\stackrel{\mathrm{def}}{=}\frac{1}{4}\sum _{i=1}^3{\mathbf{r}}_{i,n}$$

And the centroid $\mathrm{V}$ of the total volume with respect to the canopy origin $\mathrm{O}$:

$${\mathbf{r}}_{\mathrm{V}/\mathrm{O}}=\frac{1}{v}\sum _{n=1}^N{v}_n{\bar{\mathbf{v}}}_n$$

Lastly, calculating the inertia tensor of the volume can be simplified by computing the inertia tensor of a prototypical or “canonical” tetrahedron and applying an affine transformation to produce the inertia tensor of each individual volume.

First, given the covariance matrix of the “canonical” tetrahedron:

$$\begin{array}{r}\hat{\mathbf{\Sigma }}\stackrel{\mathrm{def}}{=}\left[\begin{array}{ccc}\frac{1}{60}& \frac{1}{120}& \frac{1}{120}\\ \frac{1}{120}& \frac{1}{60}& \frac{1}{120}\\ \frac{1}{120}& \frac{1}{120}& \frac{1}{60}\end{array}\right]\end{array}$$

Use the points in each triangle to define:

$$\begin{array}{r}{\mathbf{T}}_n\stackrel{\mathrm{def}}{=}\left[\begin{array}{ccc}|& |& |\\ {\mathbf{r}}_{1,n}& {\mathbf{r}}_{2,n}& {\mathbf{r}}_{3,n}\\ |& |& |\end{array}\right]\end{array}$$

The covariance of each tetrahedron volume is then:

$${\mathbf{\Sigma }}_n=\left|{\mathbf{T}}_n\right|{\mathbf{T}}_n^T\hat{\mathbf{\Sigma }}{\mathbf{T}}_n$$

And the covariance matrix of the complete volume:

$${\mathbf{\Sigma }}_v=\sum _{n=1}^N{\mathbf{\Sigma }}_n$$

And at last, the inertia tensor of the volume about the origin $O$ can be computed directly from the covariance matrix:

$${\mathbf{J}}_{v/O}=\mathrm{trace}\left({\mathbf{\Sigma }}_v\right){\mathbf{I}}_3-{\mathbf{\Sigma }}_v$$

Apparent mass of a parafoil

This section presents Barrows’ method [24] for estimating the apparent mass matrix of a wing with circular arc anhedral. (For a discussion of apparent mass effects, see Apparent Mass.) The equations have been adapted to use the standard notation of this paper.

The purpose of the equations is estimate several terms that allow the paraglider system dynamics model to calculate the apparent inertia matrix with respect to the dynamics reference point, so the apparent mass can be taken into account when calculating the canopy acceleration. The necessary terms are:

• ${\mathbf{A}}_{a/R}$: apparent inertia matrix with respect to some reference point $R$. This matrix is comprised of a translational inertia part ${\mathbf{M}}_a$ and a rotational inertia part ${\mathbf{J}}_{a/R}$.

• ${\mathbf{r}}_{RC/R}$: roll center with respect to $R$

• ${\mathbf{r}}_{PC/RC}$: pitch center with respect to the roll center $RC$

Some notes about Barrows’ development:

• It assumes the foil is symmetric about the $xz$-plane (left-right symmetry) and about the $yz$-plane (fore-aft symmetry).

• It requires that the dynamics reference point $R$ lies in the $xz$-plane

• It assumes the canopy arc is circular.

• It assumes a constant chord length over the entire span.

• It assumes constant thickness over the entire span.

• It assumes no chordwise camber.

• It assumes the chords are all parallel to the x-axis (which also means no geometric twist). This mostly isn’t a problem since our coordinate system is defined by the central chord, the geometric torsion angles tend to be quite small, and twist tends to occur over segments which represent negligible volume compared to the bulk of the wing.

Fig. 2 Geometry for Barrow’s apparent mass equations.

Some initial definitions:

$$\begin{array}{r}\begin{array}{rl}t& =\text{Airfoil thickness.}\\ h^{\ast }& =\frac{h}{b}\end{array}\end{array}$$

First, the apparent mass terms for a flat wing of a similar volume, from Barrows’ equations 34-39:

$$\begin{array}{r}\begin{array}{rl}{m}_{f11}& =k_A\pi \left(t^{2}b/4\right)\\ m_{f22}& =k_B\pi \left(t^{2}c/4\right)\\ m_{f33}& =\left[\mathrm{AR}/(1+\mathrm{AR})\right]\pi \left(c^{2}b/4\right)\\ \\ I_{f11}& =0.055\left[\mathrm{AR}/(1+\mathrm{AR})\right]b{S}^2\\ I_{f22}& =0.0308\left[\mathrm{AR}/(1+\mathrm{AR})\right]c^{3}S\\ I_{f33}& =0.055b^3{t}^2\end{array}\end{array}$$

Where $k_A$ and $k_B$ are the “correction factors for three-dimensional effects”:

$$\begin{array}{r}\begin{array}{rl}{k}_A& =0.85\\ k_B& =1.0\end{array}\end{array}$$

Assuming the parafoil arc is circular and with no chordwise camber, use Barrows equations 44 and 50 to compute the pitch center $PC$ and roll center $RC$ as points directly above the confluence point $C$ of the arc:

$$\begin{array}{r}\begin{array}{rl}{z}_{PC/C}& =-\frac{r\sin \left(\mathrm{\Theta }\right)}{\mathrm{\Theta }}\\ z_{RC/C}& =-\frac{z_{PC/C}{m}_{f22}}{m_{f22}+I_{f11}/r^2}\\ z_{PC/RC}& =z_{PC/C}-z_{RC/C}\end{array}\end{array}$$

Modifying the apparent mass terms from the flat wing to approximate the terms for the arched wing, Barrows equations 51-55:

$$\begin{array}{r}\begin{array}{rl}{m}_{11}& =k_A[1+\left(\frac{8}{3}\right){h^{\ast }}^2]\pi \left(t^{2}b/4\right)\\ m_{22}& =\frac{r^2{m}_{f22}+I_{f11}}{z_{PC/C}^2}\\ m_{33}& =m_{f33}\\ \\ I_{11}& =\frac{z_{PC/RC}^2}{z_{PC/C}^2}{r}^2{m}_{f22}+\frac{z_{RC/C}^2}{z_{PC/C}^2}{I}_{f11}\\ I_{22}& =I_{f22}\\ I_{33}& =0.055(1+8{h^{\ast }}^2)b^3{t}^2\end{array}\end{array}$$

The apparent mass and apparent moment of inertia matrices are then defined in Barrows equations 1 and 17:

(3)$$\begin{array}{r}{\mathbf{M}}_a\stackrel{\mathrm{def}}{=}\left[\begin{array}{ccc}{m}_{11}& 0& 0\\ 0& m_{22}& 0\\ 0& 0& m_{33}\end{array}\right]\end{array}$$

(4)$$\begin{array}{r}{\mathbf{I}}_a\stackrel{\mathrm{def}}{=}\left[\begin{array}{ccc}{I}_{11}& 0& 0\\ 0& I_{22}& 0\\ 0& 0& I_{33}\end{array}\right]\end{array}$$

Define two helper matrices:

$$\begin{array}{r}{\mathbf{S}}_2\stackrel{\mathrm{def}}{=}\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]\end{array}$$

$$\mathbf{Q}={\mathbf{S}}_2{\left[{\mathbf{r}}_{PC/RC}\right]}^{\times }{\mathbf{M}}_a{\left[{\mathbf{r}}_{RC/R}\right]}^{\times }$$

Where ${\left[\mathbf{x}\right]}^{\times }$ is the cross-product matrix operator.

Using the helper matrices, use Barrows equation 25 to write the rotational part of the apparent inertia matrix:

$${\mathbf{J}}_{a/R}\stackrel{\mathrm{def}}{=}\mathbf{I}-{\left[{\mathbf{r}}_{RC/R}\right]}^{\times }{\mathbf{M}}_a{\left[{\mathbf{r}}_{RC/R}\right]}^{\times }-{\left[{\mathbf{r}}_{PC/RC}\right]}^{\times }{\mathbf{M}}_a{\left[{\mathbf{r}}_{PC/RC}\right]}^{\times }{\mathbf{S}}_2-\mathbf{Q}-{\mathbf{Q}}^T$$

And the corresponding angular momentum of the apparent mass about $R$, using Barrows equation 24:

$${\mathbf{h}}_{a/R}=({\mathbf{S}}_2{\left[{\mathbf{r}}_{PC/RC}\right]}^{\times }+{\left[{\mathbf{r}}_{RC/R}\right]}^{\times }){\mathbf{M}}_a{\mathbf{v}}_{R/e}+{\mathbf{J}}_{a/R}\omega$$

And finally, the completed apparent inertia matrix with respect to the reference point $R$, from Barrows equation 27:

(5)$$\begin{array}{r}{\mathbf{A}}_{a/R}=\left[\begin{array}{cc}{\mathbf{M}}_a& -{\mathbf{M}}_a({\left[{\mathbf{r}}_{RC/R}\right]}^{\times }+{\left[{\mathbf{r}}_{PC/RC}\right]}^{\times }{\mathbf{S}}_2)\\ ({\mathbf{S}}_2{\left[{\mathbf{r}}_{PC/RC}\right]}^{\times }+{\left[{\mathbf{r}}_{RC/R}\right]}^{\times }){\mathbf{M}}_a& {\mathbf{J}}_{a/R}\end{array}\right]\end{array}$$

Plus the vectors necessary to incorporate ${\mathbf{J}}_{a/R}$ into the final dynamics:

$${\mathbf{r}}_{PC/RC}=\left[\begin{array}{ccc}0& 0& z_{PC/RC}\end{array}\right]$$

Linear momentum of the apparent mass:

(6)$${\mathbf{p}}_{a/e}={\mathbf{M}}_a\cdot ({\mathbf{v}}_{R/e}-{\left[{\mathbf{r}}_{RC/R}\right]}^{\times }{\omega }_{b/e}-{\left[{\mathbf{r}}_{PC/RC}\right]}^{\times }{\mathbf{S}}_2\cdot {\omega }_{b/e})$$

Angular momentum of the apparent mass about $R$:

(7)$${\mathbf{h}}_{a/R}=({\mathbf{S}}_2\cdot {\left[{\mathbf{r}}_{PC/RC}\right]}^{\times }+{\left[{\mathbf{r}}_{RC/R}\right]}^{\times })\cdot {\mathbf{M}}_a\cdot {\mathbf{v}}_{R/e}+{\mathbf{J}}_{a/R}\cdot {\omega }_{b/e}$$

Refer to ParagliderWing in glidersim for an implementation.

Paraglider system models

Model 6a

This section describe a paraglider dynamics model with 6 degrees of freedom. It uses a rigid-body assumption, and incorporates the effects of apparent mass. The dynamics are computed with respect to the riser midpoint $RM$ instead of the wing center of mass $B$ because it avoids needing to recompute the apparent inertia matrix whenever B changes. In this derivation all vectors are in the canopy coordinate system $c$, so the vector coordinate systems are implicit in the notation.

The derivation develops the equations of motion by starting with derivatives of linear and angular momentum. The derivation is largely based on the excellent [11], although this section uses this paper’s version of Stevens’ notation (see Notation and Symbols).

An implementation of this model is available as Paraglider6a in the glidersim package. The glidersim package also includes Paraglider6b and Paraglider6c, which decouple the translational and angular equations of motion by choosing the glider center of gravity for the dynamics reference point, but do not incorporate the apparent mass matrix.

Real mass only

Start with the equations for the translational and angular momentum of the body $b$ about the reference point $RM$ as observed by the inertial reference frame $e$:

(8)$$\begin{array}{r}\begin{array}{rl}{\mathbf{p}}_{b/e}& =m_b{\mathbf{v}}_{B/e}\\ & =m_b({\mathbf{v}}_{RM/e}+{\omega }_{b/e}\times {\mathbf{r}}_{B/RM})\end{array}\end{array}$$

(9)$${\mathbf{h}}_{b/RM}=m_b{\mathbf{r}}_{B/RM}\times {\mathbf{v}}_{RM/e}+{\mathbf{J}}_{b/RM}\cdot {\omega }_{b/e}$$

Compute the momentum derivatives in the inertial frame ${\mathcal{F}}_e$ in terms of derivatives in the body frame ${\mathcal{F}}_b$:

(10)$$\begin{array}{r}\begin{array}{c}\begin{array}{rl}{}^e{\dot{\mathbf{p}}}_{b/e}& ={}^b{\dot{\mathbf{p}}}_{b/e}+{\omega }_{b/e}\times {\mathbf{p}}_{b/e}\\ & =m_b({}^b{\dot{\mathbf{v}}}_{RM/e}+{}^b{\dot{\omega }}_{b/e}\times {\mathbf{r}}_{B/RM}+{\omega }_{b/e}\times {\cancel{{}^b\dot{{\mathbf{r}}_{B/RM}}}}^0)+{\omega }_{b/e}\times {\mathbf{p}}_{b/e}\\ & =m_b({}^b{\dot{\mathbf{v}}}_{RM/e}+{}^b{\dot{\omega }}_{b/e}\times {\mathbf{r}}_{B/RM})+{\omega }_{b/e}\times {\mathbf{p}}_{b/e}\\ \begin{array}{c}\end{array}\\ {}^e{\dot{\mathbf{h}}}_{b/RM}& ={}^b{\dot{\mathbf{h}}}_{b/RM}+{\omega }_{b/e}\times {\mathbf{h}}_{b/RM}\\ & =m_b({\cancel{{}^b\dot{{\mathbf{r}}_{B/RM}}}}^0\times {\mathbf{v}}_{RM/e}+{\mathbf{r}}_{B/RM}\times {}^b\dot{{\mathbf{v}}_{RM/e}})+{\mathbf{J}}_{b/RM}\cdot {}^b{\dot{\omega }}_{b/e}+{\omega }_{b/e}\times {\mathbf{h}}_{b/RM}\\ & =m_b{\mathbf{r}}_{B/RM}\times {}^b\dot{{\mathbf{v}}_{RM/e}}+{\mathbf{J}}_{b/RM}\cdot {}^b{\dot{\omega }}_{b/e}+{\omega }_{b/e}\times {\mathbf{h}}_{b/RM}\end{array}\end{array}\end{array}$$

Relate the derivatives of momentum with respect to the inertial frame to the net force on the body ${\mathbf{f}}_b$ and the net moment on the body about the reference point ${\mathbf{g}}_{b/RM}$:

(11)$$\begin{array}{r}\begin{array}{rl}{}^e{\dot{\mathbf{p}}}_{b/e}& ={\mathbf{f}}_b\\ {}^e{\dot{\mathbf{h}}}_{b/RM}+{\mathbf{v}}_{RM/e}\times {\mathbf{p}}_{b/e}& ={\mathbf{g}}_{b/RM}\end{array}\end{array}$$

Where

$$\begin{array}{r}\begin{array}{rl}{\mathbf{f}}_b& ={\mathbf{f}}_{b,\text{aero}}+{\mathbf{f}}_{b,\text{weight}}\\ {\mathbf{g}}_{b/RM}& ={\mathbf{g}}_{b,\text{aero}}+{\mathbf{r}}_{B/RM}\times {\mathbf{f}}_{b,\text{weight}}\end{array}\end{array}$$

Combining (10) and (11) gives the final equations for the dynamics of the real mass (solid mass plus the enclosed air) in terms of ${}^b{\dot{\mathbf{v}}}_{RM/e}$ and ${}^b{\dot{\omega }}_{b/e}$.

(12)$$\begin{array}{r}\begin{array}{c}\begin{array}{rl}{m}_b{}^b{\dot{\mathbf{v}}}_{RM/e}+m_b{}^b{\dot{\omega }}_{b/e}\times {\mathbf{r}}_{B/RM}& ={\mathbf{f}}_b-{\omega }_{b/e}\times {\mathbf{p}}_{b/e}\\ m_b{\mathbf{r}}_{B/RM}\times {}^b{\dot{\mathbf{v}}}_{RM/e}+{\mathbf{J}}_{b/RM}\cdot {}^b{\dot{\omega }}_{b/e}& ={\mathbf{g}}_{b/RM}-{\omega }_{b/e}\times {\mathbf{h}}_{b/RM}-{\mathbf{v}}_{RM/e}\times {\mathbf{p}}_{b/e}\end{array}\end{array}\end{array}$$

Rewriting the equations as a linear system:

(13)$$\begin{array}{r}{\mathbf{A}}_{r/RM}\left[\begin{array}{c}{}^b{\dot{\mathbf{v}}}_{RM/e}\\ {}^b{\dot{\omega }}_{b/e}\end{array}\right]=\left[\begin{array}{c}{\mathbf{b}}_1\\ {\mathbf{b}}_2\end{array}\right]\end{array}$$

Where:

$$\begin{array}{r}\begin{array}{rl}{\mathbf{A}}_{r/RM}& =\left[\begin{array}{cc}{m}_b{\mathbf{I}}_3& -m_b{\left[{\mathbf{r}}_{B/RM}\right]}^{\times }\\ m_b{\left[{\mathbf{r}}_{B/RM}\right]}^{\times }& {\mathbf{J}}_{b/RM}\end{array}\right]\\ \\ {\mathbf{b}}_1& ={\mathbf{f}}_b-{\omega }_{b/e}\times {\mathbf{p}}_{b/e}\\ {\mathbf{b}}_2& ={\mathbf{g}}_{b/RM}-{\omega }_{b/e}\times {\mathbf{h}}_{b/RM}-{\mathbf{v}}_{RM/e}\times {\mathbf{p}}_{b/e}\end{array}\end{array}$$

Real mass + apparent mass

Writing the dynamics in matrix form not only makes it straightforward to solve for the state derivatives, it also makes it easy to incorporate the apparent inertia matrix from Apparent mass of a parafoil. Adding the apparent inertia into the system matrix and accounting for the translational and angular apparent momentum produces:

(14)$$\begin{array}{r}\left[\begin{array}{c}{\mathbf{A}}_{r/RM}+{\mathbf{A}}_{a/RM}\end{array}\right]\left[\begin{array}{c}{}^b{\dot{\mathbf{v}}}_{RM/e}\\ {}^b{\dot{\omega }}_{b/e}\end{array}\right]=\left[\begin{array}{c}\begin{array}{r}{\mathbf{b}}_3\\ {\mathbf{b}}_4\end{array}\end{array}\right]\end{array}$$

$$\begin{array}{r}\begin{array}{rl}{\mathbf{b}}_3& ={\mathbf{b}}_1-{\omega }_{b/e}\times {\mathbf{p}}_{a/e}\\ {\mathbf{b}}_4& ={\mathbf{b}}_2-{\mathbf{v}}_{RM/e}\times {\mathbf{p}}_{a/e}-{\omega }_{b/e}\times {\mathbf{h}}_{a/RM}+{\mathbf{v}}_{RM/e}\times ({\mathbf{M}}_a\cdot {\mathbf{v}}_{RM/e})\end{array}\end{array}$$

Where ${\mathbf{A}}_{a/RM}$ is the apparent inertia matrix of the canopy from (5), ${\mathbf{M}}_a$ is the apparent mass matrix from (3), and ${\mathbf{p}}_{a/e}$ and ${\mathbf{h}}_{a/RM}$ are the linear and angular apparent momentums from (6) and (7). The extra term ${\mathbf{v}}_{RM/e}\times \left({\mathbf{M}}_a{\mathbf{v}}_{RM/e}\right)$ in ${\mathbf{b}}_4$ is necessary to avoid double counting the aerodynamic moment already accounted for by the section pitching coefficients.

Model 6b

Following the same logic as Model 6a, but targeting ${}^b{\mathbf{v}}_{B/e}$ and using the momentum about the body center of mass $B$ produces a simpler model with a diagonal system matrix, but at the cost of requiring the body center of mass to be determined before computing the apparent inertia matrix with respect to that point. For that reason the apparent mass is neglected here, although if $B$ lies in the xz-plane then the method described in Apparent mass of a parafoil could be used.

The main purpose of this model is for validating model implementations. An implementation of this model is available as Paraglider6b in the glidersim package.

(15)$${\mathbf{p}}_{b/e}=m_b{\mathbf{v}}_{B/e}$$

(16)$${\mathbf{h}}_{b/B}={\mathbf{J}}_{b/B}\cdot {\omega }_{b/e}$$

Computing the inertial derivatives with respect to the body frame:

(17)$$\begin{array}{r}\begin{array}{rl}{}^e{\dot{\mathbf{p}}}_{b/e}& =m_b{}^b{\dot{\mathbf{v}}}_{B/e}+{\omega }_{b/e}\times {\mathbf{p}}_{b/e}\\ \\ {}^e{\dot{\mathbf{h}}}_{b/B}& ={\mathbf{J}}_{b/B}\cdot {}^b{\dot{\omega }}_{b/e}+{\omega }_{b/e}\times {\mathbf{h}}_{b/B}\end{array}\end{array}$$

Using the body center of mass as the reference point simplifies the equation for angular momentum:

(18)$$\begin{array}{r}\begin{array}{rl}{}^e{\dot{\mathbf{p}}}_{b/e}& ={\mathbf{f}}_b\\ {}^e{\dot{\mathbf{h}}}_{b/B}& ={\mathbf{g}}_{b/B}\end{array}\end{array}$$

Combining (17) and (18): and rewriting as a linear system:

(19)$$\begin{array}{r}\left[\begin{array}{cc}{m}_b& 0\\ 0& {\mathbf{J}}_{b/B}\end{array}\right]\left[\begin{array}{c}{}^b{\dot{\mathbf{v}}}_{B/e}\\ {}^b{\dot{\omega }}_{b/e}\end{array}\right]=\left[\begin{array}{c}{\mathbf{f}}_b-{\omega }_{b/e}\times {\mathbf{p}}_{b/e}\\ {\mathbf{g}}_{b/B}-{\omega }_{b/e}\times {\mathbf{h}}_{b/B}\end{array}\right]\end{array}$$

Model 6c

Another option is to target ${}^b{\dot{\mathbf{v}}}_{RM/e}$ directly, but again using the momentum about the body center of mass $B$. Like Model 6b this also produces a simpler dynamics model, but again at the cost of making it less convenient to precompute the apparent inertia matrix.

The main purpose of this model is for validating model implementations. An implementation of this model is available as Paraglider6c in the glidersim package.

Computing the inertial derivatives with respect to the body frame:

(20)$$\begin{array}{r}\begin{array}{rl}{}^e{\dot{\mathbf{p}}}_{b/e}& =m_b({}^b{\dot{\mathbf{v}}}_{RM/e}+{}^b{\dot{\omega }}_{b/e}\times {\mathbf{r}}_{B/RM})+{\omega }_{b/e}\times {\mathbf{p}}_{b/e}\\ {}^e{\dot{\mathbf{h}}}_{b/B}& ={\mathbf{J}}_{b/B}\cdot {}^b{\dot{\omega }}_{b/e}+{\omega }_{b/e}\times {\mathbf{h}}_{b/B}\end{array}\end{array}$$

Using the body center of mass as the reference point simplifies the equation for angular momentum:

(21)$$\begin{array}{r}\begin{array}{rl}{}^e{\dot{\mathbf{p}}}_{b/e}& ={\mathbf{f}}_b\\ {}^e{\dot{\mathbf{h}}}_{b/B}& ={\mathbf{g}}_{b/B}\end{array}\end{array}$$

Combining (20) and (21): and rewriting as a linear system:

(22)$$\begin{array}{r}\left[\begin{array}{cc}{m}_b& -m_b{\left[{\mathbf{r}}_{B/RM}\right]}^{\times }\\ 0& {\mathbf{J}}_{b/B}\end{array}\right]\left[\begin{array}{c}{}^b{\dot{\mathbf{v}}}_{RM/e}\\ {}^b{\dot{\omega }}_{b/e}\end{array}\right]=\left[\begin{array}{c}{\mathbf{f}}_b-{\omega }_{b/e}\times {\mathbf{p}}_{b/e}\\ {\mathbf{g}}_{b/B}-{\omega }_{b/e}\times {\mathbf{h}}_{b/B}\end{array}\right]\end{array}$$

Model 9a

Similar to Model 6a, this design uses the riser connection midpoint RM as the reference point for both the body and the payload, which simplifies incorporating the apparent mass matrix. However, this model treats the body and payload as separate components, connected by a rotational spring-damper model that adds an additional three degrees-of-freedom. A similar 9DoF model derivation can be found in [35] (9DoF, but relative roll and pitch are unconstrained).

An implementation of this model is available as Paraglider9a in the glidersim package. The glidersim package also includes Paraglider9b, which uses the centers of mass as the reference points for the body and payload dynamics; that choice simplifies the derivatives for angular momentum (because it eliminates the moment arms), but prohibits incorporating the effects of apparent mass.

Real mass only

Start with the equations for the translational and angular momentum of the body $b$ about the reference point $RM$ as observed by the inertial reference frame $e$:

(23)$$\begin{array}{r}\begin{array}{rl}{\mathbf{p}}_{b/e}& =m_b{\mathbf{v}}_{B/e}\\ & =m_b({\mathbf{v}}_{RM/e}+{\omega }_{b/e}\times {\mathbf{r}}_{B/RM})\end{array}\end{array}$$

(24)$$\begin{array}{r}\begin{array}{rl}{\mathbf{p}}_{p/e}& =m_p{\mathbf{v}}_{P/e}\\ & =m_p({\mathbf{v}}_{RM/e}+{\omega }_{b/e}\times {\mathbf{r}}_{P/RM})\end{array}\end{array}$$

(25)$${\mathbf{h}}_{b/RM}=m_b{\mathbf{r}}_{B/RM}\times {\mathbf{v}}_{RM/e}+{\mathbf{J}}_{b/RM}\cdot {\omega }_{b/e}$$

(26)$${\mathbf{h}}_{p/RM}=m_p{\mathbf{r}}_{P/RM}\times {\mathbf{v}}_{RM/e}+{\mathbf{J}}_{p/RM}\cdot {\omega }_{p/e}$$

Compute the two momentum derivatives:

(27)$$\begin{array}{r}\begin{array}{c}\begin{array}{rl}{}^e{\dot{\mathbf{p}}}_{b/e}& ={}^b{\dot{\mathbf{p}}}_{b/e}+{\omega }_{b/e}\times {\mathbf{p}}_{b/e}\\ & =m_b({}^b{\dot{\mathbf{v}}}_{RM/e}+{}^b{\dot{\omega }}_{b/e}\times {\mathbf{r}}_{B/RM})+{\omega }_{b/e}\times {\mathbf{p}}_{b/e}\\ {}^e{\dot{\mathbf{h}}}_{b/RM}& ={}^b{\dot{\mathbf{h}}}_{b/RM}+{\omega }_{b/e}\times {\mathbf{h}}_{b/RM}\\ & =m_b{\mathbf{r}}_{B/RM}\times {}^b{\dot{\mathbf{v}}}_{RM/e}+{\mathbf{J}}_{b/RM}\cdot {}^b{\dot{\omega }}_{b/e}+{\omega }_{b/e}\times {\mathbf{h}}_{b/RM}\\ {}^e{\dot{\mathbf{p}}}_{p/e}& ={}^p{\dot{\mathbf{p}}}_{p/e}+{\omega }_{p/e}\times {\mathbf{p}}_{p/e}\\ & =m_p({}^p{\dot{\mathbf{v}}}_{RM/e}+{}^p{\dot{\omega }}_{p/e}\times {\mathbf{r}}_{P/RM})+{\omega }_{p/e}\times {\mathbf{p}}_{p/e}\\ & =m_p({}^b{\dot{\mathbf{v}}}_{RM/e}+{\omega }_{b/p}\times {\mathbf{v}}_{RM/e}+{}^p{\dot{\omega }}_{p/e}\times {\mathbf{r}}_{P/RM})+{\omega }_{p/e}\times {\mathbf{p}}_{p/e}\\ {}^e{\dot{\mathbf{h}}}_{p/RM}& ={}^p{\dot{\mathbf{h}}}_{p/RM}+{\omega }_{p/e}\times {\mathbf{h}}_{p/RM}\\ & =m_p{\mathbf{r}}_{P/RM}\times {}^p\dot{{\mathbf{v}}_{RM/e}}+{\mathbf{J}}_{p/RM}\cdot {{}^p\dot{\omega }}_{p/e}+{\omega }_{p/e}\times {\mathbf{h}}_{p/RM}\\ & =m_p{\mathbf{r}}_{P/RM}\times ({}^b\dot{{\mathbf{v}}_{RM/e}}+{\omega }_{b/p}\times {\mathbf{v}}_{RM/e})+{\mathbf{J}}_{p/RM}\cdot {{}^p\dot{\omega }}_{p/e}+{\omega }_{p/e}\times {\mathbf{h}}_{p/RM}\end{array}\end{array}\end{array}$$

Derivatives of the payload momentums are computed in terms of the body velocity derivative in the body frame to allow writing the dynamics as a single system of equations. First, compute the net external forces and moments:

(28)$$\begin{array}{r}\begin{array}{rl}{\mathbf{f}}_b& ={\mathbf{f}}_{b,\text{aero}}+{\mathbf{f}}_{b,\text{weight}}\\ {\mathbf{g}}_{b/RM}& ={\mathbf{g}}_{b,\text{aero}}+{\mathbf{g}}_{b,\text{weight}}\\ {\mathbf{f}}_p& ={\mathbf{f}}_{p,\text{aero}}+{\mathbf{f}}_{p,\text{weight}}\\ {\mathbf{g}}_{p/RM}& ={\mathbf{g}}_{p,\text{aero}}+{\mathbf{g}}_{p,\text{weight}}\end{array}\end{array}$$

And equate them to the derivatives of momentum with respect to the inertial frame:

(29)$$\begin{array}{r}\begin{array}{rl}{}^e{\dot{\mathbf{p}}}_{b/e}& ={\mathbf{f}}_b-{\mathbf{f}}_{RM}\\ {}^e{\dot{\mathbf{h}}}_{b/RM}+{\mathbf{v}}_{RM/e}\times {\mathbf{p}}_{b/e}& ={\mathbf{g}}_{b/RM}-{\mathbf{g}}_{RM}\\ {}^e{\dot{\mathbf{p}}}_{p/e}& ={\mathbf{f}}_p+{\mathbf{f}}_{RM}\\ {}^e{\dot{\mathbf{h}}}_{p/RM}+{\mathbf{v}}_{RM/e}\times {\mathbf{p}}_{p/e}& ={\mathbf{g}}_{p/RM}+{\mathbf{g}}_{RM}\end{array}\end{array}$$

The spring-damper connection produces forces and moments shared by the body and the payload. There are six variables but only three degrees of freedom. Both systems have the riser connection point $RM$ at a fixed position, and the force only exists to maintain the fixed relative positioning.

(30)$$\begin{array}{r}{\mathbf{g}}_{RM}=\left[\begin{array}{c}\begin{array}{rl}{\kappa }_{\varphi }\varphi & +{\kappa }_{\dot{\varphi }}\dot{\varphi }\\ {\kappa }_{\theta }\theta & +{\kappa }_{\dot{\theta }}\dot{\theta }\\ {\kappa }_{\gamma }\gamma & +{\kappa }_{\dot{\gamma }}\dot{\gamma }\end{array}\end{array}\right]\end{array}$$

Where ${\omega }_{p/b}^p=⟨\varphi ,\theta ,\gamma ⟩$ are the angular rates of the payload, ${}^p{\dot{\omega }}_{p/b}^p=⟨\dot{\varphi },\dot{\theta },\dot{\gamma }⟩$ are the angular accelerations of the payload, and the $\kappa$ are the stiffness and dampening coefficients of the spring-damper model.

This is a very simple model. A better model would need to account for the coupling between dimensions, and should really be a function of the riser strap width.]]

Combining equations (27) and (29) and rewriting as a linear system provides the dynamics of the real mass (solid mass plus the enclosed air) in terms of ${}^b{\dot{\mathbf{v}}}_{RM/e}$, ${}^b{\dot{\omega }}_{b/e}$, ${}^b{\dot{\omega }}_{p/e}^p$, and ${\mathbf{f}}_{RM}^b$:

(31)$$\begin{array}{r}{\mathbf{A}}_{r/RM}\left[\begin{array}{c}{}^b{\dot{\mathbf{v}}}_{RM/e}^b\\ {}^b{\dot{\omega }}_{b/e}^b\\ {}^p{\dot{\omega }}_{p/e}^p\\ {\mathbf{f}}_{RM}^b\end{array}\right]=\left[\begin{array}{c}{\mathbf{b}}_1^b\\ {\mathbf{b}}_2^b\\ {\mathbf{b}}_3^p\\ {\mathbf{b}}_4^p\end{array}\right]\end{array}$$

Where:

$$\begin{array}{r}{\mathbf{A}}_{r/RM}=\left[\begin{array}{cccc}{m}_b{\mathbf{I}}_3& -m_b{\left[{\mathbf{r}}_{B/RM}^b\right]}^{\times }& {\mathbf{0}}_{3\times 3}& {\mathbf{I}}_3\\ m_b{\left[{\mathbf{r}}_{B/RM}^b\right]}^{\times }& {\mathbf{J}}_{b/RM}^b& {\mathbf{0}}_{3\times 3}& {\mathbf{0}}_{3\times 3}\\ m_p{\mathbf{C}}_{p/b}& {\mathbf{0}}_{3\times 3}& -m_p{\left[{\mathbf{r}}_{P/RM}^p\right]}^{\times }& -{\mathbf{C}}_{p/b}\\ m_p{\left[{\mathbf{r}}_{P/RM}^p\right]}^{\times }{\mathbf{C}}_{p/b}& {\mathbf{0}}_{3\times 3}& {\mathbf{J}}_{p/RM}^p& {\mathbf{0}}_{3\times 3}\end{array}\right]\end{array}$$

(32)$$\begin{array}{r}\begin{array}{rl}{\mathbf{b}}_1^b& ={\mathbf{f}}_b^b-{\omega }_{b/e}^b\times {\mathbf{p}}_{b/e}^b\\ {\mathbf{b}}_2^b& ={\mathbf{g}}_b^b-{\mathbf{g}}_{RM}^b-{\mathbf{v}}_{RM/e}^b\times {\mathbf{p}}_{b/e}^b-{\omega }_{b/e}^b\times {\mathbf{h}}_{b/RM}^b\\ {\mathbf{b}}_3^p& ={\mathbf{f}}_p^p-{\omega }_{p/e}^p\times {\mathbf{p}}_{p/e}^p-m_p{\omega }_{b/p}^p\times {\mathbf{v}}_{RM/e}^p\\ {\mathbf{b}}_4^p& ={\mathbf{g}}_b^p+{\mathbf{g}}_{RM}^p-{\mathbf{v}}_{RM/e}^p\times {\mathbf{p}}_{p/e}^p-{\omega }_{p/e}^p\times {\mathbf{h}}_{p/RM}^p-m_p{\mathbf{r}}_{P/RM}^p\times ({\omega }_{b/p}^p\times {\mathbf{v}}_{RM/e}^p)\end{array}\end{array}$$

Real mass + apparent mass

As with the 6-DoF system, the effects of apparent mass on the canopy can be accounted for by adding the apparent inertia matrix from Apparent mass of a parafoil to the components of the system matrix associated with the translational and angular acceleration of the body and accounting for the translational and angular apparent momentum:

(33)$$\begin{array}{r}({\mathbf{A}}_{r/RM}+\left[\begin{array}{cc}{\mathbf{A}}_{a/RM}& {\mathbf{0}}_{6\times 6}\\ {\mathbf{0}}_{6\times 6}& {\mathbf{0}}_{6\times 6}\end{array}\right])\left[\begin{array}{c}{}^b{\dot{\mathbf{v}}}_{RM/e}^b\\ {}^b{\dot{\omega }}_{b/e}^b\\ {}^p{\dot{\omega }}_{p/e}^p\\ {\mathbf{f}}_{RM}^b\end{array}\right]=\left[\begin{array}{c}{\mathbf{b}}_5^b\\ {\mathbf{b}}_6^b\\ {\mathbf{b}}_3^p\\ {\mathbf{b}}_4^p\end{array}\right]\end{array}$$

$$\begin{array}{r}\begin{array}{rl}{\mathbf{b}}_5^b& ={\mathbf{b}}_1^b-{\omega }_{b/e}\times {\mathbf{p}}_{a/e}\\ {\mathbf{b}}_6^b& ={\mathbf{b}}_2^b-{\mathbf{v}}_{RM/e}\times {\mathbf{p}}_{a/e}-{\omega }_{b/e}\times {\mathbf{h}}_{a/RM}+{\mathbf{v}}_{RM/e}\times ({\mathbf{M}}_a\cdot {\mathbf{v}}_{RM/e})\end{array}\end{array}$$

Where ${\mathbf{A}}_{a/RM}$ is the apparent inertia matrix of the canopy from (5), ${\mathbf{M}}_a$ is the apparent mass matrix from (3), and ${\mathbf{p}}_{a/e}$ and ${\mathbf{h}}_{a/RM}$ are the linear and angular apparent momentums from (6) and (7). The extra term ${\mathbf{v}}_{RM/e}\times \left({\mathbf{M}}_a{\mathbf{v}}_{RM/e}\right)$ in ${\mathbf{b}}_6^b$ is necessary to avoid double counting the aerodynamic moment already accounted for by the section pitching coefficients.