Excerpt

## Contents

1. Simulink model . . 8

1.1. General overview . . 8

1.2. UAV + wind + track . . 9

1.3. UAV overview . . 11

1.4. Actuator dynamics . . 12

1.5. Aerodynamics . . 13

1.6. Aerodynamic velocities . . 14

1.7. Aerodynamic force unit . . 14

1.8. Body . . 15

1.9. Vanes . . 16

1.10. Shadowing factor of the aerodynamic vanes . . 18

1.11. Kinetics . . 19

1.12. Translational velocity differential equation . . 20

1.13. Position differential equation . . 21

1.14. Rotational velocity differential equation . . 21

1.15. Attitude differential equation . . 21

1.16. Wind . . 22

1.17. Translational wind . . 22

1.18. Turbulence . . . 23

1.19. Rotational wind . . 23

1.20. Track . . 23

1.21. Attitude and altitude controller . . 24

1.22. Position controller . . 26

1.23. Position command . . 29

1.24. 2-norm of a vector . . 30

1.25. Rotation about x-axis . . 31

1.26. Transformation from inertial frame to body-fixed frame . . 32

1.27. Transformation from Euler frame to body-fixed frame . . 33

1.28. Euler frame to body frame transformation matrix . . 34

1.29. Displays . . 35

1.30. Diver . . 35

1.31. Real-Time Pacer . . 36

1.32. sky diver dat.m . . 37

2. Animation . . 40

2.1. Animation overview . . 40

2.2. sd sfun.m . . 43

2.2.1. setup . . 43

2.2.2. start . . 45

2.2.2.1. Figure and Axes . . 45

2.2.2.2. Hull of the UAV . . 46

2.2.2.3. Vanes . . 47

2.2.2.4. Hull display . . 48

2.2.2.5. *x*-axis line . . 49

2.2.2.6. Field of view . . 49

2.2.2.7. Skydiver . . 50

2.2.2.8. Video . . 50

2.2.2.9. Communication structure . . 51

2.2.3. revolve.m . . 51

2.2.4. update . . 52

2.2.4.1. Hull . . 53

2.2.4.2. Vanes . . 53

2.2.4.3. *x*-axis line . . 54

2.2.4.4. Skydiver . . 54

2.2.4.5. Camera . . 55

2.2.4.6. Field of view (FOV) . . 56

2.2.4.7. Visibility . . 57

2.2.4.8. Video . . 58

2.2.5. fov test.m . . 58

2.2.6. terminate . . 61

2.2.7. vane rotate . . 61

2.2.7.1. Roll . . 61

2.2.7.2. Yaw . . 62

2.2.8. m fg . . 64

2.2.9. rotation about arbitrary axis . . 65

2.3. Skydiver model . . 65

3. Skydiver flight data . . 70

3.1. sky diver dat.m . . 70

4. trimmod . . 73

4.1. Documentation . . 73

4.1.1. Syntax . . 73

4.1.2. Description . . 73

4.1.3. Arguments . . 74

4.1.4. Example . . 75

4.1.5. Menu . . 75

4.1.6. Algorithm . . 77

4.2. Trimmod overview for UAV and skydiver . . 80

A. Aerodynamic frame of an axially symmetric body falling down . . 83

## 1. Simulink model

The purpose of the UAV is provided in [1].

### 1.1. General overview

The main Simulink window including all subsystems is shown in ﬁgure 1.1.

[Figure is omitted from this preview]

Figure 1.1.: General overview

The top central subsystem Sky (the name Sky is used as a synonym for the UAV including the camera throughout this paper) + Wind + Track (section 1.2) holds the model of the UAV, its own wind process and a block computing the spherical components of the UAV’s ﬂight path vector. The bottom central subsystem Diver + Wind + Track contains the corresponding models of the skydiver.

Scopes for the main signals can be found in Sky displays and Diver displays (section 1.29). An S-Function in the lower left of ﬁgure 1.1 is used to display an animation of the UAV and the diver [2] during the simulation. The “yellow” Real-Time Pacer block makes sure that the simulation runs in real-time if the hardware is fast enough.

While the skydiver is uncontrolled and reacts only to its own wind inputs, the UAV is controlled by a cascade control system. The inner (secondary, slave) Attitude and altitude controller measures the attitude and the altitude of the UAV and uses the actuators to maintain their values. The outer (primary, master) Position controller measures the position of the UAV and uses the inner loop as its actuators by commanding an attitude setpoint to the inner controller. The position setpoint for the outer controller and for the altitude is generated in the Position command block.

Additionally, ﬁgure 1.1 contains a dummy input, a sum and an output block that are only used during the trim process.

### 1.2. UAV + wind + track

The top level UAV block (ﬁgure 1.2) contains its mathematical model (section 1.3) including actuator dynamics, aerodynamics, and kinetics, its wind process (section 1.16), and the computation of the spherical components of its ﬂight path vector (section 1.20).

[Figure is omitted from this preview]

Figure 1.2.: UAV + wind + track

The translational ﬂight path velocity vector **V _{K}**
(i. e. the velocity of the UAV with respect to the ground) is the sum of
the aerodynamic velocity or airspeed vector VA (i. e. the velocity of the
UAV with respect to the air) and the

**V**(i. e. the velocity of the air with respect to the ground):

_{W}
*V _{K} = V_{A} + V_{W}*

The same is true for the rotational velocity vectors:

*Ω _{K} = Ω_{A} + Ω_{W}*

[Figure is omitted from this preview]

Figure 1.3.: Relation between **V _{K}**,

**V**, and

_{A}**V**[3]

_{W}Since we need the airspeed vectors in the body-ﬁxed frame, we have to transform the wind from the inertial frame (index g) to the body-ﬁxed frame (index f) with the help of a transformation block (section 1.26):

*V _{Af} =V_{Kf} · V_{Wf}*

*=V _{Kf} M_{fg} · V_{Wg}*

_{}

The transformation matrix **M _{fg}** utilizes the
Euler angle vector (attitude)

[Formula is omitted from this preview]

which therefore has to be fed back into the transformation blocks as well.

### 1.3. UAV overview

The UAV subsystem (ﬁgure 1.4) contains an Actuator dynamics block that
saturates and rate limits the actuators, an Aerodynamics block that
computes the aerodynamic forces **R _{Af}** and moments

**Q**, and a Kinetics block that integrates the forces and moments into motion, i. e. the rotational ﬂight path velocity vector in the body-ﬁxed frame

_{Af}**Ω**, the Euler angle vector (attitude)

_{Kf}**Φ**, the position vector in the inertial frame sg, and the translational ﬂight path velocity vector in the body-ﬁxed frame

**V**

_{Kf}_{ }. Additionally, the ﬂight path velocity vector is returned in the inertial frame

**. The limited actuator deﬂections are tunneled to the animation block in General overview via a Goto-block.**

*V*_{Kg}[Figure is omitted from this preview]

Figure 1.4.: UAV overview

### 1.4. Actuator dynamics

The UAV has three vanes (ﬁgure 1.5) that can be deﬂected individually
(elevators η_{1}, η_{2}, and η_{3}) about a
horizontal axis to produce a pitching (or rolling) moment and collectively
(rudder ζ) about a vertical axis to produce a yawing moment.

[Figure is omitted from this preview]

Figure 1.5.: Actuator dynamics [3]

Each elevator deﬂection is positive, saturated

[Formula is omitted from this preview]

and rate limited:

[Formula is omitted from this preview]

The rudder has symmetrical limits:

[Formula is omitted from this preview]

### 1.5. Aerodynamics

The aerodynamic force vector in the body-ﬁxed frame **R _{Af}** is computed as the product of the
corresponding aerodynamic force coeﬃcient vector

**C**and the Aerodynamic force unit E in ﬁgure 1.6:

_{RAf}
**R _{Af}**
=

*E*·

**C**

_{RAf}
For the moments we have to multiply the force by a reference length (mean
aerodynamic cord) l_{μ}:

** Q_{Af}**
= l

_{μ}·

*E*·

*C*_{QAf}[Figure is omitted from this preview]

Figure 1.6.: Aerodynamics

In the subsystem Aerodynamic velocities we compute the spherical components
(V_{A}, α, and μ) from the airspeed vector in the body-ﬁxed frame **V _{Af}** and make the rotational aerodynamic
velocity vector

**Ω**dimensionless. The aerodynamic coeﬃcients are computed as the sum of the coeﬃcients of the Body, the Vanes and a linear damping:

_{Af}
**
C_{RAf} = C_{RAfbody} + C_{RAfvanes}
**

**
C_{QAf} = C_{QAfbody} + C_{QAfvanes} + C _{QAfdamp}
**

where the linear damping coeﬃcient **C _{QAfdamp}** is
the product of the diagonal damping matrix and the dimensionless rotational
aerodynamic velocity vector

**Ω**

^{∗}**:**

_{Af}[Formula is omitted from this preview]

### 1.6. Aerodynamic velocities

In the subsystem depicted in ﬁgure 1.7 we compute the spherical components
(V_{A}, α, and μ) of the ﬂight path vector from its Cartesian
representation **V _{Af}** according to equation (A.7)
and equation (A.8).

[Figure is omitted from this preview]

Figure 1.7.: Aerodynamic velocities [3]

Additionally, we normalize the rotational velocity vector by a “time unit” in order to make it dimensionless:

[Formula is omitted from this preview]

1.7. Aerodynamic force unit

The aerodynamic force unit (ﬁgure 1.8) represents a force that is
proportional to the air density ρ, the square of the air speed V _{A} and a reference area S:

[Formula is omitted from this preview]

[Figure is omitted from this preview]

Figure 1.8.: Aerodynamic force unit [3]

It is used in section 1.5 to compute the aerodynamic forces and moments from the dimensionless coeﬃcients.

### 1.8. Body

In ﬁgure 1.9 we compute the aerodynamic force and moment coeﬃcients of the
UAV’s body with respect to the angle of attack α and the aerodynamic yaw
angle μ. According to appendix A the aerodynamic forces**C _{Rabody}** and moments

**C**of a symmetrical body only depend on the angle of attack in the aerodynamic frame. The aerodynamic yaw angle μ is then used (together with α) to transform the forces and moments to the body-ﬁxed frame via equation (A.4).

_{Qabody}[Figure is omitted from this preview]

Figure 1.9.: Aerodynamics body

In this simple linear derivative aerodynamics, the lift coeﬃcient C *Lbody* (pointing towards the negative x-axis of the aerodynamic
frame) is proportional to the angle of attack with a constant lift slope
derivative C_{Lα}. Due to the axial symmetry of the body there is
no lift for α = 0. Also, the pitch moment coeﬃcient C*m* only
depends on the angle of attack, with the pitch stability derivative C _{mα} as a proportionality factor:

C_{m} = C_{mα} · α

With the drag coeﬃcient C_{Dbody} (pointing towards the negative
z-axis of the aerodynamic frame) we have to be a little bit more ﬂexible;
Not only do we have to consider a drag coeﬃcient C_{D0} for α = 0
because of the face area, but we also have to take into account that the
induced drag is proportional to the square of the lift. Therefore, we
assume a full quadratic dependence of the drag coeﬃcient CD on the angle of
attack:

C_{D} = C_{D0} + C_{Dα} · α + C_{Dα2} · α ^{2}

In the aerodynamic frame deﬁned in appendix A there is no aerodynamic side force, roll moment, or yaw moment.

### 1.9. Vanes

While the aerodynamics of the body is modeled in the aerodynamic frame and later transformed into the body-ﬁxed frame, we compute the aerodynamic forces and moments (coeﬃcients) of the vanes directly in the body-ﬁxed frame (ﬁgure 1.10).

[Figure is omitted from this preview]

Figure 1.10.: Aerodynamic vanes

The arrangement of the three vanes of the UAV are depicted in ﬁgure 1.11 looking down in positive zf -direction.

[Figure is omitted from this preview]

Figure 1.11.: Vanes and forces as seen from above

When deﬂected by its “elevator” deﬂection angle (η_{1}, η_{2}, η_{3}) each vane generates a force (F_{1}, F _{2}, F_{3}) in its deﬂection direction and a moment about
the perpendicular axis. Additionally, all vanes generate drag forces in
(negative) z_{f} -direction.

In order to use the minimum number of derivatives we decompose the vane
deﬂection angles into their eﬀective angles in x_{f}-and y _{f} -direction.

The computation of the eﬀective angles according to ﬁgure 1.11 is done in the upper middle part of ﬁgure 1.10:

[Formulas are omitted from this preview] (1.1) (1.2)

Additionally, we compute an eﬀective collective angle for the drag:

[Formula is omitted from this preview] (1.3)

Now we can compose the aerodynamic force coeﬃcient vector in the body-ﬁxed frame:

[Formula is omitted from this preview] (1.4)

The moments generated by the vanes are computed accordingly: the eﬀective
x-vane deﬂection generates a pitching moment about the y_{f} -axis
while an eﬀective y-vane de- ﬂection results in a rolling moment about the
x_{f} -axis.

[…]

[1] D. How, B. Barbarich-Bacher, and K. Stol, “Design and Analysis of a UAV for Skydiving.” IEEE Int. Conf. on Unmanned Aircraft Systems (ICUAS’15), 2015.

[2] G. Vallabha. (2016) Real-Time Pacer for Simulink. [Online]. Available: https://de. mathworks.com/matlabcentral/ﬁleexchange/29107-real-time-pacer-for-simulink

[3] J. J. Buchholz. (2016) Skript Regelungstechnik und Flugregler. [Online]. Available: http://prof.red/rtfr/skript/skript10.pdf

- Quote paper
- Prof. Dr.-Ing. Jörg Buchholz (Author), 2017, Skydiver following camera UAV, Munich, GRIN Verlag, https://www.grin.com/document/366451

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