Skydiver following camera UAV

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.

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

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_A, and V_W [3]

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_Af, 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 Ω_Kf , the Euler angle vector (attitude) 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 V_Kg. The limited actuator deflections are tunneled to the animation block in General overview via a Goto-block.

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Figure 1.4.: UAV overview

1.4. Actuator dynamics

The UAV has three vanes (figure 1.5) that can be deflected individually (elevators η₁, η₂, and η₃) 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_RAf and the Aerodynamic force unit E in figure 1.6:

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

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 forcesC_Rabody and moments C_Qabody 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).

[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 Cm 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).

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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.

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Figure 1.11.: Vanes and forces as seen from above

When deflected by its “elevator” deflection angle (η₁, η₂, η₃) each vane generates a force (F₁, F ₂, F₃) 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.

[…]

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[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