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## Kinematic and dynamic modeling

In this section, the most commonly used model for indoor blimp is presented, it is basically a simplified form derived from the airship nonlinear model with some modifications [Gomes, 1990].

### Choice of Inertial and Body Frames

The reference frames for the blimp model are shown in Figure 2.1. The frame Fn is the local navigation frame which is tangent to the Earth surface, its direction is North-East-Down (NED). Since we consider only the operation of blimp robot in indoor environment, the movement of the Earth is ignored, thus the navigation frame Fn is assumed to be an inertial frame (Galilean reference frame). We can also denote it as Fi , but in order to distinguish the navigation frame which is only inertial in the specified application scenario, we will keep to use Fn in this chapter. The body-fixed frame Fb locates its origin at the CB of the blimp, which is also the CV of the hull, the direction of Fb is forward-right-down. Due to the fact that the gondola with actuators and other electrical com-ponents are mounted on the bottom of the hull, the center of gravity (CG) is located on the Zb axis of body-fixed frame, therefore denote its coordinate in Fb as rGb = h 0 0 zG iT. The instantaneous linear and angular velocities of the blimp are described in Fb as b = h (vb)T (!b)T iT = h vxb vyb vzb !xb !yb !zb iT.

whereas the position and orientation of the blimp with respect to Fn are ex-pressed as n = h ( 1n)T ( 2n)T iT = h xn yn zn iT. where , and are roll, pitch and yaw angle, respectively.

#### Choice of Tait-Bryan angles

Definition 2.1. RFF 0 of dimension 3 3 is the rotation matrix from frame F to frame F 0. It is a matrix whose columns are the vectors of the final frame expressed in the initial frame. Here we choose the z y0 x00 Tait-Bryan angle to transform from inertial frame Fn to body-fixed frame Fb, which means first Fn is rotated by an angle (yaw) around Zn-axis to get an intermediate frame F1 = X0Y 0Z 0, the rotation matrix from frame Fn to F1 is expressed as 3 Rn1 = 2 sin cos 0

then this frame F1 is rotated by an angle (pitch) around the Y 0-axis to ob-tain another frame F2 = X00Y 00Z 00, the rotation matrix from frame F1 to F2 is expressed as 2 cos 0 sin 3 4 sin 0 cos 5.

**Restoring forces and moments**

The lifting force of the blimp is aerostatic, which means it is independent of the flight speed thanks to the helium gas inside the balloon. From Archimedes’ principle, the buoyancy force of the blimp is equal to the weight of the air that the balloon displaces. In Figure 2.1, it is shown that, as a result of the gondola installation at the bottom of the balloon, the CG is below CB. In practice, the resultant force of buoyancy fB and gravity fG will keep the airship upright, thus it is called the restoring force. In addition the gravitational force fG acts on the CG which is at rGb = h 0 0 zG iT.

**Table of contents :**

List of Figures

List of Tables

Acronyms

Notations

**1 Introduction **

1.1 Background and Motivation

1.2 State of the art

1.2.1 Modeling of blimp

1.2.2 Sensors used for UAVs

1.2.3 Controller for blimp

1.3 Contribution

1.4 Outline of the thesis

**2 Modeling and Parameter Identification **

2.1 Introduction

2.2 General hypotheses

2.3 Kinematic and dynamic modeling

2.3.1 Choice of Inertial and Body Frames

2.3.2 Choice of Tait-Bryan angles

2.3.3 Kinematic model

2.3.4 Dynamic model

2.3.4.1 Restoring forces and moments

2.3.4.2 Propulsion forces and moments

2.3.4.3 Damping forces and moments

2.3.4.4 Inertia matrix

2.3.4.5 Coriolis and centripetal forces and moments

2.4 Simplified model

2.4.1 Hypotheses for simplified model

2.4.2 Simplified altitude movement model

2.4.3 Simplified planar movement model

2.5 Sensors

2.6 Parameter identification

2.6.1 For altitude movement nominal model

2.6.1.1 On NON-A Blimp Prototype

2.6.1.2 On NON-A Blimp V2

2.6.2 For planar movement nominal model

2.7 Conclusion

**3 Altitude Control **

3.1 Introduction

3.2 System description

3.2.1 Integral control – a first approach

3.2.2 Disturbance compensation based controller – the selected approach

3.3 Observer design

3.3.1 High gain differentiator

3.3.2 High-order sliding mode differentiator

3.3.3 Homogeneous finite-time differentiator

3.3.4 Comparison of differentiators

3.4 Controller design

3.4.1 Disturbance estimation

3.4.2 Predictor-based controller design

3.4.3 Determination of controller gain

3.5 Simulation

3.5.1 Simulation parameter setting

3.5.2 Simulation test 1

3.5.3 Simulation test 2

3.5.4 Simulation test 3

3.6 Conclusion

**4 Horizontal Plane Movement Control **

4.1 Introduction

4.2 System description

4.2.1 Under-actuated system

4.2.1.1 Dynamic extension

4.2.1.2 Coordinate transformation

4.3 Disturbance estimation

4.4 Controller design

4.5 Simulation

4.5.1 Simulation parameter setting

4.5.2 Point stabilization

4.5.2.1 Simulation test 1

4.5.2.2 Simulation test 2

4.5.2.3 Simulation test 3

4.5.3 Trajectory tracking

4.6 Conclusion

**5 Implementation and Results **

5.1 Introduction

5.2 Hardware design

5.2.1 Blimp robot system overall analysis

5.2.2 NON-A blimp robot: Two generations

5.2.3 NON-A blimp V2 robot: Electric circuit design

5.2.4 NON-A blimp V2 robot: Structure design

5.3 Testing environment setup

5.3.1 Implementation with OptiTrack

5.4 Altitude stabilization control

5.4.1 On NON-A blimp prototype

5.4.1.1 Real test 1

5.4.1.2 Real test 2

5.4.2 On NON-A blimp V2

5.5 Validation of the complete motion controller

5.6 Conclusion

Conclusion and Perspectives

**Bibliography**