Role of Accelerometers in the control of Fully Actuated UAV

ROLE OF ACCELEROMETERS IN THE CONTROL OF A FULLY ACTUATED UAV’S

B.T. (Bhanu) Chidura

MSC ASSIGNMENT

Committee:

dr. A. Franchi Y.A.L.A. Aboudorra, MSc dr. F.C. Nex

June, 2021

036RaM2021

Robotics and Mechatronics

EEMathCS

University of Twente

P.O. Box 217

7500 AE Enschede

The Netherlands

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Summary

In recent years there has been an exponential growth in Aerial Robot technology leading to many new applications in performing active and passive tasks like package delivery, surveillance, geo-mapping and tasks requiring physical interaction with objects and humans. This has led to a research in devel- oping fully actuated UAV that can overcome the shortcomings of dependent degrees of freedom in an under actuated UAV. Fully actuated Unmanned Aerial Vehicles(UAVs) can produce a full wrench and control each degrees of freedom independently. This also has an additional benefit of improving safety; as it is a major concern in flying aerial robots in human environment.

Most of the UAVs measure parameters from Micreolectromechanical Systems(MEMS) Inertial Mea- surement Unit(IMU) for the state estimation and control. The data from the measurements are fused with filters to estimate the attitude. Accelerometer measures specific acceleration which is the dif- ference between the accelerations and gravity in body frame. Under actuated UAVs like quadrotors use accelerometers with the filtering method to act as an inclinometer. The use of accelerometer as inclinometer in a close loop structure is validated by adding a non-gravitational force (which is an aerodynamic force) called rotor drag. The use of rotor drag in dynamic model has improved the state estimation and control of under actuated UAVs.

An aerial vehicle experiences multiple forces to manoeuvre in the 3D space where the dominating force being the gravitational force and the other significant non gravitational force is the Rotor drag; which is experienced in low and medium flight speeds before the body drag is induced. These forces and the accelerometer data is used for the state estimation. Understanding the role of accelerometers in an fully actuated UAV with tilted rotors by incorporating the rotor drag is the focus of this assignment.

Rotor Drag is the function of the linear velocity produced by the body and the angular speed of

the specific rotor. Due to the specific alignment of the rotors in a fully actuated UAV the rotor drag

force results in components acting in multiple axis of the body frame. This force is captured by the

accelerometer during motion. By using parameter estimation techniques the rotor drag coefficient

for the fully actuated UAV can be estimated. A fully actuated hexa-rotor with tilted propellers called

FiberThex is used for the experiments and simulation setup in this work. This can further be used in

developing a new control/estimation techniques for the fully actuated UAVs.

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Contents

1 Introduction 1

1.1 Related Work . . . . 2

1.2 Research Goal . . . . 4

1.3 Thesis Structure . . . . 4

2 Background 5 2.1 Under Actuated UAV’s . . . . 5

2.2 Fully Actuated UAVs . . . . 6

2.3 Rotor Drag . . . . 7

2.4 Usage of Acclerometer in Under actuated UAV . . . . 8

3 Modelling of Rotor Drag in Fully Actuated Vehicle 11 3.1 Modelling with Rotor Drag . . . . 11

3.2 Analytic Understanding . . . . 14

3.3 Numerical Understanding . . . . 15

3.4 Simulation Setup . . . . 19

4 Analysis of the Model 22 4.1 Rotor Drag in FA,FA-UA,HEX . . . . 23

4.2 Rotor Drag with Body Orientation in FA . . . . 30

4.3 Rotor Drag from Rotor Frame to Body Frame . . . . 45

5 Parameter Identification 52 5.1 Lab Experiments . . . . 52

5.2 Testing . . . . 53

5.3 Estimation of Rotor Drag Coefficient . . . . 54

6 Conclusion and Future Work 61 6.1 Conclusion . . . . 61

6.2 Future work . . . . 62

A Appendix 63 A.1 FiberTHex Specifications . . . . 63

A.2 Lab Experiments FiberTHex parameters . . . . 64

A.3 Genom Components list . . . . 65

Bibliography 66

List of acronyms

UAV Unmanned Aerial Vehicle

EKF Extended Kalman Filter UKF Unscented Kalman Filter IMU Inertial Measurement Unit LBF Laterally Bounded Force

DoF Degrees of Freedom

CoM Center of Mass

FA Fully Actuated UAV

FA-UA Fully Actuated UAV- Under Actuated mode

MEMS Microelectromechanical systems

1 Introduction

Technological advancements in Unmanned Aerial Vehicle(UAV) have led to applications not only in passive tasks like photography, mapping,surveillance etc but also in active tasks like environment manipulation and physical interaction . With mechanical simplicity and easier control they have reshaped the way we think about aviation.

Precision control of multi-rotor aerial vehicles has enabled their use in indoor applications, in closed confined spaces where it is difficult to localise itself using extroceptive sensors, and perform tasks like scanning and tracking objects. High degrees of stability and control are key factors which are achieved by using complex control algorithms and estimation filters that are always optimised. Active tasks, where interaction with the environment is involved, require exceptional control and stability which poses as a major challenge for the existing control algorithms. Also, safe operation is one the major concern to use them for daily activities.

In general, multi-rotor aerial vehicles are designed as under actuated systems where the multi-rotor has to tilt(change attitude) in order to move in the lateral directions as all the rotors around the body axis are perpendicular to the vertical axis. The figure 1.1 shows an quad-rotor, from figure it is seen the opposite propellers rotate in same direction. By changing the speeds of each rotor the thrust and moment produced by that rotor varies which helps the UAV to pitch, roll or yaw. Figure 1.2 shows the intensities of thrust from each rotor to achieve different orientations and motion.

Figure 1.1: Quad-rotor Figure 1.2: Thrust based control of Quad-rotor, from (3)

The IMU measurement data along with filtering techniques are used in control algorithms. In this

case, the data from the accelerometers are used to compute the roll and pitch angles- basically

acting as inclinometer. In general acclerometer measures specific acceleration, to validate its use as

inclinometer some significant non gravitational force called rotor drag is included in the model by the

authors in (12) that validates its role. This is further developed to improve the state estimation process

as discussed in (1).

On the other hand a fully actuated multi rotor system has the ability to maintain a desired attitude of the main structure while moving in lateral directions. The UAV can generate thrust in a manner where there is no requirement of tilting the main body and only exploiting the thrust from motors to move in the specified direction. This is possible due to the specific alignment of rotor around the body axis, this creates a vector component force in horizontal axis to get lateral force and vertical force to counter gravity. To use the abilities of both under-actuated and fully-actuated systems in active tasks a new configuration that morphs between both modes are introduced and one such configuration is discussed in (16) with one extra actuated to morph the system.

The state estimation in fully-actuated system needs extroceptive sensors like indoor motion capture systems to close the loop. Similar to under-actuated systems with improved dynamics by adding non gravitational force such as Rotor Drag improves the estimation process. Rotor Drag is significant only in lower speeds up to 5-8m/s as in the higher speeds rotor drag force is dominated by the body drag.

As most of the indoor and human interaction tasks are done at low speeds by understanding rotor drag in fully-actuated system improves estimation similar to under-actuated systems.

This work deals with the understanding and validating the rotor drag force from each orientated rotor and its effect on the body. The forces measured by the acclerometer during flight support better state estimation and avoids dependency on extroceptive senors except the IMU. Various motions are discussed to conclude the effects and to estimate the rotor drag coefficient of a Fully actuated system called FiberTHex.

1.1 Related Work 1.1.1 Fully Actuated UAV

With the cost-effective engineering solutions UAVs have seen a potential growth in the society. Conven- tional multi-rotors with under-actuated dynamics depend on the roll and pitch to translate forward and sideways. (15) describes the different fully-actuated UAV designs in the literature. With fixed-tilt, variable-tilt concepts from quad rotors to octa-rotors and rotor configuration of each structure is discussed. In conclusion authors have mentioned hexa-rotor with only cant angle shown in figure 1.3 and hexa-rotor with cant and dihedral angle shown in figure 1.4 as widely used concepts for their mechanical simplicity and low number of design parameters transforming the conventional hexa-rotor into a fully actuated vehicle.

Figure 1.3: Hexa-rotor with Cant angle (8)

Figure 1.4: Hexa-rotor with cant and dihe- dral angle (14)

In (7) authors have reviewed the multi-rotors literature based on the atomic actuation units and system

abilities like hovering, trajectory tracking, physical interaction. Each atomic actuation unit section

describes the structural configurations , number of degrees of freedom with the level of development

and the feasible wrench space. The paper also discussed few unique set of multi-rotor configuration

that are fully or over-actuated system which were not forwarded as a prototype from simulation.

From the review authors descried common traits in multi-rotors like Platform symmetry- for simpler stability, ignored traits like interaction of Aerodynamic forces and actuator limitations.

In (5) paper proposes a novel method for controlling under-actuated and fully actuated multi-rotors while taking into account the lateral input force. In laterally bounded force (LBF) multi-rotors the total thrust along the lateral direction is lower than principal direction thrust, this LBF rotors can be under-actuated or fully actuated multi-rotors. This paper presents a geometric controller in SE(3) that is able to track any feasible full-pose (6-D) trajectory. Depending upon the reference feasibility and system configuration the controller implicitly priorities the position trajectory over orientation. This controller is practically tested on a FiberTHex fully actuated hexa-rotor to perform various experiments which validated the full pose controller in under-actuated and fully-actuated configurations.

1.1.2 Role of Accelerometer

The work of (12) is one of the first research to included rotor drag in the quad copter model to study the role of accelerometer in the control algorithm. The model consists of aerodynamic efforts from the propellers resolved into forces and torques at the CoM. With a linearised model for independent subsystems authors validated the outputs from acclerometer to work in the closed loop feedback as an inclinometer.With an experimental setup the force model is validated from interpretation of accelerometer measurements and the implications of traditional control behaviour.

The work in (10) builds upon the earlier model to show the improved estimation performance using acclerometer measurements. Here the drag is lumped into single variable and an observer is designed to estimate the altitude, velocity reducing the dependency on position updates and drag coefficient.

The authors also describe why accelerometers measure rotor drag in quad rotor flight and the tradi- tional altitude method for measuring gravity vector are flawed. Multiple filters are used to show the improvements in altitude estimations with rotor drag in the model.

The aim of (17) is to estimate various quad parameters like Angular velocity, drag coefficients using

only IMU and motor speed feedback. The significance of Rotor Drag in rotational dynamics are

demonstrated with experiments to estimate moment of inertia of the system. The authors presented

evidence of blade flapping due the differential lift and a model to capture its variation with motor

speeds. First, it covers the design and implementation of an unscented Kalman filter used to estimate

inertial and aerodynamic parameters using only IMU and motor speed measurements for sensing,

showed that modelling the blade-flapping moment is critical for the UKF to estimate the moment of

inertia accurately.

1.2 Research Goal

In this thesis, the effort is made towards understanding how acclerometer data is processed in Fully Actuated UAV, which can be used in estimation. The focus is to understand and evaluate the addition of rotor drag in the equation of motions. This resulted into following research questions

• How is the Rotor Drag in the body frame of Under-actuated UAV (UA), fully-actuated UAV used in under-actuated mode (FA-UA), fully-actuated UAV (FA) for the same trajectory motion.

– Does UA and FA-UA mode shows similarities?

– Having a body orientation in FA system can be similar to an under actuated mode?

• How does rotor drag from each propeller affect on to the body frame.

– Does this forces sum or negate each other due to symmetry?

– Is it generic to any conical trajectories.

• Is estimation of Rotor Drag coefficients from the simulations and real experiments possible?

• Can the measurements from acclerometer used for state estimation?

– Is it possible to estimate the velocity by understanding rotor drag in the body in a fully- actuated UAV with tilted propellers ?

1.3 Thesis Structure

The organisation of this thesis is divided into six chapter: Chapter 2 provides the background theory

of Under actuated systems, Fully actuated systems , Rotor Drag , Role of acclerometers with rotor

drag. Chapter 3 describes the rotor drag in a fully actuated UAV with analytical and mathematical

representation. Chapter 4 shows the simulations to validate the analysis of rotor drag in each configu-

ration and the drag from each rotor frame to body frame in various trajectories. Chapter 5 shows the

parameter estimation of the rotor drag coefficient from real and simulated experiments.Finally, the

conclusions drawn from the research and future work are presented in Chapter 6.

2 Background

2.1 Under Actuated UAV’s

In robotics the systems that cannot achieve all the six degrees of freedom are called under actuated.

This under actuation could be the result of less number of actuators or the mechanical constraints that does not allow body to perform certain actions. Quad rotors are one of the well know Unmanned Aerial Vehicles that are under actuated. Quad rotor move in lateral axis by pitch or roll this shows the depended degrees of freedom present in the system due the structural orientation of the rotors but it is differentially flat with flat output vector of position of CoM [x,y,z], orientation [yaw] described in (4).

The article (11) discusses the modelling , estimation and control of a simple quad rotor.

In an under actuated UAV all the rotors are perpendicular the body horizontal frame and the opposite pair rotors rotate in same direction and the adjacent propellers rotate opposite in order to balance the torques produced due to high speed propellers. Conceptually the control of a under actuated UAV is done by controlling the speeds of each rotor to achieve specific pose. Multiple control algorithms are designed to achieve high stability and optimal energy to control a under actuated UAV. With the property of under actuation the research on this UAV have bought in multiple advancements like collaborative swarms, autonomous indoor mapping etc.

A generic hexa-rotor is an standard under actuated UAV used for passive tasks like photography and surveillance. It has six rotors place symmetrically around the CoM, all rotor pointing opposite to the gravity in the body plane as shown in figure 2.1.The authors in (6), (18) have achieved environment interaction using under actuated UAVs with cables and using the whole body as an end effector to do the task. With the increase in focus on complex physical interaction with environment and human the drawback of under actuated UAVs needs to be overcome. This has led to the development of fully actuated UAVs as in (13), (16) that can independently control each degrees of freedom, these systems are discussed in the next section.

Figure 2.1: Generic Hexarotor

2.2 Fully Actuated UAVs

Fully actuated systems are the platforms that can control each degree of freedom independently. With the increase in need for physical interaction and environment manipulation in coordination with human using unmanned aerial vehicle the research and development of fully actuated UAVs which can over come the shortcomings of the under actuated UAVs has increased. Six degrees of freedom and full actuation of a body in 3D space is possible using different concepts like a tilt rotor where vectoring the thrust from a rotor can give a body to produce force linearly or a body with extra actuators producing thrust in linear axis or fixed orientation of rotors radially or tangentially to produce forces laterally as a vector component. The article (15) details multiple configurations with fixed and variable tilt rotor for full actuation in 3D space and authors of (7) classify and categorise different platform in the literature based on the abilities, maturity and degrees of freedom.

The platform with ’n’ number of actuators can be defined as multi directional thrust system or fully actuated system or over actuated or omni direction thrust system, any platform can be differentiated in to these classes easily by knowing the allocation matrix of the system. Allocation matrix relates the set of feasible wrenches with controllable input variables, this can be rotor angular speeds or servo tilt angles. The rank of the control allocation [CA] matrix for a particular platform gives the class at which it belongs to as shown in the table 2.1.

Rank Platform

5 ≤ rank[CA] ≤ 6 Multi Directional Thrust

rank[CA]= 6 Fully Actuated

rank[CA]= 6 and actuators >6 Over Actuated rank[CA]=6 and can produced forces

in spherical ball Omni Directional

Table 2.1: Platform configurations

Fixed orientation and unidirectional thrust rotor can also achieve full actuation and omni directionality.

The article (14) describes an fully actuated UAV with tilted propellers shown in figure 1.4 and authors in (2) and (19) have achieved omni directional platform shown in figures 2.2, 2.3 by orienting uni directional propellers around the body frame.

Figure 2.2: Omni Directional UAV from (2) Figure 2.3: Omni Directional UAV from (19)

Each of the configuration under actuation and full actuation have their own edge, to utilise the

advantages in both, the authors in (16) have presented an novel concept to smoothly change the

configuration from under actuation to fully actuated by using only one additional motor that tilts all

propellers at the same time. In this work the fully actuated UAV with fix tilt rotors is used which will be

detailed in section 5.1.1.

2.3 Rotor Drag

Drag is the force acting opposite to the relative motion of the body. This can be acting between two fluids or a fluid and a solid, similar to fiction in mechanical systems but unlike the physical contact area this force depends on the velocity of the fluid over its surface. There are multiple types of drag like Parasitic drag, Interference Drag, Induced drag etc. Parasitic Drag is the combination of viscous pressure drag and drag due to surface roughness. Interference Drag is the drag created due to presence of multiple bodies in close proximity. Induced drag is the negative force created due to the lift force generated in a body.

In the case of a Multi rotors with small propellers rotating at high speeds and body moving in compar- atively low and medium speeds, the effect of parasitic and interference drag is not significant. These drag forces on a structure of the multi rotors can be analysed by using a virtual wind tunnel as shown in figure 2.4, which shows drag at different speeds. The figure 2.4a shown the drag due to structure at a speed of 2m/s which is comparative small to drag when the body speed is 15m/s as shown in figure 2.4d

(a) Drag at 2m/s (b) Drag at 2m/s

(c) Drag at 2m/s (d) Drag at 2m/s

Figure 2.4: Virtual Wind Tunnel Testing - Structural drag

The propellers in multi rotors are lift generating bodies which are rotated at high angular velocity to generated perpendicular lift force proportional to the angular velocity of the rotor. These are unique shapes with specific pitch and diameter designed for range of loads the motor can handle. As induced drag is present due to lift generating bodies, the drag generated by the propeller as called rotor drag is significant in multi rotors.

Rotor drag discussed in articles (12), (4),(10), (17) is significant at low and medium speeds of the multi

rotor and will be dominated by the body drag or parasitic drag in higher speeds. These drag forces

are present in any body moving in 3D space, these forces are measured by the acclerometer and can

be categorised into non -gravitational forces which also include thrust for the rotors. Rotor drag is

proportional to the angular velocity at which the propellers are rotating and the velocity at which

the body is translating. Figure 2.5 shows the rotor drag force in an rotor frame of multi rotor. Rotor drag always acts in the rotor frame perpendicular to the lift direction and opposite to the direction of motion projected into the rotor frame. The effects of rotor drag in the dynamic model in a quad rotor is described in (4) where rotor drag is treated as a disturbance and better control and tracking is achieved.

Figure 2.5: Rotor Drag in Rotor Frame

The effects of rotor drag in a fully actuated uav with tilted propellers is analysed in the next sections of this work.

2.4 Usage of Acclerometer in Under actuated UAV

The multi rotors rely on the MEMS sensors for measurement of the current state of the system. IMU being the basic and important sensor having accelerometer and gyroscope which help in closed loop feedback control using EKF, UKF filters.

Accelerometer measures specific acceleration of the body that is the difference between the accelera- tions and gravity in the body frame. In a general feedback control of a quad rotor an accelerometer is considered as an inclinometer to reconstruct the roll and pitch angles.

Conceptually accelerometer is a combination of a spring, a proof mass and a damper, when this system experiences accelerations the mass is moved to the point that it can push the spring. Measuring spring compression measures accelerations. When an accelerometer is placed at an angle say θ and is moving, the acclerometer measurement is given in equation 2.1, the variables used in the equations are explained in table 2.2.

y

acc

= ¨ x + g si n(θ) (2.1)

From the above equation the angle θ can be calculated but considering a case where the body is not

moving then the accelerations are zero ( ¨ x = 0) so the equation becomes as given in 2.2. In this case the

accelerometer can be used a inclinometer.

y

acc

= g si n(θ) =⇒ θ = si n

⁻¹

( y

acc

g ) (2.2)

Considering the non gravitational forces acting on the body at angle Ψ from the body frame and the body is oriented with angle θ as shown in figure 2.6. The acclerometer axis is aligned with body axis and holds angle θ with respect to world horizontal. In this case the measurements of acclerometer from figure 2.7 is given in the equation 2.4 from equation 2.1 and 2.3

f cos( Ψ) − Mg sin(θ) = M =⇒ ¨x = f

M cos( Ψ) − g sin(θ) (2.3)

y

_acc

= f

M cos( ψ) (2.4)

Variable Description

y

_acc

Accelerometer reading in y axis (horizontal axis) of the body frame

¨

x Accelerations of the body

g Gravity

θ Angle between the world horizontal and body horizontal axis

f Non gravitational forces

Ψ Angle between the body horizontal axis and the non gravitational forces

Table 2.2: Variables used

Figure 2.6: body with an orientation and components of forces

Similarly when the non-gravitational forces balance the gravity as shown in figure 2.8 the acclerometer measures just the component of gravity and can be used as inclinometer as shown in equation 2.5.

But if the non-gravitational forces are inline with the acclerometer axis where Ψ = 0 it measures the forces in the axis from equation 2.4 the accelerometer measurement becomes given in 2.6. Similarly when ψ = 90 the acclerometer in perpendicular axis measure zero.

y

acc

= g si n(θ)∀ f = −M g =⇒ θ = si n

⁻¹

( y

acc

g ) (2.5)

Figure 2.7: Components of forces in accelerometer axis

y

acc

= f

M (2.6)

Figure 2.8: Non Gravitational forces balancing gravitational forces

In general acclerometer measures the projection of non gravitational forces on its axis scales by the

mass of the system. It does not tell anything about he body orientation. In an under actuated UAV all

the force act perpendicular to the body plane and the acclerometer in the horizontal axis of the body

does not measure any forces. But an under actuated UAV achieves stable flight with this measurement

in the closed loop. This was insignificant until the authors in (12) describe why the accelerometer

measurements can be used as inclinometer and achieves a stable flight. The updated dynamics

with inclusion of rotor drag validated the closed loop dynamics of an under actuated UAV. This work

extends the outcomes of (12) on to the fully actuated UAVs with tilted propellers.

3 Modelling of Rotor Drag in Fully Actuated Vehicle

This chapter introduces Rotor drag as a component that is the function of rotor speeds and linear ve- locity of the body. The equations of rotor drag are defined and are discussed with analytical,numerical analysis with few assumptions.

3.1 Modelling with Rotor Drag

A multi rotor can be descried in the Newton-Euler form equations of motion.That relates the transla- tional and rotational dynamics of the body in the 3D space. Equation 3.1 shows the general form of translational dynamics described in world frame, counter-acting the gravity with the forces produced by the propellers.

a = ((

^W

R

B

F

1

u)/m) − g (3.1)

Equation 3.2 shows the general form of Rotational dynamics describe in the body frame. Figure 3.1 shows the illustration of the parameters used in rotational and translational dynamics.

ω = J ˙

⁻¹

(− − −ω × Jω + (F

2

u)) (3.2)

Figure 3.1: Illustration of parameters used

The component of rotor drag from (12) is describe in equation 3.3, here the aerodynamic drag from

a rotor is divide into into 4 parts where the first part with λ

1

as the coefficient is the rotor drag due

to linear velocity and rotational speed, the part with λ

2

is the drag due to the component of angular

velocity with the direction of thrust, λ

3

is the component of linear velocity with the direction of thrust

depends in the direction of rotor and λ

4

is the projection of body angular velocity. Out of all these

aerodynamic forces Rotor Drag as discussed in section 2 is significant, hence only one part is used in this work. Other aerodynamic components can be added to the dynamics in further stages.

Aer od ynami cDr ag = −ω

i

( λ

1

v

^⊥_A

i

− λ

2

Ω × k

b

) + ²

i

ω

i

( λ

3

v

Ai

− λ

4

Ω

^⊥

) (3.3) Rotor Drag is caused by the rotation of propeller in high speed, where each propeller has a coefficient of rotor drag produced which is the function of the linear velocities produced in the body. Rotor drag is given in equation 3.4.

B

rd

i

= ω

i

λ

1

v

^⊥_A

i

(3.4)

From (10) the linear velocity of the rotor hub is given by 3.5

B

v

Ai

=

^w

v +

^B

ω

B

×

^B

p

_i

(3.5)

All the variables used are described in table 3.1.

The linear velocity on rotor hub expressed in world frame is given by equation 3.6.

B

v

Ai

=

^B

R

WW

v +

^B

Ω

B

×

^B

p

_i

(3.6)

The velocity in the rotor hub of i

^{t h}

propeller should be projected on the rotor plane, the projection of a vector on to a plane is given in equation 3.7.

pr o j ec t i on

_s

= s − (s · z

pi

)

|z

pi

|

²

z

pi

(3.7)

where s is a vector.

z

pi

is the normal to the rotor plane expressed body frame.

Using the projection formula the Rotor Drag from the i

^{t h}

rotor with the angular speed of that rotor is given by

B

rd

i

=

^B

v

^⊥_Ai

ω

i

λ

i

(3.8)

Rotor Drag force on to the body(COM) is given by by the summation for Rotor drag from each propeller and a drag force coefficient λ

1

B

rd

f

= λ

1 6

X

i =1

B

rd

i

(3.9)

Moment Drag (Moment Rotor Drag) from each rotor frame on to the body frame is given by

B

rd

m

= µ

1

X

6 i =1

B

p

_i

×

^B

rd

f

(3.10)

Translational Dynamics with Rotor Drag in World Frame

a = ((

^W

R

B

F

1

u)/m) − g − ((

^W

R

BB

rd

f

)/m) (3.11) Rotational Dynamics with Rotor Drag in Body Frame

ω = J ˙

⁻¹

(− − −ω × Jω + (F

2

u) −

^B

rd

m

) (3.12)

W

R

B

Rotation Matrix expressing body with respect to world SO3 p

_i

Position vector from the COM to the rotor expressed in body frame R

³

ω Angular velocity of the body expressed in body frame

v Linear velocity of the body expressed in world frame.

J Inertia of the body expressed in body frame.

F1 Force part of the Allocation Matrix R

^3×6

F2 Moment part of the Allocation Matrix R

^3×6

u Propeller Speeds Square

m mass

B

v

Ai

linear velocity of Rotor hub i expressed in body frame.

W

v linear velocity of body expressed in world frame.

λ

1

coefficient of Rotor Drag Force.

µ

1

coefficient of Rotor Drag Moment ω

i

angular speeds of the i

^{t h}

rotor.

l length of the arm

F

W

World frameO

w

, X

W

, Y

W

, Z

W

F

B

Body Frame O

B

, X

B

, Y

B

, Z

B

O

W

Origin of World Frame

O

_B

Origin and CoM of Body Frame

g Gravity

p Position ofO

_B

in world frame O

R₃

Origin of rotor 3

Table 3.1: Variables in modellings

3.2 Analytic Understanding

From the section 3.1 analytically analysing the projection of Rotor Drag from each rotor frame on to the body frame shows the each rotor produces drag in all axis due to it alignment (pose). This works deals with a fully-actuated UAV with tilted propeller called FiberTHex. It an Hex-rotor with rotors around the CoM placed 60°from each other and having a radial tilt of 20°, the figure 3.2 shows the real model of FiberTHex. The detailed hardware and software specifications of the UAV are explained in section 5.1.1.

Figure 3.2: FiberTHex UAV

The figure 3.3 shows the projections of rotor drag from the rotor frame to body frame of FiberTHex for

P2,P5 which are opposite to each other.

Figure 3.3: Rotor Drag Projections from P2 and P5

In the figure 3.3 the direction of motion is considered in to be only in positive X direction. The projection of velocity on the rotor plane is shown in black dash line and the drag in rotor frame is shown in red colour line. The transformation of rotor drag into body frame from rotor frame. The drag from propeller 2 produces a drag in -X, -Y ,+Z of the body frame and drag from propeller 5 produces a drag in -X, -Y, +Z. As these 2 propellers are opposite to each and almost have same propeller speeds.

The sum of P2,P5 in body frame become -2X, -2Y, 2Z. The adjacent pair of propellers P3,P6 produced drag in -X,+Y,+Z, sum becomes -2X, +2Y, +2Z.

By summing the rotor drag produced in body frame by P2,P3,P5,P6 the dominant rotor drag force is present in negative X axis that is opposite to the direction of motion and in Z axis due to the orientation of the rotors. The magnitude of drag depends upon the linear velocities produced in the body frame.

3.3 Numerical Understanding

From the equations in section 3.1, the variables can be written as follows. The linear velocity produced

in world frame,angular velocity in body frame, position of rotor frame expressed in body frame, the

rotation matrix between body and world expressed in world frame are shown below. l is the length of

the rotor arm from body CoM.

w

v

B

=



 v

x

v

y

v

z





B

ω

B

=



 ω

x

ω

y

ω

z





B

p

_i

=



 p

x

p

y

p

z



 =





l si n( θ

i

) l cos( θ

i

)

0





w

R

B

=





1 0 0 0 1 0 0 0 1





The rotation matrix between body and rotor frame expressed in body frame is given by the rotation in z,y,x axis respectively. Considering cant angle ( β) as 0°. θ

i

is the position of rotor around z axis and α

i

is the tilt angle of the specific rotor.

B

R

pi

=





cos( θ

i

) −cos(α

i

)si n( θ

i

) si n( α)sin(θ

i

) si n( θ

i

) cos( α

i

)cos( θ

i

) −si n(α

i

)cos( θ

i

)

0 si n( α

i

) cos( α

i

)



 From equation 3.6 the velocity of rotor hub expressed in body frame

B

v

Ai

=





1 0 0 0 1 0 0 0 1







 v x v y v z



 + cr oss(



 ω

x

ω

y

ω

z







 p

xi

p

yi

p

z_i



)

B

v

_Ai

=



 v

x

v

y

v

_z



 +





−p

yi

ω

z

p

x_i

ω

z

p

_yi

ω

x

− p

xi

ω

y





B

v

Ai

=





v

x

− (p

yi

ω

z

) v

y

+ (p

x_i

ω

z

) v

z

+ (p

yi

ω

x

− p

xi

ω

y

)





Using the projection formula given in equation 3.7.

Projv

_Ai

=





v

x

− (p

yi

ω

z

) v

y

+ (p

xi

ω

z

) v

_z

+ (p

yi

ω

x

− p

xi

∗ ω

y

)



 −





v

x

− (p

y_i

ω

z

) v

y

+ (p

xi

ω

z

) v

z

+ (p

yi

ω

x

− p

xi

ω

y

)



.





si n( α

i

)si n( θ

i

)

−si n(α

i

)cos(θ

i

) cos( α

i

)





nor m|





si n( α

i

)si n( θ

i

)

−si n(α

i

)cos( θ

i

) cos( α

i

)



 |





si n( α

i

)si n( θ

i

)

−si n(α

i

)cos( θ

i

) cos( α

i

)





Projv

_Ai

=





v

x

− (p

y_i

ω

z

) v

_y

+ (p

xi

ω

z

) v

z

+ (p

yi

ω

x

− p

xi

ω

y

)





− (v

x

− (p

xi

ω

z

)si n( α

i

)si n( θ

i

) − (v

y

+ (p

xi

ω

z

)si n( α

i

)cos( θ

i

) + cos(α

i

)(v

z

− (p

xi

ω

y

+ p

yi

ω

x

) cos( α

i

)

²

+ (si n(α

i

)si n( θ

i

))

²

+ (si n(α

i

)si n( θ

i

))

²





si n( α

i

)si n( θ

i

)

−si n(α)cos(θ

i

) cos( α

i

)





Projv

_Ai

=





v

_x

− (p

yi

ω

z

) − [X si n(α

i

)si n( θ

i

)]

v

y

+ (p

xi

ω

z

) + [X si n(α

i

)cos( θ

i

)]

v

z

+ (p

yi

ω

x

− p

xi

ω

y

) + [X cos(α

i

)])





where X is the scalar quantity defined by

X = (v

x

− (p

xi

ω

z

)si n( α

i

)si n( θ

i

) − (v

y

+ (p

xi

ω

z

)si n( α

i

)cos( θ

i

) + cos(α

i

)(v

z

− (p

xi

ω

y

+ p

yi

ω

x

) cos(α

i

)

²

+ (si n(α

i

)si n(θ

i

))

²

+ (si n(α

i

)si n(θ

i

))

²

The rotor drag from i

^{t h}

rotor given in equation 3.8.

rd

i

= −ω

i

Projv

_Ai

λ

1

Understanding the above equation it can be stated that the Rotor Drag is the function of velocity in the rotor plane frame and the rotor speeds. There is an influence of the pose of the rotor frame in the direction of drag produced.

Computing the above equation numerically to validate the conclusion of subsection 3.2 with few assumptions listed below.

• As its a fully actuated UAV there will be no body orientations so angular velocity of body ω

x

, ω

y

, ω

z

can be zeros.

• Pair of propellers have same speed. ω

i

=56

• Motion only in Y axis so velocities produced only Y so [v

x

= 0, v

y

= 1, v

z

= 0].

Taking an example of P2 and P5 where the tilt angle α is +20°for P2 and -20°for P5 with right hand convention with positions around CoM θ

2

= 60, θ

5

= 240.

Rotor Drag for P2 becomes

rd

2

=





−u(v

x

− 0.877v

x

+ 0.05707v

y

− 0.2783v

z

)

−u(v

y

+ 0.507v

x

− 0.0292v

y

+ 0.1607v

z

)

−u(v

z

− 0.2783v

x

+ 0.1607v

y

− 0.8830v

z

)



 (3.13)

Rotor Drag for P5 becomes

rd

5

=





−u(v

x

− 0.877v

x

+ 0.0507v

y

− 0.2783v

z

)

−u(v

y

+ 0.0507v

x

− 0.0292

y

+ 0.1607v

z

)

−u(v

z

− 0.2783v

x

+ 0.1607v

y

− 0.8830v

z

)



 (3.14)

The Rotor Drag produced by P2,P5 on the body frame are

rd

2

=





−2.8366

−54.3623

−8.9990





rd

5

=





−2.8366

−54.3623

−8.9990





Here The Rotor Drag produced by P2,P5 is equal, calculating the Rotor Drag produced By P1,P3,P4,P6 on body frame are

rd

1

=



 0

−49.4492 17.9981





rd

3

=





2.8366

−54.3623

−8.9990





rd

4

=



 0

−49.4492 17.9981





rd

6

=





2.8366

−54.3623

−8.9990





From the Rotor drag produced by each rotor at the instance the summation in X axis goes to zero and

summation on Z axis leaves a small residue of +0.0362. This means motion in Y axis produces a drag

on the body in -Y which is opposite to the direction of motion and a small drag on the Z axis with no

effect in X axis in body frame, this validates the analytical understanding with the assumptions given.

3.4 Simulation Setup

A simulink model is setup for the FiberTHex which has six rotors are placed around the CoM with angle between 2 consecutive rotors is 60°and each rotor frame has a tilt angle of 20°(Clock wise (-) for odd rotors and Counter Clock wise(+) for even rotors). The frame alignment of FiberThex is give in figure 3.4 where red line shows x axis, green line shows y axis and z shows the z axis of the particular frame and R1 ..R6 are the rotors. Figure 3.5 shows the top view of the rotor frames.

Figure 3.4: Alignment of frames in FiberThex

Figure 3.5: Top view of Rotor frames in FiberThex

As seen from the frame alignment each rotor frame has a opposite frame, like rotors 1,2,3 are opposite to rotors 4,5,6 and have z axis(blue) parallel to each other.

The figure 3.6 shows the full simulink block designed to simulation the dynamics of a multi rotor with 2 controllers position and attitude, a UAV model mask. The controllers position, attitude use the concepts from the paper (5) to get a more solid understanding the reader can refer it. The position controller takes in the reference data like the position, velocity, acceleration and orientation with a feed back loop of current orientation and velocity to calculate the forces required to achieve the desired. The attitude controller takes input of the desired forces, orientation from position controller, desired angular velocity with current orientation, current angular velocity to calculate the rotor speeds for desired inputs.

Figure 3.6: Simulink model for simulating a multi rotor

The calculated rotor speeds in given as the input UAV_model block which contains equation of motion

of the multi rotor in the function dynamics seen in figure 3.7. The dynamics function block outputs

the acceleration and angular acceleration of the body which can be integrated to get velocity and

position which can be fed back to the controller. The rotational and translation dynamics used in the

dynamics function block is given in equation 3.1, 3.2. This simulink model is extended to include

rotor drag in the equations of motion as discussed in section 3.1.

Figure 3.7: UAV_model mask

The model is further developed by adding a three axis acclerometer for the aerospace tool box of simulink library. This acclerometer can generated acclerometer readings by taking accelerations in body frame , angular velocity, angular acceleration from the model as input. The model acclerometer can be tuned according to the requirement by changing natural frequency and damping ratio. Further this block allows to add noise in the readings. This model acclerometer is used in parameter estimation toolbox which will be discussed in 5.3.2.

The table 3.2 summarises all the inputs, outputs and flow of data in the simulink model.

Simulink Blocks Inputs Outputs

Position

Controller User Input Reference Position Forces required

Reference Velocity Desired Orientation

Reference Acceleration Reference Orientation

Feedback Current Position. Velocity, Orientation Attitude

Controller User Input Desired Angular Velocity Forces required

Desired Orientation rotor speeds

Feedback Current Orientation and angular velocity

UAV dynamics Rotor speeds Acceleration

Feedback angular velocity, orientation, Angular accelerations Model Acclerometer Accelerations in body body frame Acclerometer data

Angular velocity of the body Angular accelerations

Table 3.2: Input Output parameters Simulink

4 Analysis of the Model

The goal of this chapter is to detail multiple simulations in various motions to understand the affect of rotor drag force from each propeller on to the body frame in a Fully Actuated UAV with tilted rotors.

The chapter also gives out a clear comparison of rotor drag between 3 configurations of vehicles- Fully Actuated UAV (FA), Fully Actuated UAV Under Actuated mode (FA-UA) and a Generic Hexacopter (HEX)r with no tilted propellers as shown in figure 2.1. As FiberThex show symmetry in positioning of the rotor around the body frame, the motions by the body can be simplified into 2 basic motions.

Motion axis between the rotors and motion axis inline with the rotor as shown in figure 4.1. These can also be described as the orthogonal frame at the CoM. There would be 6 configuration where one of the axis of the orthogonal body frame passes thorough the rotors and other between the rotors. This will help to understand the effect of rotor pose and symmetricity in the body.

Figure 4.1: Basic motion with respect to symmetry of the body

For easy understanding and analysis the X axis of the body frame passes through rotor 1 and rotor 4 and the Y axis being perpendicular passes between the rotors is considered, the body frame is shown in figure 4.2.

Figure 4.2: Body Frame alignment FiberTHex

4.1 Rotor Drag in FA,FA-UA,HEX

This section compares the rotor drag in the body frame of 3 configurations FA, FA-UA, HEX and analyses various parameters for understanding rotor drag. The reference trajectory used is s simple point to point motion moving continuously or easily a sin wave with a specific frequency shown in figure 4.3. This trajectory help to understand the motion in two direction of the body and affect of velocity change in the rotor drag. The analysis is divided based on the two simple motions between or inline with the rotors.

Figure 4.3: Reference Trajectory

4.1.1 Motion axis In-line with the Rotors

The motion axis aligns with the X axis of the body frame as described earlier with same reference trajectory . The forces computed by the controller (lift forces) in the body to achieve a the motion in-line with rotors is shown in figure 4.4.

Figure 4.4: Lift Forces produced in body frame

As the motion is along the X axis of the body frame, from the figure the FA produces significant varying forces in X axis for motion, a varying force in Y axis and a constant force in Z axis. Both FA-UA, HEX does not not produce any force in x and y axis of the body. Figure 4.5 shows the total rotor drag in the body frame.

Figure 4.5: Rotor Drag produced in body frame

From the figure the FiberTHex in both FA and UA modes produces significant drag in X axis of the

body frame where as HEX produces very less. In Y axis on the FA produces drag where as FA-UA ,HEX

is zero. The rotor drag in Z axis is varying between 3 systems, FA produces the highest drag and HEX

does not have any magnitude. This is due to the alignment of rotor perpendicular to the around the

body plane. The rotor drag is always opposite to the direction of motion in the body frame. Figure 4.6

and figure 4.7 shows the rotor speeds and linear velocities of the systems. As rotor drag is the function

of rotor speeds and liner velocities, from the figure it is seen the rotor speeds of HEX is smaller than

the FA systems, linear velocities produced in all 3 bodies is same.

Figure 4.6: Rotor Speeds of Inline motors motion

Figure 4.7: Liner Velocities

As discussed in the section 3.3 one the assumptions is all rotors have same speeds, but in reality from figure 4.6 only the opposite pair of rotors have same speed so there is no perfect cancellation of rotor drag from adjacent pair of propellers in perpendicular axis to axis of motion. Hence there is a rotor drag component in other axis in the plane perpendicular(Yaxis) to the axis of motion(Xaxis). The figure 4.8 shows the body orientations of 3 multi rotors.

Figure 4.8: Body Orientations for motion axis inline with the rotors.

4.1.2 Motion between the Rotors

In this case the direction of motion is between the rotors or Y axis of the body frame. The forces produced by the controller in the body frame of 3 configurations in shown in figure 4.9.

In contrast to the earlier case of motion inline of rotors there is no forces produced in perpendicular

axis to the axis of motion (X axis) and FA produces significant varying forces in Y axis and a constant

force in Z axis. Figure 4.10 shows the forces rotor drag forces produced in body from of 3 systems.

Figure 4.9: Forces Produced in the body frame

Figure 4.10: Rotor Drag in the body frame

Similar to force case there is no drag in X axis when the motion is is in Y axis of the body. FA and FA-UA produces similar vector of rotor drag in different modes of operation but FA produces more drag in the Z axis of the body frame. This is similar to previous case; the dominant drag is in the axis of motion and a residue in Z axis of the body. In Hex the drag is only present in axis of motion. Figure 4.11 shows the rotor speeds of this motion.

From the figure it is seen that opposite pair of rotors P2,P5,P6,P3 have same speed which is different from the earlier case. This semi proves the assumption of same speeds for all rotors then the perpen- dicular axis of motion the rotor drag is zero. To achieve the reference trajectory the UA systems needs to tilt, figure 4.12 shows the body orientations in the motion axis between the rotors.

Figure 4.11: Rotor Speeds between the rotors motion axis

Figure 4.12: Body Orientations

From the figure HEX and FA-UA does the same roll of around 3°to achieve the reference trajectory and the linear velocities in the world frame are shown in the figure 4.13. Here the velocities are produced only in Y axis as per the reference trajectory with a small residue in Z axis.

Figure 4.13: Liner Velocities

From the above simulation it can be seen that motion axis inline with rotors have different affects of

rotor drag on the body frame then the motion axis between the rotors.

4.2 Rotor Drag with Body Orientation in FA

In this section a FA system holding a body orientation (tilt) though out the flight is compared to a UAV with no body tilt and a UA system with same tilt to understand how body orientation in FA system effect the rotor drag in body frame.

4.2.1 FA with 3°Tilt

This analysis is also done with motion axis inline the rotors and axis in between the rotors.

Motion axis Inline with Rotors

Using the simple trajectory as reference in motion axis inline with the rotors (X axis of the body) for a body holding 3°pitch and a body with no pitch the rotor drag is analysed. Figure 4.14 shows the lift forces produced in the body frame.

Figure 4.14: Lift Force in body frame.

Here with 3°pitch the forces in the body are varied. The motion is in X axis but the force produced is less this is due to the pitch which vectors thrust in that direction, but a very small amount variation in the magnitude of other axis. Figure 4.15 shows the rotor drag in the body frame.

Figure 4.15: Rotor Drag in body frame.

With the orientation in the body also there is not change in the drag produced in the axis of motion

but the pitch of the body effects the other 2 axis with a minimal amount of increase in drag, overall it

does not affect. Figure 4.16 shows the rotor speeds for this inline rotors axis motion.

Figure 4.16: Rotor Speeds in motion axis inline to the rotors.

From figure it is seen that to hold the pitch the opposite pair if rotors reduce speed and other pair increases but there is not change in the speeds of the rotors that are inline with pitch or axis of motion.

Figure 4.17 shows the body orientations for this motion.

Figure 4.17: Body Orientations

Motion axis between the rotors

To do a motion in axis between the rotors the body needs to hold roll angle. The lift force produced in the body are shown in figure 4.18.

Figure 4.18: Forces in the body frame.

Similar to the earlier case there is no force produced in the axis perpendicular to axis of motion and

the forces in the direction of motion increase in the body tilt body. This is to hold the roll angle in

flight. Figure 4.19 shows the rotor drag comparison for this motion.

Figure 4.19: Rotor Drag in body frame.

Here the rotor drag in tilt body and no tilt body is same in the axis of motion that is the axis between the rotors and a little increase in the Z axis of the body frame. Figure 4.20 shows the rotors speeds in this motion.

Figure 4.20: Rotor Speeds in the motion axis between the rotors .

Similar to earlier case in the motion axis between the rotors, 4 rotors which on the either side of the axis of motion have same speed and figure 4.21 shows the body orientations for this motion.

Figure 4.21: Body Orientations .

For the above analysis it can be inferred that small body orientation does not effect the total drag produced in the body frame in any axis of motion.

4.2.2 FA with 3°tilt and FA-UA

This analysis shows if the FA system with constant tilt in body have similarities with an FA-UA system.

This analysis is also done for motion axis inline with the rotors and motion axis between the rotors.

4.2.3 Motion axis inline with rotors

The figure 4.22, figure 4.23, figure 4.24 shows the forces in the body frame , total rotor drag in the

body and the rotor speeds between the a FA system with 3°pitch and an FA-UA system given the same

reference trajectory of point to point motion.

Figure 4.22: Forces in the body frame.

Figure 4.23: Total Rotor Drag in the body.

Figure 4.24: Rotor Speeds in the motion axis inline with the rotors.

This comparison does not show any difference in the rotor drag produced in the body frame except

some variations in residuals in Y and Z axis as the motion is in X axis of the body. The forces produced

and rotor speeds are little different in both the systems but does not effect the total drag. Figure 4.25

shows the linear velocities of the systems in the world frame and figure 4.26 shows the orientations in

the body.

Figure 4.25: Linear velocity of the body with motion axis inline with the rotors.

Figure 4.26: Orientations in the body

Motion axis Between the rotors

This comparison between a tilted(roll) FA and FA-UA does not have any significant difference in the Rotor drag in the body frame in motion axis between the rotors. The plots of forces in the body, Rotor drag in the body frame and rotor speeds for this motion is shown in the figure 4.27, figure 4.28, figure 4.29 respectively.

Figure 4.27: Forces in the body frame.

Figure 4.28: Total Rotor Drag in the body.

Figure 4.29: Rotor Speeds in the motion axis in between the rotors.

Hold a 3 °roll constantly the body does point to point motion, the figure 4.30 shows the body orienta- tions for this configuration comparison.

Figure 4.30: Orientations in the body

4.2.4 Canonical Trajectory for FA,FA-UA,HEX

Given a canonical trajectory that has multiple movements in Y and Z axis of the body frame (motion axis between the rotors and vertical axis) simultaneously shown in figure 4.31 as the reference to 3 configurations this section shows multiple variables and the rotor drag in the body frame.

Figure 4.31: Canonical Trajectory given as reference.

The figure 4.32 shows the rotor drag in the body frame by the given reference motion. Here the most significant drag is present in the Y axis of the body frame and drag in Z axis varies with the motion.

The HEX body does not produced any significant drag in the body frame except some residues in the Y

axis as that is the axis of motion. FA, FA-UA follow similar drag pattern despite having different modes

of actuation. This validates that body orientation does not effect the total rotor drag in body frame

notable way. Figure 4.33 shows the body orientation for this motion and both the FA-UA and HEX

have similar tilt angles at required.

Figure 4.32: Total Rotor Drag in the body.

Figure 4.33: Orientations of the body.

The rotor speeds show similar variation in pairs for FA and FA-UA for the full trajectory but in some case the FA deviates for the FA-UA shown in the figure 4.34.

Figure 4.34: Rotor Speeds

4.3 Rotor Drag from Rotor Frame to Body Frame

From the previous section it is seen that the total rotor drag in the body frame varies by the motion axis. The rotor drag in the body frame is the summation of rotor drag in each rotor frame. In this section the rotor drag in each axis by each rotor is analysed and a conclusion is derived on how the drag in each axis of body frame from each rotor is summed or negates one other. Considering the similar trajectory of point to point motion as earlier and a FA system.

4.3.1 Motion axis inline with rotors

This motion axis is also the X axis for the body frame. Figure 4.35 shows the total rotor drag in the body frame. From figure it is seen that rotor drag is present in all axis, significant being the X axis as it the motion direction but as per numerical understanding the rotors in pairs should negate the drag in axis perpendicular to the motion axis which is not present here. Figure 4.36 shows the rotor drag from each rotor in X axis of the body on to the body frame. Seen from the figure the drag produced by each rotor is in the same direction hence the summation on the body gives a significant value.

Figure 4.35: Rotor Drag force in the body frame

Figure 4.36: Rotor Drag by each rotor on body frame in X axis

Figure 4.37 shows the rotor drag from each rotor in the Y axis of the body frame. This axis is also

perpendicular to axis of motion hence it should negated by the pair of rotors. From the figure its is

seen at time 19.16 seconds the sum of P2,P5 is greater then the sum of P1,P3,P4,P6, hence there is a

small residue present. The residue is present as all the opposite pair of rotor does not rotate at the

same speed given in figure 4.38.

Figure 4.37: Rotor Drag by each rotor on body frame in Y axis

Figure 4.38: Rotor Speeds

The figure 4.39 shows the rotor drag in each propeller on to the Z axis of the body frame. Here also the sum of P2,P5 is greater than the sum of P1,P3,P4,P6, hence a residue is present in the body rotor drag.

The motion in body frame is only in the axis inline with the rotor but the drag in other axis because of the rotor orientation with respect to the body frame. FiberTHex has a 20 radial tilt for all the rotor as discussed in earlier sections. Here when the motion axis in inline with the rotors there is drag present in all the axis of the body frame due to uneven rotor speeds for the motion.

Figure 4.39: Rotor Drag by each rotor on body frame in Z axis

Motion axis between the rotors

In this case the motion axis is in between the rotors also its the Y axis of the body frame as considered.

The total rotor drag in body frame is shown in figure 4.40. From the figure it is seen that the rotor drag

is only present in Y and Z axis of the body frame which is different from earlier motion axis inline with

rotor case. Figure 4.41 shows the rotor speeds of each rotor for this motion, it is clear that 4 rotors

namely P2,P3,P5,P6 rotor at the same speed for this motion.