## vol1-is1-pp5

of Fixed-Pitch Coaxial Rotor Helicopter

Satoshi Suzuki†, Takahiro Ishii‡, Gennai Yanagisawa§, Kazuki Tomita£, Yasutoshi Yokoyama§†Young Researchers Empowerment Center, Shinshu UniversityTokida, Ueda-shi, Nagano, Japan‡Graduate School of Science and Technology, Shinshu UniversityTokida, Ueda-shi, Nagano, Japan§Gen CorporationSasaga, Matsumoto-shi, Nagano, Japan£Engineering SystemSasaga, Matsumoto-shi, Nagano, Japan

*Abstract*— In this study, a mathematical model of a fixed-pitch co-axial rotor unmanned helicopter is derived by using a multi-body dynamics modeling technique. First, we consider the helicopter as a rigid body system which consists of 2 rigid bodies. The velocity transformation method, which is one of the multi body dynamics technique, is applied to derive equation of motion of the helicopter. All the forces and moments impressed into the helicopter are derived in consideration of the aerodynamics of co-axial rotors. Derived mathematical model of the helicopter is verified by comparing the flight experiment data with model output. Finally, fundamental motion analysis using the model is conducted to establish the motion characteristics of the helicopter.

*Keywords*— co-axial rotor helicopter, multi-body dynamics modeling, motion analysis.

** I.INTRODUCTION **

In recent years, unmanned helicopter has become to be developed and used for various practical purposes such as aerial photography, surveillance, and crop dusting. Unmanned helicopters are safer and more convenient than manned helicopters, and they can be potentially employed in a wide range of applications. There are many types of unmanned helicopters, which have various sizes, weights, and have various mechanisms. For example, single-rotor type helicopter [1], tilt-rotor type [2], co-axial rotor type [3], and multi-rotor (quad or more) type [4]. In this study, we propose a fixed-pitch co-axial rotor unmanned helicopter, which has specific mechanisms. The fixed-pitch co-axial rotor helicopter has some advantages compared with any variable-pitch rotor type or single-rotor type helicopters. For example, the simplicity of the mechanisms, well maintainability, and well energy conversion efficiency are part of the advantages of our helicopter. Because of these advantages, fixed-pitch co-axial rotor mechanism has been adopted for world smallest manned helicopter shown in **Figure 1**.

Figure 1 Co-axial rotor manned helicopter

For the practical operation, it is important to know how the helicopter behaves in flight, and it is also important to know how we should coordinate the mechanical parameters to improve the behavior of the helicopter. Moreover, precise mathematical model of the helicopter is necessary to design the autonomous control system for our helicopter. However, modeling and motion analysis of fixed-pitch co-axial rotor helicopter has not been studied so far because the helicopter has some specific mechanisms such as fixed-pitch co-axial rotor, rotor tilting mechanism, and a mechanism for shifting the center of the mass.

In this paper, a mathematical model of fixed-pitch co-axial rotor unmanned helicopter is derived to know how the helicopter behaves in flight. A multi-body dynamics modeling technique named velocity transformation method is adopted to derive an equation of motion. Numerical simulation is performed to compare the output of the derived model with experimental data, and validity of the model is shown. In the simulation, all the forces and moments impressed into the helicopter are derived in consideration of the aerodynamics of co-axial rotors. Finally, fundamental motion analysis is performed using the model to establish the relation between the motion and the mechanical specification of the helicopter. The main contribution of this paper is that the fixed-pitch co-axial rotor unmanned helicopter with novel mechanical specification is proposed and modeled. The rest of this paper is organized as follows. The overview of our fixed-pitch co-axial rotor unmanned helicopter is introduced in section II. The equation of motion of the helicopter is derived and verified in section III. In section III.A, a fundamental motion analysis is performed using the derived model. Finally, the conclusion of this paper is shown in section V.

**II. FIXED-PITCH CO-AXIAL ROTOR UNMANNED HELICOPTER**

**Figure 2** shows an overview of our co-axial rotor unmanned helicopter. The main specifications of this helicopter are listed in **Table 1**. This helicopter has four 125 cc gasoline engines for a source of power and could lift large payload approximately 100 kg because it was originally designed for manned helicopter. Because of this large payload capacity, the helicopter is expected to be applied to various industrial tasks not only information gathering such as aerial photography but also transportation of heavy goods. Most important features of this helicopter are to have a fixed-pitch co-axial rotor as shown in **Figure 3**. The mechanisms of the rotor segment are simpler than which of any other co-axial rotor helicopters [6] by this feature. This simplicity conducts a reduction of parts count, and improvement of the maintainability. Moreover, the size of this helicopter is small, and the energy conversion efficiency is better than single-rotor type helicopters because it does not have tail rotor. Meanwhile, the fixed-pitch helicopter has a disadvantage in forward flight, called pitch-up phenomenon. In forward flight, relative wind blows into the rotor disc, and causes imbalance of lift force on the disc. Then, this imbalance generates aerodynamic moment, which lifts the nose of the helicopter. In the case of variable-pitch helicopter, the imbalance can be canceled out by cyclic pitch control. By contrast, fixed-pitch helicopter cannot cancel out this imbalance, and the nose of the helicopter is lifted, and then the forward velocity decreases. Therefore, the maximum cruise speed of the fixed-pitch helicopter may be smaller than the speed of the variable-pitch helicopter. It is a serious problem for practical tasks. To solve the above mentioned problem, we have proposed the mechanism for shifting the center of mass of the helicopter.

The entire mechanism of our helicopter is shown in **Figure 4**. The helicopter is divided into two parts, the mission unit and the frame unit, and two units are connected by the universal joint. The mission unit consists of co-axial rotor and gasoline engine, the frame unit consists of fuel tanks and the electric devices for control. There are two mechanisms near the universal joint. The tilting mechanism directly tilts the rotor disc by using four servo motors to generate the horizontal force. The shifting mechanism changes horizontal position of the frame unit relative to the mission unit, and so the center of the mass of the helicopter shifts, and the moment of a couple is generated by gravitational force and the rotor thrust. This moment is used for canceling out the pitch-up moment in forward flight.

Figure 2 Co-axial rotor unmanned helicopter Figure 3 Fixed-pitch co-axial rotors

Figure 4 Mechanisms of the helicopter

Table 1 Specifications of Unmanned Helicopter

Rotor diameter |
4000 [mm] |

Height |
1600 [mm] |

Weight |
166 [kg] |

Source of power |
Gasoline Engine |

Endurance |
1 [h] |

Payload |
Approximately 100 [kg] |

**III. MODELING**

In this section, we derive the mathematical model of our fixed-pitch co-axial rotor unmanned helicopter. To derive the mathematical model of the helicopter, we consider the helicopter as a rigid model which consists of 2 rigid bodies. We apply the velocity transformation method to derive nonlinear equations of motion of the helicopter. We assume that all the forces and moments impressed into the helicopter are generated by co-axial rotors, and we derive them in consideration of rotor aerodynamics. First, the velocity transformation method is briefly introduced, and we apply this method to our helicopter, and the nonlinear equation of motion is derived.

**A. Velocity Transformation Method**

The velocity transformation method is one of the multi-body dynamics technique proposed by Tajima ([7]), and it is similar to a traditional velocity transformation technique. In the following, the outline of the velocity transformation method is described. The detail of the method is presented in [7], and the details of the velocity transformation technique also appear in [8]-[10]. First, the generalized velocity of a rigid body without constraint is considered as ** H**. Then the equation of motion of the rigid body is represented as

Here, is a symmetric matrix representing the inertia of the rigid body, ..is the external force, which is impressed on the rigid body. If some mechanical constraints are added to the rigid body, is rewritten as

** (2****)**** **

Here, represents the constraint force. On the other hand, a generalized velocity of a rigid body with constraint is considered as ** S**, and the equation of the motion is obtained as

Here, represents the inertia matrix, and represents the external force vector. Now, in the case of holonomic and simple non-holonomic system, the relation between generalized velocities ** H **and

**is obtained as follows:**

*S*** ****(3****)**** **

Here, and are appropriate matrices. Calculating the time derivatives of , and substituting it into , following equation could be obtained.

** ****(4****)**** **

According to [7], it can be easily shown that is equal to zero. Therefore, comparing and , and is expressed by using and as follows:

If and were given and the relation could be derived, the equation of motion of rigid bodies with constraints could be obtained by using , , and . This method includes the velocity transformation , therefore the method is called velocity transformation method.

**B. Equation of Motion of Co-Axial Rotor Helicopter**

The equation of motion of the co-axial rotor helicopter is derived. Coordinate systems used in this section are defined in **Figure 5**. The ** O-**frame is an inertial frame fixed at an arbitrary point on the ground,

**-frame is fixed on the mission unit, and**

*A***-frame is fixed on the frame unit.**

*B***-frame and**

*A***-frame are movable coordinate system which moves in conjunction with the motion of each unit. Moreover, the notation**

*B***denotes the universal joint. In these coordinate systems, represents the position of the mission unit relative to the**

*C***frame, and it is expressed as the vector on**

*O-***frame. On the other hand, represents the same vector which is expressed on**

*O-***frame. This manner is also applied to linear velocity, angular velocity, and force vector. Moreover, in the case of the frame unit, relative position vector is represented as and . If there was no constraint by the universal joint, the mission unit and the frame unit are regarded as free rigid bodies. Thus, the equation of motion of each unit is represented as follows:**

*A-*** ****(7****)**** **

** ****(8****)**

Here, denotes the mass of each unit, , the inertia matrix, and , the linear and angular velocity, and , denote the external force and torque vector. Additionally, denotes the skew symmetric matrix which represents vector product. Now, we choose the generalized velocity without constraint as

** ****(****11)**

Then, and in are obtained from - as

** ****(****12)**

** **

** ****(****13)**

We consider the constraint of angular velocity as

Here, is the coordinate transformation matrix which transforms a vector from ** B**-frame to

**-frame, is the relative angular velocity of the mission unit to the frame unit, and it is determined by the motion of the tilting mechanism. The constraint of the position is obtained as**

*A*Here, and are the coordinate transformation matrices. And represent the position of the universal joint, and defined as

** ****(****16)**

is a distance between center of gravity of the frame unit and universal joint, is the distance between the mission unit and the joint (**Figure 5**). and are determined by the motion of the shifting mechanism. Time derivatives of is obtained as

** ****(****17)**

Similarly, is obtained as

Here, we used to derive . Considering the generalized velocity with constraint as

** ****(****19)**

and are obtained as follows by using , , and

** ****(20****)**

** ****(2****1)**

Finally, the equation of motion of the co-axial rotor helicopter is derived as . Here, and are obtained as follows:

** ****(22****)**** **

** ****(23****)**

** ****(24****)**

Additionally, is defined as

** (25)**

Table 2 Specifications of Unmanned Helicopter

Mass of mission unit |
66 [kg] |

Inertia moment of mission unit |
diag(50, 50, 50) [kg m^2] |

Mass of frame unit |
100 [kg] |

Inertia moment of frame unit |
diag(50, 50, 50) [kg m^2] |

Distance between COG of mission unit and joint |
0.206 [m] |

Distance between COG of frame unit and joint |
0.693 [m] |

Figure 6 Comparison of pitch angular velocity Figure 7Comparison of pitch angle

Figure 8Comparison of forward velocity Figure 9Comparison of roll angular velocity

Figure 10 Comparison of roll angle Figure 11 Comparison of rightward velocity

Figure 12 Examination of the position universal joint

To verify the validity of derived model, a simulation was conducted by using the equation of motion. In the simulation, the outputs of the model are compared with the experimental data. Model parameters used in this simulation are listed in **Table 2**. All the forces and moments in -, which areimpressed into the helicopter, were derived in consideration of the aerodynamics of co-axial rotors. *Blade Element Theory* and *Momentum Theory* in **[11]**-**[12]** were used to obtain the thrust and torque of the rotors. The gravity was also included in -. Moreover, , and the time derivatives of them were determined by the motion of the tilting and shifting mechanisms. They are considered as the inputs of the model.

**Figure 6**** **- **Figure 11** show the results of the simulation for horizontal motion. In these figures, the solid line represents the output of the mathematical model; dashed line represents flight experiment data measured by sensors, which are mounted on the helicopter. **Figure 6** and **Figure 9** show the pitching and rolling angular velocity of the helicopter measured by using gyro sensor. **Figure 7** and **Figure 10** show the pitching and rolling attitude angle measured by the attitude sensor. **Figure 8** and **Figure 11** show the horizontal velocity measured by GPS. From **Figure 6**, **Figure 7**, **Figure 9**, and **Figure 10**, the model outputs accords with the experimental data in the case of angular velocity and attitude angle. Accordingly, it seems that attitude dynamics of the helicopter could be expressed precisely by the mathematical model. However, in the case of horizontal velocity, the model outputs don’t accord with experimental data well. The reason of mismatch seems to be the influence of aerodynamics which was not considered in the simulation. In fact, we have not yet considered the imbalance of lift force on the disc which causes pitch-up phenomenon. Moreover, we also have not yet considered vertical and yawing motion. Therefore, we should implement them into the simulation in future works.

**IV. MOTION ANALYSIS**

To know the relation between the motion and mechanical specification of the helicopter, we conducted motion analysis by using derived mathematical model. Although the dynamics of horizontal velocity have not yet been expressed well by the model, we conclude** **that the attitude model is precise enough to use motion analysis. Therefore, only attitude motion of the helicopter is analyzed in this section. For operation of the helicopter, reducing the vibration of the frame unit is the most important because some payloads such as a camera, sensors for application, and a product is mounted on the frame unit. For this reason, we examine the optimal position of the universal joint to reduce vibration of the frame unit by simulation. In the simulation, we impressed the input to the tilting mechanism as sinusoidal signal, and the input to shifting mechanism is zero. Simultaneously, we record the amplitude of the attitude of the frame unit in steady state. This simulation was carried out repeatedly while changing the frequency of the input to the tilting mechanisms and the position of the universal joint. Meanwhile, the distance between COG of the frame unit and the COG of the mission unit (*l* + *d *in **Figure 5**) was fixed at 1 m in the simulation.

**Figure 12** shows the examination result of the optimal position of the universal joint. In this figure, the lateral axis represents the frequency of the input to the tilting mechanism, the longitudinal axis represents the amplitude of the attitude of the frame unit in steady state. Moreover, notation *d *represents the distance between COG of the frame unit and the universal joint as shown in **Figure 5** . From the figure, it is shown that the amplitude of the attitude of the frame unit is largest when the input frequency is between 0.1-0.2 Hz. It means that the helicopter has the resonance frequency at 0.1-0.2 Hz. Moreover, it is clear from the figure that the amplitude of the frame unit is affected drastically by the position of the universal joint, and the maximum of the amplitude could be suppressed by changing the position of the universal joint. To find the optimal position of the universal joint, examination was carried out in detail between 0.1-0.2 Hz. The result is shown in **Figure 13** . From the figure, the maximum of the amplitude of the frame unit could be smallest when the distance between COG of the frame unit and the universal joint is 0.2 m. As a result, we conclude that the optimal position of the universal joint for reducing vibration of the frame unit is 0.2 m from COG of the frame unit.

**V. CONCLUDING REMARKS**

In this paper, we derived the mathematical model of fixed-pitch co-axial rotor unmanned helicopter by using multi-body dynamics modeling technique. We applied the velocity transformation method to derive the equation of motion of the helicopter. Verification of the model is conducted by comparing the flight experiment data with model output. Finally, fundamental motion analysis for reducing vibration of the frame unit was performed by using derived mathematical model.

In future woks, we implement the aerodynamic effect of the co-axial rotor which cannot be considered in current model such as the interference of each rotor, vertical motion, and yaw motion to the mathematical model to realize more precise simulation and motion analysis.

**ACKNOWLEDGMENT**

This work was partly supported by JSPS KAKENHI Grant Number 24760185.

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