# Center-of-Gravity Variation-Driven Spherical UAV System and Its Control Law

Authors: Donghoon Kim; Sungwook Yang
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

Most of the spherical unmanned aerial vehicles (SUAVs) use control surfaces, which are functions of aileron and an elevator, to generate control torque. The work proposes a new conceptual design of an SUAV system controlled through center-of-gravity (CG) variations with its path-tracking control law designed for the system. Compared to the one using control surfaces, the concept suggested is beneficial in the aspect of the expandability of building lighter and smaller SUAVs, especially. A CG variation principle by actuating a pendulum type of a moving part is considered as a methodology for both translational and rotational motion control of an SUAV. Since variations of the moment-of-inertia (MOI) elements which resulted from the motion of the moving part affect the performance of the suggested method, the variations of MOI analysis are performed for all angular ranges of the moving part. As a result, certain angular ranges for the moving part to prevent the degradation of the path-tracking performance by the effect of the MOI changes are found. By considering the findings, numerical studies are performed for hovering, ascent, descent, and horizontal tracking missions. The applicability of the proposed SUAV system and the corresponding controller to achieve the path-tracking missions is demonstrated through the numerical simulation.


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## Body

## 1. Introduction

Researches for the various shapes of unmanned aerial vehicles (UAVs) have been performed continuously because of the variety of UAVs’ application areas: reconnaissance, delivery, rescue, monitoring, etc. [1–4]. Among the several types of UAVs, spherical UAVs (SUAVs) do not require using multiple rotors. In this regard, the SUAVs have various advantages in terms of compactness, lightness, inexpensiveness, etc. [1].Malandrakis et al. and Loh and Jacob demonstrated SUAVs consist of a contrarotating (parts of a mechanism rotate in opposite directions about a common axis) motor and several control vanes [5, 6]. The contrarotating motor is used to minimize the reaction torque generated by the motor rotation, and a classical proportional-integral-derivative (PID) control law is applied for attitude control. Position control is not considered for these two SUAVs. In Ref. [6], an ultrasonic sensor is applied for altitude control. Sato Humiyuki and Technical Research and Development Institute of Japan’s Ministry of Defense developed an SUAV called Japan Flying Sphere where the propulsion system consists of a single motor and eight control vanes [7]. Kim et al. developed a rotary-wing micro aerial vehicle that has the same concept of operations as an SUAV using a single motor, four antitorque vanes, and three control vanes [8]. In these aerial vehicles, control vanes play a role as an aileron, an elevator, and a rudder of fixed-wing aerial vehicles to generate torque for attitude control. As one of the techniques for generating the control torque, the moving center-of-gravity (CG) was studied and considered in attitude and position controls. Bouabdallah et al. and Bermes et al. suggested a CG shifting system, which is designed to control horizontal motions, for an indoor coaxial helicopter [9, 10]. This research proposed a conventional independent proportional-derivative (PD) controller for attitude control. In Refs. [9, 10], a compensation method is introduced for altitude control using attitude angles and a distance measured by an infrared sensor. Seisan designed a coaxial helicopter including the moving mass system [11]. In Ref. [11], a nonlinear control law and a PD control law are applied to manipulate position and attitude, respectively. All these SUAVs have vanes to control their attitude. These vanes are usually designed according to how much aerodynamic forces generate torques to control attitudes. Since these forces depend on the size of vanes, the size of the SUAV might be determined by the size of the vane. The main idea of our approach is to develop a smaller platform than a former SUAV by diminishing the size of the vane in the SUAV as possible.The main objective of this paper is to show the feasibility of the proposed conceptual design for the SUAV and its operation. To do so, a control method for the SUAV is suggested by combining a CG variation technique, which generates appropriate torques, and a path-tracking controller. Equations of motion of the SUAV are derived, and the variations of CG are calculated by using control inputs. The applicability of the suggested method is affected by the varying moment-of-inertia (MOI) components, which vary with position and mass of the moving part. To generate more torques for large-angle maneuvers, a mass of the moving part or distance between a pivot point and the moving part must be increased. Varying these two factors, however, results in the variation of MOI components, which is not negligible. The analyses of MOI variations are conducted to see how MOI components are varied with respect to two factors. To avoid the large variation of MOI, two factors are constrained for generating small amount of torque, which leads to small-angle maneuvers. The applicability of the suggested methodology is validated through numerical simulation studies.


## 2. A Spherical UAV Control Technique

### 2.1. A Center-of-Gravity Variation Technique

An SUAV can be controlled by manipulating a thrust force and CG variations [12]. For a hovering purpose, both constant thrust force and torque are applied to maintain the position and attitude. A thrust force is only applied for vertical (altitude) movements. A translational motion along the X (north) and Y (east) axes in the inertial coordinate is related to θ and ϕ, respectively [13, 14]. The movement along the Z (down) axis in the inertial coordinate with respect to the altitude is controlled by the thrust force from a rotor. An attitude controller of the SUAV is able to attain a desired position on the X and Y axes, since the movements along the X and Y axes are accompanied by controlling θ and ϕ. The combination of the frame area of the SUAV and the reaction torque produced by the motor speed is used for ψ control [5]. The key in this work is the control of θ and ϕ using the torque generated by varying the CG. For rotating θ (or ϕ) with respect to a translation along the X (or Y) axis, the CG is moved along the X (or Y) axis, and uθ (or uϕ) can be generated from a thrust force on the Z axis and the weight of a moving part in the SUAV. As the implementation strategy of the proposed method, one considers using a moving part, which is composed of motors and a pendulum, to change the CG of the SUAV. The position of the moving part can be manipulated by two motors: one is equipped on the top of the moving part, and the other is equipped on the frame of the moving part (see Figure 1). The upper and lower motors are in charge of manipulating the elevation angle α and the azimuth angle β, respectively. Since the space on the top of the SUAV is small to put in a sensor for measuring β, a stepping motor with a driver is considered as the upper motor, and this motor has a role in the control of β. The motor for controlling α is fitted on the rod that is connected to the motor for controlling β. There is enough space in order that a motor with a sensor for measuring α is equipped. Therefore, a DC motor with an encoder or a potentiometer could be an option for α control. When operating the moving part, in the case of varying attitude, the sequence of the control for the moving part is to rotate β and then α. On the other hand, attitude control for the SUAV uses the reverse sequence of the moving part control. This is to prevent a reaction torque generated by the moving part in a case where α is not zero. Figure 1 describes an operation concept for the suggested control method.Figure 1
Operation concept using the suggested method.Forψ control, uψ is generated by using the combination of the frame area of the SUAV and the main rotor speed. To rotate ψ, it is necessary to increase or decrease the rotor speed. The frame area creates the drag force, and this force generates the moment to prevent the reaction torque. Basically, the design of the frame area is based on the condition of the hovering status. Therefore, when increasing the rotor speed, the reaction torque is greater than the moment generated by the drag force and vice versa. Furthermore, the frame width generates a drag with respect to an angular velocity of a rotor in order that a moment of Z axis created by this method could curtail a reaction torque. The moment for a reduction of a reaction torque can be expressed as
(1)Mz=12Cdhwl3ωz2,where Mz is the moment generated by the drag, Cd is the drag coefficient of a plate, h is the frame height, w is the frame width, l is the distance between the rotor and the center of the frame, and ωz is the angular velocity of the Z axis in the body-fixed frame, respectively.


### 2.2. Calculation of Moment-of-Inertia Variation

Since the CG variation technique results in varying the mass properties, especially inertia properties, the dynamics of the SUAV has the parametric uncertainty caused by the proposed method. In other words, the angle of the moving part varies the inertia properties of the SUAV so the characteristic of the rotation dynamics is also changed by the varying inertia properties. Thus, the analysis of how much the inertia properties are varied by the movement of a moving part is necessary to determine system parameters which reduce the effect of the parametric uncertainty. Figure2 describes a configuration of the SUAV including a moving part for varying the CG and coordinates to derive equations of motion for the SUAV. The angles α and β for a moving part can be controlled by two motors, and the operational angular ranges are as follows: (i) β operates all angular ranges and (ii) the angular operation range for α is limited to α≤90° because of the rod length L and the pivot position of the moving part zl. The rotations of α and β result in the variations of the CG, which generate torques for attitude control of the SUAV. For simplification in the formulation development, the body-fixed coordinate expression B· and body description ·b for any variable are expressed by omitting the superscript B and the subscript b. Relatively much smaller product-of-inertia elements, which are negligible, are assumed. Then, the MOI elements of the body in the body-fixed coordinate are calculated by
(2)Jk=Js,k+Jp,k+mprk2,where m is the mass, the subscript s denotes the sphere structure except a moving part, the subscript p denotes the moving part, and rk is the perpendicular distance between the moving part and each axis (k=x, y, and z) in the body-fixed coordinate. It means that rx is the distance between the X axis of rotation ϕ and the position of the moving part on the XY surface in the body-fixed coordinate and ry and rz can also be calculated in the same manner.Figure 2
Configuration of the SUAV.The relationship between the body-fixed and the moving part coordinates can be expressed by the direction cosine matrixCPB, which transforms from P (moving part) to B (body-fixed) frame given the β-α transformation, as follows:
(3)CPB=cαcβ−sβsαcβcαsβcβsαsβ−sα0cα,and rp is expressed in the body-fixed coordinate as
(4)rp≡rp,xrp,yrp,z=CPBPrp+op=LsαcβLsαsβLcα−zl,where rpP=0,0,LT, op=0,0,−zlT, and s and c are the sine and cosine functions, respectively. Using equation (4), rk is calculated as
(5)rk=rx=rp,y2+rp,z2=L2s2αs2β+Lcα−zl2,ry=rp,z2+rp,x2=L2s2αc2β+Lcα−zl2,rz=rp,x2+rp,y2=Lsα,and substituting equation (5) into equation (2) yields
(6)Jk=Jx=Js,x+Jp,x+mpL2s2αs2β+Lcα−zl2,Jy=Js,y+Jp,y+mpL2s2αc2β+Lcα−zl2,Jz=Js,z+Jp,z+mpL2s2α.The variation of MOI elements with respect toα and β is defined as
(7)ΔJk=ΔJx=Jx−Jx,idle=mpL1−cα2zl−L1+cαc2β,ΔJy=Jy−Jy,idle=mpL1−cα2zl−L1+cαs2β,ΔJz=Jz−Jz,idle=mpL2s2α=mpL21−c2α,where the MOI elements at the idle position denote the static condition, and they are calculated at α=β=0. From equation (7), ΔJx, ΔJy, and ΔJz are close to zero for small α, but the effect of the variation of MOI elements must be considered when the angular operation range of α is larger.From equation (7), the variation of MOI elements is the function of variables mp, L, zl, α, and β. The variables mp, α, and β can be determined from the structure design of the SUAV while L and zl are determined by considering the variation of MOI elements.In equation (7), ΔJx and ΔJy have the maximum value of 2mpLzl at α≤90° and ∀β. In the case of ΔJz, the maximum value is found as mpL2 at the same condition of both α and β. The difference between the maximum values of ΔJx and ΔJz can be calculated as follows:
(8)ΔJxmax−ΔJzmax=mpL2zl−L.Since2zl−L in equation (8) is greater than zero according to zl≥L from the configuration of the SUAV, the maximum value of ΔJz does not exceed the maximum value of both ΔJx and ΔJy.By defining a variableh=zl/L, ΔJx in equation (7) can be expressed as
(9)ΔJx=mpzl2h21−cα2h−1+cαc2β,and the first and second derivatives of equation (9) with respect to h are found as
(10)∂ΔJx∂h=2mpzl2h21−cαc2βh1+cα−1,∂2ΔJx∂h2=2mpzl2h31−cα2−3c2βh1+cα−1.The maximum value ofΔJx can be found at ∂ΔJx/∂h=0 satisfying ∂2ΔJx/∂h2<0 from equation (10) except the case when cα equals to 1, which results in ΔJx=0 as well. That is, h is found as
(11)h=zlL=1+cαc2β,and the range of L with respect to zl can be obtained as
(12)zl2≤L≤zl≤R,where 0≤c2β≤1 at ∀β and 1≤1+cα≤2 at α≤90°. Equation (12) can be also derived by using ΔJy in the same manner.Substituting equation (11) into equation (9) yields
(13)ΔJx=mpzl2t2α21c2β,where t denotes the tangent function. The range of ΔJx can be expressed as
(14)0≤ΔJx≤mpzl2,where 0≤t2α/2≤1 at ∣α∣≤90°. Therefore, mp and zl can be calculated by using the maximum value of ΔJx=mpzl2 and vice versa. Note that the maximum value of ΔJx, which is negligibly small, is determined by design parameters of the SUAV to avoid the large variation of MOI.


### 2.3. Path-Tracking Controller Design

Mathematical models for a rigid SUAV with respect to the attitude and the position in the body-fixed coordinate are generally expressed as [15]
(15)MV̇+ω×MV=T,
(16)Jω̇+ω×Jω+J̇ω=τ,where M≡diagm,m,m, J≡diagJx,Jy,Jz, V∈ℛ3 is the velocity, ω∈ℛ3 is the angular velocity, T∈ℛ3 is the thrust, and τ∈ℛ3 is the torque. As shown in Figure 2, α and β are required to control the SUAV, and the attitude angles are calculated by using control inputs. Note that the following assumptions are applied to design a path-tracking controller for the SUAV [13]: (i) in equation (15) T≫ω×MV; (ii) in equation (16) τ≫ω×Jω; (iii) in equation (16), J̇ω is relatively smaller than other terms; and (iv) ϕ and θ are relatively small. Because the propulsion system of the SUAV consists of one rotor and one propeller only, the thrust force for the SUAV is much smaller than the one for a UAV (e.g., a quadcopter) with the number of both motors and propellers. The low thrust force limits the total weight for the SUAV, and this leads to the mass budget limitation for a moving part. Thus, the SUAV only can generate a small amount of torque by controlling the moving part, and this limits the agile movement level. Also, the relatively lower speed of the SUAV helps to eliminate the effect of inertia component variation.Applying (i), (ii), and (iii) into equations (15) and (16), leads to
(17)NV̇=CBNM−1T+g,ζ¨=J−1τ,where g=0,0,gT, g is the gravity acceleration, ζ=φ,θ,ψT, T=0,0,uthT, and τ=uϕ,uθ,uψT. The matrix CBN, which transforms from B to N frame given the Euler 3-2-1 (ψ-θ-ϕ) transformation, is introduced as [16]
(18)CBN=cθcψsϕsθcψ−cϕsψcϕsθcψ+sθsψcθsψsϕsθcψ+cϕcψcϕsθsψ−sϕcψ−sθ−sϕcθcϕcθ,where the superscript N denotes the inertial frame. Applying (iv) into equation (17), leads to
(19)Nx¨=θcψ+ϕsψuthm,Ny¨=θsψ−ϕcψuthm,
(20)Nz¨=g+uthm,
(21)ϕ¨=uϕJx,θ¨=uθJy,ψ¨=uψJz,where u is the control command, and the subscript th denotes the thrust.The desired accelerationad=x¨d,y¨d,z¨dT for the path-tracking task is generated from the classical PID control laws given as
(22)Nad=Kae+Daė+Ia∫edt,where the subscript d denotes the desired state, e∈ℛ3 is the position error in the inertial coordinate with respect to the desired path. The gain matrices are defined as mass functions as follows: Ka=kaM, Da=daM, and Ia=iaM, where ka, da, and ia are scalars.From equations (20) and (22), uth is expressed as
(23)uth=m2kaez+daėz+ia∫ezdt−gm.The rapid manipulation of the moving part leads to the drastic attitude change for the SUAV, and this generates the overshoot of the system response. To generate the faster-damped response for the system, PD control law rather than PID control law that shows lesser damping is considered for the SUAV attitude control. Thus, the control input is defined as(24)τ=Kτζd−ζ+Dτζ̇d−ζ̇,and the gain matrices are defined as MOI functions as follows: Kτ=kτJ and Dτ=dτJ, where kτ and dτ are scalars. Substituting ϕd, θd, and ψd into equations (19) and (22) yields [17]
(25)ϕdθd=muthsψd−cψdcψdsψdNx¨dNy¨d,and ϕ̇d and θ̇d are obtained by using equations (22) and (25) as
(26)ϕ̇d=−muthkaėy+dae¨y+iaey,θ̇d=muthkaėx+dae¨x+iaex,where ψd and ψ̇d are the required constant value in accordance with the path-tracking mission, and they are simply assumed as zeros without loss of generality. The control of ψ is distinct from the control of ϕ and θ since ψ is not related to the translational motion. For the translational motion along the X and Y axes, it is requisite for controlling ϕ and θ from equations (25) and (26).


### 2.4. Calculation ofα and β for Generating τ

Figure3 describes a control scheme for the SUAV. The CG variation has no effect on uth with respect to T. The combination of uth and the weight of the moving part, however, generates τ for controlling attitude. In equation (24), τ for the attitude control in the body-fixed coordinate generated from the position of the moving part in equation (4), uth, and the weight of the moving part is expressed as
(27)τ=rp×mpCNBg−T,where r∈ℛ3 is the position and CNB=CBNT. From equation (27), the desired position of the moving part in the body-fixed coordinate can be expressed as
(28)rp,d=T−mpCNBg×−1τ,where η× denotes a skew-symmetric matrix for an arbitrary vector η=η1,η2,η3T as follows:
(29)η×=0−η3η2η30−η1−η2η10.Figure 3
Control structure for the CG variation control scheme.From equations (4) and (28), αd and βd to follow the desired path are obtained as
(30)βd=atan2rp,d,y,rp,d,x,αd=s−1rp,d,xLcβdNote that the double-variable-based arctangent function is used for calculatingβd. Around the hovering or only vertical movement positions, T−mpCNBg× in equation (28) would be the row rank matrix, and αd cannot be determined from rp,d,z. To resolve this problem and obtain the stable solution with respect to αd, αd is calculated from rp,d,x.


## 2.1. A Center-of-Gravity Variation Technique

An SUAV can be controlled by manipulating a thrust force and CG variations [12]. For a hovering purpose, both constant thrust force and torque are applied to maintain the position and attitude. A thrust force is only applied for vertical (altitude) movements. A translational motion along the X (north) and Y (east) axes in the inertial coordinate is related to θ and ϕ, respectively [13, 14]. The movement along the Z (down) axis in the inertial coordinate with respect to the altitude is controlled by the thrust force from a rotor. An attitude controller of the SUAV is able to attain a desired position on the X and Y axes, since the movements along the X and Y axes are accompanied by controlling θ and ϕ. The combination of the frame area of the SUAV and the reaction torque produced by the motor speed is used for ψ control [5]. The key in this work is the control of θ and ϕ using the torque generated by varying the CG. For rotating θ (or ϕ) with respect to a translation along the X (or Y) axis, the CG is moved along the X (or Y) axis, and uθ (or uϕ) can be generated from a thrust force on the Z axis and the weight of a moving part in the SUAV. As the implementation strategy of the proposed method, one considers using a moving part, which is composed of motors and a pendulum, to change the CG of the SUAV. The position of the moving part can be manipulated by two motors: one is equipped on the top of the moving part, and the other is equipped on the frame of the moving part (see Figure 1). The upper and lower motors are in charge of manipulating the elevation angle α and the azimuth angle β, respectively. Since the space on the top of the SUAV is small to put in a sensor for measuring β, a stepping motor with a driver is considered as the upper motor, and this motor has a role in the control of β. The motor for controlling α is fitted on the rod that is connected to the motor for controlling β. There is enough space in order that a motor with a sensor for measuring α is equipped. Therefore, a DC motor with an encoder or a potentiometer could be an option for α control. When operating the moving part, in the case of varying attitude, the sequence of the control for the moving part is to rotate β and then α. On the other hand, attitude control for the SUAV uses the reverse sequence of the moving part control. This is to prevent a reaction torque generated by the moving part in a case where α is not zero. Figure 1 describes an operation concept for the suggested control method.Figure 1
Operation concept using the suggested method.Forψ control, uψ is generated by using the combination of the frame area of the SUAV and the main rotor speed. To rotate ψ, it is necessary to increase or decrease the rotor speed. The frame area creates the drag force, and this force generates the moment to prevent the reaction torque. Basically, the design of the frame area is based on the condition of the hovering status. Therefore, when increasing the rotor speed, the reaction torque is greater than the moment generated by the drag force and vice versa. Furthermore, the frame width generates a drag with respect to an angular velocity of a rotor in order that a moment of Z axis created by this method could curtail a reaction torque. The moment for a reduction of a reaction torque can be expressed as
(1)Mz=12Cdhwl3ωz2,where Mz is the moment generated by the drag, Cd is the drag coefficient of a plate, h is the frame height, w is the frame width, l is the distance between the rotor and the center of the frame, and ωz is the angular velocity of the Z axis in the body-fixed frame, respectively.


## 2.2. Calculation of Moment-of-Inertia Variation

Since the CG variation technique results in varying the mass properties, especially inertia properties, the dynamics of the SUAV has the parametric uncertainty caused by the proposed method. In other words, the angle of the moving part varies the inertia properties of the SUAV so the characteristic of the rotation dynamics is also changed by the varying inertia properties. Thus, the analysis of how much the inertia properties are varied by the movement of a moving part is necessary to determine system parameters which reduce the effect of the parametric uncertainty. Figure2 describes a configuration of the SUAV including a moving part for varying the CG and coordinates to derive equations of motion for the SUAV. The angles α and β for a moving part can be controlled by two motors, and the operational angular ranges are as follows: (i) β operates all angular ranges and (ii) the angular operation range for α is limited to α≤90° because of the rod length L and the pivot position of the moving part zl. The rotations of α and β result in the variations of the CG, which generate torques for attitude control of the SUAV. For simplification in the formulation development, the body-fixed coordinate expression B· and body description ·b for any variable are expressed by omitting the superscript B and the subscript b. Relatively much smaller product-of-inertia elements, which are negligible, are assumed. Then, the MOI elements of the body in the body-fixed coordinate are calculated by
(2)Jk=Js,k+Jp,k+mprk2,where m is the mass, the subscript s denotes the sphere structure except a moving part, the subscript p denotes the moving part, and rk is the perpendicular distance between the moving part and each axis (k=x, y, and z) in the body-fixed coordinate. It means that rx is the distance between the X axis of rotation ϕ and the position of the moving part on the XY surface in the body-fixed coordinate and ry and rz can also be calculated in the same manner.Figure 2
Configuration of the SUAV.The relationship between the body-fixed and the moving part coordinates can be expressed by the direction cosine matrixCPB, which transforms from P (moving part) to B (body-fixed) frame given the β-α transformation, as follows:
(3)CPB=cαcβ−sβsαcβcαsβcβsαsβ−sα0cα,and rp is expressed in the body-fixed coordinate as
(4)rp≡rp,xrp,yrp,z=CPBPrp+op=LsαcβLsαsβLcα−zl,where rpP=0,0,LT, op=0,0,−zlT, and s and c are the sine and cosine functions, respectively. Using equation (4), rk is calculated as
(5)rk=rx=rp,y2+rp,z2=L2s2αs2β+Lcα−zl2,ry=rp,z2+rp,x2=L2s2αc2β+Lcα−zl2,rz=rp,x2+rp,y2=Lsα,and substituting equation (5) into equation (2) yields
(6)Jk=Jx=Js,x+Jp,x+mpL2s2αs2β+Lcα−zl2,Jy=Js,y+Jp,y+mpL2s2αc2β+Lcα−zl2,Jz=Js,z+Jp,z+mpL2s2α.The variation of MOI elements with respect toα and β is defined as
(7)ΔJk=ΔJx=Jx−Jx,idle=mpL1−cα2zl−L1+cαc2β,ΔJy=Jy−Jy,idle=mpL1−cα2zl−L1+cαs2β,ΔJz=Jz−Jz,idle=mpL2s2α=mpL21−c2α,where the MOI elements at the idle position denote the static condition, and they are calculated at α=β=0. From equation (7), ΔJx, ΔJy, and ΔJz are close to zero for small α, but the effect of the variation of MOI elements must be considered when the angular operation range of α is larger.From equation (7), the variation of MOI elements is the function of variables mp, L, zl, α, and β. The variables mp, α, and β can be determined from the structure design of the SUAV while L and zl are determined by considering the variation of MOI elements.In equation (7), ΔJx and ΔJy have the maximum value of 2mpLzl at α≤90° and ∀β. In the case of ΔJz, the maximum value is found as mpL2 at the same condition of both α and β. The difference between the maximum values of ΔJx and ΔJz can be calculated as follows:
(8)ΔJxmax−ΔJzmax=mpL2zl−L.Since2zl−L in equation (8) is greater than zero according to zl≥L from the configuration of the SUAV, the maximum value of ΔJz does not exceed the maximum value of both ΔJx and ΔJy.By defining a variableh=zl/L, ΔJx in equation (7) can be expressed as
(9)ΔJx=mpzl2h21−cα2h−1+cαc2β,and the first and second derivatives of equation (9) with respect to h are found as
(10)∂ΔJx∂h=2mpzl2h21−cαc2βh1+cα−1,∂2ΔJx∂h2=2mpzl2h31−cα2−3c2βh1+cα−1.The maximum value ofΔJx can be found at ∂ΔJx/∂h=0 satisfying ∂2ΔJx/∂h2<0 from equation (10) except the case when cα equals to 1, which results in ΔJx=0 as well. That is, h is found as
(11)h=zlL=1+cαc2β,and the range of L with respect to zl can be obtained as
(12)zl2≤L≤zl≤R,where 0≤c2β≤1 at ∀β and 1≤1+cα≤2 at α≤90°. Equation (12) can be also derived by using ΔJy in the same manner.Substituting equation (11) into equation (9) yields
(13)ΔJx=mpzl2t2α21c2β,where t denotes the tangent function. The range of ΔJx can be expressed as
(14)0≤ΔJx≤mpzl2,where 0≤t2α/2≤1 at ∣α∣≤90°. Therefore, mp and zl can be calculated by using the maximum value of ΔJx=mpzl2 and vice versa. Note that the maximum value of ΔJx, which is negligibly small, is determined by design parameters of the SUAV to avoid the large variation of MOI.


## 2.3. Path-Tracking Controller Design

Mathematical models for a rigid SUAV with respect to the attitude and the position in the body-fixed coordinate are generally expressed as [15]
(15)MV̇+ω×MV=T,
(16)Jω̇+ω×Jω+J̇ω=τ,where M≡diagm,m,m, J≡diagJx,Jy,Jz, V∈ℛ3 is the velocity, ω∈ℛ3 is the angular velocity, T∈ℛ3 is the thrust, and τ∈ℛ3 is the torque. As shown in Figure 2, α and β are required to control the SUAV, and the attitude angles are calculated by using control inputs. Note that the following assumptions are applied to design a path-tracking controller for the SUAV [13]: (i) in equation (15) T≫ω×MV; (ii) in equation (16) τ≫ω×Jω; (iii) in equation (16), J̇ω is relatively smaller than other terms; and (iv) ϕ and θ are relatively small. Because the propulsion system of the SUAV consists of one rotor and one propeller only, the thrust force for the SUAV is much smaller than the one for a UAV (e.g., a quadcopter) with the number of both motors and propellers. The low thrust force limits the total weight for the SUAV, and this leads to the mass budget limitation for a moving part. Thus, the SUAV only can generate a small amount of torque by controlling the moving part, and this limits the agile movement level. Also, the relatively lower speed of the SUAV helps to eliminate the effect of inertia component variation.Applying (i), (ii), and (iii) into equations (15) and (16), leads to
(17)NV̇=CBNM−1T+g,ζ¨=J−1τ,where g=0,0,gT, g is the gravity acceleration, ζ=φ,θ,ψT, T=0,0,uthT, and τ=uϕ,uθ,uψT. The matrix CBN, which transforms from B to N frame given the Euler 3-2-1 (ψ-θ-ϕ) transformation, is introduced as [16]
(18)CBN=cθcψsϕsθcψ−cϕsψcϕsθcψ+sθsψcθsψsϕsθcψ+cϕcψcϕsθsψ−sϕcψ−sθ−sϕcθcϕcθ,where the superscript N denotes the inertial frame. Applying (iv) into equation (17), leads to
(19)Nx¨=θcψ+ϕsψuthm,Ny¨=θsψ−ϕcψuthm,
(20)Nz¨=g+uthm,
(21)ϕ¨=uϕJx,θ¨=uθJy,ψ¨=uψJz,where u is the control command, and the subscript th denotes the thrust.The desired accelerationad=x¨d,y¨d,z¨dT for the path-tracking task is generated from the classical PID control laws given as
(22)Nad=Kae+Daė+Ia∫edt,where the subscript d denotes the desired state, e∈ℛ3 is the position error in the inertial coordinate with respect to the desired path. The gain matrices are defined as mass functions as follows: Ka=kaM, Da=daM, and Ia=iaM, where ka, da, and ia are scalars.From equations (20) and (22), uth is expressed as
(23)uth=m2kaez+daėz+ia∫ezdt−gm.The rapid manipulation of the moving part leads to the drastic attitude change for the SUAV, and this generates the overshoot of the system response. To generate the faster-damped response for the system, PD control law rather than PID control law that shows lesser damping is considered for the SUAV attitude control. Thus, the control input is defined as(24)τ=Kτζd−ζ+Dτζ̇d−ζ̇,and the gain matrices are defined as MOI functions as follows: Kτ=kτJ and Dτ=dτJ, where kτ and dτ are scalars. Substituting ϕd, θd, and ψd into equations (19) and (22) yields [17]
(25)ϕdθd=muthsψd−cψdcψdsψdNx¨dNy¨d,and ϕ̇d and θ̇d are obtained by using equations (22) and (25) as
(26)ϕ̇d=−muthkaėy+dae¨y+iaey,θ̇d=muthkaėx+dae¨x+iaex,where ψd and ψ̇d are the required constant value in accordance with the path-tracking mission, and they are simply assumed as zeros without loss of generality. The control of ψ is distinct from the control of ϕ and θ since ψ is not related to the translational motion. For the translational motion along the X and Y axes, it is requisite for controlling ϕ and θ from equations (25) and (26).


## 2.4. Calculation ofα and β for Generating τ

Figure3 describes a control scheme for the SUAV. The CG variation has no effect on uth with respect to T. The combination of uth and the weight of the moving part, however, generates τ for controlling attitude. In equation (24), τ for the attitude control in the body-fixed coordinate generated from the position of the moving part in equation (4), uth, and the weight of the moving part is expressed as
(27)τ=rp×mpCNBg−T,where r∈ℛ3 is the position and CNB=CBNT. From equation (27), the desired position of the moving part in the body-fixed coordinate can be expressed as
(28)rp,d=T−mpCNBg×−1τ,where η× denotes a skew-symmetric matrix for an arbitrary vector η=η1,η2,η3T as follows:
(29)η×=0−η3η2η30−η1−η2η10.Figure 3
Control structure for the CG variation control scheme.From equations (4) and (28), αd and βd to follow the desired path are obtained as
(30)βd=atan2rp,d,y,rp,d,x,αd=s−1rp,d,xLcβdNote that the double-variable-based arctangent function is used for calculatingβd. Around the hovering or only vertical movement positions, T−mpCNBg× in equation (28) would be the row rank matrix, and αd cannot be determined from rp,d,z. To resolve this problem and obtain the stable solution with respect to αd, αd is calculated from rp,d,x.


## 3. Numerical Simulation

The established path is composed of the following four flight conditions: ascent, horizontal movement, descent, and hovering as listed in Table1. The trajectory is mainly designed to verify whether the proposed method works properly under the two following conditions. First, the moving part operates properly in the lateral motion to maneuver the SUAV in the combination of four cardinal directions. Second, the yaw angle does not change largely when increasing the altitude at the same time as the operation of the moving parts. If the SUAV proposed met both conditions, the feasibility of the CG-driven UAV system could be confirmed.Table 1
Desired trajectory.
Index
Time (sec)
Target position (m)
Phase
—
0
(0, 0, 0)
Initial position
A
0<t≤10
(0, 0, -10)
Ascent
B
10<t≤15
(0, 0, -10)
Hovering
C
15<t≤25
(10, 10, -10)
Horizontal movement
D
25<t≤30
(10, 10, -10)
Hovering
E
30<t≤40
(20, 0, -20)
Ascent and horizontal movement
F
40<t≤45
(20, 0, -20)
Hovering
G
45<t≤55
(10, -10, -10)
Descent and horizontal movement
H
55<t≤60
(10, -10, -10)
Hovering
I
60<t≤70
(0, 0, -10)
Horizontal movement
J
70<t≤80
(0, 0, 0)
DescentThe simulation conditions are shown in Table2. The stability of the closed-loop system suggested is stable because all poles are located on the left hand plane as follows: −0.0521 and −1.5118±1.1839i in altitude, −0.0028,−1.7353×103 in roll (ϕ) and pitch (θ), and −0.0028,−1.7569×103 in yaw (ψ), respectively. Figures 4–6 represent the maximum and the minimum changes of ΔJx, ΔJy, and ΔJz with respect to mp and L variations, respectively. Note that the logarithmic scale is used to facilitate classification between the maximum and the minimum changes of MOI elements. As shown, ΔJx, ΔJy, and ΔJz vary up to about 153.6%, 153.6%, and 63.8% at most and 0.16%, 0.16%, and 0.48% at least, respectively. Figure 7 represents the variations of the MOI elements with respect to mp and L variations as an example. Figures 4–7 are obtained by using equation (7) at α≤90° and ∀β. As shown in Figure 7, the mass change of the moving part affects the variation of Jx and Jy mainly, and all inertia properties are affected by the variation of the rod length of the moving part. Since both the mass and the rod length of the moving part are related to the amount of the generating torque for attitude control, it is necessary to increase the mass of the moving part for a large amount of generating torque and a low variation of the inertia properties. In other words, the MOI elements are affected by both mp and L, and L is the more dominant factor to influence the change of the MOI elements.Table 2
Simulation parameters.
Parameter
Symbol
Value
Simulation time (sec)
t
80
Time interval (sec)
Δt
0.10
Total mass (kg)
m
0.51
Moving part mass (kg)
mp
0.01
Radius of the SUAV (m)
R
0.25
Pivot position of the moving part (m)
zl
0.25
Rod length (m)
L
0.125
Inertia components (kg·m2)
Jx
126.57×10−5
Jy
126.57×10−5
Jz
125.02×10−5
Scalar gains for control
ka
1.9608
da
1.5686
ia
0.0980
kτ
6.2000
dτ
2.2000Figure 4
ΔJxw.r.t.mp and L variation.Figure 5
ΔJyw.r.t.mp and L variation.Figure 6
ΔJzw.r.t.mp and L variation.Figure 7
ΔJkw.r.t.mp and L variation.Figures8 and 9 represent the three-dimensional (3D) trajectory of the SUAV captured with the desired path and the velocities of the SUAV, respectively. It is shown that the SUAV with the designed control law follows the designated path listed in Table 1. Figure 10 and Table 3 show that the trajectory errors in all directions are less than 0.84 m. One reason for such a result is that the SUAV system proposed is the underactuated system. The SUAV only generates a thrust force that has an upward direction with respect to the body-fixed frame. In the case of lateral motion, it is necessary to combine the attitude and the thrust. Although the control law is derived by using this relationship between the attitude and the thrust, the decelerating force is not enough to come to a halt at each waypoint in time. This path-tracking error could be improved by applying a type of nonlinear adaptive controller, but this is not the scope of this work at this time. The maximum absolute error (MAE) and the root mean square error (RMSE) in Table 3 are calculated as
(31)MAE=maxek,RMSE=ekTekn,where k=x,y, and z, and n is the total number of the data.Figure 8
3D trajectory of the SUAV.Figure 9
Time history of velocities.Figure 10
Time history of position error.Table 3
Error between desired path and trajectory.
Parameter
North
East
Altitude
MAE (m)
0.8353
0.8292
0.7170
RMSE (m)
0.2926
0.2850
0.2517The time histories of Euler angles, which are the attitude of the SUAV, and the corresponding control inputs to achieve both the desired position and attitude of the SUAV are described in Figures11 and 12. Note that ψ is not equal to zero in spite of uψ being equal to zero because the torque of BZ axis for rotating ψ is generated from the weight of the moving part according to the attitude of the SUAV. Figure 13 represents the time history of α and β of the moving part for the SUAV control, and the corresponding CG variation is shown in Figure 14. The CG variations do not exceed 1 centimeter over time, and this surely indicates that inertia variation is small enough to ignore the effect of parametric uncertainty. Figure 15 shows the position of the moving part for the path-tracking mission according to the phases listed in Table 1. During phases A and B, which are the ascent and hovering modes, the moving part does not need to behave since no attitude changes are required. In the case of phases C, E, G, and I, the moving part is operated to follow the desired path. In the case of phases D, F, H, and J, the moving part is operated for braking purposes for about 0.1 sec and then for keeping all attitude equals to zero.Figure 11
Time history of Euler angles.Figure 12
Time history of control inputs.Figure 13
Time history ofα and β.Figure 14
3D position of the CG for the SUAV.Figure 15
Position of the moving part w.r.t. phases.


## 4. Conclusion

This paper presents a conceptual design of a spherical unmanned aerial vehicle (SUAV) with a center-of-gravity (CG) variation-based path-tracking control law, and it works with the following three steps: (1) the control inputs are calculated from the path-tracking controller; (2) the required position of the moving part is calculated from the control inputs; and (3) the rotating angles of the moving part are calculated from the required position of the moving part. The position of the moving part, which varies the CG, is controlled by a stepping motor and a DC motor providing two degrees-of-freedom such as elevation and azimuth angles. The control torques for attitude control with respect to roll and pitch motions can be generated from the combination of the position, the mass of the moving part, and the thrust force from a rotor. The changed position of the moving part results in the variation of MOI, and this variation has an effect on the performance of the suggested method. The variation of MOI analysis is performed under angular ranges of elevation and azimuth. Based on the analysis, the rod length and the moving part mass are determined to avoid the effect of the MOI variation on the control performance. Numerical simulation results demonstrate the feasibility of operations of the SUAV proposed with the path-tracking control law.



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*Source: 5754205-2020-04-07.xml*