Dynamics and Control of Spacecraft with a Generalized Model of ...

Dynamics and Control of Spacecraft with a Generalized Model of Variable Speed Control Moment Gyroscopes Sasi Prabhakaran Viswanathan∗ Frederick Leve Amit K. Sanyal† Research Aerospace Engineer Mechanical and Aerospace Engineering Air Force Research Laboratory New Mexico State University Space Vehicles Directorate Las Cruces, NM 88003 Kirtland Air Force Base, NM 87117 Email:[email protected]∗ , Email:[email protected] Email:[email protected]† N. Harris McClamroch Professor Emeritus Aerospace Engineering University of Michigan Ann Arbor, MI 48109-2140 Email:[email protected]

ABSTRACT The attitude dynamics model for a spacecraft with a variable speed control moment gyroscope (VSCMG), is derived using the principles of variational mechanics. The resulting dynamics model is obtained in the framework of geometric mechanics, relaxing some of the assumptions made in prior literature on control moment gyroscopes. These assumptions include symmetry of the rotor and gimbal structure, and no offset between the centers of mass of the gimbal and the rotor. The dynamics equations show the complex nonlinear coupling between the internal degrees of freedom associated with the VSCMG and the spacecraft base body’s rotational degrees of freedom. This dynamics model is then further generalized to include the effects of multiple VSCMGs placed in the spacecraft base body, and sufficient conditions for non-singular VSCMG configurations are obtained. General ideas on control of the angular momentum of the spacecraft using changes in the momentum variables of a finite number of VSCMGs, are provided. A control scheme using a finite number of VSCMGs for attitude stabilization maneuvers in the absence of external torques and when the total angular momentum of the spacecraft is zero, is presented. The dynamics model of the spacecraft with a finite number of VSCMGs is then simplified under the assumptions that there is no offset between the centers of mass of the rotor and gimbal, and the rotor is axisymmetric. As an example, the case of three VSCMGs with axisymmetric rotors, placed in a tetrahedron configuration inside the spacecraft, is considered. The control scheme is then numerically implemented using a geometric variational integrator. Numerical simulation results with zero and non-zero rotor offset between centers of mass of gimbal and rotor are presented.

1

Introduction Internal momentum exchange devices can be used as spacecraft attitude actuators to stabilize a desired attitude or track a desired attitude profile. A Control Moment Gyroscope (CMG) is a momentum exchange device consisting of a rotor running at a constant angular speed, mounted on a gimbal frame structure that can be rotated at a controlled rate. CMGs are used for attitude control of spacecraft [1], including attitude stabilization of agile spacecraft [2]. They can be categorized as Single Gimbal Control Moment Gyroscopes (SGCMGs) or Double Gimbal Control Moment Gyroscopes (DGCMGs). Further, a

∗ PhD

Student Professor, Address all correspondence to this author

† Asst.

Variable Speed Control Moment Gyroscope, abbreviated as VSCMG, combines the features of a constant speed SGCMG with a rotor whose angular speed can be varied. Defining features and comparisons between SGCMGs and DGCMGs are given in the book by Wie [3]. Theoretical aspects of VCSMGs for attitude control are given in [4, 5], while experiments using VSCMGs for this purpose are reported in [6]. Single Gimbal CMGs are known to produce larger torques for the same actuator mass compared to reaction wheels [4, 7]. However, the dynamics of CMGs is more complex than that of reaction wheels, with inherent singularities in the map between the inputs (gimbal rates) and outputs (angular momentum or torque acting on spacecraft bus). A detailed analysis of singularities and steering laws for SGCMGs, obtained for different geometrical configurations, is found in [8]. A coordinate-free analysis of singular states of SGCMGs, with analysis of the geometry of singular momentum surfaces and angular momentum envelopes of SGCMG clusters, is given in [7]. Most of the available steering laws that locally produce instantaneous control torques, have difficulties avoiding singularities in minimally redundant SGCMG systems [9]. An early treatment of singular configurations of SGCMG clusters and singularity avoidance is provided in [10]. A hybrid approach that takes into account the form of singularities by avoiding hyperbolic singularities and escaping elliptic singularities for a four-SGCMG rooftop arrangement, is provided in [11]. The formulation of the equations of motion of spacecraft with SGCMGs, which have also been referred to as gyrodynes in prior literature, was presented in Newton-Euler form in [12]. Most of the existing (VS)CMG dynamics models use some simplifying assumptions [3,5,13,14] in their formulation. These assumptions (e.g., axisymmteric rotor rotating about its axis of symmetry) are explicitly stated in section 5 of this paper. Variations in the inertia matrix of a spacecraft with CMGs under these assumptions, were considered in [15]. In this paper, a more general dynamics model of a spacecraft with VSCMG is obtained without using most of these assumptions. Since the configuration space of attitude motion of a spacecraft with internal actuators is a nonlinear manifold, the global dynamics of this system is treated using the formulation of geometric mechanics [16, 17]. This model is obtained using variational mechanics, and in the framework of geometric mechanics on the nonlinear state space of this system. The dynamics model is thereafter generalized to a spacecraft with a finite number of VSCMGs. Attitude control of a spacecraft with these generalized VSCMG actuators, under the assumption that there are no external moments acting on the spacecraft, is treated. Prior research on dynamics and control of spacecraft with internal momentum exchange devices includes the treatment of the dynamics and control of a rigid spacecraft with axisymmetric internal rotors (reaction wheels) in [17]. Bloch et al. treated the geometry and control of gyroscopic systems for rigid body attitude tracking and stabilization [18–20]. The dynamics model obtained here is applicable to VSCMGs with nonaxisymmetric rotors and gimbals, where the rotation axis of the rotor may be offset from the center of mass of the gimbal structure. While it is true that existing VSCMG designs may not need these generalizations when manufactured, it is possible that during installation within a spacecraft bus or during the course of operations they become misaligned, in which case this generalized model and the control schemes based on it would still apply. Sufficient conditions are provided under which a cluster of VSCMGs in a spacecraft will not have any singularities in producing a three-component angular momentum vector and a three-component torque vector acting on the spacecraft bus. To the best of the authors’ knowledge, there has been no complete formulation of these equations of motion in all this generality and a study of the attitude control thereof, in the published literature on spacecraft attitude dynamics and control with control moment gyroscopes. The structure of this paper is as follows: the kinematics of the rotational motion of a spacecraft with an internal VSCMG actuator is given in section 2. Following that, the dynamics of this system in the absence of external moments acting on the spacecraft, using the framework of geometric mechanics is derived in section 3. The relation between the total angular momentum of the spacecraft system and the momentum quantities associated with the internal VSCMG states are also obtained in this section. Generalization of this attitude dynamics model to the case of multiple (finite) VSCMGs inside the spacecraft, is provided in section 4. Section 5 gives an attitude control scheme based on this dynamics model, for a pointing maneuver for the spacecraft base body actuated by multiple VSCMGs. Section 6 provides details on the application of some commonly used assumptions for VSCMGs to the generalized VSCMG model, which results in the more familiar model for (VS)CMGs seen in the literature. Sufficient conditions for placing such VSCMGs to avoid singularities in the transformations between VSCMG angular rates to the momentum and torque acting on the base spacecraft control are given in section 7. Section 8 discretizes the dynamics model obtained in section 3 using discrete geometric mechanics, which results in a geometric variational integrator that preserves the total angular momentum of the system as well as the geometry of the state space. Numerical simulations are carried out using this variational integrator in Section 9, to verify the performance of the control law. Observations on the performance of this control scheme for a spacecraft’s attitude dynamics actuated by VSCMGs with zero and non-zero rotor offset are drawn. Finally, section 10 provides concluding remarks and possible future research directions.

2

Attitude Kinematics of a Spacecraft with one VSCMG The dynamics model of a spacecraft with a single VSCMG is formulated. The gimbal axis of the VSCMG is orthogonal to the rotor’s axis of rotation, which passes through its center of mass. There is a scalar offset along the rotor axis between the centers of mass of the gimbal and the rotor; this offset is denoted as σ. A schematic CAD rendering of a VSCMG assembly,

Z Spacecraft CoM Y

g × η(t)

X ρg

ρr Gimbal CoM

Rotor CoM

η(t)

σ g

(a)

(b)

Fig. 1: (a) CAD Rendering of a VSCMG design with an axisymmetric rotor; (b) Vector representation of VSCMG axes, with rotor offset in the axial direction η(t) from the center of mass of the gimbal about its axis g.

is shown in fig. 1a. 2.1

Coordinate Frame Definitions Consider a right-handed coordinate frame fixed to the center of mass of the spacecraft base body (spacecraft bus), which has a VSCMG as an internal attitude actuator. The attitude of the spacecraft is given by the rotation matrix from this base body-fixed coordinate frame to an inertially fixed coordinate frame. Consider the gimbal of the VSCMG to rotate about an axis fixed with respect to this spacecraft base body coordinate frame. Let g ∈ S2 denote the fixed axis of rotation of the gimbal, and let α(t) represent the gimbal angle, which is the rotation angle of rotor axis η about the gimbal axis g. Let η(α(t)) ∈ S2 denote the instantaneous direction of the axis of rotation of the rotor, which depends on the gimbal angle α at time instant t. Both g and η(α(t)) are expressed as unit vectors in the base body coordinate frame, and η(α(t)) is normal to the fixed gimbal axis g. Let σ be the distance of the center of rotation of the rotor from the center of mass of the gimbal; this distance is assumed to be along the vector η(α(t)). Define a VSCMG gimbal-fixed coordinate frame with its first coordinate axis along g, its second axis along η(α(t)) and its third axis along g × η(α(t)), to form a right-handed orthonormal triad of basis vectors as depicted in fig. 1b. 2.2

Rotational Kinematics If η0 denotes the initial rotor axis direction vector with respect to the base body frame, the rotor axis at time t > 0 is given by η(α(t)) = exp(α(t)g× )η(0) = cos α(t)η0 + sin α(t)(g × η0 ),

(1)

using Rodrigues’ rotation formula. Let Rg denote the rotation matrix from the VSCMG gimbal-fixed coordinate frame to the spacecraft base body coordinate frame, which can be expressed as   Rg (α(t)) = g η(α(t)) g × η(α(t)) .

(2)

The time rate of change of the rotor axis η(α(t)) is given by   × ˙ ˙ ˙ ˙ η(α(t), α(t)) = α(t) cos α(t)(g × η0 ) − sin α(t)η0 = α(t)g η(α(t)).

(3)

where the map (·)× : R3 → so(3) is the vector space isomorphism ∈ R3 and ∈ so(3), given by:  ×   gx 0 −gz gy g× = gy  =  gz 0 −gx  . gz −gy gx 0 The time derivative of Rg is then given by
    ˙ ˙ ˙ ˙ ˙ × η(α(t)) | g × η(α(t)) × η(α(t)) | g × α(t)g × η(α(t)) . R˙ g (α(t), α(t)) = 0 | α(t)g = 0 | α(t)g

(4)

  Using the vector triple product identity a× b× c = (aT c)b − (aT b)c in the last step above, the above can also be expressed as   × ˙ ˙ ˙ ˙ R˙ g (α(t), α(t)) = Rg (α(t)) 0 | α(t)e 3 | −α(t)e 2 = α(t)R g (α(t))e1 ,

(5)

where e1 = [1 0 0]T , e2 = [0 1 0]T and e3 = [0 0 1]T are the standard basis vectors (as column vectors) of R3 . Since e1 denotes the gimbal axis in the gimbal-fixed frame, the angular velocity of the gimbal-fixed frame with respect to the base body, expressed in the base body frame, is ˙ = α(t)g. ˙ ωg (t, α)

(6)

Let θ(t) denote the instantaneous angle of rotation of the rotor about its symmetry axis η(α(t)); this is the angle between the first axis of a VSCMG rotor-fixed coordinate frame and the gimbal axis, where the rotor axis forms the second coordinate ˙ is time-varying and a constant rotor rate axis of this coordinate frame. For a VSCMG, the angular speed of the rotor θ(t) corresponds to a CMG. The rotation matrix from this VSCMG rotor-fixed coordinate frame to the base body coordinate frame is
Rr (α(t), θ(t)) = Rg (α(t)) exp θ(t)e× 2 .

(7)

The time rate of change of the rotation matrix Rr is obtained as follows:

× × ˙ ˙ ˙ ˙ R˙ r (α(t), θ(t), α(t), θ(t)) =R˙ g (α(t), α(t)) exp θ(t)e× 2 + Rg (α(t)) exp θ(t)e2 θ(t)e2

× × × ˙ ˙ =α(t)R + Rr (α(t), θ(t))θ(t)e r (α(t), θ(t)) exp −θ(t)e2 e1 2, using the fact that RT a× R = (RT a)× for R ∈ SO(3) and a ∈ R3 [21]. It can be verified that the following holds for φ ∈ S1 :
exp φe× 2 e1 = (cos φ)e1 − (sin φ)e3 . Making use of this identity, the time derivative of Rr can be expressed as  ×
˙ ˙ ˙ ˙ R˙ r (α(t), θ(t), α(t), θ(t)) = Rr (α(t), θ(t)) α(t) (cos θ(t))e1 + (sin θ(t))e3 + θ(t)e . 2 Therefore, the angular velocity of the VSCMG rotor with respect to the base body, expressed in the base body frame, is  
˙ ˙ ˙ ˙ ωr (α(t), α(t), θ(t)) = Rr (α(t), θ) α(t) (cos θ(t))e1 + (sin θ(t))e3 + θ(t)e 2 ˙ ˙ ˙ ˙ = α(t)R + θ(t)η(α(t)). g (α(t))e1 + θ(t)Rg (α(t))e2 = α(t)g

(8)

Let R(t) denote the rotation matrix from the base body-fixed coordinate frame to an inertial coordinate frame. If Ω(t) is the total angular velocity of the base body expressed in the base body frame, then the attitude kinematics of the base body is given by ˙ = R(t)Ω(t)× . R(t)

(9)

Let ρg denote the position vector from the center of mass of the base-body to the center of mass of the gimbal, expressed in the base body frame. The position vector from the center of mass of the base-body to the center of mass of the rotor is therefore given by ρr (α(t)) = ρg + ση(α(t)), in the base body frame. These vectors expressed in the inertial frame are rg (R(t)) = R(t)ρg and rr (R(t), α(t)) = R(t)ρr (α(t)). Thus, the inertial velocities of these centers of masses with respect to the base-body center of mass are given by r˙g (R(t), Ω(t)) = R(t)Ω(t)× ρg and
× ˙ ˙ r˙r (R(t), Ω(t), α(t), α(t)) = R(t) Ω(t)× ρr (α(t)) + σα(t)g η(α(t)) .

(10)

The spacecraft with a VSCMG has five rotational degrees of freedom, which are described by the variables α, θ and R. The configuration space Q of this system has the structure of a principal (fiber) bundle with base space B = T2 = S1 × S1 and fiber G = SO(3) [22].

3 Lagrangian Dynamics 3.1 The Lagrangian of a Spacecraft with one VSCMG The dynamics model of a spacecraft with a VSCMG is developed in this section. The generalized model of a VSCMG consists of a rotor of mass mr and inertia matrix Jr about its center of mass; this rotor is mounted on a gimbal structural frame of mass mg and inertia matrix Jg about its center of mass. For notational convenience, the time-dependence of variables are ˙ = r˙r (α(t), α), ˙ etc. not explicitly denoted in the rest of this paper, i.e., Ω = Ω(t), Rg (α) = Rg (α(t)), r˙r (α, α) Therefore, the total rotational kinetic energy of the gimbal (alone) is ˙ = Tg (Ω, α, α)


T
1 1 ˙ ˙ + mg r˙gT r˙g , Ω + ωg (α) Rg (α)Jg Rg (α)T Ω + ωg (α) 2 2

(11)

and the total rotational kinetic energy of the rotor (alone) is ˙ = ˙ θ, θ) Tr (Ω, α, α,



1 ˙ T Rr (α, θ)Jr Rr (α, θ)T Ω + ωr (α, α, ˙ + 1 mr r˙r (α, α) ˙ θ) ˙ θ) ˙ T r˙r (α, α). ˙ Ω + ωr (α, α, 2 2

(12)

Let Jb denote the inertia matrix of the base body expressed in the base body frame. The rotational kinetic energy of the base body of the spacecraft is 1 Tb (Ω) = ΩT Jb Ω. 2

(13)

Therefore, the total rotational kinetic energy of the system is ˙ = Tg (Ω, α, α) ˙ + Tb (Ω). ˙ θ, θ) ˙ + Tr (Ω, α, α, ˙ θ, θ) T (Ω, α, α,

(14)

In a compact notation, the total kinetic energy (14) of a spacecraft with a VSCMG is expressed as 1 T (γ, χ) = χT J (γ)χ, 2

(15)

where,       Λ(γ) B(γ) α(t) Ω(t) γ= , χ= and J (γ) = . θ(t) γ˙ (t) B(γ)T Jgr (γ)

(16)

Associated quantities of equation (16) are: 2 Λ(γ) = Λ0 + IT (γ); Λ0 = Jb − (mg + mr )(ρ× g) ,

IT (γ) = Jc (γ) + Ic (α) − mr ση(α)× ρ× g,

(17)

Jc (γ) = Rg (α)Jg Rg (α)T + Rr (γ)Jr Rr (γ)T , 
 × × 2 , Ic (α) = −mr σ ρ× g η(α) + σ η(α)   3×2 B(γ) = (Jc (γ) + Ic (α))g Rr Jr RT , r η(α) ∈ R
2 Ir (α) = −mr ση(α)× ,
 T  g Jc (γ) + Ir (α) g gT Rr Jr RT η(α) r Jgr (γ) = ∈ R2×2 . T T η(α)T Rr Jr RT r g η(α) Rr Jr Rr η(α)

(18) (19) (20)

(21)

Consider the spacecraft system to be in the absence of any gravitational potential. In this case, the Lagrangian for the spacecraft with a VSCMG is given by

L (γ, χ) = T (γ, χ).

(22)

Note that the Lagrangian is independent of the attitude R of the base body in this case. 3.2

Formulation of the Equations of Motion The equations of motion for a spacecraft with a VSCMG are obtained using variational mechanics, taking into account the geometry of the configuration space. The action functional over a time interval [0, T ] is defined as S(L (γ, χ)) =

Z T 0

L (γ, χ)dt,

(23)

˙ θ and θ˙ in this interval. Note that η, η˙ are uniquely specified by which depends on the time evolution of the states R, Ω, α, α, α, α˙ and the constant vector g. In the presence of non-conservative external and internal torques to the spacecraft base body and the VSCMG, the Lagrange-d’Alembert principle is used to obtain the dynamics equations of motion for the system. Considering only non-conservative internal torques acting on the gimbal τg and rotor τr of the VSCMG, then Lagranged’Alembert principle for the system can be written as Z T

δS +





τ (t) τT δγdt = 0, where τ = g . τr (t)

0

(24)

Before obtaining the first variation of the action functional, the first variations of the variables R ∈ SO(3), Ω ∈ R3 , η ∈ S2 , and η˙ ∈ Tη S2 are expressed as follows: δR = RΣ× , δΩ = Σ˙ + Ω × Σ, ×

×

(25) ×˙

˙ η + δαg η, δη = δαg η, δη˙ = δαg

(26)

where Σ× ∈ so(3) gives a variation vector field on SO(3); thus, δR is tangent to SO(3) at the point R [16,17]. Using equation (3), the vector triple product identity, and the fact that g and η are orthogonal vectors, the second of equations (26) can be expressed as ˙ × η − αδαη. ˙ δη˙ = δαg

(27)

These variations must be fixed endpoint variations, such that: Σ(0) = Σ(T ) = 0, δα(0) = δα(T ) = 0, and δθ(0) = δθ(T ) = 0. Define the following angular momentum quantities, which depend on the kinetic energy (15): ∂L ∂T = = Λ(γ)Ω + B(γ)˙γ, ∂Ω ∂Ω ∂T ∂L = = B(γ)T Ω + Jgr (γ)˙γ. p= ∂˙γ ∂˙γ

Π=

(28) (29)

The equations of motion obtained from applying the Lagrange-d’Alembert principle (24) are as follows: dΠ =Π × Ω, dt dp ∂T = + τ. dt ∂γ

(30) (31)

For an axis-symmetric rotor rotating about its axis of symmetry, the Lagrangian is independent of the angle θ and the scalar angular momentum pθ would be conserved in the absence of any friction or torques acting on the rotor. 3.3

Relations Between Total Angular Momentum and VSCMG Momenta The angular momenta (pα and pθ ) corresponding to the two degrees of freedom of the VSCMG, are related to the ˙ θ˙ and spacecraft base body’s angular velocity vector Ω as follows: angular rates α,
 T     α˙ pα − gT (Jc (γ) + Ic (α))Ω g Jc (γ) + Ir (α) g gT Rr Jr RT r η(α) = . T T θ˙ pθ − η(α)T Rr Jr RT η(α)T Rr Jr RT r Ω r g η(α) Rr Jr Rr η(α)

(32)

Equation (32) can be expressed more compactly as Jgr (γ)˙γ = p − B(γ)T Ω.

(33)

Using the notation defined in (21)-(20), the total angular momentum of the spacecraft is expressed as: Spacecraft basebody momenta

Π=

z

}|
{ Λ0 + IT (γ) Ω

+

B(γ)˙γ. | {z }

(34)

VSCMG momenta

As described earlier, this configuration space has the structure of a fiber bundle with its base space coordinates given by the VSCMG coordinates α and θ and its fiber space described by the attitude R of the base body. The total angular momentum of a spacecraft with a VSCMG can be expressed in a compact way as,     Π Ω = J (γ) . p γ˙

4

Spacecraft with multiple VSCMGs The spacecraft equations of motion with one VSCMG developed in section 3 are extended to the case of a spacecraft with multiple (n > 1) VSCMGs in this section. Generalization of the relationship between internal (VSCMG) momenta and the base body’s overall angular momentum for a spacecraft with n VSCMGs of the type depicted in fig. 1b, where n > 1 is looked upon. For such a spacecraft, the configuration space Q of the dynamics is a principal bundle with base space B = T2 × T2 × · · · T2 (n-fold product) or B = (T2 )n , and fiber space F = SO(3). The quantities corresponding to the ith VSCMG in the set of n VSCMGs are denoted by using the superscript (·)i . For this system, the mass matrix is of the form  Λ(Γ) B1 (γ1 ) . . . Bn (γn ) 1 (γ1 ) . . . B1 (γ1 )T Jgr 0    J(Γ) =  . , . ..  .. ... 0  n (γn ) Bn (γn )T 0 . . . Jgr 

(35)

where, n

Λ(Γ) = Λ0 + ∑ ITi (γi ),

(36)

i=1 n 

 × Λ0 = Jb − ∑ (mig + mir )([ρig ] )2 ,

(37)

i=1

n 
 ITi (γi ) = ∑ Rig Jg (Rig )T + Rir Jr (Rir )T − mir σi σi (ηi (αi )× )2 − ηi ((αi )× (ρig )× + (ρig )× ηi (αi )× ,

(38)

i=1

and γi = [αi θi ]T ∈ T2 denotes the vector of angles of the ith VSCMG. The structure of the mass matrix J(Γ) ∈ R(3+2n)×(3+2n) shows that the angular momentum contributions from each individual VSCMG to the base body’s total angular momentum, are given by the Bi (γi ). The total angular momentum of the spacecraft with multiple VSCMGs is given by   n n ˙ Π = Λ0 + ∑ ITi (γi ) Ω + ∑ Bi (γi )˙γi = Λ(Γ)Ω + B (Γ)Γ, i=1

 T where Γ = γ1 k γ2 k · · · k γn =

i=1 nn

i=1

 i T γ ∈ R2n ,

  and B = B1 (γ1 ) k B2 (γ2 ) k · · · k Bn (γn ) ∈ R3×2n .

(39) (40) (41)

The angular momentum of the ith VSCMG, denoted pi , is given by i (γi )˙γi . pi = Bi (γi )T Ω + Jgr

(42)

The vectors pi can be concatenated to form a vector P ∈ R2n

P=

nn  T pi .

(43)

i=1

The relation between all angular momentum and angular velocity quantities for a spacecraft with n VSCMGs is given by combining equations (35), (40) and (43) as,   Π

  Ω = J(Γ) ˙ . P Γ

(44)

5

Attitude Control using VSCMG For spacecraft attitude stabilization or tracking using VSCMGs, one needs to ensure that the angular velocity Ω of the spacecraft base body, or alternately the base-body momentum Πb , is controlled using the internal momentum u, which ˙ One can express the total angular momentum of the spacecraft with n VSCMGs, depends on the VSCMG rotation rates Γ. using equations (36) and (41) as follows ˙ Π = Πb + u, where Πb = Λ(Γ)Ω and u = B (Γ) Γ. ˙ b = Πb × Ω + (u × Ω − u) Therefore, Π ˙ = Πb × Ω + τcp ,

(45) (46)

where τcp = u × Ω − u˙ is the control torque generated by the “internal” momentum u from the VSCMGs. Note that from (46) and the attitude kinematics (9), the total momentum in an inertial frame, ΠI = RΠ = R(Πb + u) is conserved. There are several control problems of interest for a spacecraft with n VSCMGs. For most applications, the common spacecraft attitude maneuvers are slewing to rest, pointing , and attitude tracking maneuver. The results and discussion for a slew to rest attitude control scheme (de-tumbling maneuver) for a spacecraft with multiple VSCMGs are presented in [23, 24]. 5.1

Pointing Maneuver of Spacecraft with n VSCMGs The objective of a pointing maneuver of spacecraft with n VSCMGs is to point a body-fixed imaging, communication or other instrument in a specified direction in the reference frame, where rotation about that specified body-fixed direction are not important. The control problem of pointing the spacecraft to a desired attitude is considered. To achieve this, a continuous feedback controller given by τcp (R, Ω) : SO(3) × R3 → R3 , is selected that asymptotically stabilizes the desired attitude equilibrium given by the desired attitude Rd ∈ SO(3) and the angular velocity, Ω(t) = 0. The continuous feedback control torque for this system given by, τcp = −Lv Ω − L p Ξ(R)

(47)

where Lv , L p ∈ R3×3 are positive definite matrices and the map 3

Ξ(R) , ∑ ai ei × (RT d R.ei ),

(48)

i=1

   T with e1 e2 e3 ≡ I3×3 and a = a1 a2 a3 where a1 , a2 , and a3 are distinctive positive integers, asymptotically stabilizes the attitude and angular velocity states to (R, Ω) = (Rd , 0), for a given desired attitude, Rd ∈ SO(3). The almost global asymptotically stable control law given in equation (47) brings the spacecraft base-body from an arbitrary initial attitude and angular velocity to a rest attitude [25]. The control inputs are the VSCMG angular rates, obtained by taking the pseudo-inverse of B in equation (45), which gives the gimbal rates (α˙ i ) and rotor rates (θ˙ i ) of all VSCMGs: Γ˙ = B † u,

(49)

where u is obtained by integrating τcp = u × Ω − u. ˙ The VSCMGs need to be positioned in a way such that: B ∈ Rm×n in † equation (49) always has full row rank; i.e., BB is non-singular. Then B † is the right inverse of B , such that BB † = Im . The pseudo-inverse is explicitly given by,

B † = B T (BB T )−1 . Sufficient conditions for B to be non-singular are given in section 7.

5.2

Null Motion of VSCMG The rotor is often required to maintain a nominal angular rate at its breakaway velocity [26], to minimize the effect of static friction on the motor and bearings. This can be achieved by spinning the VSCMG gimbals and rotors, such that the vector of VSCMG rates Γ˙ is in the null-space of B ∈ R3×2n ; this is termed the null motion of the VSCMG assembly. The projection matrix to the null-space of B is given by

P = (I2n×2n − B T (BB T )−1 B ),

(50)

such that P 2 = P and BP = 0. Let N ∈ R2n be the vector of preferred or nominal VSCMG rates. Then the vector of VSCMG rates with null motion is given by Γ˙ = B † u + Γ˙ n such that Γ˙ n = P N .

(51)

6

Simplification of the Generalized Dynamics Model under certain Assumptions In this section, the generalized dynamics model formulated in Section 3, is simplified by some assumptions that are commonly used in VSCMG literatures. The following subsection will provide the mathematical treatment to obtain a generalized expression for VSCMG momenta (34). 6.1

Generalized VSCMG Momenta The momentum contribution from a VSCMG to the total angular momentum of the spacecraft (45) is given as
˙ u = B(γ)˙γ = Jc (γ) + Ic (α) α˙ g + Rr Jr RT r θ η(α).

(52)

The inertia terms (18) and (19) are given as,
T × × × 2 Jc (γ) = Rg Jg RT g + Rr Jr Rr , Ic (α) = −mr σ ρg η(α) + σ(η(α) ) , and therefore ˙ r Jr RT ˙ c (γ)g + αI ˙ c (α)g + θR u = αJ r η(α).

(53)


Note that Rr = Rg exp θe× 2 . The first term on the expression for u can be simplified to T Jc (γ)g = Rg Jg RT g g + Rr Jr Rr g,


= Rg Jg e1 + Rr Jr cos θe1 + sin θe3 .

(54)

The second term in the equation (53) can be written as,
× × × 2 ˙ c (α)g = α[−m ˙ αI r σ ρg η(α) + σ(η(α) ) ]g × ˙ = mr σ[ρ× g + ση(α) ]η(α),

(55)

and the last term of equation (53) is,
T × Rr Jr RT r η(α) = Rr Jr exp − θe2 Rg η(α)
= Rr (t)Jr exp − θ(t)e× 2 e2 = Rr Jr e2 .

(56)

Using equations (54) − (56), the expression (53) for u leads to,  
× × ˙ r Jr e2 . u = α˙ [Rg Jg e1 + Rr Jr cos θe1 + sin θe3 ] + mr σ[ρ× + ση(α) ]g η(α) + θR g Any assumptions to simplify the dynamics the VSCMG, will result in simplifications of the expression (57).

(57)

6.2

Commonly used Assumptions The expression (57) for the angular momentum contribution of the VSCMGs can be reduced to the well-known expression for the angular momentum contribution from a set of CMGs, by imposing some commonly used assumptions. Assumptions 1. To summarize, the common assumptions for VSCMGs and CMGs are listed below: 1. The offset between the rotor center of mass and the gimbal axis in the direction of the rotor’s rotation axis is zero i.e., σ = 0. 2. Both the gimbal and the rotor-fixed coordinate frames are their corresponding principal axes frames, and the rotor is axisymmetric. Both rotor and gimbal inertias are about their respective center of masses i.e., Jr = JrCoM and Jg = JgCoM . 3. The gimbal frame structure has “negligible” inertia i.e., Jg ' 0. 4. The angular rate of the gimbal frame is much smaller (“negligible”) compared to the rotor angular rate about its ˙ ˙  θ(t). symmetry axis i.e., α(t) 5. For a CMG, the speed of the rotor θ˙ is constant, i.e., the VSCMG is operated as a standard SGCMG. The following analysis shows how this series of assumptions leads to the common model for the control torque on a spacecraft due to a standard control moment gyroscope. Here, the VSCMG momenta given in the expression (53) is reduced by applying the above-summarized assumptions, one by one in sequence. Assumption 1.1: VSCMG rotor center of mass is perfectly aligned with gimbal center of mass along the gimbal axis g, i.e., setting σ = 0 in equation (53). This leads to the following simplification,
u = Rg Jg e1 + Rr Jr cos θe1 + sin θe3 + Rr Jr e2 .

(58)

Assumption 1.2: By using the assumption that both the gimbal and the rotor inertias are about their respective center of masses i.e., Jr = JrCoM and Jg = JgCoM ; both the gimbal and the rotor fixed coordinate frames are their corresponding principal axes frames, and that the rotor is axisymmetric; thus, Jg = diag(Jg1 , Jg2 , Jg3 ), Jr = diag(Jr1 , Jr2 , Jr1 ). This simplifies the expression (54) to: Jc (γ)g = Jg1 g + Jr1 Rr (cos θe1 + sin θe3 ).

(59)

Then expressing Rr e1 and Rr e3 as follows:
Rr e1 = Rg exp θe× 2 e1
= g cos θ − g × η(α) sin θ,
and, Rr e3 = Rg exp θe× 2 e3
= cos θ g × η(γ) + sin θg. Substituting these in the expression for Jc (γ)g in equation (59), gives
Jc (γ)g = Jg1 + Jr1 g,

(60)

which shows that this is a constant vector that is a scalar multiple of g. Further, reducing the last term of the expression (53) under the same assumption as, Rr Jr RT r η(α) = Rr Jr e2 = Jr2 Rr e2
= Jr2 Rg exp θe× 2 e2 = Jr2 Rg e2 = Jr2 η(α).

(61)

Substituting these simplified expressions under the assumptions that: (1) the offset σ = 0, and (2) both the gimbal and the rotor fixed coordinate frames are their corresponding principal axes frames, and that the rotor is axisymmetric; the angular momentum due to the rotational motions of the gimbal and rotor can be expressed as:
˙ r2 η(α). u = α˙ Jg1 + Jr1 g + θJ

(62)

The rate of change of this angular momentum, which is part of the torque acting on the spacecraft’s attitude dynamics due to the VSCMG, is given by
¨ r2 η(α) + α˙ θJ ˙ r2 g× η(α). τu = u˙ = α¨ Jg1 + Jr1 g + θJ

(63)

Note that even under these assumptions on the symmetric inertia distribution of the VSCMG rotor and principal coordinate frame for gimbal and rotor, the torque τu has components along all three axes of the coordinate frame fixed to the VSCMG gimbal. These two assumptions are commonly used, although they may not be explicitly stated. Assumption 1.3: The expressions for u in equation (62) and its time rate of change can be further simplified under the additional assumptions that the gimbal inertia is “negligible” [6], i.e., Jg ≡ Jg1 ' 0,
˙ r2 η(α). u = α˙ Jr1 g + θJ

(64)

Assumption 1.4: The angular rate of the gimbal axis is much smaller (“negligible”) compared to the rotor rate about its ˙ ˙  θ(t)). symmetry axis (i.e., α(t) Under these assumptions, the contribution to the total angular momentum of the spacecraft from the VSCMG is obtained as, ˙ r2 η(α). u = θJ

(65)

The time derivative of u(t), which is part of the control torque on the spacecraft due to the VSCMG, is evaluated under these assumptions. This is evaluated to be

τu =

du ¨ ˙ r2 g× η(α), = θJr2 η(α) + α˙ θJ dt

(66)

which shows that under these additional assumptions, this torque component is on the plane normal to the gimbal axis g. Assumption 1.5: Finally, the assumption that the speed of the rotor θ˙ is constant, which makes the VSCMG a standard single gimbal CMG. Under the set of assumptions 1.1 through 1.5, the time rate of change of u is found to be given by

τu =

du ˙ ˙ r2 g× η(α) = καg ˙ × η(α), = θαJ dt

(67)

˙ r2 is a constant scalar. where κ = θJ Note that assumptions 1.3 and 1.4 eliminates the torque component of τu along the gimbal axis, as can be seen from expression (66), and this component can certainly help from a control theoretic perspective. It is important to note that these assumptions and their validity have a large effect on the condition number of B (Γ) as defined by (41). Substantial research has been directed at avoiding “singularities” of CMG arrays or clusters, where the matrix B (Γ) becomes instantaneously rank deficient. In the light of expressions (62) and (63) for u and τu under the assumptions 1.1 and 1.2 for VSCMGs, it is clear that the torque due to a VSCMG can be made to equal any torque vector in R3 by suitable choice of α, θ and their first two time derivatives, provided that actuator constraints on these time derivatives are not violated. Under assumptions 1.1, 1.2 and 1.5 for standard CMGs, it is clear that τu has no component along the η(α) axis, which is the axis of rotation (also axis of symmetry) of the flywheel. However, even in this case, unless assumptions 1.3 and 1.4 are added, the torque output τu can have a non-zero component along the gimbal axis. In fact, in this case, an additional assumption that the angular acceleration α¨ ' 0 has to be appended to Assumption 1.4; note that α˙  θ˙ does not imply that α¨ is “negligible.” From a practical viewpoint, assumptions 1.1 and 1.2 are justifiable for most (VS)CMG designs, and under these assumptions, the expressions for the angular momentum contribution from the (VS)CMGs and its time derivative in the spacecraft body frame, are given by (62) and (63). The next section deals with singularity analysis of B (Γ) under these first two assumptions.

7

Singularity Analysis of Spacecraft with n VSCMGs under Assumptions 1.1 and 1.2 Expressions for the angular momentum contribution from n VSCMGs, and the time derivative of this angular momentum, are provided here. As can be seen from these expressions, the “control influence” matrices are non-singular when either this angular momentum or its time derivative are considered as control inputs, under simple and easy-to-satisfy conditions on the gimbal axis orientations of n VSCMGs, when n ≥ 2 [27]. This shows that with as few as two VSCMGs, one can obtain a desired output angular momentum or output torque from the VSCMGs, provided that angular rate and angular acceleration constraints on the gimbal and rotor are satisfied. Consider n VSCMGs, satisfying assumptions 1.1 and 1.2; in which case the angular momentum contribution of each k denote the inertia VSCMG is given by (62) and the time rate of this momentum for each VSCMG is given by (63). Let Jg1 k , J k , J k ) denote of the gimbal frame of the kth VSCMG about the gimbal axis (its symmetry axis), and let Jrk = diag(Jr1 r2 r1 the inertia matrix of the axisymmetric rotor of the kth VSCMG, where k = 1, . . . , n for n ≥ 2. Then equation (62) for the momentum output of the VSCMGs in the base body coordinate frame can be generalized to 1 1 1 n n n B (Γ)Γ˙ = u, where B (Γ) = [(Jg1 + Jr1 )g1 Jr2 η1 (α1 ) · · · (Jg1 + Jr1 )gn Jr2 ηn (αn )] ∈ R3×2n ,

and Γ = [α1 θ1 · · · αn θn ]T ∈ R2n .

(68)

The following result easily follows from the structure of the influence matrix B (Γ) in the above expression. Theorem 1. For a spacecraft with two or more VSCMGs (n ≥ 2) satisfying assumptions 1.1 and 1.2 as stated earlier, the control influence matrix B (Γ) defined by (68) has full rank if g j is neither parallel nor orthogonal to gk for any pair of VSCMGs ( j, k = 1, . . . , n, j 6= k). Proof. Without loss of generality, consider the case that j = 1 and k = 2 in the above statement. Then the control influence matrix for the VSCMG array has its first four columns given by 1 1 1 n n 2 (Jg1 + Jr1 )g1 , Jr2 η1 (α1 ), (Jg2 + Jr2 )g2 and Jr2 η2 (α2 ).

Since g1 and g2 are neither parallel nor orthogonal to each other, and since η1 (α1 ) is always orthogonal to g1 , the triad of vectors g1 , η1 (α1 ) and g2 are either non-coplanar or planar. When they are non-coplanar, they span R3 and hence rank(B (Γ)) = 3. At instants when g1 , η1 (α1 ) and g2 are co-planar,
η2 (α2 ) is normal to this plane since η2 (α2 ) is always orthogonal to g2 . Therefore, in this case, Span g1 , η1 (α1 ), g2 , η2 (α2 ) = R3 and the result follows. The above result shows that it is possible to obtain any given angular momentum output with just a pair of VSCMGs, provided that constraints on the angular rates of the gimbal and rotor are satisfied. Now consider the time derivative of this angular momentum, which is the torque output from the VSCMGs acting on the spacecraft base body. For a single VSCMG, this torque output is given by the expression (63). For n ≥ 2 VSCMGs, this torque output equation is given by

C (Γ)ϒ = u, ˙ where n n n n × 1 1 1 × 1 g1 η1 (α1 ) · · · (Jg1 + Jr1 )gn Jr2 ηn (αn ) Jr2 gn ηn (αn )] ∈ R3×3n , + Jr1 )g1 Jr2 η1 (α1 ) Jr2 C (Γ) = [(Jg1

(69)

and ϒ = [α¨ 1 θ¨ 1 α˙ 1 θ˙ 1 · · · αn θn α˙ n θ˙ n ]T ∈ R3n .

Note that the control influence matrix C (Γ) for the torque output has the column vectors of the influence matrix B (Γ) for the angular momentum output, in addition to additional columns that are the third axes of each of the VSCMG gimbal-fixed coordinate frames. This observation leads to the following result, which is a corollary of Theorem 1. Corollary 2. For a spacecraft with two or more VSCMGs satisfying assumptions 1.1 and 1.2 as given earlier, the control influence matrix C (Γ) defined by (69) has full rank if g j is neither parallel nor orthogonal to gk for any pair of VSCMGs ( j, k = 1, . . . , n, j 6= k). Theorem 1 holds even without Assumption 1.1, in which case σ 6= 0 provides additional directions for control that lead to better controllability properties. In Section 9, attitude control of a spacecraft using VSCMGs with and without Assumption 1.1 is compared.

8

A Geometric Variational Integrator for Spacecraft with VSCMGs The dynamics of the multibody system consisting of the spacecraft with VSCMGs can be discretized for numerical implementation, using a variational integrator that preserves the geometry of the state space and the conserved norm of the total angular momentum. The idea behind variational integrators is to discretize the variational principles of mechanics: Hamilton’s principle for a conservative system or the Lagrange-d’Alembert principle for a system with nonconservative forcing [28]. Variational integrators are known to show good energy behavior over exponentially long time intervals and maintain conserved momentum quantities during numerical integration for systems with conserved momenta [29]. Geometric variational integrators combine variational integrators with geometric (e.g., Lie group) methods that preserve the geometry of the configuration upon discretization [16,30]. In a geometric variational integrator for multibody systems, rigid body attitude is globally represented on the Lie group of rigid body rotations. General multi-purpose numerical integration methods do not necessarily preserve first integrals or the geometry of the configuration space. Many interesting dynamical systems, including rigid body and multibody systems, evolve on Lie groups or fiber bundles on Lie groups. Lie group methods in numerical integration techniques preserve this geometry of the configuration space by discretizing the dynamics directly on the Lie group [31]. Treatment of Lie group variational integrators for discretization of rigid body dynamics is given in [32–34]. Since geometric variational integrators are variational by design, they conserve momentum quantities when the system exhibits symmetry, like the magnitude of the total angular momentum Π of the spacecraft with VSCMGs. 8.1

Discretization of Equations of Motion In this section, the discrete equations of motion for a spacecraft with one VSCMG in the form of a geometric variational integrator, corresponding to the equations of motion (30) and (31) are obtained. This is followed by generalizing these discrete equations to the case when the spacecraft has multiple VSCMGs. 8.1.1 Discrete Lagrangian The discrete Lagrangian function Ld : Q × Q → R approximates the continuous Lagrangian using quadrature rules [35]. Let h 6= 0 denote a fixed time step size, i.e., tk+1 − tk = h, and consider the discrete Lagrangian

Ld ≈

tk+1 Z

h 2

L dt = L (Ωk , Rk , γk , γ˙ k )h = χTk J (γk )χk ,

tk

(70)

   h  T T  Λ(γk ) B(γk ) Ωk = Ωk γ˙ k . B(γk )T Jgr (γk ) γ˙ k 2 For notational convenience, define Λ(γk ) = Λk , B(γk ) = Bk and Jgr (γk ) = Jgr,k . 8.1.2 Discretization of Kinematics Consider the discrete analogue of the continuous spacecraft attitude kinematics equation (9) given by, Rk+1 = Rk Fk , such that,

Fk = exp(hΩ× k ).

(71) (72)

× The first variation of both sides of equation (72) is obtained under the assumption that δΩ× k commutes with Ωk , i.e., the Lie × × bracket [δΩk , Ωk ] = 0, as follows: × × δFk = hδΩ× k exp(hΩk ) = hδΩk Fk . × The variation of Fk can be computed from equations (25) and (71) as δFk = −Σ× k Fk + Fk Σk+1 , then

1 1 × × T T δΩ× k = h δFk Fk = h (−Σk Fk + Fk Σk+1 )Fk ,
1 = − Σ× + (Fk Σk+1 )× , k h 1 δΩk = (Fk Σk+1 − Σk ). h

(73)

The VSCMG angular rates are approximated using the trapezoidal rule as: γ˙ k '

γk+1 − γk 1 and its variation, δ˙γk = (δγk+1 − δγk ). h h

(74)

8.1.3 Discretization of Dynamics Consider the discrete Lagrangian (70) and the associated action sum Sd = ∑N−1 k=0 Ld , then the discrete Lagrange-d’Alembert principle states that N−1

∑


δLd + hτT d δγk = 0.

(75)

k=0

The first variation of the discrete Lagrangian (70) is     T T  Λk Bk ∂ δΩk {Ld }δγk , δLd = h Ωk γ˙ k + δ˙ γ BT J ∂γ k k k gr,k  T  T ˙ k δ˙γk + Mk (γk , γ˙ k , Ωk )δγk , = h Λk Ωk + Bk γ˙ k δΩk + h BT k Ωk + Jgr,k γ T γ + M δγ , = hΠT k k k k δΩk + hpk δ˙

(76)

where the discrete total angular momentum of the spacecraft (Πk ) is obtained using equation (74) as Πk = Λk Ωk + Bk γ˙ k = Πb,k + uk ,

(77)

and the internal momentum quantities are given by ˙ k , Mk = pk = BT k Ωk + Jgr,k γ

∂Ld , ∂γk


1 and uk = Bk γk+1 − γk . h Substituting equations (73), (74) and (76) into the Lagrange-d’Alembert principle (75) gives, N−1 h

∑

k=0

i
T δγ = 0. T δγ + M δγ + hτ ΠT (F Σ − Σ ) + p − δγ k k k k k+1 k k+1 k γ,k k k

(78)

Using the fact that the variations Σk and δγk vanish at the endpoints (for k = 0 and k = N) and re-indexing the discrete action sum, equation (78) can be re-expressed as

N−1 h

∑

k=1

i T T T T (ΠT k−1 Fk−1 − Πk )Σk + (pk−1 − pk + Mk + hτγ,k )δγk = 0.

(79)

Since the above expression is true for arbitrary Σk and δγk , one obtains the following relations for the discrete dynamics: Πk+1 = FkT Πk ,

(80)

pk+1 = pk + Mk+1 + hτγ,k+1 .

(81)

This discrete dynamics of the spacecraft with one VSCMG, given by equations (80) and (81), is generalized to the case of multiple VSCMGs in a manner similar to that in section 4. The discrete analog of the spacecraft attitude dynamics with multiple VSCMGs is obtained from equations (77) and (80), as follows: Πb,k+1 = FkT Πb,k + FkT uk − uk+1 .

(82)

8.2

Geometric Variational Integrator Algorithm The algorithm for the geometric variational integrator (GVI) routine is as follows:

1. 2. 3. 4. 5. 6. 7.

˙ Initialize Fk = exp(hΩ× k ) and Γk+1 = hΓk + Γk ; Compute Bk+1 and Λk+1 as functions of Γk+1 ;
Equate, FkT uk − uk+1 = hτcp,k = −Lv Ωk − L p ai ei × (RT d Rk ei ) and substitute in equation (82); Compute, uk+1 = FkT uk − hτcp,k ;
† † Update VSCMG rates one time step forward as: Γ˙ k+1 = Bk+1 uk+1 + I2n×2n − Bk+1 Bk+1 N ; Obtain Πb,k+1 from equation (82) and Ωk+1 = Λ−1 k+1 Πb,k+1 ; Loop through steps 1 to 7 for all k.

This algorithm provides an explicit geometric variational integrator, which is used in the following section to numerically simulate the dynamics of a spacecraft with multiple VSCMGs, while maintaining a constant magnitude for the total angular momentum (kΠk+1 k = kΠk k by equation (80)) and preserving the geometry of the configuration space.

9

Numerical Simulation Results

In this section, the feedback control law given by equation (47) is numerically validated on the attitude dynamics of a spacecraft with three VSCMGs in a tetrahedron configuration.

9.1

Example Case of a Spacecraft with three VSCMG in Tetrahedron Configuration

Spacecraft with three VSCMG in tetrahedron configuration as shown in the figs. 2a and 2b, is considered in this section. Each VSCMG has identical gimbal and rotor mass, i.e. m1g = m2g = m3g = mg and m1r = m2r = m3r = mr , respectively. From fig. 2c, the rotor axes η1,2,3 and gimbal axes g1,2,3 corresponding to VSCMG1,2,3 , respectively are resolved in the spacecraft base body frame. The center of mass of the spacecraft base body is kept aligned to the plane containing the intersection of g and η. For the tetrahedron mount of three single-gimbal VSCMGs with skew angle of β as illustrated in fig. 2, the gimbal-axis vectors can be represented as,

      Cβ −SΘCβ −SΘCβ g1 =  0  ; g2 =  CΘCβ  and g3 = −CΘCβ . Sβ Sβ Sβ

(a)

(b)

(83)

(c)

Fig. 2: (a) CAD Rendering of the three VSCMG in tetrahedron configuration; (b) Satellite with three VSCMG actuators; (c) Vector representation of three VSCMGs, in tetrahedron formation with respect to the base body coordinate frame [XY Z].

Similarly, the rotor axis η(0) at time t = 0 can be represented as,       −Sβ −SβSΘ SβSΘ η1 (0) =  0  ; η2 (0) =  CΘSβ  and η3 (0) = CΘSβ . Cβ Cβ Cβ

(84)

Here, S(α/β/Θ) ≡ sin(α/β/Θ) and C(α/β/Θ) ≡ cos(α/β/Θ), π π 2π Θ = | − |, for tetrahedron, n = 3 ⇒ Θ = radian or 30◦ . 2 n 6 The time varying η(t), as a function of time about g is obtained using equation (1) and rotation matrices Rig and Rir are given as,  
Rig = gi ηi gi × ηi and Rir = Rig exp θi e× 2 ; i = 1, 2, 3. From Corollary 2, the tetrahedron configuration of VSCMG in the spacecraft base body ensures the existence of nonsingular B matrix for all time t ≥ 0. The skew angle β is chosen, such that g j is neither parallel nor orthogonal to gk for all three VSCMGs ( j, k = 1, 2, 3; j 6= k). 9.2

Results for a Simulated Pointing Maneuver The geometric variational integrator scheme given in section 8 and the simulation parameters from Table 1 of Appendix A are used to obtain numerical results for a simulated pointing maneuver. The initial attitude is chosen to be a 1 radian rotation about the inertial axis [−1/3 1/3 0.8/3]T . The full attitude pointing maneuver transfers the initial attitude to a ˙ desired final attitude, asymptotically. The control inputs are provided in terms of VSCMG angular rates, i.e. the vector Γ. This vector of VSCMG rates is obtained using the GVI algorithm given in section 8.2. The VSCMG angular momentum contribution converges to zero as the gimbals and rotors converge to their prescribed nominal rates, i.e., Γ˙ converges to Γ˙ n in the null space of B ; this helps in avoiding “zero crossing” of the gimbal and rotor drive motors. In the first set of simulation results shown in fig. 3d, control schemes based on Assumption 1.1 (σ = 0) are applied when σ = 0 and when σ 6= 0, and the results are compared. For this maneuver, fig. 3a compares the time response of the norm of spacecraft total angular velocity for these two cases, and fig. 3b compares the control torque norms generated by the control law. The importance of this

kτcp k (Nm)

kΩk (rad/s)

0.08

kΩσ=0 k kΩσ6=0 k

0.03 0.02 0.01 0

0

100

200

300

400

500

0.04 0.02 0

600

kτcpσ=0k kτcpσ6=0k

0.06

0

100

200

t (s) (a)

400

500

600

(b) 0.06

kuk (Nm-s)

0.8

Φ

0.6 0.4 0.2 0

300

t (s)

Φσ=0 0

100

200

300

t (s) (c)

400

Φσ6=0 500

600

kuσ=0 k kuσ6=0 k

0.04

0.02

0

0

100

200

300

400

500

600

t (s) (d)

Fig. 3: Plots of: (a) norm of angular velocity of the spacecraft base-body, (b) norm of control torque, (c) attitude stabilization ˙ error, and (d) norm of VSCMG angular momentum, output u = B Γ.

40

α˙ i (rad/s)

α˙ i (rad/s)

40 20 0 −20 −40

α˙ 1 0

100

200

300

α˙ 2 400

α˙ 3 500

20 0 −20 −40

600

α˙ 2

α˙ 1 0

100

200

t (s)

300

400

α˙ 3 500

600

t (s)

(a)

(b)

Fig. 4: (a) VSCMG gimbal rate at σ = 0, and (b) VSCMG gimbal rate at σ = 0.005 m. 40

θ˙ i (rad/s)

θ˙ i (rad/s)

40 20 0 −20 −40

θ˙ 2

θ˙ 1 0

100

200

300

400

θ˙ 3 500

20 0 −20 −40

600

θ˙ 2

θ˙ 1 0

100

200

t (s)

300

400

θ˙ 3 500

600

t (s)

(a)

(b)

Fig. 5: (a) VSCMG rotor rate at σ = 0, and (b) VSCMG rotor rate at σ = 0.005 m. 2

θ˙ei (rad/s)

α˙ ie (rad/s)

2 0 −2 −4 −6

α˙ e2

α˙ e1 0

100

200

300

400

α˙ e3 500

600

0 −2 −4 −6

θ˙e2

θ˙e1 0

100

200

300

t (s)

t (s)

(a)

(b)

400

θ˙e3 500

600

Fig. 6: (a) Error in gimbal rate, α˙ ie = α˙ iσ=0 − α˙ iσ6=0 and (b) Error in rotor rate, θ˙ ie = θ˙ iσ=0 − θ˙ iσ6=0 . generalized VSCMG model is demonstrated by fig. 3c; the spacecraft actuated by VSCMG with non-zero rotor offset is not stabilized, as the gimbal and rotor are not spinning in the null space of Bσ6=0 (note that Bσ6=0 6= Bσ=0 ). As (VS)CMG are mainly used for precise pointing application of the spacecraft, even a small change in the dynamics can lead to a considerable deviation in pointing accuracy. Figure 3d shows the VSCMG angular momenta with and without rotor offset. From these results, it is inferred that a small misalignment of the rotor CoM with gimbal CoM (σ = 0.005 m) can greatly affect the dynamics leading to failure in achieving the control objectives. Figures 4a and 4b compare the time evolution of each VSCMG’s gimbal rates for rotor offsets of σ = 0 and σ = 0.005 m, respectively. The difference in these gimbal rates is given in fig. 6a. Similarly, VSCMG rotor rates with zero rotor offset (fig. 5a) and non-zero offset (fig. 5b) are plotted and the difference in the rotor rates for these two cases is plotted in fig. 6b. These results confirm the reduction in the gimbal and rotor rates of the VSCMG with non-zero rotor offset to that of the VSCMG with zero rotor offset. Based on the numerical results, one can see that the generalized dynamics model can lead to better control performance when certain simplifying assumption(s) do not hold in practice.

10

Conclusion The dynamics model of a spacecraft with a generalized model of one VSCMG with an offset between the rotor center of mass and gimbal center of mass along the gimbal axis is presented. The model is then generalized to a spacecraft with n such VSCMGs. Equations of motion are expressed in terms of angular momentum variables of the spacecraft’s

attitude motion and the rotational degrees of freedom of the VSCMGs. Thereafter, certain commonly used assumptions are applied to the generalized VSCMG model, which reduces it to the conventional model. Sufficient conditions for nonsingular VSCMG steering under some of these assumptions are stated. An attitude pointing control scheme for a spacecraft using only VSCMGs is obtained and the control scheme is compared for VSCMGs with zero and non-zero rotor offset. The overall dynamics is numerically simulated, when there is no gravity or external moment, and when the overall angular momentum of the system is zero. The continuous equations of motion for this system are discretized in the form of a geometric variational integrator, for numerical simulation of the feedback system. The numerical simulation results confirm the stabilizing properties of the VSCMG-based attitude control and show improved pointing control performance when the VSCMG model takes into account an existing rotor offset. Future work will look at the efficiency of these VSCMG in terms of the ratio of output torque to input power.

References [1] Lappas, V., Steyn, W., and Underwood, C., 2002. “Torque amplification of control moment gyros”. Electronics Letters, 38(15), jul, pp. 837 – 839. [2] Defendini, A., Faucheux, P., Guay, P., Bangert, K., Heimel, H., Privat, M., and Seiler, R., 2003. “Control Moment GYRO CMG 15-45 S: a compact CMG product for agile satellites in the one ton class”. In 10th European Space Mechanisms and Tribology Symposium, R. A. Harris, ed., Vol. 524 of ESA Special Publication, pp. 27–31. [3] Wie, B., 2008. Space Vehicle Dynamics and Control, 2nd ed. American Institute of Aeronautics and Astronautics, Reston, VA. [4] Schaub, H., and Junkins, J. L., 2009. Analytical Mechanics of Space Systems, 2nd ed. AIAA Education Series, Reston, VA, October. [5] McMahon, J., and Schaub, H., 2009. “Simplified singularity avoidance using variable speed control moment gyroscope nullmotion”. AIAA Journal of Guidance, Control, and Dynamics, 32(6), Nov. – Dec., pp. 1938–1943. [6] Yoon, H., and Tsiotras, P., 2002. “Spacecraft adaptive attitude and power tracking with variable speed control moment gyroscopes”. AIAA Journal of Guidance, Control and Dynamics, 25, pp. 1081–1090. [7] Margulies, G., and Aubrun, J. N., 1978. “Geometric theory of single-gimbal control moment gyro systems”. Journal of the Astronautical Sciences, 26(2), pp. 159–191. [8] Kurokawa, H., 1998. A geometric study of single gimbal control moment gyros (singularity problems and steering law). Tech. Rep. 175, Agency of Industrial Technology and Science, Japan. [9] Paradiso, J., 1991. A search-based approach to steering single gimbaled CMGs. Tech. Rep. CSDL-R-2261, Draper Laboratory Technical Report, Cambridge, Massachusetts, August. [10] Schiehlen, W. ., 1972. Two different approaches for a control law of single gimbal control moment gyros. Tech. Rep. NASA-TM-X-64693, NASA-Marshall Space Flight Center, Alabama. [11] Leve, F., and Fitz-Coy, N., 2010. “Hybrid steering logic for single-gimbal control moment gyroscopes”. Journal of guidance, control, and dynamics, 33(4), pp. 1202–1212. [12] Nebylov, A., 2004. Automatic Control in Aerospace, 1st ed. Elsevier. [13] Hughes, P., 1986. Spacecraft attitude dynamics. J. Wiley. [14] Tokar, E., and Platonov, V., 1979. “Singular surfaces in unsupported gyrodyne systems”. Cosmic Research, 16, pp. 547–555. [15] Sun, Z., Zhang, L., Jin, G., and Yang, X., 2010. “Analysis of inertia dyadic uncertainty for small agile satellite with control moment gyros”. In International Conference on Mechatronics and Automation, pp. 813–818. [16] Marsden, J. E., and Ratiu, T. S., 1999. Introduction to Mechanics and Symmetry, 2 ed. Springer Verlag, New York. [17] Bloch, A. M., 2003. Nonholonomic Mechanics and Control. No. 24 in Interdisciplinary Texts in Mathematics. Springer Verlag. [18] Marsden, J. E. Lectures on Mechanics. Cambridge University Press. [19] Bloch, A. M., Krishnaprasad, P. S., Marsden, J. E., and de Alvarez, G. S., 1992. “Stabilization of rigid body dynamics by internal and external torques”. Automatica, 28(4), pp. 745–756. [20] Wang, L.-S., and Krishnaprasad, P. S., 1992. “Gyroscopic control and stabilization”. Journal of Nonlinear Science, 2, pp. 367–415. [21] Bullo, F., and Lewis, A. D., 2004. Geometric Control of Mechanical Systems. No. 49 in Texts in Applied Mathematics. Springer Verlag. [22] Bredon, G. E., 1993. Topology and Geometry. No. 139 in Graduate Texts in Mathematics. Springer Verlag. [23] Sanyal, A., Prabhakaran, V., Leve, F., and McClamroch, N., 2013. “Geometric approach to attitude dynamics and control of spacecraft with variable speed control moment gyroscopes”. In 2013 IEEE International Conference on Control Applications (CCA), pp. 556–561. [24] Viswanathan, S. P., Sanyal, A., Leve, F., and McClamroch, H. N., 2013. “Geometric mechanics based modeling of the

[25] [26] [27] [28] [29]

[30] [31] [32] [33] [34] [35]

attitude dynamics and control of spacecraft with variable speed control moment gyroscopes”. In ASME Dynamical System Control Conference, no. DSCC 2013-4033. Chaturvedi, N. A., Sanyal, A. K., and McClamroch, N. H., 2011. “Rigid-body attitude control using rotation matrices for continuous, singularity-free control laws”. IEEE Control System Magazine, June. Virgala, I., and Kenderov, Peter Frankovsky, M., 2013. “Friction effect analysis of a dc motor”. American Journal of Mechanical Engineering, 1(1), pp. 1–5. Yoon, H., and Tsiotras, P., 2004. “Singularity analysis of variable speed control moment gyros”. AIAA Journal of Guidance, Control and Dynamics, 27(3), pp. 374–386. Marsden, J., and West, M., 2001. “Discrete mechanics and variational integrators”. Acta Numerica, 10, pp. 357–514. Lee, T., McClamroch, N., and Leok, M., 2005. “A lie group variational integrator for the attitude dynamics of a rigid body with applications to the 3d pendulum”. In Control Applications, 2005. CCA 2005. Proceedings of 2005 IEEE Conference on, pp. 962–967. Hairer, E., Lubich, C., and Wanner, G., 2000. Geometric Numerical Integration. Springer Verlag. Iserles, A., Munthe-Kaas, H. Z., N¨orsett, S. P., , and Zanna, A., 2000. “Lie-group methods”. Acta Numerica, 9, pp. 215–365. Hussein, I., Leok, M., Sanyal, A., and Bloch, A., 2006. “A discrete variational integrator for optimal control problems on SO(3)”. In Decision and Control, 2006 45th IEEE Conference on, pp. 6636 –6641. Sanyal, A. K., Lee, T., Leok, M., and McClamroch, N. H., 2008. “Global optimal attitude estimation using uncertainty ellipsoids”. Systems and Control Letters, 57(3), pp. 236 – 245. Bloch, A., Hussein, I., Leok, M., and Sanyal, A., 2009. “Geometric structure-preserving optimal control of a rigid body”. Journal of Dynamical and Control Systems, 15, pp. 307–330. Davis, P. J., and Rabinowitz, P., 1975. Methods of numerical integration; 1st ed. Computer science and applied mathematics. Academic Press, New York, NY.

Appendix A: Simulation Parameters

Table 1: Simulation parameters Parameter

Value

Units

Description

n Jb

3 1.7 × I3×3 h i diag 0.6 1 0.6 × 10−3 h i diag 1 0.1 1 × 10−3

kg-m2

no. of VSCMG base-body inertia

kg-m2

rotor inertia in its body frame

kg-m2

gimbal inertia in its body frame

mb mr mg β

29.42 0.88 0.65 30◦

kg kg kg degree

base-body mass rotor mass gimbal mass Skew angle

Γ˙ p

h iT 0 −60 0 45 0 55

rad/s

Preferred VSCMG rate

tf h

600 0.001

sec -

final time time step

Jr Jg