Directional passability and quadratic steering logic for pyramid-type single gimbal control moment gyros

Acta Astronautica 102 (2014) 103–123

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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro

Directional passability and quadratic steering logic for pyramid-type single gimbal control moment gyros Katsuhiko Yamada a,n, Ichiro Jikuya b a b

Department of Mechanical Engineering, Osaka University, Osaka 565-0871, Japan Department of Aerospace Engineering, Nagoya University, Aichi 464-8603, Japan

a r t i c l e i n f o

abstract

Article history: Received 18 November 2013 Received in revised form 7 March 2014 Accepted 19 May 2014 Available online 28 May 2014

Singularity analysis and the steering logic of pyramid-type single gimbal control moment gyros are studied. First, a new concept of directional passability in a specified direction is introduced to investigate the structure of an elliptic singular surface. The differences between passability and directional passability are discussed in detail and are visualized for 0H, 2H, and 4H singular surfaces. Second, quadratic steering logic (QSL), a new steering logic for passing the singular surface, is investigated. The algorithm is based on the quadratic constrained quadratic optimization problem and is reduced to the Newton method by using Gröbner bases. The proposed steering logic is demonstrated through numerical simulations for both constant torque maneuvering examples and attitude control examples. & 2014 IAA. Published by Elsevier Ltd. All rights reserved.

Keywords: Control moment gyro Singularity Passability Attitude control Steering law Gröbner bases

1. Introduction A Control Moment Gyro (CMG) system is a momentum exchange actuator used for the attitude control of a spacecraft [1,2]. This system is composed of multiple CMGs installed in the spacecraft, where each CMG contains a wheel spinning at high speed and gimbal rotating structures, and is classified into various types such as Single Gimbal CMG (SGCMG) [3–27], Double Gimbal CMG (DGCMG) [28,29], Variable Speed CMG (VSCMG) [30–33], and Double Gimbal Variable Speed CMG (DGVSCMG) [34] according to the variants of the wheel spinning speed and gimbal rotating structures. An advantage of CMG systems is their efficient torque-producing capability: a relatively small gimbal torque input produces a large torque output on the spacecraft according to the conservation of angular momentum. This makes CMG systems popular for reorienting

n

Corresponding author. E-mail address: [email protected] (K. Yamada).

http://dx.doi.org/10.1016/j.actaastro.2014.05.022 0094-5765/& 2014 IAA. Published by Elsevier Ltd. All rights reserved.

large space structures and for agile maneuvering of satellites. A disadvantage of CMG systems is the difficulty in designing a CMG steering logic because the gimbal angles change during attitude control. In particular, a CMG system singularity is one of the major obstacles for designing a CMG steering logic. Studies on the singularity problem can be roughly classified into singularity analysis, singularity avoidance, and passability of a singular surface. Singularity analyses have been carried out by Margulies and Aubrun [3] and by Tokar [4–7] for SGCMG systems. Continuous studies have been conducted by Bedrossian et al. [8], Kurokawa [9,10], Wie [11], Sands et al. [12], and Yamada et al. [13], for SGCMG systems and by Yoon and Tsiotras [32] for VSCMG systems (see [9–11] for a comprehensive survey). In order to analyze the singularity of a CMG system, the angular momentum of the CMG system is expanded to a series of gimbal angles. The CMG system, as well as the angular momentum and gimbal angles, is in a singular state if the Jacobian matrix, the first-order coefficient matrix of the series expansion, is deficient. Thus, no control torque can be produced in the sense of the

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first-order approximation along a certain direction, where a unit vector pointing in this direction is called a singular vector. The set of all of the angular momentum values in the singular states constitutes the singular surfaces in the angular momentum space. Singular surfaces are classified as external singular surfaces and internal singular surfaces. A singular surface is called an external singular surface if the magnitude of the angular momentum on the surface is larger than that of the other surfaces along each direction; otherwise, the surface is called an internal singular surface. Additional angular momentum for its direction is not possible on an external singular surface by definition; therefore, the external singular surfaces are impassable for any trajectory of the gimbal angular velocity. In contrast, both passable and impassable cases are possible on the internal singular surfaces. Because an internal singular surface exists for any direction inside the external singular surfaces, the trajectory of the angular momentum vector might approach the internal singular surface during the CMG steering. If the angular momentum vector approaches an impassable singular surface, the CMG steering might stick on the surface. Therefore, it is necessary to take into account the internal singular surfaces when designing the CMG steering logic. Singularity avoidance has been extensively studied [12,14–27,35]. The singularity robust inverse [14–17,35] is simple and suitable for real-time calculations, but the torque error added near a singular surface is an obstacle to highly accurate control. The concept of a constrained workspace [21] is also valid for singularity avoidance, but the restriction of the available angular momentum might degenerate the control performance such as the settling. Feedforward control based on path planning [24–27] is effective in the case of the rest-to-rest maneuver because it is not necessary to perform a back calculation of the gimbal angular velocities from the torque. However, singularity avoidance is difficult in a case where the attitude changes with tracking over the whole trajectory. Singularity avoidance is also difficult in the case of feedback control for the wide-range ground-surface observation by changing the attitude of the spacecraft. Hence, the passability of a singular surface is an important problem for attitude changes with high speed and high accuracy. The passability of a singular surface is essential in singularity analysis [3–11]. By the definition of a singular state, a singular surface is not passable in the direction of a singular vector in the sense of a first-order approximation. The passability of a singular surface has been evaluated in the sense of a second-order approximation; specifically, a hyperbolic singular surface is passable in the direction of a singular vector by using null motion. An elliptic singular surface is another major obstacle in passing a singular surface. Although the elliptic singular surface is impassable in the direction of a singular vector by using null motion, it might be possible to pass the elliptic singular surface in a certain passing direction by relaxing it from null motion to arbitrary motion. An example of passing along a specific direction is constant torque maneuvering, which is often considered for an agile attitude change. This leads to a discussion on the directional passability in a specific direction. Once the surface is judged to be

directionally passable, it is necessary to compute a trajectory for the gimbal angular velocities from a trajectory of the angular momentum; however, a concrete steering logic that generates a precise control torque up to second-order approximation has not been presented in previous studies. This also leads to a discussion of a new steering logic for passing a singular surface. In this study, these problems are addressed for the most typical pyramid-type SGCMG systems. Our first objective is to study the directional passability of a singular surface. The singular surface is directionally passable if the angular momentum of CMG systems passes the singular surface along a given direction by using an arbitrary motion. This directional passability will be evaluated in the sense of the second-order approximation. First, the condition for directional passability is obtained for a general direction. The relation for the condition for passability is then discussed. Subsequently, the conditions for directional passability are obtained for the following cases: the direction of a singular vector and the opposite direction of the angular momentum. It will be shown that the singular surface is always directionally passable in the opposite direction of the angular momentum and that the singular surface is passable in any direction if it is passable in the same direction as the angular momentum. This reveals the difficulty of passing a singular surface in the same direction as the angular momentum. The differences between the passability and the directional passability will be visualized in detail for 0H, 2H, and 4H singular surfaces. The second objective is to investigate a new steering logic, referred to as quadratic steering logic (QSL), for passing a singular surface. QSL is based on a quadratic constrained quadratic optimization problem, i.e., the problem to generate the gimbal angular velocities in response to the control torque up to a second-order approximation such that the magnitude of the gimbal angular velocities is suppressed. It should be noted that the well-known pseudo-inverse steering logic is based on a linear constrained quadratic optimization problem; therefore, QSL is a natural extension of pseudo-inverse steering logic. QSL is computed via numerical optimization. A key idea is to represent the codependence between one free gimbal angle and the other three gimbal angles in the sense of the second-order approximation by using Gröbner bases. Then, the minimization of the quadratic cost of the gimbal angular velocities can be numerically calculated by the Newton method. The effectiveness of the proposed logic will be demonstrated through numerical simulations for two examples. One is for constant torque maneuvering examples for both directionally passable and directionally impassable cases. The other is for attitude control examples, and the logic is compared with singularity avoidance/ escape steering logic [17]. It should be noted that the magnitude of the gimbal angular velocities can be sufficiently suppressed for a sufficiently small control time interval in directionally passable cases; however, the magnitude of the gimbal angular velocities cannot be always suppressed in directionally impassable cases. This requires accounting for the saturation of the gimbal angular velocities for implementation; therefore, the modified logic is included in attitude control examples.

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105

2. Pyramid-type SGCMG system

where τ^ i , A, and θ are expressed as follows:

2.1. Attitude control torque

τ^ i ¼

Consider a pyramid-type SGCMG system that consists of four SGCMGs in a pyramid-type arrangement, as shown in Fig. 1, where the i-th CMG is denoted by CMG-i (i ¼ 1; 2; 3; 4). Let g^ i denote a unit vector in the direction of the gimbal axis of CMG-i (i ¼ 1; 2; 3; 4). The unit vectors g^ i are expressed in the body-fixed coordinates as follows:

The matrix A is the Jacobian matrix of the total angular momentum h with respect to the gimbal angular velocities θ_ i divided by the magnitude of the angular momentum of each CMG hw. The attitude control torque to the spacecraft, τ c , is given by τ c ¼  τ.

2 6 g^ 1 ¼ 4

sin β 0

3 7 5;

cos β 2 6 g^ 3 ¼ 4

2

0

3 0 6 7 g^ 4 ¼ 4  sin β 5; cos β

2

ð1Þ

6 h^ 1 ¼ 4

 sin θ1 cos β cos θ1 sin θ1 sin β

2 6 h^ 3 ¼ 4

sin θ3 cos β  cos θ3 sin θ3 sin β

3 7 5;

3 7 5;

2

3  cos θ2 6 7 h^ 2 ¼ 4  sin θ2 cos β 5; sin θ2 sin β 2 3 cos θ4 6 sin θ cos β 7 ^ h4 ¼ 4 4 5; sin θ4 sin β

ð2Þ

ð3Þ

i

where hw denotes the magnitude of the angular momentum of each CMG. The torque τ generated from the CMG system is expressed as follows: ∂h^ _ _ τ ¼ ∑hw h^ i ¼ hw ∑ i θ_ i ¼ hw Aθ; i i ∂θ i

Fig. 1. Pyramid-type arrangement (θ1 ¼ θ2 ¼ θ3 ¼ θ4 ¼ 0).

The CMG system as well as the angular momentum and the gimbal angles is in the singular state if the matrix A is deficient, i.e., rank A o 3. The gimbal angles in the singular state are characterized by the solutions of the following equation: ð5Þ

Using the Binet–Cauchy theorem, this determinant is expanded as follows [11]: detðAAT Þ ¼ ∑ detðAi Þ2 ;

ð6Þ

i

where Ai denotes the square matrix that is obtained by eliminating the i-th column of A. It follows that the singular state is characterized by the gimbal angles θi ði ¼ 1; …; 4Þ satisfying the following four equations:

where θi denotes the gimbal angle of CMG-i, and it is initialized, i.e., θi ¼ 0, when h^ i lies in the xy-plane of the body-fixed coordinates (i ¼ 1; 2; 3; 4). The total angular momentum of the CMG system, h, is expressed as follows: h ¼ ∑hw h^ i ;

θ ¼ ½θ1 θ2 θ3 θ4 T :

detðAAT Þ ¼ 0:

where β denotes an inclination angle of the pyramid. Let h^ i denote the unit vector along the angular momentum of CMG-i (i ¼ 1; 2; 3; 4). The unit vectors h^ i are expressed in the body-fixed coordinates as follows: 2

A ¼ ½τ^ 1 τ^ 2 τ^ 3 τ^ 4 ;

2.2. Singular surface

3

6 7 g^ 2 ¼ 4 sin β 5; cos β

3  sin β 7 0 5; cos β

∂h^ i ; ∂θi

ð4Þ

sin θ2 sin θ3 cos θ4  sin θ2 cos θ3 cos θ4 cos β þ cos θ2 sin θ3 sin θ4 þ cos θ2 cos θ3 sin θ4 cos β þ 2 cos θ2 cos θ3 cos θ4 cos 2 β ¼ 0;

ð7Þ

sin θ1 cos θ3 sin θ4 þ sin θ1 cos θ3 cos θ4 cos β þ cos θ1 sin θ3 sin θ4  cos θ1 sin θ3 cos θ4 cos β þ 2 cos θ1 cos θ3 cos θ4 cos 2 β ¼ 0;

ð8Þ

sin θ1 sin θ2 cos θ4 þ sin θ1 cos θ2 sin θ4 þ cos θ1 sin θ2 cos θ4 cos β  cos θ1 cos θ2 sin θ4 cos β þ 2 cos θ1 cos θ2 cos θ4 cos 2 β ¼ 0;

ð9Þ

 sin θ1 sin θ2 cos θ3 þ sin θ1 cos θ2 cos θ3 cos β  cos θ1 sin θ2 sin θ3  cos θ1 cos θ2 sin θ3 cos β  2 cos θ1 cos θ2 cos θ3 cos 2 β ¼ 0:

ð10Þ

Because the rank of A in the singular state is generally 2 in the case of pyramid-type SGCMG systems, two of Eqs. (7)–(10) are independent and sufficient to characterize the singular state. For example, by solving the first and third equations with respect to tan θ1 and tan θ3 , the following relations are obtained: tan θ1 ¼ 

cos βð tan θ2  tan θ4 þ2 cos βÞ ; tan θ2 þ tan θ4

ð11Þ

tan θ3 ¼ 

cos βð  tan θ2 þ tan θ4 þ 2 cos βÞ : tan θ2 þ tan θ4

ð12Þ

Eqs. (11) and (12) are both satisfied in the singular state. In the singular state, the CMG system cannot generate the attitude control torque τ c in arbitrary directions. Let u^ denote a unit vector that is orthogonal with each column of the matrix A. This vector u^ is called the singular vector;

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the CMG system cannot generate the attitude control torque along u^ in the sense of the first-order approximation. The singular vector u^ is obtained by the singularvalue decomposition of matrix A, as described later. A set of all of the angular momentum values in the singular state constitutes the singular surfaces in the angular momentum space. Singular surfaces exist for any direction in the angular momentum space and are classified as external singular surfaces or internal singular surfaces. The singular states of the pyramid-type CMG system are classified into the following three cases by the signs of the inner products u^  h^ i (i ¼ 1; …; 4): 4H, where all of the inner products have the same signs; 2H, where one of the inner products has a different sign than the others; and 0H, where two of the inner products have positive signs and two of the inner products have negative signs. All of the 4H singular states and a portion of the 2H singular states compose the external singular surfaces; the other portion of the 2H singular states and all of the 0H singular states compose the internal singular surfaces. 3. Directional passability of singular surface

The CMG system in the singular state cannot generate the attitude control torque along u^ in the sense of the firstorder approximation but may generate it in the sense of a higher-order approximation. This leads to the concept of “passability” by using null motion. Let Δθi denote the variations of the gimbal angles ði ¼ 1; …; 4Þ and Δh denote the corresponding variations of the angular momentum from the singular surface. It follows from Eq. (2) that each h^ i depends on θi but does not depend on θj ðj a iÞ. Hence, the second-order approximation of Δh with respect to Δθi is given by ð13Þ

ð14Þ

Δθ ¼ ½Δθ1 Δθ2 Δθ3 Δθ4 T

ð15Þ

Δ

Δθ22

Δθ23

Δθ24 T :

ð16Þ

Consider the singular value decomposition of A as follows: T

A ¼ UΣV T ¼ ∑si u^ i v^ i ;

ð17Þ

i

where si is the singular value of the matrix A, U is a 3  3 orthogonal matrix, and V is a 4  4 orthogonal matrix, u^ i and v^ i are the i-th column vectors of U and V, respectively. In the case of the pyramid-type arrangement, the rank of A is 2 in the singular state; therefore, the singular values si satisfy s1 Z s2 4 0, s3 ¼ 0. It follows that the range space of matrix A is spanned by vectors u^ 1 and u^ 2 , whereas u^ 3 ¼ u^ without loss of generality. By premultiplying Eq. (13) by U T , the projections of Δh on u^ i ði ¼ 1; 2; 3Þ are given by   ð18Þ Δh1 ¼ Δh  u^ 1 ¼ hw s1 α1  12 αT G1 α

ð20Þ

α ¼ ½α1 α2 α3 α4 T ¼ ½v^ 1  Δθ v^ 2  Δθ v^ 3  Δθ v^ 4  ΔθT ð21Þ and Gi ði ¼ 1; 2; 3Þ are given by Gi ¼ V T S i V

ð22Þ

h i S i ¼ diag u^ i  h^ 1 ; u^ i  h^ 2 ; u^ i  h^ 3 ; u^ i  h^ 4 :

ð23Þ

A variation of the gimbal angles, Δθi , is called null motion if it does not generate the torque; in other words, it generates a zero variation of the angular momentum Δh ¼ 0 in the sense of the first-order approximation. Eqs. (18)–(20) are helpful to analyze null motion. Nonzero α1 and α2 generate nonzero Δh but nonzero α3 and α4 generate zero Δh in the sense of the first-order approximation. Hence, the null motion is characterized by the following equation: ð24Þ

A singular surface is passable if the angular momentum passes the singular surface from one side to the other side by using the null motion. Because α1 and α2 generate null motion, α3 and α4 are available to generate Δh3 in the sense of the second-order approximation. Divide Gi into 2  2 submatrices as follows: " # Gi11 Gi12 Gi ¼ ; ð25Þ GTi12 Gi22 where

2

Gi11 ¼ 4

Gi12 ¼ 4

H ¼ ½h^ 1 h^ 2 h^ 3 h^ 4 

θ ¼ ½Δθ21

hw T α G3 α; 2

ð19Þ

where αi ði ¼ 1; …; 4Þ is the projection of Δθ on v^ i given by

2

where

2

Δh3 ¼ Δh  u^ 3 ¼ 

Δθ ¼ v^ 3 α3 þ v^ 4 α4 :

3.1. Concept of directional passability

∂h ∂2 h hw Δh C ∑ Δθi þ ∑ Δθi Δθj ¼ hw AΔθ  HΔ2 θ; 2 i ∂θ i i;j ∂θ i ∂θ j

  Δh2 ¼ Δh  u^ 2 ¼ hw s2 α2  12 αT G2 α

2 Gi22 ¼ 4

T v^ 1 T v^ 2 T v^ 1 T v^ 2 T v^ 3 T v^ 4

3 5S i ½v^ 1 v^ 2 

ð26Þ

3 5S i ½v^ 3 v^ 4 

ð27Þ

3 5S i ½v^ 3 v^ 4 

ð28Þ

The second-order condition for this second-order approximation is that the eigenvalues of G322 have both positive and negative signs, which is referred to as the hyperbolic singularity. Because G322 is a 2  2 matrix, this condition for the passability can be rewritten as follows: det G322 o0:

ð29Þ

Then, the condition for passability is reduced to hyperbolic singularity; the hyperbolic singular surface is passable along both u^ and  u^ by using null motion. Only the elements of α3 and α4 are available in null motion. The other elements of α1 and α2 will also be available if the gimbal angles are not restricted to null motion. It should be noted that not all the singular surfaces are hyperbolic. An elliptic singular surface is another major obstacle in passing a singular

K. Yamada, I. Jikuya / Acta Astronautica 102 (2014) 103–123

surface. Hence, this paper introduces a new concept of “directional passability” to investigate the structure of an elliptic singular surface in detail. An internal singular surface is directionally passable if the angular momentum passes the singular surface in the ^ which is not restricted to 7 u. ^ An specified direction p, external singular surface is directionally passable if the angular momentum escapes from the singular surface in ^ which is not also restricted to the specified direction p, ^ but is restricted to directly inside the singular surface. 7 u, The latter definition is properly introduced because the angular momentum cannot be located outside an external singular surface. In the next subsection, the conditions for directional passability are studied for an arbitrary direction. Then, the conditions are further investigated for specific directions. 3.2. Directional passability analysis 3.2.1. Directional passability along arbitrary direction Consider the variation of the angular momentum ^ Δh ¼ εp;

ð30Þ

where ε 40 denotes a scalar variation, and p^ is a fixed unit vector specifying the direction of the variation. When the angular momentum passes the singular surface, p^ is not tangent to the singular surface. Hence, the vector p^ is supposed to be p^  u^ a 0 without loss of generality. By substituting Eq. (30) into Eqs. (18) and (19), α1 and α2 are determined in the sense of the first-order approximation. By substituting Eq. (30) and u^ 3 ¼ u^ into Eq. (20), the following equation is obtained: εp^  u^ ¼ 

hw T α G3 α: 2

ð31Þ

Divide α as follows: " # " # α1 α3 α1 ¼ ; α2 ¼ : α2 α4

ð32Þ

Then, Eq. (31) is expressed as follows: hw T hw ^ α G322 α2 þhw αT1 G312 α2 þ αT1 G311 α1 ¼  εp^  u: 2 2 2

ð33Þ

Diagonalize G322 by using a 2  2 orthogonal matrix P as " # λ1 0 G322 ¼ P T P; ð34Þ 0 λ2 where λ1 and λ2 are real eigenvalues of G322 . Then, α2 is transformed into ξ1 and ξ2 as " # ξ1 ð35Þ ¼ Pα2 : ξ2 Eq. (33) is expressed as follows: λ1 ξ21 þ λ2 ξ22 þ c1 ξ1 þc2 ξ2 ¼ c0 ; where " # c1 ¼ 2PGT312 α1 ; c2

c0 ¼ 

2ε p^  u^ αT1 G311 α1 : hw

ð36Þ

ð37Þ

107

The problem here is to find ξ1 and ξ2 from Eq. (36). If the sign of c0 is the same as that of λ1 or λ2, Eq. (36) has a real solution with respect to ξ1 and ξ2. Because α is the firstorder variation, the sign of c0 is determined by that of ^ Therefore, if the sign of either λ1 or λ2 is opposite to  p^  u. ^ there are real solutions ξ1 and ξ2 in Eq. (36). that of p^  u, Hence, the condition for the directional passability along p^ is reduced to the following condition: the sign of either λ1 ^ i.e., or λ2 is opposite to that of p^  u, λ1 p^  u^ o0

and=or

λ2 p^  u^ o 0:

ð38Þ

The essence of directional passability is backward calculation from Δh to Δθ by combining the first-order approximation in Eqs. (18) and (19) and the second-order approximation in Eq. (36), where the class of gimbal motions is not restricted to null motion and is arbitrary. For implementation, it is important to generate sufficiently small Δθ from sufficiently small Δh. Consider the directionally passable case. Then, if ε in Eq. (30) converges to 0, α1 and α2 determined from Eqs. (18) and (19) and α3 and α4 determined from Eq. (36) converge to 0. It follows that there exists a sequence of Δθ that converges to 0 when Δh converges to 0. On the other hand, consider the directionally impassable case. Then, if ε in Eq. (30) converges to 0, α1 and α2 determined from Eqs. (18) and (19) converge to 0 but α3 and α4 are not determined from Eq. (36). It follows that there is no sequence of Δθ that converges to 0 when Δh converges to 0. In this sense, the directional passability and the continuity of Δθ with respect to Δh are equivalent. It is clear that directional passability is essential for smooth and precise steering of CMGs. In the case of hyperbolic singularity, the condition in Eq. (29) corresponds to λ1 λ2 o0. The condition in Eq. (38) ^ therefore, the hyperbolic singular is satisfied for any p; surface is directionally passable in any direction. Here, the relation between passability and directional passability in any direction is not obvious for the following reasons. Because motions of gimbal angles are not restricted to null motion in the definition of directional passability, the directional passability in any direction does not immediately imply passability. Because a passing direction is not specified in the definition of passability, passability also does not immediately imply the directional passability in any direction. However, it is interesting to observe that both concepts of passability and directional passability in any direction are actually equivalent because the hyperbolic singular surface is passable [19]. In the case of elliptic singularity, the condition of det G322 4 0 corresponds to λ1 λ2 40. The condition in ^ Because λ1 and Eq. (38) is not always satisfied for any p. λ2 have the same sign, Eq. (38) is reduced to λ1 p^  u^ o 0 (and equivalently λ2 p^  u^ o 0). It follows that if the condition λ1 p^  u^ o 0 (and equivalently λ2 p^  u^ o0) is satisfied, the elliptic singular surface is directionally passable along ^ Otherwise, the elliptic singular surface is directionally p. ^ It is interesting to observe that if the impassable along p. ^ elliptic singular surface is directionally passable along p, it is directionally impassable along  p^ and vice versa. The relations between the directional passability and the related conditions are summarized in Table 1.

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3.2.2. Directional passability along singular vector u^ In this subsection, the directional passability along the singular vector u^ is further investigated. Consider the variation of the angular momentum ^ Δh ¼ εu;

ð39Þ

^ By substituting Eq. (39) into where p^ is specified as p^ ¼ u. Eqs. (18) and (19), α1 and α2 are determined to be α1 ¼ 0 and α2 ¼ 0 in the sense of the first-order approximation. By substituting Eq. (39) and u^ 3 ¼ u^ into Eq. (20), the following equation is obtained: ε¼ 

hw T α G322 α2 : 2 2

ð40Þ

Here, let us consider the 4H singularity surface, on which the inner products u^  h^ i have common signs ði ¼ 1; …; 4Þ. Because the 4H singular surface is an external singular surface, it is not passable but might be directionally passable from the 4H singularity surface to its inside. By the definition of G322 in Eq. (28), G322 is definite. There are real solutions α3 and α4 of Eq. (40) where ε and the eigenvalues of G322 have opposite signs. However, there are no real solutions α3 and α4 of Eq. (40) where ε and the eigenvalues of G322 have the same sign. The analysis here proves that the 4H singularity surface is not passable, but it composes the momentum envelope. In contrast, this also proves that the 4H singularity surface can be directionally passable into its inside. 3.2.3. Directional passability in opposite direction of angular momentum h^ In this subsection, the directional passability in the opposite direction of the angular momentum h^ ¼ h= J h J is further investigated. Consider the variation of the angular momentum ^ Δh ¼ εh;

ð41Þ

^ Then, it can be shown that where p^ is specified as p^ ¼  h. the singularity surface is always directionally passable in ^ The the opposite direction of the angular momentum h. proof of this claim is quite long and therefore is given in the Appendix. The directional passability in the opposite direction of the angular momentum h^ is nontrivial and interesting. Moreover, its consequence is helpful for analyzing the singular surface. Suppose that the singularity surface is directionally passable in the same direction of the angular ^ The sign of one eigenvalue of G momentum h. 322 is ^ Because the sign of one opposite to that of h^  u. Table 1 Classification of singularity (D.P. and D.Imp. denote directionally passable and directionally impassable, respectively). Singularity

Condition

Passability

Continuity

Hyperbolic

λ1 λ2 o 0

Yes

Elliptic

λ1 λ2 4 0 and λ1 p^  u^ o 0 λ1 λ2 4 0 and λ1 p^  u^ 4 0

Passable D.P. in any direction D.P. along p^ D.Imp. along p^

No

Yes

Table 2 Directional passability along 7 h^ (D.P. denotes directionally passable). Along

Assertion

p^ ¼ h^

^ the singular state is hyperbolic and passable If D.P. along h,

p^ ¼  h^ Always D.P. along  h^

^ the eigeneigenvalue of G322 is identical to that of h^  u, values of G322 have positive and negative signs. Hence, if the singularity surface is directionally passable in the same ^ the singular state is direction of the angular momentum h, hyperbolic and therefore is passable in any direction of the singular surface. The results of the directional passability in the directions of h^ and  h^ are summarized in Table 2.

4. Quadratic steering logic 4.1. Quadratic constrained quadratic optimization approach The directional passability in the previous section merely guarantees the existence sufficiently small Δθ for sufficiently small Δh. For practical purposes, an explicit _ in procedure to generate the gimbal angle velocities, θ, response to the control torque, τ c , is required. The objective of this subsection is to propose a new steering logic, referred to as quadratic steering logic (QSL), which is based on the quadratic constrained quadratic optimization problem. Consider a variation in the gimbal angles, Δθ, over a time interval, Δt. Then, a variation in the control torque is given by Δh, whose second-order approximation with respect to Δθ is given by Eq. (13). The problem here is to minimize the quadratic cost function   J ¼ 12 Δθ21 þ Δθ22 þ Δθ23 þ Δθ24 ; ð42Þ subject to the quadratic constraints given by Eq. (13). It should be noted that if the second-order terms in Eq. (13) are neglected, the problem is reduced to the linear constrained quadratic optimization problem, minθ J subject to Δh ¼ hw AΔθ [17]. Hence, QSL is an extension of the well-known pseudo-inverse steering logic, Δθ ¼ A† Δh=hw , where A† is the pseudo-inverse of A. Although QSL is not analytically solvable, it has the potential to pass the singular surface and approximate the control torque accurately rather than the linear constrained problem.

4.2. Numerical algorithm For simplicity, QSL is divided into three steps: (1) construction of the Gröbner bases, (2) selection of the free parameter, and (3) the Newton method.

4.2.1. Construction of the Gröbner bases Eq. (13) consists of three equations and Δθ consists of four elements; therefore, one element in Δθ becomes a free parameter. By selecting Δθ4 , for instance, for this free

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represented by the function of Δθ4 . The variation in the gimbal angles, Δθ, satisfying both Eq. (44) and Δθ4 ¼ 0 may not be unique but is represented by the solution of

parameter, Eq. (13) is rewritten as follows: hw Δh hw τ^ 4 Δθ4  h^ 4 Δθ24 2 2 3 2 23 Δθ1 Δθ1 h 6 Δθ 7 6 Δθ2 7 w ¼ hw A4 4 2 5  H 4 4 2 5; 2 Δθ3 Δθ23

ð43Þ

where A4 and H 4 are 3  3 matrices obtained by removing the 4-th column of matrices A and H, respectively. Here, the gimbal angles can be determined by the pseudoinverse logic when A4 is full column rank. It can be shown that either A4 or H 4 are full column rank except for isolated points; therefore, those exceptional points are negligible in numerical computations. Because the first order solutions are obtained when A4 is full rank, H 4 is supposed to be column full rank in the following numerical algorithm. From Eq. (43), Δθ21 , Δθ22 , and Δθ23 are expressed as linear equations with respect to Δθ1 , Δθ2 , and Δθ3 , respectively. Simultaneous second-order equations with respect to Δθ1 , Δθ2 , and Δθ3 can be solved by using the Gröbner bases because H 4 is full column rank. By premultiplying H †4 , Eq. (43) is transformed to an equation of the form 2 23 2 3 2 3 Δθ1 b1 Δθ1 6 Δθ2 7 6 Δθ 7 6 b 7 ¼ B þ ð44Þ 4 25 4 2 5 4 2 5; Δθ23

Δθ3

109

b3

where B, b1, b2, and b3 are determined from the coefficients of Eq. (43). By choosing the pure lexical order d1 4d2 4d3 , the Gröbner bases g1, g2, g3 are obtained from Eq. (44) by symbolic computations. The explicit forms of g1, g2, and g3 are omitted here for simplicity. Then, Eq. (44) is substituted for the following equations: g 3 ðΔθ3 ; Δθ4 Þ ¼ 0

ð45Þ

g 2 ðΔθ2 ; Δθ3 ; Δθ4 Þ ¼ 0

ð46Þ

g 1 ðΔθ1 ; Δθ3 ; Δθ4 Þ ¼ 0:

ð47Þ

In Eq. (45), g 3 ðΔθ3 ; Δθ4 Þ ¼ 0 becomes an eighth-order algebraic equation with respect to Δθ3 . From this equation, the value of Δθ3 can be obtained when the value of arbitrary variable Δθ4 is set. Eqs. (46) and (47), i.e., g 2 ðΔθ2 ; Δθ3 ; Δθ4 Þ ¼ 0 and g 1 ðΔθ1 ; Δθ3 ; Δθ4 Þ ¼ 0, are linear equations with respect to Δθ2 and Δθ1 , respectively. When the value of Δθ3 is obtained, the values of Δθ2 and Δθ1 are easily obtained by solving these equations. Hence, once the eighth-order algebraic equation g 3 ðΔθ3 ; Δθ4 Þ ¼ 0 is solved, all solutions of Eq. (44) can be easily obtained. 4.2.2. Selection of free parameter The parameterization utilizing the Gröbner bases is available for decreasing the norm of Δθ with respect to the parameter. The remaining problems are the selection of the parameter from Δθ1 to Δθ4 and the optimization with respect to the selected parameter. Because simultaneous optimizations are complicated, parameter selection is carried out first. The free parameter is specified as Δθ4 at first. The variation in the gimbal angles, Δθ, satisfying Eq. (44) is

g 3 ðΔθ3 ; 0Þ ¼ 0

ð48Þ

g 2 ðΔθ2 ; Δθ3 ; 0Þ ¼ 0

ð49Þ

g 1 ðΔθ1 ; Δθ3 ; 0Þ ¼ 0:

ð50Þ

Eq. (48) has at most eight real solutions, and then, Eqs. (49) and (50) have the same number of real solutions. The minimum norm of Δθ satisfying both Eq. (44) and Δθ4 ¼ 0 is determined among possible real solutions for the free parameter Δθ4 . In a similar way, the free parameter is specified as Δθj for each j ¼ 1; …; 4. The minimum norm of Δθ satisfying both Eq. (44) and Δθj ¼ 0 is determined among possible real solutions for the free parameter Δθj . For the purpose of numerically efficient optimization, the free parameter Δθj is selected such that the minimum norm of Δθ with Δθj ¼ 0 is minimal. 4.2.3. Newton method Once the free parameter Δθj is selected, the obtained Δθ is selected as the initial value in the numerical optimization; then, the remaining problem is to minimize the norm of Δθ with respect to the selected parameter. Because the derivatives of the cost function with respect to the free parameter are available by using the properties of the Gröbner bases, the numerical optimization is carried out based on the Newton method. Consider the following cost function J in Eq. (42) which is used to minimize the norm of Δθ with respect to the selected parameter, where the selected parameter is supposed to be θ4 without loss of generality. The Newton method consists of the following iteration:  J0  ; ð51Þ Δθ4ðn þ 1Þ ¼ Δθ4n   J″ Δθ4 ¼ Δθ4n where Δθ4n denotes the n-th iteration with the initial value Δθ40 ¼ 0 and the superscript 0 denotes the derivative with respect to parameter Δθ4 , that is, J0 ¼

dJ ; dΔθ4

2

J″ ¼

d J : dΔθ24

ð52Þ

First, the derivatives J 0 and J″ are necessary to compute the right-hand side of Eq. (51). The differentiations of the Gröbner bases, g3, g2, and g1, with respect to Δθ4 are computed in order to compute J 0 . It follows from Eqs. (45) to (47) that 2 32 0 3 2 3 Δθ1 0 0 g 33 g 34 6 0 g 6 0 7 6 7 g 23 7 ð53Þ 4 54 Δθ2 5 ¼  4 g 24 5; 22 Δθ03 g 11 0 g 13 g 14 where gij denotes the derivative of gi with respect to Δθj . The derivative J 0 is obtained by substituting Eq. (53) into the following equation: J 0 ¼ Δθ1 Δθ01 þΔθ2 Δθ02 þ Δθ3 Δθ03 þ Δθ4 :

ð54Þ

Next, the second differentiations of the Gröbner bases, g1, g2, and g3, with respect to Δθ4 are computed in order to

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K. Yamada, I. Jikuya / Acta Astronautica 102 (2014) 103–123

It follows from Eqs. (45) to (47) that 32 ″ 3 g 33 Δθ1 6 Δθ″ 7 g 23 7 54 2 5 g 13 Δθ″3 3 g 344 þ 2g 334 Δθ03 þg 333 Δθ02 3 6 g þ 2g Δθ0 þ 2g Δθ0 þ g Δθ02 7 ¼  4 244 224 234 233 2 3 3 5 g 144 þ 2g 114 Δθ01 þ 2g 134 Δθ03 þ g 133 Δθ02 3

equation:

compute J″. 2 0 0 6 0 g 4 22 g 11 0 2

^ q^_ ¼ 12 q^  ω:

ð59Þ

The Euler parameters, qe , and the vector, ωe , representing the error attitude and the error attitude rate, respectively, are expressed as ð55Þ

where gijk denotes the second partial derivative of gi with respect to Δθj and Δθk . Note that g 222 ¼ g 223 ¼ g 311 ¼ g 313 ¼ 0 by definition. The second derivative J″ is obtained by substituting Eqs. (53) and (55) into the following equation: ″ 02 ″ 02 ″ J″ ¼ Δθ02 1 þ Δθ 1 Δθ 1 þΔθ 2 þ Δθ 2 Δθ 2 þΔθ 3 þ Δθ 3 Δθ 3 þ1:

ð56Þ

†

qe ¼ q^  q

ð60Þ

^  ω; ωe ¼ ω

ð61Þ

† ^ The following where q^ is the conjugate quaternion of q. control torque

τ c ¼ kd ωe þ kp Vðqe Þ

ð62Þ

will be considered in the numerical simulations, where kd and kp are control gains and Vðqe Þ is the vector part of qe .

Finally, the Newton method can be executed by substituting Eqs. (54) and (56) into Eq. (51).

5.2. CMG steering logic

5. Application to attitude control

The objective of CMG steering logic is to determine the _ in response to the control gimbal angular velocities, θ, torque, τ c , satisfying

5.1. Attitude control example

_ τ c ¼  τ ¼ hw Aθ;

This section presents an attitude control example of a spacecraft by using QSL. The equations of motion of the spacecraft are expressed as

where the second relation is due to Eq. (4). Let Δt denote the sampling interval of control. Then, the variation in attitude angular momentum on each sampling interval is given by

_ þ ω  hT ¼ τ c ; Jω

ð57Þ

where J is the inertia matrix of the spacecraft, ω is the angular velocity of the spacecraft, hT is the total angular momentum of the spacecraft, and τ c is the control torque of the spacecraft. The matrix and vectors are expressed in body-fixed coordinates. Let q denote the Euler parameters representing the attitude of the spacecraft body with respect to the inertia coordinates. The kinematics are expressed as q_ ¼ 12 q  ω;

ð58Þ

where  is the quaternion multiplication. Let q^ denote the Euler parameters representing the target attitude of the ^ denote the target angular velocity spacecraft body and ω ^ satisfy the following of the spacecraft, where q^ and ω

Δh ¼ τ c Δt:

ð63Þ

ð64Þ

By substituting Δh into Eq. (43), Δθ is computed by QSL. The numerical algorithm presented in the previous section is applicable not only to the directionally passable singular state but also to the other cases, i.e., the directionally impassable singular state and the non-singular state. In the directionally impassable case, it is not always possible to find sufficiently small Δθ of Eq. (13) for sufficiently small Δh. In the directionally passable singular state and in the non-singular state, it is possible to find sufficiently small Δθ of Eq. (13) for sufficiently small Δh; however, it is not always possible to find sufficiently small Δθ of Eq. (13) for a specific fixed value of Δt. The rate saturation of gimbals needs to be satisfied for implementation.

Fig. 2. Singular surface (β ¼ π=4 rad). (a) 4H singular surface, (b) 2H singular surface and (c) 0H singular surface. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

K. Yamada, I. Jikuya / Acta Astronautica 102 (2014) 103–123

Then, θ_ is implemented as follows: 8 Δθ > > if J ΔθJ o θ_ max Δt < _θ ¼ Δt Δθ _ > > θ max if J ΔθJ Z θ_ max Δt; : J ΔθJ

111

6. Numerical examples 6.1. Visualization of directionally passable singular surface ð65Þ This subsection demonstrates the classification of the singular surfaces into the 4H, 2H, and 0H singular surfaces for a pyramid-type SGCMG system with the inclination angle β ¼ π=4 rad. In order to investigate the structure of

where θ_ max 4 0 is the threshold that accounts for the _ practically allowable magnitude of θ.

Fig. 3. Singular surface (β ¼ π=4 rad, directional passability along þz-axis). (a) 4H singular surface, (b) 2H singular surface and (c) 0H singular surface. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

3

θ1 θ2 θ3 θ4

2

x y z

Momentum / hw

Gimbal Angle [rad]

3

1

0

2

1

0

-1 0

0.2

0.4

0.6

0.8

-1

1

0

0.2

Momentum Ratio

0.6

0.8

1

3

θ1 θ2 θ3 θ4

2

x y z

Momentum / hw

Gimbal Angle [rad]

3

1

0

-1

0.4

Momentum Ratio

2

1

0

0

0.2

0.4

0.6

1 - (Momentum Ratio)

0.8

1

-1

0

0.2

0.4

0.6

0.8

1

1 - (Momentum Ratio)

Fig. 4. Motion planning of total angular momentum of CMGs via gimbal angles (Case B1). (a) Gimbal angles in incremental motion, (b) angular momentum in incremental motion, (c) gimbal angles in decremental motion and (d) angular momentum in decremental motion.

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K. Yamada, I. Jikuya / Acta Astronautica 102 (2014) 103–123

of the 4H singular surface is shown in blue, which shows the directional impassability along the þz-axis. The 2H singular surface is shown in Fig. 3(b). The outer upper parts of the 2H singular surface are shown in blue, which shows the directional impassability along þz-axis, and the other parts of the 2H singular surface are shown in green, which shows the directional passability along the þz-axis. The 0H singular surface is shown in Fig. 3(c). The upper edge of the 0H singular surface is shown in blue, which shows the directional impassability along þz-axis, and the other parts of the 0H singular surface are shown in green, which shows the directional passability along the þz-axis. It is interesting to observe that compared to the impassable surfaces (see Fig. 2), the directionally impassable surfaces along the þz-axis (see Fig. 3) are biased toward the region where the z-component of the angular momentum has a þ sign. This observation is consistent with the analytical results in Section 3.2.3; the singular surface is always directionally passable in the oppo^ site direction of the angular momentum h.

passability, each singular surface is visualized by using the relation expressed in Eqs. (11) and (12). The passability/ impassability is examined by using Eq. (29) at each point on the singular surface and is displayed using green/blue color, respectively, where the value of the angular momentum is divided by hw. The 4H singular surface is shown in Fig. 2(a). The whole 4H singular surface is shown in blue, which shows the impassability. This is consistent with the fact that the external singular surface is always impassable. The 2H singular surface is shown in Fig. 2(b). The inner parts of the 2H singular surface are shown in green, which shows the passability, and the outer parts of the 2H singular surface are shown in blue, which shows the impassability. The 0H singular surface is shown in Fig. 2(c). The outer edge of the 0H singular surface is shown in blue, which shows the impassability, and the other inner parts of the 0H singular surface are shown in green, which shows the passability. For comparison, the directional passability along the þz-axis is considered, namely, p^ ¼ ½0 0 1T in Eq. (30). The directional passability/impassability along the þz-axis is examined by using Eq. (36) at each point on the singular surface and is displayed in green/blue color, respectively. The 4H singular surface is shown in Fig. 3(a). The lower part of the 4H singular surface is shown in green, which shows the directional passability along the þz-axis, and the upper part

6.2. Constant torque maneuvering examples This subsection demonstrates constant torque maneuvering examples for the pyramid-type arrangement of four CMGs with β ¼ π=4 rad. The total angular momentum of CMGs and the gimbal angles are discretized for implementation. The 4

3

x y z

2

Momentum / hw

Gimbal Angle [rad]

3 1 0 -1 -2

θ1 θ2 θ3 θ4

-3 -4 0

2

1

0

0.2

0.4

0.6

0.8

-1

1

0

0.2

Momentum Ratio

0.4

0.6

0.8

4

3

x y z

2

Momentum / hw

Gimbal Angle [rad]

3 1 0 -1 -2

θ1 θ2 θ3 θ4

-3 -4

1

Momentum Ratio

0

0.2

0.4

0.6

1 - (Momentum Ratio)

0.8

2

1

0

1

-1

0

0.2

0.4

0.6

0.8

1

1 - (Momentum Ratio)

Fig. 5. Motion planning of total angular momentum of CMGs via gimbal angles (Case B2). (a) Gimbal angles in incremental motion, (b) angular momentum in incremental motion, (c) gimbal angles in decremental motion and (d) angular momentum in decremental motion.

K. Yamada, I. Jikuya / Acta Astronautica 102 (2014) 103–123

in Fig. 4(a). When the condition number of A is larger than a threshold, e.g., 10, the CMG system is evaluated to be in the singular state. As shown in these figures, the gimbal angles encounter the directionally impassable singular state at the end of an incremental motion. Fig. 4(c) shows the gimbal angles of the CMGs in a decremental motion, where the final gimbal angles in the incremental motion are set to the initial angles in the decremental motion. Fig. 4(d) shows the corresponding total angular momentum of the CMGs computed from the gimbal angles in Fig. 4(c). Similar to an incremental motion, the gimbal angles encounter the directionally passable singular state at the beginning of an incremental motion. The gimbal angles continuously change to the final gimbal angle, where the final gimbal angles in the decremental motion recover to the initial angles in the incremental motion. The second case, referred to as Case B2, demonstrates that the total angular momentum of the CMGs encounters the directionally impassable singular surface in the incremental motion and encounters the directionally passable singular surface in the decremental motion. Consider

total angular momentum is gradually increased from 0 to the external singular surface using Δh in the fixed direction (called incremental motion). Similarly, the total angular momentum is gradually decreased from the external singular surface to 0 using Δh in the opposite fixed direction (called decremental motion). The symbol ○ shows the passable singular state and □/  shows the directionally passable/ impassable singular state in the specified direction. The following three cases are demonstrated. The first case, referred to as Case B1, demonstrates that the total angular momentum of CMGs does not encounter the singular surface in either the incremental or the decremental motions. Consider Δh ¼ hmax1 α^ 1 =N

ð66Þ

where α^ 1 ¼ ½0:2673 0:5345 0:8018;

hmax1 ¼ 2:85hw ;

N ¼ 500: ð67Þ

The difference of the gimbal angles, Δθ, is calculated based on QSL at each point in the incremental and decremental motions. Fig. 4 shows the relation between the angular momentum of the CMGs and the gimbal angles. Fig. 4(a) shows the gimbal angles of the CMGs in an incremental motion. Fig. 4(b) shows the corresponding total angular momentum of the CMGs computed from the gimbal angles

Δh ¼ hmax2 α^ 2 =N

α^ 2 ¼ ½0:2277 0:8726 0:4321;

N ¼ 500:

4 x y z

3 1

Momentum / hw

Gimbal Angle [rad]

hmax2 ¼ 2:92hw ;

ð69Þ

2

0 -1 -2

θ1 θ2 θ3 θ4

-3 -4 0

2

1

0

0.2

0.4

0.6

0.8

-1

1

0

0.2

Momentum Ratio

0.4

0.6

0.8

1

Momentum Ratio

4

3 2

x y z

3

1

Momentum / hw

Gimbal Angle [rad]

ð68Þ

where

3

0 -1 -2

θ1 θ2 θ3 θ4

-3 -4

113

0

0.2

0.4

0.6

1 - (Momentum Ratio)

0.8

2 1 0

1

-1

0

0.2

0.4

0.6

0.8

1

1 - (Momentum Ratio)

Fig. 6. Motion planning of total angular momentum of CMGs via gimbal angles (Case B3). (a) Gimbal angles in incremental motion, (b) angular momentum in incremental motion, (c) gimbal angles in decremental motion and (d) angular momentum in decremental motion.

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the theoretical result that the singular state is always directionally passable in the decremental motion. The final gimbal angles in the decremental motion recover to the initial angles in the incremental motion. The third case, referred to as Case B3, demonstrates that the total angular momentum of the CMGs encounters the directionally impassable singular surface in the incremental motion and encounters the passable singular surface in the decremental motion. Consider

Fig. 5 shows the relation between the angular momentum of the CMGs and the gimbal angles in Case B2. Fig. 5(a) shows the gimbal angles of the CMGs in an incremental motion. Fig. 5(b) shows the corresponding total angular momentum of the CMGs computed from the gimbal angles in Fig. 5(a). As shown in these figures, the gimbal angles encounter the passable singular state at approximately 74% of the incremental motion. The difference of the gimbal angles, Δθ, smoothly changes in this passable singular state and therefore, it is acceptable for implementation. The gimbal angles also encounter the directionally impassable singular state at the end of the incremental motion. Fig. 5(c) shows the gimbal angles of the CMGs in a decremental motion, where the final gimbal angles in the incremental motion are set to the initial angles in the decremental motion. Fig. 5(d) shows the corresponding total angular momentum of the CMGs computed from the gimbal angles in Fig. 5(c). As shown in these figures, the gimbal angles encounter the directionally passable singular state at the beginning of the decremental motion and the passable singular state at approximately 26% of the decremental motion. The difference of the gimbal angles, Δθ, smoothly changes in this passable singular state. This supports

where α^ 3 ¼ ½1 0 0;

x y z

10

0

-1.6

15

0

5

x y z

5

10

Body Rate Error [deg/s]

Euler Parameters Error

x 10-4

0

10

15

CMG Momentum [Nms]

Gimbal Rates [rad/s]

0

Time [s]

5

15

4

0

0

5

10

0

-4

15

θ1 θ2 θ3 θ4

0

5

10 x y z

60 40 20

0

5

10

Time [s]

10

15

Time [s]

80

0

10

Time [s]

x y z

-0.015

15

θ1 θ2 θ3 θ4

5

0

Time [s]

1

0

x y z

-10

15

0.015

Time [s]

-1

0

Time [s]

0

-1

10

Gimbal Angles [rad]

5

ð71Þ

10 x y z

15

Condition Number

0

N ¼ 500:

Fig. 6 shows the relation between the angular momentum of the CMGs and the gimbal angles in Case B3. Fig. 6(a) shows the gimbal angles of the CMGs in an incremental motion. Fig. 6(b) shows the corresponding total angular momentum of the CMGs computed from the gimbal angles in Fig. 6(a). As shown in these figures, the gimbal angles encounter the directionally impassable singular surface at approximately 40% of the incremental motion, and they then discontinuously

Time [s]

1

hmax2 ¼ 3:41hw ;

Torque [Nm]

0

-0.14

ð70Þ

1.6

Body Rate [deg/s]

Euler Parameters

0.14

Δh ¼ hmax3 α^ 3 =N

10

10

2

1

0

0

5

10

15

Time [s]

Fig. 7. Attitude control example for singularly escape/avoidance steering logic (Case C1). (a) Vector part of attitude, (b) attitude rate, (c) control torque, (d) vector part of error attitude, (e) error attitude rate, (f) gimbal angles of CMGs, (g) gimbal angle rates of CMGs, (g) angular momentum of CMGs and (i) condition number of A.

K. Yamada, I. Jikuya / Acta Astronautica 102 (2014) 103–123

with β ¼ π=4 rad. The simulation parameters in the equations of motion in Eq. (57) and the control torque in Eq. (62) are specified as follows: 2 3 2000 0 0 6 7 2000 0 5 kg m2 ; hw ¼ 30 N m s; J ¼4 0 ð72Þ 0 0 2000

change. The difference of the gimbal angles, Δθ, is large and is not acceptable for implementation. Further implementation issues will be demonstrated and discussed for attitude control examples in the next subsection. The gimbal angles also encounter the directionally impassable singular state at the end of the incremental motion. Fig. 6(c) shows the gimbal angles of CMGs in a decremental motion, where the final gimbal angles in the incremental motion are set to the initial angles in the decremental motion. Fig. 6(d) shows the corresponding total angular momentum of the CMGs computed from the gimbal angles in Fig. 6(c). As shown in these figures, the gimbal angles encounter the directionally passable singular state at the beginning of the decremental motion and the passable singular state at approximately 60% of the decremental motion. This supports the theoretical result that the singular state is always directionally passable in the decremental motion. The gimbal angles smoothly change around the singular state and are therefore implementable.

kp ¼ 50; 000 N m=rad;

qð0Þ ¼ 1;

where ω^ i is the i-component of ω, ω0i is the i-component of ^ is ω0 , and ωfi is the i-component of ωf . The target attitude, q, then computed from Eq. (59).

10

0

-1.6

15

10 x y z

Torque [Nm]

Body Rate [deg/s]

0

5

0 x y z

5

10

10

Time [s]

15

CMG Momentum [Nms]

Gimbal Rates [rad/s]

0

0

0

5

10

15

0

-4

15

θ1 θ2 θ3 θ4

0

5

10 x y z

60 40 20

0

5

10

Time [s]

10

15

Time [s]

80

0

10

4 x y z

-0.015

15

θ1 θ2 θ3 θ4

5

5

Time [s]

1

0

0

Time [s]

0.015

Time [s]

-1

x y z

-10

15

Gimbal Angles [rad]

x 10-4

0

0

Time [s]

Body Rate Error [deg/s]

Euler Parameters Error

Time [s]

10

15

Condition Number

Euler Parameters

x y z

-1

ð74Þ

1.6

0

1

^ qð0Þ ¼ 1:

fi

0.14

5

ð73Þ

^ is gradually increased from the The target angular velocity, ω, initial target velocity, ω0 , to the final target velocity, ω0 þ ωf , and is then fixed to the final target velocity, i.e., 8 < ωfi t t oT T ði ¼ x; y; zÞ; T ¼ 10 s; ð75Þ ω^ i ¼ ω0i þ :ω t ZT;

This subsection demonstrates attitude control examples for a spacecraft equipped with a pyramid-type SGCMG system

0

kd ¼ 10; 000 N m s=rad:

^ The initial values of the attitude, q, and the target attitude, q, are specified as follows:

6.3. Attitude control example

-0.14

115

10

10

2

1

0

0

5

10

15

Time [s]

Fig. 8. Attitude control example for quadratic steering logic (Case C1). (a) Vector part of attitude, (b) attitude rate, (c) control torque, (d) vector part of error attitude, (e) error attitude rate, (f) gimbal angles of CMGs, (g) gimbal angle rates of CMGs, (h) angular momentum of CMGs and (i) condition number of A.

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K. Yamada, I. Jikuya / Acta Astronautica 102 (2014) 103–123

To implement the control torque, τ c , in Eq (62), QSL isapplied to generate θ_ satisfying Eq. (63). The variation in attitude angular momentum in each sampling interval is given by Eq. (64), where the sampling interval is Δt ¼ 0:01 s. By substituting Δh into Eq. (43), Δθ is obtained, as presented in Section 4.1. Then, in accordance with Section 4.2, θ_ is implemented as Eq. (65), where the threshold θ_ max is a parameter and is selected as θ_ max ¼ 5 rad=s. For comparison, the singularity escape/avoidance steer_ SEASL has the ing logic (SEASL) [17] is tested to generate θ. following form: 1 θ_ ¼  A# τ c hw

2

6 R ¼ λ4 ϵ3 ϵ2

5

10

0

-2

15

10

Body Rate Error [deg/s]

x y z

5

0

5

10

15

λ

λ

W3

λ

λ

3

7 λ 7 7 λ 7 5 W4

ð78Þ

λ ¼ λ0 expð  μ detðAAT ÞÞ:

ð80Þ

x y z

0

5

10

15

Time [s]

3 x y z

0

0

5

10

0

-3

15

θ1 θ2 θ3 θ4

0

5

10

40 20 0 -20

x y z

-40 0

5

10

Time [s]

10

15

Time [s]

60

-60

λ

0

-30

15

Condition Number

CMG Momentum [Nms]

0

Time [s]

10

0.1

-0.1

15

θ1 θ2 θ3 θ4

5

W2

Time [s]

5

0

λ

Time [s]

x 10-4

0

λ

30

Torque [Nm]

Body Rate [deg/s]

Euler Parameters Euler Parameters Error

1

x y z

Time [s]

Gimbal Rates [rad/s]

ϵ2

Gimbal Angles [rad]

0

0

-5

ϵ1 7 5;

W1 6 6 λ W ¼6 6 λ 4 λ

θð0Þ ¼ ½  0:424 1:765 3:357 0:998T rad

Time [s]

-5

1

2

x y z

5

2

3

The parameters ϵ0, ν, ϕ1, ϕ2, ϕ3, λ0, μ, W1, W2, W3, and W4 must be appropriately selected. Three cases are demonstrated for this attitude control example. The first case, referred to as Case C1, demonstrates the hyperbolic singularity where two eigenvalues of G322 are positive and negative. Case C1 starts at the initial gimbal angles, θð0Þ, and the initial angular velocity, ωð0Þ, the initial target angular velocity, ω0 , and ends at the final target angular velocity, ωf , as follows:

ð77Þ

0

-0.2

ϵ2

ð79Þ

where

0.2

ϵ3

ϵi ¼ ϵ0 sin ðνt þ ϕi Þ ði ¼ 1; 2; 3Þ;

ð76Þ

A# ¼ WAT ðAWAT þ RÞ  1

1

15

10

10

10

3

2

1

0

0

5

10

15

Time [s]

Fig. 9. Attitude control example for singularly escape/avoidance steering logic (Case C2). (a) Vector part of attitude, (b) attitude rate, (c) control torque, (d) vector part of error attitude, (e) error attitude rate, (f) gimbal angles of CMGs, (g) gimbal angle rates of CMGs, (h) angular momentum of CMGs and (i) condition number of A.

K. Yamada, I. Jikuya / Acta Astronautica 102 (2014) 103–123

ωð0Þ ¼ ω0 ¼ ½ 0:752  1:537  0:727T deg=s

ð81Þ

ωf ¼ ½0:438 0:894 0:423T deg=s

ð82Þ

smoothly passes the hyperbolic singular surface. The attitude error and the error attitude rate are sufficiently suppressed during transition in each steering logic. It should be noted that the hyperbolic singular surface is directionally passable to any direction, as shown in Section 3.2.1 and is therefore classified to be tractable. Hence, these observations show that the performances of SEASL and QSL are similar in the passable singularity. The second case, referred to as Case C2, demonstrates the elliptic singularity where two eigenvalues of G322 are positive. Case C2 starts at the initial gimbal angles, θð0Þ, and the initial angular velocity, ωð0Þ, the initial target angular velocity, ω0 , and ends at the final target angular velocity, ωf , as follows:

The simulation parameters in SEASL are selected as ϵ0 ¼ 0:1, ν ¼ 1 rad=s, ϕ1 ¼ ϕ2 ¼ ϕ3 ¼ π=2 rad, λ0 ¼ 0:01, μ¼10, W 1 ¼ 1, W 2 ¼ 1, W 3 ¼ 1, and W 4 ¼ 1. These parameters have been selected by trial and error so that the CMGs pass the singular surface as smoothly as possible. Figs. 7 and 8 show the results of attitude control based on SEASL and QSL in Case C1. In each figure, the first subfigure shows the vector part of the attitude, VðqÞ. The second shows the attitude rate, ω. The third shows the control torque, τ c . The fourth shows the vector part of the error attitude, Vðqe Þ. The fifth shows the error attitude rate, ωe . The sixth shows the gimbal angles, θ. The seventh shows _ The eighth shows the total angular the gimbal angle rates, θ. momentum of the CMG system, h. The ninth shows the condition number of A. The results of SEASL and QSL are analogous for this hyperbolic singularity case. The peaks of the condition number of A are sharp in each steering logic. It follows that the angular momentum of CMGs quickly and

x y z

5

10

0

5

x 10-4

0

5

10

10

15

0

5

15

3 x y z

0

0

5

10

0

-3

15

θ1 θ2 θ3 θ4

0

5

10

40 20 0 -20

x y z

-40 0

5

10

Time [s]

10

15

Time [s]

60

-60

10

Time [s]

Condition Number

CMG Momentum [Nms]

0

Time [s]

-30

15

0.1

-0.1

15

θ1 θ2 θ3 θ4

5

x y z

Time [s]

5

0

10

0

Time [s]

Body Rate Error [deg/s]

Euler Parameters Error

ð85Þ

30

Time [s]

Gimbal Rates [rad/s]

ωf ¼ ½ 0:5 1 0T deg=s

0

-2

15

x y z

-5

ð84Þ

Gimbal Angles [rad]

0

0

-5

ωð0Þ ¼ ω0 ¼ ½  1:096 0:405 0:035T deg=s

x y z

Time [s]

5

ð83Þ

Torque [Nm]

0

-0.2

θð0Þ ¼ ½ 1:25 0:2 1  0:15T rad

2

Body Rate [deg/s]

Euler Parameters

0.2

117

15

10

10

10

3

2

1

0

0

5

10

15

Time [s]

Fig. 10. Attitude control example for quadratic steering logic (Case C2). (a) Vector part of attitude, (b) attitude rate, (c) control torque, (d) vector part of error attitude, (e) error attitude rate, (f) gimbal angles of CMGs, (g) gimbal angle rates of CMGs, (h) angular momentum of CMGs and (i) condition number of A.

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The simulation parameters in SEASL are common to those in Case C1. Figs. 9 and 10 show the results of attitude control based on SEASL and QSL in Case C2. The peak of SEASL is much broader than that of QSL. It follows that the passing time of QSL is relatively shorter than that of SEASL. The magnitudes of the error attitude and the error attitude rate are also relatively suppressed in QSL, as compared to those of SEASL. It should be noted that this elliptic singular surface is directionally passable according to the condition in Section 3.2.1 and is therefore classified to be relatively tractable. Hence, these observations show that the performance of QSL is improved from that of SEASL in the directionally passable singularity. The third case, referred to as Case C3, also demonstrates the elliptic singularity; in particular, the gimbal angles θ1 ¼  π=2 rad, θ2 ¼ 0 rad, θ3 ¼ π=2 rad, θ4 ¼ 0 rad, which are known to be difficult to pass. Case C3 starts at the initial gimbal angles, θð0Þ, and the initial angular velocity, ωð0Þ, the initial target angular velocity, ω0 , and

ends at the final target angular velocity, ωf , as follows: θð0Þ ¼ ½0 0 0 0T rad

ð86Þ

ωð0Þ ¼ ω0 ¼ ½0 0 0T deg=s

ð87Þ

ωf ¼ ½  2 0 0T deg=s

ð88Þ

The simulation parameters in SEASL are selected as ϵ0 ¼ 0:1, ν ¼ 1 rad=s, ϕ1 ¼ ϕ2 ¼ ϕ3 ¼ π=2 rad, λ0 ¼ 0:01, μ¼10, W 1 ¼ 10  3 , W 2 ¼ 1, W 3 ¼ 1, and W 4 ¼ 1. Some of these parameters have been reselected because it is difficult to find common parameters that attain smooth passing for all cases. Figs. 11 and 12 show the results of attitude control based on SEASL and QSL in Case C3. The peaks of SEASL are much broader than that of QSL. It follows that the passing time of QSL is relatively shorter compared to that of SEASL. The magnitudes of the error attitude and the error attitude rate are relatively suppressed in QSL compared to those of SEASL. It should be

0

5

10

x y z

-2.5

15

Torque [Nm]

x y z

0

0

5

1.5

x 10-3

0

-1.5

x y z

0

5

10

5

3

0

0

5

0

5

10

Time [s]

10

θ1 θ2 θ3 θ4

0

5

15

10 x y z

60 40 20 0 -20

0

5

10

Time [s]

10

15

Time [s]

100 80

15

0

-3

15

Condition Number

θ1 θ2 θ3 θ4

10

Time [s]

x y z

-0.5

15

CMG Momentum [Nms]

Gimbal Rates [rad/s]

0

Time [s]

15

0

x y z

-100

15

0.5

Time [s]

-15

0

Time [s]

Body Rate Error [deg/s]

Euler Parameters Error

Time [s]

10

Gimbal Angles [rad]

-0.2

Body Rate [deg/s]

Euler Parameters

100 0

15

10

10

2

1

0

0

5

10

15

Time [s]

Fig. 11. Attitude control example for singularly escape/avoidance steering logic (Case C3). (a) Vector part of attitude, (b) attitude rate, (c) control torque, (d) vector part of error attitude, (e) error attitude rate, (f) gimbal angles of CMGs, (g) gimbal angle rates of CMGs, (h) angular momentum of CMGs and (i) condition number of A.

K. Yamada, I. Jikuya / Acta Astronautica 102 (2014) 103–123

119

x y z

0

5

10

x y z

-2.5

15

0

5

1.5

x 10

x y z

0

5

10

0

5

10

15

Time [s]

0

-0.5

15

3 x y z

0

5

5

-3

15

θ1 θ2 θ3 θ4

0

5

10

15

100 x y z

80 60 40 20 0 -20

0

Time [s]

5

10

Time [s]

10

15

Time [s]

Condition Number

0

CMG Momentum [Nms]

θ1 θ2 θ3 θ4

0

10

0

Time [s]

4

Gimbal Rates [rad/s]

-40

15

0.5

Time [s]

-4

x y z

-3

0

-1.5

0

Time [s]

Body Rate Error [deg/s]

Euler Parameters Error

Time [s]

10

Gimbal Angles [rad]

-0.2

0

Torque [Nm]

Body Rate [deg/s]

Euler Parameters

40 0

15

10

3

10

2

10

1

10

0

0

5

10

15

Time [s]

Fig. 12. Attitude control example for quadratic steering logic (Case C3). (a) Vector part of attitude, (b) attitude rate, (c) control torque, (d) vector part of error attitude, (e) error attitude rate (f) gimbal angles of CMGs, (g) gimbal angle rates of CMGs, (h) angular momentum of CMGs and (i) condition number of A.

noted that this elliptic singular surface is directionally impassable according to the condition in Section 3.2.1 and is therefore classified to be intractable. Hence, these observations show that the performance of QSL better than that of SEASL in the directionally impassable singularity. Apart from specific cases, one thousand examples have been tested for passing a singular surface based on both SEASL and QSL. Then, the averaged computation time of QSL is about 39 times as long as that of SEASL. Hence, reducing the computation time is desirable for implementing QSL in an on-board computer of the spacecraft. 7. Conclusions In this paper, a new concept of directional passability was introduced to investigate the structure of a singular surface of a pyramid-type single gimbal control moment gyro (CMG) system. A condition for directional passability

in a specific direction was obtained based on the secondorder approximation of the angular momentum vector. It follows that the hyperbolic singular surface is always passable and directionally passable in any direction. It also follows that the elliptic singular surface is impassable and that it is directionally passable in a specific direction if the above condition is satisfied. Directional passability is equivalent to the continuity of the generated gimbal angular velocities with respect to the desired torque; therefore, this directional passability is inevitable for smooth and precise steering of CMGs. Directional passability was further investigated for special directions. The singular surface is always directionally passable in the opposite direction of the angular momentum of CMGs. In contrast, the singular surface is not always directionally passable in the same direction as the angular momentum of CMGs. The results of these analyses reveal the difficult aspects of passing a singular surface. A new steering logic, called quadratic steering logic (QSL), was also developed

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based on the quadratic constrained quadratic optimization problem. The numerical optimization was carried out by the Newton method, using Gröbner bases. The effectiveness of QSL was demonstrated through numerical simulations for constant torque maneuvering examples and attitude control examples. In particular, QSL is effective for directionally impassable singularities; the rate saturations of gimbal velocities can be taken into account by modifying the logic. Further, the passing time can be shortened and the error attitude and error attitude rate can be suppressed. Hence, QSL will be effective for the stable maneuvering of a spacecraft with pyramid-type CMGs even in the presence of a gimbal-rate limit. Reducing the computational time as well as the implementation issue will be one of the important future topics of this research.

This appendix is devoted to proving the directional passability of the singular state when Δh is in the opposite direction of the angular momentum.

A.1. Representation of singular state in gimbal angle Consider the CMGs in the singular state and characterize the gimbal angles θ1, θ3 by the gimbal angles θ2, θ4 by using Eqs. (11) and (12). Because the rank of A is 2, the singular vector u^ satisfies AT u^ ¼ 0. For simplicity, the following vector u is introduced: 2 3 2 sin β cos β cos θ2 cos θ4 6 7 sin β sin ðθ2 þ θ4 Þ u¼4 ðA:1Þ 5  cos β sin ðθ2  θ4 Þ which satisfies AT u ¼ 0, but its norm is not normalized to 1. By taking the inner products of h^ i (i ¼ 1; …; 4) and u, the following equations are obtained: h^ 1  u ¼ sin β cos θ1 sin ðθ2 þθ4 Þ

¼

f ¼ x1 x3 þ2 cos β cos θ2 2 cos β cos θ4 :

ðA:3Þ

Let V 0 denote a matrix whose column components span the kernel of A. V 0 is explicitly obtained as follows: 2 3 x1 x1 6 7 6  2 sin θ4  2 cos β cos θ4 7 7: V0 ¼ 6 ðA:4Þ 6 7 x3 x3 4 5  2 sin θ2 2 cos β cos θ2 When Δh is in the opposite direction of h, S 3 is obtained as follows: h i sin β diag x1 ; −2 cos β cos θ4 ; −x3 ; 2 cos β cos θ2 : S3 ¼ juj Then, G322 is obtained as follows:   sin β A C G322 ¼ V T0 S 3 V 0 ¼ juj C B

Appendix A. Directional passability analysis: opposite case

þ

It follows that if Δh is in the direction of h, the inner product Δh  u takes the same sign as f, which is defined by

sin β cos 2 β cos θ1 ð2 cos β cos θ2 cos θ4 þ sin ðθ2 θ4 ÞÞ2 sin ðθ2 þθ4 Þ

  sin β sin ðθ2 þθ4 Þ cos 2 θ1 1þ tan 2 θ1 ¼ sin βx1 cos θ1

h^ 2  u ¼  2 sin β cos β cos θ4

A ¼ x31  x33 þ8 cos βð sin 2 θ2 cos θ2  sin 2 θ4 cos θ4 Þ B ¼ x31  x33 þ 8 cos 3 βð cos 3 θ2  cos 3 θ4 Þ C ¼ ½x31 þ x33 þ 8 cos 2 βð sin θ2 cos 2 θ2 þ sin θ4 cos 2 θ4 Þ: Here, the problem is to prove that when Δh is in the opposite direction of the angular momentum, at least one eigenvalue of G322 has the same sign as f. The eigenvalues of G322 are determined by multiplying the scalar sin β=juj by the solutions of the polynomial equation λ2 ðA þBÞλ þAB  C 2 ¼ 0: The solutions have the same sign if and only if AB  C 2 4 0; moreover, the sign of the eigenvalues is the same as that of AþB in such a case. Hence, the problem is reduced to proving the statement: “if the condition AB  C 2 4 0 is satisfied, A þB and f have the same signs.” If this statement is valid, the singular state is not passable in the direction of the angular momentum when AB C 2 40. It follows that if the singular state is passable in the direction of the angular momentum, the condition AB C 2 o0 is satisfied (it will be shown later that the condition AB C 2 ¼ 0 implies the condition ðA þ BÞf Z 0; therefore, the singular state is impassable in the direction of the angular momentum). Because the eigenvalues of G322 consist of positive and negative signs when AB C 2 o0, the singular state is passable in any direction when the singular state is passable in the direction of the angular momentum.

h^ 3  u ¼  sin β cos θ3 sin ðθ2 þθ4 Þ 

sin β cos 2 β cos θ3 ð2 cos β cos θ2 cos θ4  sin ðθ2  θ4 ÞÞ2 sin ðθ2 þθ4 Þ

¼

  sin β sin ðθ2 þθ4 Þ cos 2 θ3 1þ tan 2 θ3 ¼  sin βx3 cos θ3

Instead of directly proving the statement: “if the condition AB C 2 4 0 is satisfied, A þB and f have the same signs,” the case where f ¼0 is considered first. In this case, the singular vector u and the total angular momentum of the CMGs are orthogonal to each other. By using Eq. (A.3), the following equation is obtained:

h^ 4  u ¼ 2 sin β cos β cos θ2

where x1 and x3 are given by x1 ¼

sin ðθ2 þθ4 Þ ; cos θ1

x3 ¼

sin ðθ2 þθ4 Þ : cos θ3

A.2. Orthogonal case in singular vector and total angular momentum vector

ðA:2Þ

x1 x3 ¼  2 cos βð cos θ2  cos θ4 Þ:

ðA:5Þ

K. Yamada, I. Jikuya / Acta Astronautica 102 (2014) 103–123

By using Eqs. (11), (12), and (A.2), the following equations are obtained: x21 ¼ sin 2 ðθ2 þθ4 Þ þ cos 2 β½ sin ðθ2  θ4 Þ þ 2 cos β cos θ2 cos θ4 2

ðA:6Þ

x23 ¼ sin 2 ðθ2 þθ4 Þ þ cos 2 β½ sin ðθ2  θ4 Þ  2 cos β cos θ2 cos θ4 2 :

ðA:7Þ

It follows that x21  x23 ¼ 8 cos 3 β cos θ2 cos θ4 sin ðθ2  θ4 Þ:

ðA:8Þ

By using Eqs. (A.5) and (A.8), the following equation is obtained: 2

4 cos 2 β cos θm ð cos 2 θm  sin θp Þ x1 þ x3 ¼ ; sin θp

2 cos β½ cos β cos θm ð cos 2 θm  sin 2 θp Þ þ sin θm sin 2 θp  x1 ¼ sin θp

ðA:9Þ 2

2

2 cos β½ cos β cos θm ð cos 2 θm  sin θp Þ  sin θm sin θp  : sin θp

ðA:10Þ By multiplying x1 by itself and equating the right-hand side of Eq. (A.6), the following equation is obtained: 2

2

2

2

2

2

½ cos βð cos θm  sin θp Þ  sin θp cos θp  ð sin 2 β sin 2 θp þ cos 2 β cos 2 θm Þ ¼ 0:

satisfied; in other words, the sign of ðA þBÞf is not reversed if the condition AB  C 2 40 is satisfied. In order to investigate the sign of ðAþ BÞf under the condition of AB  C 2 4 0, the case where AB C 2 ¼ 0 is analyzed. The aim of the analysis is to prove the condition ðAþ BÞf Z 0 in this case. If this statement is valid, it can be shown that the condition AB  C 2 40 implies the condition ðAþ BÞf 4 0 by taking into account the following two statements: the sign of ðA þBÞf is invariant under the condition of AB  C 2 4 0 and the condition AB C 2 ¼ 0 implies ðAþ BÞf Z 0. The condition AB C 2 ¼ 0 can be transformed into  3   α2 x1 a1 x33 a3 ¼  ; 4

ðA:12Þ

where

where θp and θm are defined by θp ¼ ðθ2 þ θ4 Þ=2 and θm ¼ ðθ2  θ4 Þ=2, respectively. Notice here that sin θp is supposed to be sin θp a 0. (If θp satisfies sin θp ¼ 0, cos θ2 ¼ cos θ4 is satisfied; moreover, the condition x1 ¼ x3 is satisfied. Then, B ¼0 holds. This will be included in the following analysis.) Then, x1 and x3 are represented as follows:

x3 ¼

121

ðA:11Þ

Hence, the condition f¼0 (as well as sin θp a 0) is reduced to one of the following conditions: 1. cos βð cos 2 θm  sin 2 θp Þ ¼ cos θp sin θp . 2. cos βð cos 2 θm  sin 2 θp Þ ¼  cos θp sin θp . In each case, by substituting each relation into x1 and x3, x1 and x3 are represented in θ2 and θ4 as follows: 1. x1 ¼ 2 cos β cos θ4 , x3 ¼ 2 cos β cos θ2 . 2. x1 ¼  2 cos β cos θ2 , x3 ¼ 2 cos β cos θ4 . In each case, it is easy to check f ¼0; moreover, it can be shown that B ¼0 is satisfied. Therefore, f ¼0 implies B ¼0. A.3. Sign of ðA þ BÞf Next, the relation between the signs of ðA þBÞf and AB C 2 is discussed. The sign of ðA þBÞf is reversed in the neighborhoods of A þB ¼ 0 and f ¼0. When Aþ B ¼ 0 is valid, it follows from A ¼  B that the condition AB  C 2 r 0 is satisfied. When f ¼0 is valid, it follows from B¼ 0 that the condition AB C 2 r0 is satisfied. Hence, the sign of ðA þ BÞf is reversed only if the condition AB  C 2 r 0 is

a1 ¼ 2 cos β½ 2 cos βð sin θ2 cos 2 θ2 þ sin θ4 cos 2 θ4 Þ þð cos θ2  cos θ4 Þð  1 þ sin 2 βð cos 2 θ2 þ cos θ2 cos θ4 þ cos 2 θ4 ÞÞ a3 ¼ 2 cos β½ 2 cos βð sin θ2 cos 2 θ2 þ sin θ4 cos 2 θ4 Þ ð cos θ2  cos θ4 Þð  1 þ sin 2 βð cos 2 θ2 þ cos θ2 cos θ4 þ cos 2 θ4 ÞÞ α ¼ 4 cos βð cos θ2  cos θ4 Þ½ 1 þ ð1 þ cos 2 βÞð cos 2 θ2 þ cos θ2 cos θ4 þ cos 2 θ4 Þ: A and B can be represented as follows: A ¼ x31  a1  ðx33  a3 Þ  α

ðA:13Þ

B ¼ x31 a1 ðx33 a3 Þ þ α:

ðA:14Þ

First, let us study the sign of f under the conditions of AB  C 2 ¼ 0 and A þB 4 0. It follows from A þ B 4 0 that ðx31  a1 Þ  ðx33 a3 Þ 40:

ðA:15Þ 2

When the condition AB  C ¼ 0 is valid, the condition α a0 implies that the signs of x31  a1 and x33  a3 are opposite. It follows from Eq. (A.15) that x31  a1 40 4x33 a3 :

ðA:16Þ

Hence, f satisfies the following inequality, and f is bounded from below by f0 f ¼ x1  x3 þ 2 cos β cos θ2  2 cos β cos θ4 4 f 0 1=3

1=3

f 0 ¼ a1  a3 þ 2 cos β cos θ2  2 cos β cos θ4 :

ðA:17Þ ðA:18Þ

In contrast, the condition α¼0 implies x31 ¼ a1 or x33 ¼ a3 . If the condition x31 ¼ a1 is valid, it follows from A þB 40 that the condition x33 a3 o0 is satisfied. Then, in a similar way, it can be shown that f 4 f 0 . (Remark here that if both conditions x31 ¼ a1 and x33 ¼ a3 are satisfied, the condition A þB ¼ 0 is satisfied. Hence, both conditions cannot be simultaneously satisfied when A þ B 4 0.) Conversely, let us consider the case f 4 f 0 under the condition AB  C 2 ¼ 0. It follows from f 4f 0 that 1=3

1=3

x1  a1 4 x3  a3 :

ðA:19Þ

By using AB  C 2 ¼ 0, the condition αa 0 implies that the signs of x31  a1 and x33  a3 are opposite. It follows from

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Eq. (A.19) that 1=3 1=3 x1 a1 40 4 x3 a3 ;

ðA:20Þ

namely, x31 a1 4 0 4x33  a3 :

ðA:21Þ

It turns out that A þ B ¼ ðx31  a1 Þ  ðx33 a3 Þ 40;

ðA:22Þ

that is to say, the condition f 4 f 0 implies A þ B 4 0. In contrast, the condition α¼0 implies that x31 ¼ a1 or x33 ¼ a3 . It also turns out that the condition f 4 f 0 implies A þ B 40. (Remark here that if both conditions x31 ¼ a1 and x33 ¼ a3 are satisfied, the condition f ¼ f 0 is satisfied. Hence, both conditions cannot be simultaneously satisfied when f 4 f 0 .) Finally, let us consider the case f¼0. Although B¼0 in this case, it will be shown that A¼0 by using AB  C 2 ¼ 0. If the condition α a 0 is valid, by substituting Eq. (A.12) into Eq. (A.14), B¼ 0 is transformed into 1  3 α 2 1  3 α2 x1 a1 þ ¼ 3 x3  a3  ¼ 0: B¼ 3 2 2 x1  a1 x 3  a3 ðA:23Þ x31

x33

and are represented by It follows that α α 3 3 x1 ¼ a1  ; x3 ¼ a3 þ : 2 2

ðA:24Þ

Notice that these representations are valid for both αa 0 and α¼0. In contrast, it is shown that x1 ¼ 2 cos β cos θ4 ;

x3 ¼ 2 cos β cos θ2

ðA:25Þ

or x1 ¼  2 cos β cos θ2 ;

x3 ¼ 2 cos β cos θ4

ðA:26Þ

are satisfied in the case of f¼0. For instance, the following equation is obtained: cos βð cos 2 θm  sin 2 θp Þ ¼ cos θp sin θp

ðA:27Þ

in the former case. By using this equation, the equation α ðA:28Þ x31 ¼ ð2 cos β cos θ4 Þ3 ¼ a1  2 is reduced to the following equation: cos β sin θp cos θp cos θm  4ð1 þ cos 2 βÞ sin θp cos 2 θp  sin θp

 cos β cos θp ¼ 0:

ðA:29Þ

Here, it follows from A  B ¼ 2α that B¼0 implies A ¼  2α. By using (A.27), α is obtained as follows: α ¼ 8 sin θp sin θm ½  4ð1 þ cos 2 βÞ sin θp cos 3 θp  4 cos βð1 þ cos 2 βÞ sin 2 θp cos 2 θp þ ð1 þ cos 2 βÞ sin θp cos θp þ cos β:

ðA:30Þ

It follows from Eq. (A.29) that one of the conditions sin θp ¼ 0, cos θp ¼ 0, cos θm ¼ 0, or 4ð1 þ cos 2 βÞ sin θp cos 2 θp  sin θp  cos β cos θp ¼ 0 ðA:31Þ is valid. If the condition sin θp ¼ 0 is valid, α¼0 is satisfied. If one of the conditions cos θp ¼ 0 or cos θm ¼ 0 is valid, it follows from Eq. (A.27) that one of the conditions

sin θm ¼ 0 or sin θp ð cos θp þ cos β sin θp Þ ¼ 0 is satisfied; therefore, α¼0 is also satisfied. By direct computation, the following relation is obtained:  4ð1 þ cos 2 βÞ sin θp cos 3 θp 4 cos βð1 þ cos 2 βÞ sin 2 θp cos 2 θp þð1 þ cos 2 βÞ sin θp cos θp þ cos β ¼  ½4ð1 þ cos 2 βÞ sin θp cos 2 θp  sin θp  cos β cos θp ð cos β sin θp þ cos θp Þ:

ðA:32Þ

It follows that Eq. (A.31) implies that the third component on the right-hand side of Eq. (A.30) is equal to 0. Hence, the statement “f ¼0 implies A¼ 0” is proved. Similarly, the same result is obtained in the case of Eq. (A.26). Let us summarize the discussions. Under the condition AB C 2 ¼ 0, the condition f ¼0 implies the condition A ¼ B ¼ 0; hence, A þB ¼ 0. Under the same condition AB C 2 ¼ 0, the conditions A þB 40 and f 4 f 0 are equivalent. Hence, the condition Aþ B 4 0 implies the condition f 4 0 under the condition AB C 2 ¼ 0. In a similar way, it can be shown that the condition A þ B o 0 implies f o 0 under the condition AB  C 2 ¼ 0. By taking into account the case A þB ¼ 0, it follows that the condition AB  C 2 ¼ 0 implies the condition ðA þ BÞf Z 0. Recall that the sign of ðAþ BÞf is invariant under the condition AB  C 2 4 0. In summary, it has been shown that the condition AB  C 2 4 0 implies the condition ðA þ BÞf 4 0. References [1] B. Wie, Space Vehicle Dynamics and Control, 2nd ed. AIAA, Reston, VA, 1998, 419–430. [2] H. Schaub, J.L. Junkins, Analytical Mechanics of Space Systems, AIAA Education Series, Reston, VA, 2003, 353–373. [3] G. Margulies, J.N. Aubrun, Geometric theory of single-gimbal control moment gyro systems, J. Astronaut. Sci. 26 (2) (1978) 159–191. [4] E.N. Tokar, Efficient design of powered gyrostabilizer systems, Cosmic Res. (1978) 16–23; also Kosmicheskie Issledovaniya 16(1) (1978) 22–30. [5] E.N. Tokar, Problems of gyroscopic stabilizer control, Cosmic Res. (1978) 141–147; also Kosmicheskie Issledovaniya 16(2) (1978) 179–187. [6] E.N. Tokar, Effect of limiting supports on gyro stabilizer, Cosmic Res. (1979) 413–420; also Kosmicheskie Issledovaniya 16(4) (1978) 505–513. [7] E.N. Tokar, V.N. Platonov, Singular surfaces in unsupported gyrodyne systems, Cosmic Res. (1979) 547–555; also Kosmicheskie Issledovaniya 16(5) (1978) 675–685. [8] N.S. Bedrossian, J. Paradiso, E.V. Bergmann, D. Rowell, Redundant single gimbal control moment gyroscope singularity analysis, J. Guid. Control Dyn. 13 (6) (1990) 1096–1101. [9] H. Kurokawa, A geometric study of single gimbal control moment gyros-singularity and steering law (Ph.D. Dissertation), Aeronautics and Astronautics, University of Tokyo, Tokyo, 1997; Report of Mechanical Engineering Laboratory, No. 175, 1998 〈http://staff.aist. go.jp/kurokawa-h/CMGpaper97.pdf〉. [10] H. Kurokawa, Survey of theory and steering laws of single-gimbal control moment gyros, J. Guid. Control Dyn. 30 (5) (2007) 1331–1340. [11] B. Wie, Singularity analysis and visualization for single-gimbal control moment gyro systems, J. Guid. Control Dyn. 27 (2) (2004) 271–282. [12] T. Sands, J.J. Kim, B.N. Agrawal, Nonredundant single-gimbaled control moment gyroscopes, J. Guid. Control Dyn. 35 (2) (2012) 578–587. [13] K. Yamada, I. Jikuya, O. Kwak, Rate damping of a spacecraft using two single-gimbal control moment gyros, J. Guid. Control Dyn. 36 (6) (2013) 1606–1623. [14] H.S. Oh, S.R. Vadali, Feedback control and steering laws for spacecraft using single gimbal control moment gyros, J. Astronaut. Sci. 39 (2) (1991) 183–203.

K. Yamada, I. Jikuya / Acta Astronautica 102 (2014) 103–123

[15] N.S. Bedrossian, J. Paradiso, E.V. Bergmann, D. Rowell, Steering law design for redundant single-gimbal control moment gyroscopes, J. Guid. Control Dyn. 13 (6) (1990) 1083–1089. [16] B. Wie, D. Bailey, C.J. Heiberg, Singularity robust steering logic for redundant single-gimbal control moment gyros, J. Guid. Control Dyn. 24 (5) (2001) 865–872. [17] B. Wie, Singularity escape/avoidance steering logic for control moment gyro systems, J. Guid. Control Dyn. 28 (5) (2005) 948–956. [18] H. Leeghim, H. Bang, J. Park, Singularity avoidance of control moment gyros by one-step ahead singularity index, Acta Astronaut. 64 (2009) 935–945. [19] F.A. Leve, N.G. Fitz-Coy, Hybrid steering logic for single-gimbal control moment gyroscopes, J. Guid. Control Dyn. 33 (4) (2010) 1202–1212. [20] K. Takada, H. Kojima, N. Matsuda, Control moment gyro singularityavoidance steering control based on singular-surface cost function, J. Guid. Control Dyn. 33 (5) (2010) 1442–1450. [21] H. Kurokawa, Constrained steering law of pyramid-type control moment gyros and ground tests, J. Guid. Control Dyn. 30 (3) (1997) 445–449. [22] Z. Jingrui, J. Jin, L. Zaozhen, An improved installation for control moment gyros and its applications on reconfiguration and singular escape, Acta Astronaut. 85 (2013) 93–99. [23] S. Krishnan, S.R. Vadali, An inverse-free technique for attitude control of spacecraft using CMGs, Acta Astronaut. 39 (1996) 431–438. [24] M. Asagiri, K. Yamada, I. Jikuya, Attitude maneuver planning of spacecraft with CMGs, Trans. JSASS 61 (1) (2013) 16–22. (in Japanese). [25] B.R. Hoelscher, S.R. Vadali, Optimal open-loop and feedback control using single gimbal control moment gyroscopes, J. Astronaut. Sci. 42 (1994) 189–206.

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[26] S.R. Vadali, S. Krishnan, Suboptimal command generation for control moment gyroscopes and feedback control of spacecraft, J. Guid. Control Dyn. 18 (1995) 1350–1354. [27] Y. Kusuda, M. Takahashi, Feedback control with nominal inputs for agile satellites using control moment gyros, J. Guid. Control Dyn. 34 (2011) 1209–1218. [28] J. Ahmed, D. Bernstein, Adaptive control of double-gimbal controlmoment gyro with unbalanced rotor, J. Guid. Control Dyn. 25 (1) (2002) 105–115. [29] K. Yamada, N. Takatsuka, I. Jikuya, Spacecraft attitude control using a double-gimbal control moment gyro, Trans. JSASS Aerosp. Technol. Jpn. 8 (ists27) (2011) Pd_45–Pd_52. [30] H. Shaub, S.R. Vadali, J.L. Junkins, Feedback control law for variable speed control moment gyros, J. Astronaut. Sci. 46 (3) (1998) 307–328. [31] H. Schaub, J.L. Junkins, Singularity avoidance using null motion and variable-speed control moment gyros, J. Guid. Control Dyn. 23 (1) (2000) 11–16. [32] H. Yoon, P. Tsiotras, Singularity analysis of variable speed control moment gyros, J. Guid. Control Dyn. 27 (3) (2004) 374–386. [33] K. Yamada, N. Takatsuka, T. Shima, Pointing control using a variable-speed control moment gyro, Trans. JSASS Aerosp. Technol. Jpn. 7 (ists26) (2009) Pd_1–Pd_6. [34] D. Stevenson, H. Schaub, Nonlinear control analysis of a doublegimbal variable-speed control moment gyroscope, J. Guid. Control Dyn. 34 (2012) 787–793. [35] Y. Nakamura, H. Hanafusa, Inverse kinematic solutions with singularity robustness for robot manipulator control, J. Dyn. Syst. Meas. Control 108 (3) (1986) 163–171.