Title should be in Times New Roman font, size 14

10th National Symposium and Exhibition on Aerospace and Related Mechanisms (ARMS 2016) Thiruvananthapuram, Kerala, November 18-19, 2016

ARMS-2016-115

Control Moment Gyro (CMG) Sizing and Cluster Configuration Selection for Agile Spacecraft Abhilash Mony*, Hari Hablani#, Gireesh Sharma N$ *Engineer, Space Robotics and Mechanisms Division, ISRO Inertial Systems Unit (IISU), Trivandrum, #Professor, Centre of Astronomy, IIT-Indore, $Division Head, Space Robotics and Mechanisms Division, IISU, Trivandrum E-mail:[email protected] Abstract A typical IRS spacecraft uses reaction wheels for attitude control. For low earth imaging spacecraft which demand high agility and spot-to-spot imaging, a high torque output is essential along with sufficient angular momentum. In order to meet these dual requirements, it is proposed to replace conventional reaction wheels with single gimbal control moment gyros (CMGs). CMGs are capable of generating more than 50 times the torque output of a typical reaction wheel. It also imparts adequate angular momentum to the spacecraft platform. A cluster of reaction wheels are arranged with some optimum azimuth and slant angles to meet the mission momentum requirement. However, since CMGs change the direction of the angular momentum (by gimballing the spinning wheel) unlike reaction wheels which changes the magnitude of the angular momentum (by varying the wheel speed), the CMG sizing methodology is different. In this paper, the momentum envelopes of 2-CMG and 4-CMG clusters are studied. They are compared based on the shape of the momentum envelope and the distribution of the momentum envelope about each axis. Then, these are associated to the mission requirement to arrive at the type of cluster, number of CMGs in the cluster, individual CMG angular momentum magnitude ℎ and the value of slant angle 𝛽 which satisfies the mission requirement. The mechanical constraints in sizing the gimbal wheel (GW), which is the spinning wheel inside CMG, are also accounted to arrive at a feasible actuator size. Keywords: CMGs, CMG Sizing, Momentum Envelope, Pyramid Cluster, Roof-type cluster

1 Introduction India has one of the largest fleet of remote sensing spacecraft for civilian purpose. A typical Indian Remote Sensing (IRS) spacecraft use reaction wheels (RWs) for their attitude control. RWs are simple momentum exchange actuators which change the rotating speed of the flywheel to impart torque to the spacecraft body. However, their output torque is typically limited to 1 N.m as a result of constraints in flywheel size and torque motor power. This means that to obtain high torque and angular momentum from a RW, we require both a high torque motor and a bulky flywheel which will demand high power and will have more mass. Hence, for mission demanding high torque output and angular momentum, reaction wheels are replaced by Control Moment Gyros (CMGs). However, unlike RWs which change their spin speeds but not their body-fixed directions, the gimballed CMGs change the direction of their spin axes but not their speeds (unless they are variable speed CMGs) to generate torque. CMGs are capable of generating 50 times the output torque of a typical RW and it consumes less electric power per unit output torque. It also provides adequate angular momentum to the spacecraft. The CMG angular momentum is decided by the wheel. However, it is capable of quickly exchanging the available angular momentum by generating a high torque. The sizing of any actuator is essential before starting its design. The sizing of RWs is well known. Based on the mission requirements, they are typically arranged in a tetrahedral pyramid configuration with some optimum azimuth and slant angles [1]. However, CMGs change the direction of the angular momentum vector by gimballing. Hence, the sizing of CMGs is different from that of RWs. Due to the steering of the angular momentum, the study ________________

Corresponding Author Telephone: 0471-2569396, Fax: 0471-2569419, E-mail:[email protected]

10th National Symposium and Exhibition on Aerospace and Related Mechanisms (ARMS 2016) Thiruvananthapuram, Kerala, November 18-19, 2016

ARMS-2016-115

of momentum envelope of a CMG cluster is similar to that of a robotic manipulator with its links replaced by the corresponding momentum vectors [2]. Here our aim is to size a CMG cluster based on a typical mission requirement of an agile spacecraft. Studying the momentum envelope gives the distribution as well as extremities of momentum values achievable about each body-axis. This is used to identify the configuration of the CMG cluster, number of CMGs in the cluster, value of the slant angle β and the magnitude of the individual CMG angular momentum ‘h’. There are three methods of studying the angular momentum envelope of CMGs. One is by projection of the angular momentum vectors about the direction of the singular direction unit vector [3]. The singular direction unit vectors point in the direction about which the CMG cluster cannot generate an instantaneous torque. The second method is by using a cutting plane technique used by Stocking and Meffe [4]. These methods do not use the gimbal angles explicitly. The third method developed by Wie et al uses Binet-Cauchy identity and uses the gimbal angles directly [5]. In this paper the method developed by Margulies and Auburn is used for studying the momentum envelope [3]. The paper is organized in the following way. In the Section 2, a typical agile spacecraft mission is described. The angular momentum requirement is calculated. A brief description of method proposed by Margulies and Auburn is given in Section 3. In Section 4, the expressions for angular momentum envelope of 2-CMG and 4-CMG clusters are derived. These expressions are numerically evaluated and the momentum envelope plots are given and discussed. 4-CMG cluster in pyramid and roof-type configurations are studied. The maximum angular momentum values obtained from their resultant angular momentum expression are compared with that obtained from the momentum envelope. Then, the slant angle and the individual CMG angular moments are derived. Section 5 concludes the paper.

2 Mission Overview The mission consists of an agile spacecraft in a sun-synchronous polar orbit which should be capable to taking spot-to-spot images of the targets of interest within its field of view. The maximum rates about the roll, pitch and yaw axes are 40/s, 30/s and 2.50/s respectively. The inertia matrix of the spacecraft is diag [1396 1545 1100] kg.m2. From these values, the maximum angular momentum demand is 𝑯𝒔𝒂𝒕 = [97.4 80.9 48.0] 𝑻 𝑁. 𝑚. 𝑠. The momentum values are distributed unequally about the three body-axes. Thus, the number of CMGs, the type of CMG arrangement (cluster type), the slant angle 𝛽 and the individual CMG angular moment h are to be chosen such that this requirement is satisfied.

3

Momentum Envelope Expression of CMG Cluster

For the ith CMG, let 𝐬̂𝐢 be the unit vector along the singular direction. It is parametrized using ̂ 𝑖 be the unit vector along the angular momentum of CMG and 𝒕̂𝑖 the two angles 𝜃1 , 𝜃2 . Let 𝒉 ̂ 𝑖 is the gimbal axis unit vector, we can write the unit vector along the output torque. If 𝒈 following relations [3]. 𝐬̂𝐢 = 𝑠𝑥 𝒊 + 𝑠𝑦 𝒋 + 𝑠𝑧 𝒌 (1) 𝐬̂𝐢 = sin 𝜃2 𝒊 − sin 𝜃1 cos 𝜃2 𝒋 + cos 𝜃1 cos 𝜃2 𝒌 (2) Where 𝜃1 = [0 𝜋] and 𝜃2 = [0 2𝜋] Since the torque, momentum and gimbal axis unit vector form an orthogonal set: ̂𝑖 ̂𝑖 × 𝒉 𝒕̂𝑖 = 𝒈 (3) The direction in which one CMG or a cluster of CMGs is unable to produce torque is called singular direction. In other words, under singular condition, all output torque lie on a plane and the CMG cluster will be unable to generate torque perpendicular to this plane. The

10th National Symposium and Exhibition on Aerospace and Related Mechanisms (ARMS 2016) Thiruvananthapuram, Kerala, November 18-19, 2016

ARMS-2016-115

normal to this plane is the singular direction unit vector 𝐬̂𝐢 . The above idea can be extended and we can state that 𝐠̂ 𝐢 is orthogonal to 𝐭̂ 𝐢 and 𝐬̂𝐢 is also orthogonal to 𝐭̂ 𝐢 . Consequently, 𝐠̂ 𝐢 and 𝐬̂𝐢 lie on the same plane and we can write the unit vector 𝒕̂𝑖 as 𝒕̂𝑖 =

̂ 𝑖 × 𝒔̂𝑖 𝒈 |𝒈 ̂ 𝑖 × 𝒔̂𝑖 |

Using the orthogonality of the unit vectors ̂ 𝑖 = 𝒕̂𝑖 × 𝒈 ̂𝑖 𝒉 ̂ × 𝒔̂𝑖 𝒈 ̂𝑖 = 𝑖 ̂𝑖 𝒉 × 𝒈 |𝒈 ̂ 𝑖 × 𝒔̂𝑖 |

(4)

(5)

Using the vector triple product identity [3] ̂ 𝑖 ∙ 𝒔̂𝑖 ) 𝒈 ̂𝑖 𝒔̂𝑖 − (𝒈 (6) |𝒈 ̂ 𝑖 × 𝒔̂𝑖 | There can be positive or negative projections of the angular momentum vector along the singular direction, 𝜀𝑖 = ±1. Here, 𝜀𝑖 is the projection of the angular momentum vector along the singular direction. ̂ 𝑖 ∙ 𝒔̂𝑖 > 0, 𝜀𝑖 = 1 𝒉 (7) ̂ 𝒉𝑖 ∙ 𝒔̂𝑖 < 0, 𝜀𝑖 = −1 Thus, for an N-CMG cluster and for any arbitrary direction in three dimensional space, there are 2𝑁 combinations of gimbal angles for which the CMG cluster will be singular in that direction [2,3,5]. In this paper, the method developed by Margulies et al in used for studying the saturation angular momentum envelope. The momentum envelope of a N-CMG cluster can be obtained by summation of Eq.6 from 1 to N by taking all 𝜀𝑖 ′𝑠 = 1 . The momentum envelope is the maximum angular momentum obtained from the N-CMG cluster in three dimensional space. Total saturation momentum envelope for an N-CMG cluster can be written as: ̂ 𝑖 = 𝜀𝑖 𝒉

𝑵

𝑁

̂𝑖 = ∑ 𝑯 = ∑𝒉 𝒊=𝟏

𝑖=1

̂ 𝑖 ∙ 𝒔̂𝑖 ) 𝒈 ̂𝑖 𝒔̂𝑖 − (𝒈 |𝒈 ̂ 𝑖 × 𝒔̂𝑖 |

(8)

̂ 𝑖 will be a function of the cluster slant angle 𝛽. The value Here, the gimbal axis unit vector 𝒈 of 𝒔̂𝑖 will be calculated as a function of its parametric angles 𝜃1 and 𝜃2 as shown in Eq2.

4

Momentum Envelope Studies of CMG Clusters

In section 2, the maximum momentum requirement for the mission along the three body axes were given as 𝑯𝒔𝒂𝒕 = [97.4 80.9 48.0] 𝑻 𝑁. 𝑚. 𝑠 about the Roll (X), Pitch (Y) and Yaw (Z) axis respectively. In order to size the CMG, to select the type of the cluster and slant angle 𝛽 , the chosen configuration should satisfy the mission requirement. In this section, the momentum envelopes of CMGs in different configurations are studied to arrive at an appropriate cluster design and CMG sizing. 4.1 Momentum Envelope of 2-CMGs with orthogonal gimbal axes Consider two CMGs with their gimbal axis along the X and Z axis.Let, ℎ1 = ℎ2 = 1 and ̂ 1 = [1 0 0]𝑇 and 𝒈 ̂ 2 = [0 0 1]𝑇 . Thus, Eq8 can be expanded and written as: 𝒈

10th National Symposium and Exhibition on Aerospace and Related Mechanisms (ARMS 2016) Thiruvananthapuram, Kerala, November 18-19, 2016

𝑠𝑥 ̂ ∙ 𝒔̂ 𝒈 𝑯 = 𝑒 [𝑠𝑦 ] − 1𝑒 𝑖 1 1 𝑠𝑧 1

𝑔1𝑥 𝑠𝑥 ̂ ∙ 𝒔̂ 1 𝒈 𝑔 [ 1𝑦 ] + 𝑒 [𝑠𝑦 ] − 2𝑒 𝑖 2 2 𝑔1𝑧 𝑠𝑧

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𝑔2𝑥 𝑔 [ 2𝑦 ] 𝑔2𝑧

(9)

𝑠𝑖𝑛𝜃2 √1−𝑐𝑜𝑠2 𝜃1 𝑐𝑜𝑠2 𝜃2 −𝑠𝑖𝑛𝜃2 𝑐𝑜𝑠𝜃2

𝑯=

√1−𝑐𝑜𝑠2 𝜃1 𝑐𝑜𝑠2 𝜃2

[

− 𝑠𝑖𝑛𝜃1

𝑐𝑜𝑠𝜃1

(10)

]

̂ 𝑖 × 𝒔̂𝑖 | = √1 − (𝒈 ̂ 𝑖 ∙ 𝒔̂𝑖 )2 Here 𝑒𝑖 = |𝒈 The two orthogonal CMGs produce a three axis angular momentum envelope as shown in Fig1(a). The schematic of the orthogonal CMG cluster is given in Fig1(b).

Figure 1(a): Momentum Envelope of 2-CMG cluster with orthogonal gimbal axis

Figure 1(b): Schematic of 2-CMG cluster with orthogonal gimbal angles (along X and Z)

The total angular momentum in terms of the gimbal angles 𝛿𝑖 and individual CMG angular momentum h can be written as 𝑐𝑜𝑠𝛿2 𝑯 = ℎ [𝑐𝑜𝑠𝛿1 + 𝑠𝑖𝑛𝛿2 ] 𝑠𝑖𝑛𝛿1

(11)

When 𝛿1 =00 and 𝛿2 =900, the maximum angular momentum with 2 units (if h=1) is obtained along the Y-axis. For 𝛿1 =900 and 𝛿2 =00, the maximum value about the X and Z axes are simultaneously obtained as unity. The momentum envelope shown in Fig1.1 has 4 holes of unit radius along the X and Z axis. On inspection of Fig1.1, the maximum values of 2 units is obtained along Y-axis and 1 unit along X and Z axis in agreement with the maximum values obtained from Eq11. So, in order to meet the mission angular momentum requirement, we may need a 100 N.m.s and 50 N.m.s wheel along the X and Z axes. Thus a very large wheel may be needed for realizing the required angular momentum. As a result, we need to increase the number of CMGs. In the next sub-section a cluster of 4-CMGs arranged at a slant angle 𝛽 is considered. In order to obtain the angular momentum components of each CMG along all three body-axes, they are arranged with a slant angle 𝛽 4.2 Momentum Envelope of 4-CMG cluster 4-CMGs in pyramid configuration are widely studied in literature [2,3,4,5]. In this paper, four CMGs in pyramid and roof type clusters are studied. The roof-type cluster has not been dealt

10th National Symposium and Exhibition on Aerospace and Related Mechanisms (ARMS 2016) Thiruvananthapuram, Kerala, November 18-19, 2016

ARMS-2016-115

in detail in literature. It is considered here due to the unequal momentum demands about the body-axes. The effect of the slant angle 𝛽 in the angular momentum distribution is also included. 4.2.1

4-CMG Pyramid Cluster

Figure 2a: Schematic of 4-CMG pyramid cluster with all gimbal angles zero

Figure 2b: 4-CMG pyramid cluster with actual CMG

The 4-CMG pyramid cluster is shown in Figs2a and 2b with all gimbal angles at zero degrees. The CMG gimbal angle vector 𝜹 = [𝛿1 𝛿2 𝛿3 𝛿4 ]𝑇 . The gimbal angles are assumed positive about counter clockwise direction with respect to the corresponding gimbal axis. By inspecting Fig2a, the gimbal axis unit vectors can be written in terms of the slant angle 𝛽 as shown in Eq12. 0

𝑐𝑜𝑠𝛽]𝑇

𝑠𝑖𝑛𝛽

𝑐𝑜𝑠𝛽]𝑇

̂ 1 = [𝑠𝑖𝑛𝛽 𝒈 ̂ 2 = [0 𝒈

̂ 3 = [−𝑠𝑖𝑛𝛽 𝒈

(12)

0 𝑐𝑜𝑠𝛽]𝑇

̂ 4 = [0 −𝑠𝑖𝑛𝛽 𝑐𝑜𝑠𝛽]𝑇 𝒈 In terms of the individual CMG angular momentum ℎ, gimbal angles 𝜹 and slant angle 𝛽, the total angular momentum of the 4-CMG cluster in the body-axes can be written as shown in Eq13. −𝑐𝛽𝑠𝛿1 − 𝑐𝛿2 + 𝑐𝛽𝑠𝛿3 + 𝑐𝛿4 𝑯 = ℎ [ 𝑐𝛿1 − 𝑐𝛽𝑠𝛿2 − 𝑐𝛿3 + 𝑐𝛽𝑠𝛿4 ] 𝑠𝛽𝑠𝛿1 + 𝑠𝛽𝑠𝛿2 + 𝑠𝛽𝑠𝛿3 + 𝑠𝛽𝑠𝛿4

(13)

where 𝑠𝛿𝑖 = 𝑠𝑖𝑛𝛿𝑖 , 𝑐𝛿𝑖 = 𝑐𝑜𝑠𝛿𝑖 and i =1,2,3,4. The value of h in the momentum envelope plot is unity. The maximum angular momentum of the 4-CMG pyramid cluster about the three co-ordinate axes can be obtained by substituting the appropriate gimbal angles in Eq13 such that the individual CMG angular momentums are projected maximally along the required direction. These values are calculated and given in Table1. By using Eq8 and Eq12, the equation for the angular momentum envelope for a 4-CMG pyramid cluster can be derived as given in Eq14. 𝐻𝑥 =

[𝑠𝑥 𝑐𝛽]2 −𝑠𝑧 𝑠𝛽𝑐𝛽 𝑒1

𝑠

+ 𝑒𝑥 + 2

[𝑠𝑥 𝑐𝛽]2 +𝑠𝑧 𝑠𝛽𝑐𝛽 𝑒3

𝑠

+ 𝑒𝑥

4

(14a)

10th National Symposium and Exhibition on Aerospace and Related Mechanisms (ARMS 2016) Thiruvananthapuram, Kerala, November 18-19, 2016

𝐻𝑦 =

𝐻𝑧 =

𝑠𝑦

𝑒1 [𝑠𝑧 𝑠𝛽]2 −𝑠𝑥 𝑠𝛽𝑐𝛽 𝑒1

+ +

[𝑠𝑦 𝑐𝛽]2 −𝑠𝑧 𝑠𝛽𝑐𝛽 𝑒2 [𝑠𝑧 𝑠𝛽]2 −𝑠𝑦 𝑠𝛽𝑐𝛽 𝑒2

+ +

𝑠𝑦

+

ARMS-2016-115

[𝑠𝑦 𝑐𝛽]2 +𝑠𝑧 𝑠𝛽𝑐𝛽

𝑒3 𝑒4 [𝑠𝑧 𝑠𝛽]2 +𝑠𝑥 𝑠𝛽𝑐𝛽

+

𝑒3

(14b)

[𝑠𝑧 𝑠𝛽]2 +𝑠𝑦 𝑠𝛽𝑐𝛽

(14c)

𝑒4

Here the terms 𝑒𝑖 (i=1,2,3,4) is defined as 𝑒1 = 𝜖1 √1 − [𝑠𝑥 𝑠𝛽 + 𝑠𝑧 𝑐𝛽]2 𝑒2 = 𝜖2 √1 − [𝑠𝑦 𝑠𝛽 + 𝑠𝑧 𝑐𝛽]

2

(15)

𝑒3 = 𝜖3 √1 − [−𝑠𝑥 𝑠𝛽 + 𝑠𝑧 𝑐𝛽]2 2

𝑒4 = 𝜖1 √1 − [−𝑠𝑦 𝑠𝛽 + 𝑠𝑧 𝑐𝛽]

Table 1. Maximum momentum values in pyramid cluster Gimbal Angle 𝜹

Sl No: 1

Total Angular Momentum H

[−900 1800 900 00 ]

𝐻 = [2ℎ(1 + 𝑐𝛽)

2

[00 −900 1800 900 ]

𝐻 = [0

3

[900 900 900 900 ]

0

0]

2ℎ(1 + 𝑐𝛽)

0]

𝐻 = [0

0

4ℎ𝑠𝛽]

The momentum envelope plot obtained by numerically evaluating Eq14 is given in Fig3 and Fig4. The surface has 2N holes and gives the maximum angular momentum achievable in the given configuration for h=1. We can also see from Figs4a and 4b that the envelope becomes flat when the slant angle is reduced from 𝛽=53.130 to 𝛽=300.

Figure3a. Momentum envelope of 4-CMG Cluster with 𝛽=53.130

Figure4a. Momentum envelope of 4-CMG Cluster with 𝛽=53.130 (XZ View)

Figure3b. Momentum envelope of 4-CMG Cluster with 𝛽=300

Figure4b. Momentum envelope of 4-CMG Cluster with 𝛽=300 (XZ View)

10th National Symposium and Exhibition on Aerospace and Related Mechanisms (ARMS 2016) Thiruvananthapuram, Kerala, November 18-19, 2016

4.2.2

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4-CMG Roof-type Cluster

Z

Y X Figure 5b: 4-CMG roof-type cluster with actual CMG

Figure 5a: Schematic of 4-CMG roof-type cluster with all gimbal angles zero

Similarly, for the roof-type cluster shown in Fig5a, the total angular momentum equation in terms of the gimbal angles 𝜹, individual CMG angular momentum h and slant angle 𝛽 can be written as shown in Eqs16 and 17. 𝑐𝛿1 + 𝑐𝛿2 − 𝑐𝛿3 − 𝑐𝛿4 (16) 𝑯 = ℎ [−𝑐𝛽𝑠𝛿1 − 𝑐𝛽𝑠𝛿2 + 𝑐𝛽𝑠𝛿3 + 𝑐𝛽𝑠𝛿4 ] 𝑠𝛽𝑠𝛿1 + 𝑠𝛽𝑠𝛿2 + 𝑠𝛽𝑠𝛿3 + 𝑠𝛽𝑠𝛿4 𝐻𝑥 = 2

𝑠𝑥 𝑒1 1

+

𝑠𝑥 𝑒2 1

+

𝑠𝑥 𝑒3

+

𝑠𝑥

(17a)

𝑒4

2

1

1

3

4

𝐻𝑦 = [(𝑠𝑦 − 𝑠𝑦 𝑠𝛽𝑐𝛽) − 𝑠𝑧 𝑠𝛽𝑐𝛽] [𝑒 + 𝑒 ]+[(𝑠𝑦 − 𝑠𝑦 𝑠𝛽𝑐𝛽) + 𝑠𝑧 𝑠𝛽𝑐𝛽] [𝑒 + 𝑒 ] 1

2

1

1

1

1

1

2

3

4

𝐻𝑧 = [𝑠𝑧 − 𝑠𝑦 𝑠𝛽𝑐𝛽 − 𝑠𝑧 𝑐 2 𝛽] [𝑒 + 𝑒 ]+[𝑠𝑧 + 𝑠𝑦 𝑠𝛽𝑐𝛽 − 𝑠𝑧 𝑐 2 𝛽] [𝑒 + 𝑒 ]

(17b) (17c)

Here the terms 𝑒𝑖 (i=1,2,3,4) is defined as 𝑒1 = 𝜖1 √1 − [𝑠𝑦 𝑠𝛽 + 𝑠𝑧 𝑐𝛽] 𝑒2 = 𝜖2 √1 − [𝑠𝑦 𝑠𝛽 + 𝑠𝑧 𝑐𝛽]

2

2

(18) 2

𝑒3 = 𝜖3 √1 − [−𝑠𝑦 𝑠𝛽 + 𝑠𝑧 𝑐𝛽]

2

𝑒4 = 𝜖1 √1 − [−𝑠𝑦 𝑠𝛽 + 𝑠𝑧 𝑐𝛽]

The maximum angular momentum along the three body axes for the roof-type cluster can be obtained by substituting the corresponding values of gimbal angles which maximizes the projection of the angular momentum Eq16. These are given in Table 2. Table 2. Maximum momentum values in roof-type cluster

Sl No: 1

Gimbal Angle 𝜹 [00 00 1800 1800 ]

Total Angular Momentum H 𝐻 = [4ℎ

0

2

[−900 −900 900 900 ]

𝐻 = [0

4ℎ𝑐𝛽

3

[900 900 900 900 ]

𝐻 = [0

0

0] 0]

4ℎ𝑠𝛽]

10th National Symposium and Exhibition on Aerospace and Related Mechanisms (ARMS 2016) Thiruvananthapuram, Kerala, November 18-19, 2016

ARMS-2016-115

Using Eq17 the roof-type cluster momentum envelope for 𝛽=53.130 and 𝛽=300 are computed and shown in Figs 6 and 7.

Figure6a. Momentum envelope of 4-CMG roof-type cluster with 𝛽=53.130

Figure7a. Momentum envelope of 4-CMG roof-type cluster with 𝜷=53.130 (YZ View)

Figure6b. Momentum envelope of 4-CMG roof-type cluster with 𝛽=300

Figure7b. Momentum envelope of 4-CMG roof-type cluster with 𝜷=300 (YZ View)

4.3 Momentum Envelope: Observations and Sizing The envelope for the pyramid cluster has a spherical shape (for 𝛽=53.130) and ellipsoid shape (for 𝛽=300) as shown in Fig3 and Fig4. It has 8 holes corresponding to 4 CMGs. In the rooftype cluster, the holes in the momentum envelope made by the (N-1)th angular momentum vector about the two gimbal axes merges as two gimbal axes are parallel. The momentum envelope has the shape of a roof which flattens with lower slant angles (Figs7a and 7b). As computed in Table2 and shown in Fig6, for h=1,the maximum momentum along the X-axis is 4 units for any slant angle and along Y and Z axes is 3.46 and 2 units at 𝛽=300. The corresponding values for the pyramid cluster is 3.73 units, 3.73 units and 2 units (for 𝛽=300). From the momentum distribution of the pyramid and the roof-type cluster, we can note that the pyramid cluster gives a circular distribution in the X-Y plane with the Z value decreasing with decrease in the slant angle (This is clear from Table3 as well). The peak values obtained from the momentum envelope and analytical values are matching. The roof-type cluster momentum distribution is maximum along the X axis for all slant angles. The distribution between the Y and Z axes depends upon the slant angle of the roof-type cluster. Using the expressions given in Table1 and Table2 and the maximum angular momentum requirement of the mission given in section 2, the value of h and 𝛽 can be calculated as shown below. 4.3.1

Pyramid Cluster

[𝐻𝑧 ]𝑚𝑎𝑥 4ℎ𝑠𝑖𝑛𝛽 𝛽 48.0 𝑁. 𝑚. 𝑠 = = 2𝑡𝑎𝑛 = [𝐻𝑥 ]𝑚𝑎𝑥 2ℎ(1 + 𝑐𝑜𝑠𝛽) 2 97.4 𝑁. 𝑚. 𝑠

10th National Symposium and Exhibition on Aerospace and Related Mechanisms (ARMS 2016) Thiruvananthapuram, Kerala, November 18-19, 2016

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∴ 𝛽 = 27.70 48 ℎ= 4𝑠𝑖𝑛𝛽 ∴ ℎ = 25.8 𝑁. 𝑚. 𝑠 4.3.2

Roof-type Cluster

[𝐻𝑧 ]𝑚𝑎𝑥 4ℎ𝑠𝑖𝑛𝛽 48.0 𝑁. 𝑚. 𝑠 = = 𝑡𝑎𝑛𝛽 = [𝐻𝑥 ]𝑚𝑎𝑥 4ℎ𝑐𝑜𝑠𝛽 97.4 𝑁. 𝑚. 𝑠 ∴ 𝛽 = 26.20 48 ℎ= 4𝑠𝑖𝑛𝛽 ∴ ℎ = 27.1 𝑁. 𝑚. 𝑠 4.3.3

Constraints for finalizing CMG sizing

From the mission requirement we know that spacecraft needs unequal momentum distribution about its body axes. Apart from this, the angular momentum value of the spinning wheel inside CMG is restricted to 25 N.m.s owing to constraint in its mechanical design. The maximum angular momentum value along the body-axes at h=25 N.m.s for different slant angle 𝛽 for the 4-CMG roof-type and pyramid cluster is given in Table3. Table3. Maximum momentum values in 4-CMG pyramid and roof-type cluster at h=25 N.m.s for different slant angles 𝛽 (in N.m.s) Pyramid Cluster

Roof-type Cluster

𝛽

𝐻𝑥

𝐻𝑦

𝐻𝑧

𝐻𝑥

𝐻𝑦

𝐻𝑧

65

71.1

71.1

90.6

100

42.3

90.6

60

75

75

86.6

100

50

86.6

54.73

78.9

78.9

81.6

100

57.7

81.6

53.13

80

80

80

100

60

80

45

85.4

85.4

70.7

100

70.7

70.7

40

88.3

88.3

64.27

100

76.6

64.27

35

90.96

90.96

57.4

100

81.9

57.4

30

93.3

93.3

50

100

86.6

50

25

95.3

95.3

42.3

100

90.6

42.3

22.5

96.2

96.2

38.3

100

92.4

38.3

20

96.98

96.98

34.2

100

93.96

34.2

Mission demand (in N.m.s)

97.4

80.9

48.0

97.4

80.9

48.0

On the basis of our understanding of the momentum envelope, the mission demand, mechanical constraints and momentum margin needed, a 4-CMG roof-type cluster with β=300 (Table3) meets all conditions. The excess angular momentum available along X, Y and Z axes is 2.6 N.m.s, 5.7 N.m.s and 2 N.m.s respectively. Typical earth pointing rate for a low earth sun-synchronous spacecraft will be 0.060/s. This will demand 1.72 N.m.s over and above the said demand along the Y-axis (80.9 N.m.s). This can also be met in the chosen configuration. However, the given analysis doesn’t consider the external disturbance torque.

10th National Symposium and Exhibition on Aerospace and Related Mechanisms (ARMS 2016) Thiruvananthapuram, Kerala, November 18-19, 2016

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In actual case, some part of the angular momentum will be used to counter the disturbances acting on the spacecraft thus resulting in a lower achievable peak rate.

5 Conclusions For a spacecraft with high agility and spot-to-spot imaging, a single gimbal control moment gyro (CMG) is chosen as the attitude control actuator. The mission angular momentum demand of a typical agile satellite was derived. The expressions of computing the angular momentum envelope of CMG cluster was explained. Then, the momentum envelope of 2 and 4-CMG clusters were studied. In the 4-CMG cluster, the pyramid and roof-type arrangement were studied. Based on the mission demand and angular momentum distribution, along with the constraints posed by mechanical design for the CMG wheel, a 4-CMG cluster in roof-type configuration with 𝛽=300 and h=25 N.m.s is chosen as the final configuration. The CMG is sized satisfying the mission requirement under the given constraints.

References [1] R.G. Reynolds and F. Landis Markley, “Maximum torque and momentum envelopes for reaction wheel arrays,". [2] N. S. Bedrossian, “Steering law design for redundant single gimbal control moment gyro systems," M.S Thesis, Charles Stark Draper Laboratory, 1987. [3] G. Margulies and J. Aubrun, “Geometric theory of single-gimbal control moment gyro systems," AIAA Guidance and Control Conference, San Diego, California. [4] M.Meffe and G.Stocking, “Momentum Envelope Topology of Single-Gimbal CMG Arrays for Space Vehicle Control”, Proceedings of AAS Guidance and Control Conference, Keystone, CO, Jan.-Feb. 1987, AAS Paper 87-002. [5] J.A. Dominguez and B.Wie, “Computation and Visualization of CMG Singularities”, AIAA Guidance, Navigation and Control Conference, Monterey, California, 5-8 Aug. 2002.