Optimal Steering Laws for Variable Speed Control Moment Gyros

Optimal Steering Laws for Variable Speed Control Moment Gyros

Hyunjae Lee

^∗

, Il-Hyoung Lee

^∗

Hyochoong Bang

^†

Korea Advanced Institute of Science and Technology(KAIST), Daejon, 305-701, Korea.

Singularity avoidance steering laws for control momentum gyros(CMGs) and variable speed control mo- ment gyros(VSCMGs) are addressed in this paper. The proposed methodology is based on constrained opti- mization theory. The key idea of the strategy is to minimize a cost function which consists of singularity indices and energy terms. The proposed method enables CMG/VSCMG clusters not to come across the singularity.

It turns out that the proposed optimal singularity avoidance steering law is equivalent to a general solution including the conventional null motion approach. Also, an optimal steering law is developed to overcome the possibility of wheel speed saturation during the operation of VSCMG clusters. The performance of the proposed laws is demonstrated through simulation study with a three parallel single-gimbal CMG/VSCMG cluster.

I. Introduction

N

^EXTgeneration spacecrafts will require rapid attitude reorientation capability to accomplish a variety of missions.

Control moment gyros(CMGs) along with conventional reaction wheels are representative torque producing de- vices for spacecraft attitude control. CMGs produce relatively larger control torque than the reaction wheels at the same amount of energy so that they are commonly adopted to reorient large spacecraft structures. For this reason, they are sometimes called torque amplifiers. The role of CMG clusters could be more increased than before.

One of the drawbacks of CMG clusters is due to the complex steering law to implement. This drawback may be resolved by the on-board microprocessors with enhanced performances. Another significant problem is the steering law suffering from the singular problem. The torque vectors of reaction wheels are fixed. The property of the moving torque vectors of CMGs, on the other hand, bring high possibility of the singularity. It occurs when the torque vectors of CMG clusters are aligned in the same plane or line. Under such singularity conditions, production of the control torque along arbitrary directions becomes actually impossible. Efficient avoidance of the singularity has been a major issue in recent studies.

Various attempts have been made to solve the singularity problem during the last few decades. For the enhancement of the pseudo-inverse technology, the basic logic of the steering law using SVD(Singular Value Decomposition) was addressed by Junkins and Kim.¹ A singularity robust approach was proposed in the field of robotic manipulators control.² This method is believed to influence follow-on researches on the singularity problems of CMG clusters.³⁻⁵ This law provides a singularity avoidance strategy, but introduce control torque error at the price of maintaining CMG clusters away from the singular conditions. Moreover, it does not always guarantee desired results of singularity avoidance.

Some literatures discussed the null motion into the CMG steering law for singularity avoidance.⁶⁻⁸ The null motion represents a state of the gimbals that produces no net control torque. A variety of analytical approaches to develop proper null motions has been also studied. The commonly known gradient method as an attractive choice for the null motion has received significant attention. Despite the complexity of the steering laws using the null motion, the approach could be a potential alternative to the classical CMG steering laws for the singular-free CMG clusters.

Variable speed control moment gyro(VSCMG) clusters were introduced as a new set of CMG clusters.⁹ They have extra degrees of freedom to produce the control torque by controlling wheel speeds as well as gimbal rates. While the

∗Graduate Student , Department of Aerospace Engineering, KAIST, 373-1, Kusong, Yusong, Daejon, Korea, 305-701.

†Associate Professor, AIAA member, Department of Aerospace Engineering, KAIST, 373-1, Kusong, Yusong, Daejon, Korea, 305-701.

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conventional CMG clusters use constant wheel speed, VSCMG clusters allow the wheel speed to vary continuously.

Thus, VSCMG clusters offer flexibility to escape the singularity. The advantage of VSCMG clusters opened potential possibilities to resolve the singularity problem. Consequently, the VSCMG clusters steered by the null motion strategy may provide a powerful solution for the next generation CMG clusters.

Some recent studies are dedicated to the null motion strategy using the VSCMG clusters.^7,8They made it possible to prevent the singularity in many cases except some non-realistic cases. However, it may not be desirable from the perspective of energy saving if large amount of energy is consumed to avoid the singularity. There has been no qualitative indices to measure the performance of the steering laws to VSCMG/CMG clusters. One has to consider the performance of the CMG/VSCMG clusters as well as the desired torque command. Consequently, one may not claim that the current approaches can be the best to avoid the singularity while satisfying the torque command. Also, there exist possibilities for the wheel speeds of VSCMG clusters to reach so high or low. The high wheel speed tends to cause the wheel saturation problem. But, considering the inherent characteristics of the CMGs, the low wheel speed is not always desirable.

By designing a singular-free steering law with qualitative indices, a new steering law is proposed in the present paper. The best way to solve the both objectives, singularity avoidance and performance, simultaneously is probably to consider the optimization technique. The proposed method herein is based on constraint optimization theory. The principal idea is to determine a cost function composed of singularity indices and energy terms. The proposed cost function can be the qualitative index of CMG/VSCMG clusters to measure the performance. Application of optimal theory to minimize the cost function yields new results. The closed-form solution obtained by optimization consists of two parts; the first part corresponds to the minimum norm solution, and the other is associated with the null motion approach. In this paper, the wheel regulation during the VSCMG cluster operations is also investigated. An enhanced optimal steering law by taking the wheel speed saturation problem into account is proposed.

This paper is organized as follows; First, we simply formulate the dynamic equations of motion for VSCMG clusters, and address problems of the steering laws due to the singularity. Then, a new approach to avoid the singularity for VSCMGs is introduced and the wheel speed saturation problem is discussed. Then, the new steering law is suggested to control the wheel speed. Finally, a three-parallel CMG cluster is examined to demonstrate the proposed laws using numerical simulations.

II. Problem Statement

Various types of CMG/VSCMG clusters exist for different applications. Typical pyramid-type and three-parallel type CMG clusters are illustrated in Fig. 1. For the generalization, consider a VSCMG cluster which consists ofN gimbals. Note that the magnitude of the gimbal angular rate vectorγ˙ = [ ˙γ1,γ˙2, ...,γ˙_n]^T is usually much smaller than that of the wheel speed vector$ = [$1, $2, .., $_n]^T as a reasonable assumption. As a consequence, the portion of the total momentum generated byγ˙ is assumed to be negligible. Also, the moment of inertia of the VSCMG cluster with respect to the spacecraft body frame can be assumed constant without loss of generality. Let us introduce the VSCMG coordinate system such as

ai=i-th gimbal axis unit vector bi=i-th spin axis unit vector

ci=i-th torque axis unit vector, given byci=ai×bi, whereaiandbiare always orthogonal.

From the properties of the VSCMG cluster, the derivatives of the unit vectors are given by⁸

˙

ai = 0 (1)

b˙_i = γ˙_ic_i (2)

˙

ci = −γ˙ibi (3)

The total angular momentum of the VSCMG cluster can be expressed as

h(γ, $) =

N

X

i=1

h_ib_i (4)

wherehis the total angular momentum vector,h_i=I_i$_irepresents thei-th internal momentum produced by thei-th VSCMG, andI_idenotes the moment of inertia of thei-th VSCMG wheel.

The time derivative ofhis equal to the torque generated by the VSCMGs as follows;

h˙ =

N

X

i=1

h˙ibi+

N

X

i=1

hib˙i

=

N

X

i=1

I_i$˙_ib_i+

N

X

i=1

h_iγ˙_ic_i=u (5)

whereu ∈ R³ represents the torque output vector. For three-axis torque generating devices, the vector size ofM would be three dimensional. By introducing new matrices, Eq.(5) can be simplified into

h

A(γ, $) B(γ) i

"

˙ γ

˙

$

#

=u (6)

where

A(γ, $) = [h1c1, ..., hNcN]∈R^M×N B(γ) = [I1b1, ..., INbN]∈R^M×N

Detailed derivations of the equations of motion without much simplifications is described well in the previous stud- ies.^7,10,11

For the conventional CMG clusters, by assuming that$_i is constant without loss of generality, the equations of motion can be simply as

A(γ) ˙γ=u (7)

Generally, the control torque of the CMG clusters is generated by controlling gimbal angular rates. A simple steering law to the CMG clusters is expressed as

˙

γ=A^T(AA^T)⁻¹u (8)

This relationship is generally called as the pseudo-inverse(SR) steering law. However, elements of the Jacobian matrix(A) consist of gimbal angles. Thus, existence of the pseudo-inverse of the Jacobian is not always guaran- teed; the pseudo-inverse doesn’t exist in a certain configuration. It is a mathematical singularity that is interpreted to the physical singular configuration mentioned briefly in the introduction. The singularity problem itself has been extensively investigated already in previous studies.⁵⁻⁸The main objective of this paper is to synthesize an optimal steering law for the CMG/VSCMG clusters to avoid the singularity with the qualitative optimization indices.

III. A New Sketch of Steering Laws

A. Optimal Steering Laws for CMG/VSCMGs

An optimal steering law through optimization is addressed in this section. The proposed law is derived at first by applying them to the VSCMG clusters. In the next, the optimal steering law for the conventional CMG cluster is derived by simplifying the steering law for the VSCMG clusters. Using new notations, the dynamic equation of the VSCMG cluster in Eq.(6) is rewritten as

C(χ) ˙χ=u (9)

where

C(γ, $) =h

A(γ, $) B(γ) i , χ˙ =

"

˙ γ

˙

$

#

The minimization problem to avoid the singularity is constructed by introducing a cost function such as J( ˙χ) =V(χ+ ˙χT) +1

2χ˙^TWχ˙ (10)

where the weighting matrix forχ˙ is given by

W =

"

P 0

0 Q

#

andP, Q ∈ R^N×N are the symmetric positive definite weighting matrices for γ,˙ $, respectively. The additional˙ function,V(χ+ ˙χT), is included to the cost function as a main part to be minimized.

A minimum-norm solution subject to Eq.(9) can be obtained by introducing the Lagrange multiplier vector (λ) : L( ˙χ) =J( ˙χ) +λ^T[C(χ) ˙χ−u] (11) An optimality condition is given by¹²

∂L

∂χ˙ = J( ˙χ)

∂χ˙ +λ^TC(χ) = 0 (12)

It is difficult to obtain the closed-form solution, because the proposed singularity cost is generally nonlinear. As a feasible approach, the Taylor series expansion is applied to the singularity cost as follows:

V(χ+ ˙χT) =V(χ) +T V⁰(χ)^Tχ˙+1

2T²χ˙^TV⁰⁰(χ) ˙χ+· · · (13) where

V⁰(χ) = h

∂V

∂γ

∂V

∂$

iT

(14) and

V⁰⁰(χ) =

" _∂2V

∂γ²

∂²V

∂γ∂$

∂²V

∂$∂γ

∂²V

∂$²

#

(15) For simplification, the Hessian(H) and the gradient vector(g) are defined as

H =

"

H_γ H_γ$

H$γ H$

#

≡T²V⁰⁰(χ)

and

g(χ) =h

g_γ g_$ iT

≡T V⁰(χ),

whereH_γ, H_γ$, andH_$∈R^N×Nare the splits of the Hessian, andg_γ, g_$∈R^N represent the splits of the gradient vector. By eliminating the higher order terms, the singularity cost can be rewritten as

V(χ+ ˙χT)'V(χ) +g^Tχ˙+1

2χ˙^THχ˙ (16)

By replacing Eq.(16) into Eq.(10) and applying the optimality condition in Eq.(12), the derivative of the Lagrangian with respect toχ˙ results in

∂L

∂χ˙ 'χ˙^T(H+W) +g^T+λ^TC= 0 (17) so thatχ˙is obtained by

˙

χ=−H⁻¹(C^Tλ+g) (18)

where the new matrix so-called the updated Hessian in this paper is denoted as

H =H+W (19)

Inserting Eq.(18) into Eq.(9), the Lagrange multiplier vector is obtained by

λ=−(CH⁻¹C^T)⁻¹u−(CH⁻¹C^T)⁻¹CH⁻¹g (20)

Consequently,χ˙ can be derived from

˙

χ = H⁻¹C^T(CH⁻¹C^T)⁻¹u + £

H⁻¹C^T(CH⁻¹C^T)⁻¹CH⁻¹−H⁻¹¤

g (21)

By employing new notations, the optimal steering law for the singularity avoidance of the VSCMG cluster can be expressed in a simple form

"

˙ γ

˙

$

#

=SC⁺u+ (SC⁺S^T−H⁻¹)

"

g_γ g_$

#

(22) where

S=H⁻¹C^T, C⁺=¡

CH⁻¹C^T¢⁻¹

It should be noted that, in our new approach, there is another singularity problem. The Hessian derived from the partial derivatives of the cost function could be singular, because the Hessian is also a function ofγ, $. However, from the properties of the VSCMG cluster, the singular problem can be eliminated by selecting proper weighting matrices.

Remark1 The relatively large valued weighting matrix betweenPandQwill cause small amount of energy consump- tion. The counter selection of the weighting matrices implies that the VSCMG cluster produces the control torque merely by using the gimbal rates such as general CMG clusters. Alternatively, for a large valuedP, the VSCMG cluster operates analogous to general reaction wheel assemblies. It is desirable to adjust the weighting matrices by considering the performance of the VSCMG cluster.

Remark2 Selection of the singularity-effective cost function is important to make use of the proposed optimal steering law. There may be a variety of cost functions so that several modified methods could be conceivable. Once the cost function, to minimize certain properties for a short time duration, is selected, we can adopt directly the proposed law in Eq.(22) without any further processing. Only evaluation of the Hessian and the gradient vector is needed.

B. Steering Laws Considering Wheel Speed Saturations

The optimal steering law is highly efficient to avoid the singularity by using the wheel speeds as well as the gimbal angles. In general, there are still some possibilities for the wheel speeds to be excessively high or low. The wheel saturation problem can be narrowly treated by increasing the weighting matrixQrelated to the wheel speed. Otherwise, momentum dumping between gimbal angles and wheel speeds of the VSCMG clusters could be executed to release the wheel speed. It has been reported that wheel speed is properly controlled by the null motion and preferred vector of the wheel speed.,^7? In this section, the wheel speed saturation problem is considered from optimality viewpoint.

For this objective, a new cost function is proposed as J( ˙χ) =V(χ+ ˙χT) +1

2χ˙^TWχ˙ +1

2( ˙χ−χ˙d)^TZ( ˙χ−χ˙d) (23) whereZis a positive semi-definite weighting matrix

Z =

"

Z_γ 0 0 Z_$

#

, and Z_γ, Z_$ ∈R^N×N andχ˙drepresents the preferred gradient vector given by

˙

χ_d =χ_d−χ

T (24)

whereχd = [γd, $d]^T. In this case,γdcan be set naturally toγwithout any preferred gimbal angle and$d can be selected as a vector consisting of normal wheel speeds. Adding the last term in Eq.(23) and adjusting the weighting matrix(Z), the wheel speed vector is derived into the preferred gradient vector. The Lagrangian in this section is obtained by replacing Eq.(23) into Eq.(11). Furthermore, the derivative of the Lagrangian with respect toχ˙ is given by

∂L

∂χ˙ = ˙χ^THˆ + ˆg^T+λ^TC= 0 (25)

where

Hˆ =H+W +Z, ˆg=

"

ˆ g_γ ˆ g_$

#

≡

"

g_γ−Z_γγ˙_d g_$−Z_$$˙_d

#

As a result,χ˙is obtained by

˙

χ=−Hˆ⁻¹(C^Tλ+ ˆg) (26)

Equation (26) slightly modified by the preferred gradient vector has equivalent form to Eq.(18). Consequently, the final form can be expressed as

"

˙ γ

˙

$

#

= ˆSC⁺u+ ( ˆSC⁺Sˆ^T−Hˆ⁻¹)

"

ˆ g_γ ˆ g_$

#

(27) where

Sˆ= ˆH⁻¹C^T, C⁺=³

CHˆ⁻¹C^T´⁻¹

Considering only the wheel speed saturation problem, an optimal control law can be established by replacinggˆ_γtog_γ in Eq.(27) or simply by lettingZγ = 0.

IV. Simulation study

To demonstrate the proposed steering laws, a three-parallel CMG/VSCMG cluster is examined in this section. The CMG cluster is equivalent to a two-axis control torque producing device with a redundant single-gimbal CMG. In this CMG cluster model, it is easy to show how the singularity occurs. The simulation scenario is implemented for a system with the three-parallel VSCMG cluster to develop control torque continuously from the VSCMG cluster. The required control torque command from the system is generated arbitrarily and illustrated in Fig.2.

Table 1. Weighting matrices

Symbol Case 1 Case 2 Case 3 Case 4

P 8×10⁵I 8×10⁵I 8×10⁵I 8×10⁷I Q 4×10⁴I 4×10⁵I 4×10⁵I 4×10⁷I

Z_γ 0 0 0 0

Z_$ 0 0 4×10⁵I 4×10⁷I

At first, five simulations are conducted. One simulation is for the pseudo-inverse logic in Eq.(8). Rest of sim- ulations are for the proposed optimal steering laws with different weighting matrices given in Table. 1. Note that the momentum vector is obtained from the gimbal angles through coordinate transformations. Thus, it is assumed that the gimbal angles are considered as the momentum vectors for the three-parallel VSCMG cluster without strict discrimination between them.

As can be seen in Fig. 3a for the pseudo-inverse logic, the two momentum vectors are sometimes nearly aligned in the same direction as denoted in circles. Consequently, there may be no redundant momentum vector while the two momentum vectors are aligned with the same direction. In such case, the possibility of singularity obviously increases.

It is also shown from Fig. 4that the singularity index,det(CC^T), approaches near zero. At this moment, to satisfy the torque command requirement, large gimbal angular rates in Fig.3b denoted as circles are required. Even though the configuration of momentum vectors is close to a singular condition, there is nearly no contribution from the wheel speeds of the VSCMG cluster to avoid the singularity in Fig. 3c. It is because the general pseudo-inverse logic dose not contain any weighting parameters.

Note that the weighting matrices for the optimal steering laws are selected asαIas listed in Table.1. By assigning a large value toα, it can prevent the updated Hessian from being singular. Fig. 5is for the optimal steering law in Eq.(22) with a relatively larger weighting onP thanQfor Case 1 in Table. 1. This implies that the VSCMG cluster devotes much more portion of the steering logic to the wheel speed for singularity avoidance maneuver than the gimbal

rate. We can see from Fig.5a that reasonable size of difference between the gimbal angles is maintained consistently.

The associated gimbal rates in Fig.5b are also shown to be well bounded.

However, the wheel speeds illustrated in Fig. 5c increase over about 150rpms. It is because large portion for the singularity avoidance is allocated to the wheel speeds by the weighting matrices. The increased wheel speeds cause the total angular momentum of the VSCMGs to increase. As a result, the singularity index, as the function of momentum vector, would increase too. The singularity index ofdet(CC^T)for the Case 1 is illustrated in Fig. 6. The increased singularity index may be seemingly desirable, however, it can create wheel speed saturation problem. The saturation problem may be resolved by properly adjusting the weighing matrices to release a certain amount of the wheel speed.

Case 2 in Fig.6shows that it may be possible to release the wheel speed. However, the approach cannot be guaranteed under every condition.

The next simulation illustrated in Fig. 7 is for Eq.(27) considering the wheel speeds by forcing them into the preferred vector($d) selected as the normal wheel speeds. The weighting matrices for this simulation are given as the Case 3 in Table. 1. In this caseZ$is added to account for the wheel speeds. The results illustrated in Fig. 7a,b for the optimal steering law with the wheel speeds considered show that the gimbal angles maintain suitable variations consistently and the associated gimbal angular rates are also even.

Of the most significance is the response of the wheel speeds in Fig. 7c. The wheel speeds are bounded within about 7rpms. With larger weightingZ_$, the wheel speeds could be regulated within smaller bounds. The proposed optimal steering law demonstrates great capability from the history of the singularity index of the Case 3 in Fig. 8.

Case 4 allocated with larger weighting matrices than the other cases yields poor singularity index (see Fig. 8). In other words, the magnitude of the gimbal angular rates and wheel speeds decreases to satisfy the energy saving goal producing poor singularity index.

V. Concluding Remarks

In this paper, optimization theory to solve the singularity problem of the CMG/VSCMG cluster is applied. The wheel saturation probabilities during the operations of VSCMG clusters are also addressed briefly in this paper and proposed optimal steering law for the wheel saturation problems. A closed-form solution was derived through the optimization theory. The resultant solution was given by a general form including the generally-known pseudo-inverse method as well as the null motion approach to avoid the singularity. The closed-form solution presented in this paper can be explored toward developing high performance CMG/VSCMG clusters. Moreover, the proposed laws can be also extensively applied to solve the singularity problems presented in the various engineering applications.

VI. Acknowledgements

The present work was supported by National Research Lab.(NRL) Program(2002, M1-0203-00-0006) by the Min- istry of Science and Technology, Korea.

References

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27, No. 3, 2004, pp. 374-386.

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a) Pyramid-type

b) Three-Parallel type

gimbal axis spin axis torque axis

skew angle

spin axis (momentum vector) CMG1 frame definition

CMG2 frame definition CMG3 frame definition

CMG4 frame definition

Figure 1. CMG/VSCMG cluster configurations.

0 100 200 300 400 500

-1.0 -0.5 0.0 0.5 1.0

u

1 u₂

time(sec)

N.m

Figure 2. Desired control torque commands.

a) gimbal angle responses

b) gimbal angular speeds

c) gimbal wheel speeds time(sec)

time(sec)

time(sec) degdeg/srpm

0 100 200 300 400 500

-15 -10 -5 0 5 10

15 γ^'1

γ^'2

γ^'3

0 100 200 300 400 500

-300 -200 -100 0 100 200 300

γ₁ γ₂ γ₃

0 100 200 300 400 500

1432 1433 1434

ϖ₁ ϖ₂ ϖ₃

Figure 3. Performance of the pseudo-inverse steering law.

0 100 200 300 400 500

0.00 0.05 0.10 0.15 0.20 0.25 0.30

time(sec)

Ÿ10⁶

det(CC )T

Figure 4. Singularity index of the pseudo-inverse steering law.

a) gimbal angle responses

b) gimbal angular speeds

c) gimbal wheel speeds time(sec)

time(sec)

time(sec) degdeg/srpm

0 100 200 300 400 500

-200 -100 0 100 200 300 400

γ₁ γ₂ γ₃

0 100 200 300 400 500

1450 1500 1550 1600

ϖ₁ ϖ2

ϖ₃

0 100 200 300 400 500

-15 -10 -5 0 5 10

15 γ^'1

γ^'2

γ^'3

Figure 5. Performance of the optimal steering law.

time(sec)

Ÿ10

6

det(CC )T

0 100 200 300 400 500

0.15 0.20 0.25 0.30 0.35 0.40

case 1 case 2

Figure 6. Singularity indices of the optimal steering laws.

0 100 200 300 400 500 -200

-100 0 100 200 300 400

γ₁ γ₂ γ3

a) gimbal angle responses

b) gimbal angular speeds

c) gimbal wheel speeds time(sec)

time(sec)

time(sec) degdeg/srpm

0 100 200 300 400 500

-15 -10 -5 0 5 10

15 γ^'1

γ^'2

γ^'3

0 100 200 300 400 500

1435 1440 1445

1450 ϖ₁

ϖ₂ ϖ₃

Figure 7. Performance of the optimal steering law considering wheel speeds.

time(sec)

Ÿ10⁶

det(CC )T

0 100 200 300 400 500

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

case 3 case 4

Figure 8. Singularity indices of the optimal steering law considering wheel speeds.