Steering laws analysis of SGCMGs based on singular value decomposition theory

Transcription

1 Appl. Math. Mech. -Engl. Ed., 2008, 29(8): DOI /s c Shanghai University and Springer-Verlag 2008 Applied Mathematics and Mechanics (English Edition) Steering laws analysis of SGCMGs based on singular value decomposition theory ZHANG Jing-rui ( ) (Beijing Institute of Technology, School of Aerospace Science Engineering, Beijing , P. R. China) (Communicated by CHEN Li-qun) Abstract The steering laws of single gimbal control moment gyros (SGCMGs) are analyzed and compared in this paper for a spacecraft attitude control system based on singular value decomposition (SVD) theory. The mechanism of steering laws escaping singularity, especially how the steering laws affect singularity of gimbal configuration and the output torque error, is studied using SVD theory. Performance of various steering laws are analyzed and compared quantitatively by simulation. The obtained results can be used as a reference for designers. Key words single gimbal control moment gyros, singular value decomposition, steering law, singularity Chinese Library Classification V Mathematics Subject Classification 37N35 Introduction A single gimbal control moment gyro (SGCMG) has the superior properties of small power consumption, simple mechanical structure, high reliability and large torque amplification capability, etc. Therefore, SGCMG has been widely used as the actuator for attitude control system of large-scale spacecraft, such as the Sport-5 of France, the MIR of the Former Soviet Union [1], and so on. In recent years, a class of smaller satellites which is required to execute rapidly attitude maneuver missions, for instance, the Pl iades and the BilSat, etc. [2 3], have also been equipped with SGCMG. One of the principal difficulties in using single gimbal control moment gyros (SGCMGs) is the configuration singularity. This occurs when all individual CMG torque output vectors are perpendicular to the commanded torque direction. The configuration singularity of SGCMGs depends on not only the geometric configuration but also on the designed steering law. Various gyro configurations have been proposed, such as dual parallel configuration, three parallel configuration, pyramid configuration (PC), roof type configuration, five pyramid configuration (FPC) and so on [4 6]. They have different characteristics of configuration singularities. Received Jan. 10, 2008 Project supported by the National Natural Science Foundation of China (No ), the Excellent Scholars Fund of Beijing (No D ), and the Excellent Young Scholars Research Fund of Beijing Institute of Technology (No. 2007YS0202) Corresponding author ZHANG Jing-rui, Associate Professor, zhangjingrui@bit.edu.cn

2 1014 ZHANG Jing-rui Until now, a number of steering laws have been proposed to steer SGCMGs, such as the pseudoinverse steering law [7], pseudoinverse steering law with null-motion [8], singularity-robust (SR) steering law [9], generalized singularity-robust (GSR) steering law [10], weighted singularityrobust (WSR) steering law [11], singular direction search steering law [12] and constrained steering law [13], etc. The pseudo-inverse steering law is the optimal solution of the gimbal angular velocity in the least square sense, but it does not have the capability of avoiding singularity. The proposed steering laws have their own advantages and disadvantages. Some researchers have compared these steering laws with each other [14 15]. However, their discussions are limited to the level of numerical simulation or semi-physical simulation [15]. Until now there has been little analysis and comparison of the mechanism of these steering laws. Which brings many difficulties with regards to choosing the right steering law for SGCMGs in the application. Therefore, it is necessary to compare these steering laws in mechanism so that the spacecraft s designer could choose a suitable steering law according to the space mission requirements and the actual capability of the actuators. By means of SVD theory, this paper analyzes and compares the steering laws of SGCMGs used in the spacecraft attitude control systems. The mechanism by which the steering laws affect the configuration singularity and output torque errors are particularly emphasized. The variations of singular values and singular vectors and their influence on output torque errors are compared when the steering laws escape singularity. The ways of escaping singularity are discussed. The performances of various steering laws are compared by a simulation example. The obtained results can be used as a reference when engineers design an attitude control system for a spacecraft. 1 Configuration singular problem of SGCMGs Consider the torque equation of SCMGs: C(δ) δ = 1 h T c, (1) where C(δ) R 3 n is the matrix relative to the direction of each gyro s momentum, its i-th column is the unit vector of the i-th gyro along its momentum direction, n is the number of gyros; δ and δ are, respectively, the n 1 dimension vector of the gimbal angle and the gimbal angular rate of gyros; h is the angular momentum of a single gyro, and it is assumed that all gyros have the same angular momentum; T c R 3 is the required control torque produced by actuators, which is required by the attitude control law. Treating C(δ) with singular value decomposition, one has C(δ) = V ΛU T = Σσ i v i u T i, (2) where V R 3 3, Λ R 3 n, U R n n ; Λ = [S 0], S R 3 3, 0 R 3 (n 3), S = diag(σ i ), i = 1, 2, 3; σ i is the i-th singular value of C(δ), and there is σ 1 σ 2 σ 3 0; both V and U are unitary matrices whose i-th column are respectively notated as v i and u i. 2 Analysis and comparison of steering laws This section will analyze the characteristics of each steering law and compare their advantages and disadvantages. To this aim, one needs to define beforehand a measure of singularity. There are some different definitions about the measure of singularity of SGCMGs. Here is the

3 Steering laws analysis of SGCMGs based on singular value decomposition theory 1015 most used definition based on the determinant: Γ = det(c(δ)c(δ) T ) = σ 2 1σ 2 2σ 2 3, Γ [0 n 3 /27]. (3) 2.1 Weighted pseudoinverse steering law Consider the following optimization problem with cost function min J = δ 2 P dt = δ T P δdt, (4) and constraint condition hc(δ) δ + T c = 0, (5) the Hamiltonian can be written as H = 1 2 δ T P δ + λ T (hc(δ) δ + T c ), (6) where λ is the vector of the Lagrangian multiplier and P is the symmetric and positive definite weighted matrix. According to the optimization theory, the co-state equation of the system is thereby λ = H δ = P δ + hc(δ) T λ = 0, (7) δ = hp 1 C(δ) T λ. (8) Then and Substituting Eq. (8) into the above equation yields δ = H λ = hc(δ) δ + T c = 0. (9) T c = h 2 C(δ)P 1 C(δ) T λ λ = 1 h 2 (C(δ)P 1 C(δ) T ) 1 T c. (10) Substituting Eq. (10) into Eq. (8) and denoting P 1 = Q, we can obtain a weighted pseudoinverse steering law as follows: δ = 1 h QC(δ)T [C(δ)QC(δ) T ] 1 T c. (11) Let Q = I, Eq. (11) will degenerate into a well known pseudoinverse steering law: δ = 1 h C(δ)T [C(δ)C(δ) T ] 1 T c. (12)

4 1016 ZHANG Jing-rui 2.2 Generalized weighted robust pseudoinverse steering law Consider the following optimization problem with cost function: min J = ( δ 2 P + T e 2 W)dt = ( δ T P δ + T T e WT e )dt, (13) where P,W being symmetric positive definite weighted matrixes, T e is the torque error vector and T e = hc(δ) δ + T c. Since the torque error is included in the cost function, the optimal problem is transformed to an unconstrained one. The Lagrangian function can be written as L = 1 2 ( δ T P δ + Te T WT e). (14) To obtain the extremum of the function, one has L δ = P δ + h 2 C(δ) T WC(δ) δ + hc(δ) T WT c = 0 (15) and δ = 1 h [P/h2 + C(δ) T WC(δ)] 1 C(δ) T WT c. (16) From the property of an inverse matrix, we have [P/h 2 + C(δ) T WC(δ)] 1 C(δ) T W = P 1 C(δ) T [C(δ)P 1 C(δ) T + W 1 /h 2 ] 1. Denoting P 1 = Q and W 1 /h 2 = D, there is δ = 1 h QC(δ)T [C(δ)QC(δ) T + D] 1 T c. (17) The above equation is a generalized weighted robust pseudoinverse steering law [11]. The weighed matrix is recommended in Ref. [11] is Q = w 1 ε ε ε ε w 2 ε ε ε ε w 3 ε ε ε ε w 4, D = εe, E = 1 λ 3 λ 2 λ 3 1 λ 1, λ 2 λ 1 1 where λ i is a scalar function of less than 1, λ i = λ 0 sin(ωt + φ i ), i = 1, 2, 3; ε is a scalar function which is small enough, and ε = ε 0 exp( µγ); λ 0, ω, φ i, ε 0, µ are constants chosen by the designers. The robust pseudoinverse steering law can be obtained if choosing Q = I 4, D = εi 3 in Eq. (17), δ = C(δ) T [C(δ)C(δ) T + εi 3 ] 1 T c, (18) and the generalized robust pseudoinverse steering law can be obtained if choosing Q = I 4, D = εe in Eq. (17), δ = C(δ) T [C(δ)C(δ) T + εe] 1 T c. (19)

5 Steering laws analysis of SGCMGs based on singular value decomposition theory Analysis of the mechanism of singularity avoidance 3.1 Singularity mechanism of pseudoinverse steering law Steering law (12) can be rewritten as δ = 1 h UΛT V T [V S 2 V T ] 1 T c = 1 [ ] S h U 1 V T T 0 c = 1 h n 1 σ i u i v T i T c. When T c is parallel to v j, denoting T c = βv j, where β is a constant; since v T i v j = there is { 1 (i = j) 0 (i j) δ = β hσ j u j. (20) It means that δ and u j are parallel to each other. If σ j 0, gimbal angular rates will only change along the direction of u j ; especially if j = 3 and σ 3 = 0 occur simultaneously, there is δ, which means that the steering law is unavailable. According to Eq. (20), even if σ 3 = 0 but j 3, the steering law is still available. From the above analysis, not all of the singular configurations will lead the pseudoinverse steering laws to fail. If the control torque T c is perpendicular to v 3, even if σ 3 = 0, there is still δ = 1 h 2 1 σ i u i v T i T c. (21) It means that in order to improve the singular characteristics of gyro clusters, one should take two means, increasing the singular values or changing the direction of singular vectors. 3.2 Singularity avoidance mechanism of the generalized weighted robust pseudoinverse steering law Consider the generalized weighted robust pseudoinverse steering law, Eq. (17) which can be rewritten as δ = ] 1Tc h QC(δ)T[ ( σ i v i u T i )Q( σ i u i vi T ) + D, (22) i.e., δ = 1 h QC(δ)T[ 3 j=1 3 1Tc σ j σ i v j u T j Qu ivi T + D]. (23) a) Role of the weighted matrix D { 1 (j = i) To clearly show the influence of D, denote Q = I. Since u T i u j =, Eq. (23) can 0 (j i) be rewritten as Selecting D = 3 ξ2 i v iv T i, then δ = 1 3 1Tc h C(δ)T[ σi 2 v ivi T + D]. (24) δ = 1 3 ] 1Tc h C(δ)T[ (σi 2 + ξ2 i )v ivi T. (25),

6 1018 ZHANG Jing-rui In this case, even if singularity occurs, i.e., σ 3 = 0, the above steering law can still work. If we choose D = ξ 2 3v 3 v T 3, then only the third singular value of the above equation (25) changes from 0 to ξ 3, which actually reduces the introduced torque error. The function of weighted matrix D is mostly to change the singular value of C(δ). Furthermore, if D is chosen as a general symmetric positive definite matrix, besides changing the singular value, it may have influence on the singular vectors of C(δ). However, the influence cannot be estimated, thus it is not easily used effectively. b) Role of weighted matrix Q Consider Eq. (17). To clearly show the role of Q, denote D = 0 and Q I. Equation (17) will degenerate to Eq. (11). Denoting U = [U 1 U 2 ], U 1 R n 3, U 2 R n (n 3), then Eq. (11) becomes δ = 1 h QUΛT V T [V ΛU T QUΛ T V T ] 1 T c = 1 h QUΛT [ΛU T QUΛ T ] 1 V T T c = 1 [ ] U h U T 1 QU 1 S 1 V T T 0 c = 1 [ ] S h U 1U1 T QU 1 1 V T T 0 c = 1 n 1 ζ i vi T h σ T c, i where ζ i is the i-th column vector of U 1 U T 1 QU 1. Consider the case that T c is parallel to v j. Denote T c = βv j, since v T i v j = there is { 1 (i = j) 0 (i j), δ = β hσ i ζ i. (26) Comparing Eq. (20) with Eq. (26), it is obvious that the role of matrix Q is to change the direction of the singular vectors. If σ i 0, gimbal angular rates do not have to be changed along the direction of u i. Time-variant matrix Q can be chosen to change the direction of ζ i, so that the gimbal angular rates are regulated (which regulates the configuration of the gimbal) to escape the singularity. If i = 3, when the CMGs plunge into singularity (i.e., σ 3 = 0), there is δ. That is, the steering law only using weighted matrix Q is not valid enough. Therefore, matrices D and Q should be employed simultaneously in practice. D is used to change singular values, and Q is used to change the direction of singular vectors. Although Ref. [11] recommended a method for choosing D and Q, it is not convenient for application. Furthermore, it does not discuss the relationship between the change of singular direction and the sensitivity of singular values. The problem of how to carefully select D and Q will be discussed in another of our papers. 4 Simulation results Considering SGCMGs of FP configuration Γ, used as the singularity measure defined in Eq. (3), is adopted. The characteristics of the generalized pseudoinverse steering law (GP), the robust pseudoinverse steering law (RP) and the weighted robust pseudoinverse steering law

7 Steering laws analysis of SGCMGs based on singular value decomposition theory 1019 (WRP) are compared with each other. To make the research results clear, for RP, only the choice of weighted matrix D is considered, and for WRP, only the choice of weighted matrix Q is considered. In order to verify the capability of escaping singularity, here the step signal is used as the input torque, which could make the gyros plunge into singularity rapidly. The initial condition of the simulation is The input torque is [δ 1 (0) δ 2 (0) δ 3 (0) δ 4 (0)] = [ ], [ δ 1 (0) δ 2 (0) δ 3 (0) δ 4 (0)] = [ ]. T c = [ ], and D = ξ 2 3 v 3v T 3, ξ 3 = The choice of Q is dependent on that of ζ 3. When the gyros nearly reach singularity, u 3 will rotate by small angle so that the singular direction is changed. The simulation results are shown in Figs Figure 1 shows that the GP steering law makes gyros plunge into singularity after one second, and the singularity can not be overcome any more. Figure 2 shows that the RP steering law can escape singularity by a suitable choice of D, but it will introduce an output torque error, and the gyros may plunge into singularity repeatedly. Figure 3 shows the WRP steering law can escape singularity by choosing Q suitably and the torque error introduced is small enough, but it needs larger gimbal angular rates. It can be noticed that under normal working conditions, gyros mainly absorb the periodically variational disturbance angular momentum. If gyros work under constant disturbance for a long time, they will saturate due to the accumulation of the angular momentum. In this case, the gyros need an external action to unload the saturation since they can not do it by themselves. Fig. 1 Simulation results of GP

8 1020 ZHANG Jing-rui Fig. 2 Simulation results of RP Fig. 3 Simulation results of WRP

9 Steering laws analysis of SGCMGs based on singular value decomposition theory Conclusions This paper studied the singularity of SGCMGs when they are used as the actuators for attitude control of spacecraft. The mechanism of escaping singularity and introducing output torque error were in vestigated for several popular steering laws based on the SVD theory. Analysis and comparison show the GP steering law does not have the ability of escaping singularity at all, while the WRP steering law has the preferred ability of avoiding singularity. It is not ideal to solely use the weighted matrix D or Q. Solely using D may lead to larger output torque errors, and solely using Q may lead to the requirement of unrealizable gimbal angular rates. Therefore, the process of selecting the weighted matrices D and Q is an open problem. The results of this paper can be used as a reference for engineers in designing attitude control systems of spacecraft. References [1] Branets V N, Weinberg D M, Vevestchagin V P et al. Development experience of the attitude control system using single-axis control moment gyros for long-term orbiting space station[j]. Journal of Acta Astronautica, 1988, 18: [2] Baudoin A, Boussarie E, Damilano P, Rum G, Caltagirone F. Pléiades: a multi mission and multi co-operative program[c]. IAF-01-B.1.07, 52nd International Astronautical Congress, Toulouse, France, 1-5 Oct [3] Lappas V J, Oosthuizen P, Madle P, Yuksel G, Fertin D. Design, analysis and in-orbit performance of the BILSAT-1 microsatellite twin control moment gyroscope experimental cluster[r]. AIAA , [4] Margulies G, Aubrun J N. Geometric theory of single-gimbal control moment gyro systems[j]. Journal of the Astronautic Sciences, 1978, 26(2): [5] Zhang Jinjiang. Research on configuration analysis and comparison of SGCMG system[j]. Chinese Space Science and Technology, 2003, 23(3):52 56 (in Chinese). [6] Yoshikawa T. Steering law for roof type configuration control moment gyro system[j]. Automatica, 1977, 13: [7] Farmer J E. A reactive torque control law for gyroscopically controlled space vehicles[r]. NASA TM X-64790, [8] Bedrossian N S, Paradiso J P, Bergmann E V, Rowell D. Steering laws design for redundant single gimbal control moment gyroscopes[j]. Journal of Guidance, Control and Dynamics, 1990, 13(6): [9] Ford K A, Christopher D H. Singular direction avoidance steering for control-moment gyros[j]. Journal of Guidance, Control and Dynamics, 2000, 23(4): [10] Wie B, David B, Christopher H. Singularity robust steering logic for redundant single-gimbal control moment gyros[r]. AIAA , [11] Wie B. New singularity escape/avoidance steering logic for control moment gyro systems[r]. AIAA , [12] Cornick D E. Singularity avoidance control laws for single gimbal control moment gyros[r]. AIAA , [13] Kurakawa H. Constrained steering law of pyramid-type control moment gyros and ground tests[j]. Journal of Guidance, Control, and Dynamics, 1997, 20(3): [14] Wu Zhong, Wu Hongxin. Survey of steering laws for single gimbal control moment gyroscope systems[j]. Journal of Astronautics, 2000, 21(4): (in Chinese). [15] Zhang Jinjiang, Li Jisu, Wu Hongxin. Research on physics simulation of large spacecraft attitude control system using SGCMG[J]. Journal of Astronautics, 2004, 25(4): (in Chinese).