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計測自動制御学会 産業論文集 Vol.10，No.24，198/208 (2011) 特集「移動体の誘導制御」 Singularity Robust Control Law for Periodic Maneuvering Satellites with CMGs Takuya Ohmura*, Takehiro Higuchi**, and Seiya Ueno*** Control moment gyro (CMG) is a device used as an actuator for attitude control of spacecrafts and provides high agility. This study focuses on control law that is especially efficient for periodic maneuvering satellites with single-gimbal CMGs (SGCMGs) arranged like a pyramid. The periodic maneuvers of satellites can be utilized for observation satellites to widen the swath and observe multiple targets. Minimum-energy CMG control law for periodic maneuvering satellites is formulated. Simulation results show its performance of stabilizing gimbal angles at desired optimal gimbal angles during periodic maneuvers. When the state approaches a singularity, the control law cannot steer gimbal angles toward desired optimal gimbal angles. Using Discrete Lyapunov Stability, this paper analyzes the singularity problem of proposed control law. Lyapunov analysis explains why the singularity problem occurs, and control law is modified inserting diagonal matrix to avoid this problem. The modified control law makes the system asymptotically stable at desired optimal gimbal angles. Simulation results demonstrate the useful merits of the modified control law to stabilize at desired optimal gimbal angles even if a singularity exists. The singularity robust control law will enable spacecrafts, especially for next-generation observation satellites, to perform periodic maneuvers much more effectively. Key Words : Optimal control, Control moment gyros, Periodic control, Stability, Singularity problem 1. INTRODUCTION Control moment gyro (CMG) is a device used as an actuator for attitude control of spacecrafts. CMG contains a spinning rotor with large constant angular momentum. The spinning rotor can be gimballed to produce gyroscopic torque, which is orthogonal to both spin and gimbal axes. The major benefit of utilizing CMG for the satellite control is torque amplification property that can provide satellites with high torque capability. High agility with CMG is very important particularly for earth observation satellites such as Pleiades-HR1). Pleiades-HR has remarkable observation abilities using CMGs. For example, rapid multiple target imaging as shown in Fig. 1, global observation in short period, high resolution disaster monitoring, and so on. CMG is capable of expanding the ability of observation performance for an attitude controller. Considerable studies have been conducted on control problems of CMG systems, including singularity avoidance. Kurokawa2) conducted the survey of singularity avoidance steering laws and grouped as exact methods, offline planning, and SR inverse family. As example of singularity * Department of ocean-space system engineering, Yokohama National University, Hodogaya-ku, Yokohama-shi, Kanagawa, Japan(E-mail: d09GB202@ynu.ac.jp) ** Interdisciplinary Research Center, Yokohama National University *** Research Institute of Environment and Information Sciences, Yokohama National University (Received December 3, 2010) avoidance laws, Bedrossian et al.3) applied singularity robust inverse steering law to control CMGs to avoid singularities with allowing some torque error. Vadali et al.4) proposed the method based on back integration of the gyro torque equation from desired terminal condition to determine preferred initial gimbal angles that can avoid singularities. Wie5) developed CMG steering logic that uses an additional weighting matrix to avoid explicitly singularity encounters. These results assume that the satellites with CMGs perform the one-way maneuver from the initial attitude to the specified terminal condition. This paper focuses on periodic maneuvering satellites, as shown in Fig. 1, with singlegimbal CMGs arranged like a pyramid. Periodic maneuvers of satellites can be utilized to widen the swath and observe multiple targets. Satellite Fig. 1 Schematic image of periodic control of satellite for multiple targets TRIA 024/11/1024 © 2010 SICE Ueno6) 7) has proposed optimal control law for adaptive structure with periodic constraint. It was considered in these studies that the periodic control enables the manipulators to track and capture the rotating structures, especially the space debris that is out of control. Control law in this study is well related to the approach proposed by Ueno. This study discusses the effectiveness of applying periodic optimal control strategy to satellite attitude control problems with CMGs. The purposes of this study are (1) to design minimum energy CMG control law for periodic maneuvering satellites, (2) to analyze singularity problem using the proposed control law numerically and propose singularity robust control law, and (3) to demonstrate useful merits of the singularity robust control law from simulation results. The rotational equation of motion of a rigid satellite with CMGs is described by, ω H 0. (1) H H=[Hx, Hy, Hz]T is the total angular momentum vector of the system that is sum of the satellite main body angular momentum I and the CMG angular momentum h. Time derivative of H is H . =[ x, y, z]T is the angular velocity vector of the satellite. By substituting H=I+h, Equation (1) gives ω Iω h. h Iω (2) In Equation (2), I R33 is the inertia matrix of the satellite. Quaternion q=[q1, q2, q3, q4]T is chosen to describe the attitude of the satellite. The quaternion consists of the maneuver angle and desired axis of rotation r in the body fixed frame. 2. CMG CONTROL LAW FOR PERIODIC MANEUVERING SATELLITES r l m nT , This section introduces satellite attitude dynamics with CMGs and formulates control law for periodic maneuvering satellites. This section also shows simulation results with the proposed control law. q1 q2 q3 q4 T l sin 2 q3 q2 q4 ω 2 q3 q4 q1 q2 q1 q4 2.1 Satellite Attitude Dynamics The model in this study is a rigid body with a cluster of four CMGs arranged in a pyramid configuration like Figure 2. In Fig. 2, ii=1,2,3,4is the ith gimbal angle and is the pyramid skew angle. (3) m sin q q1 1 q q2 2 . q q3 3 q4 2 n sin 2 T cos , 2 (4) (5) This study treats periodic maneuvering satellites. The maneuver angle therefore has to be satisfied with the periodic condition, (t+T)=(t). From Equations (3)-(5), is also given as periodic form, (6) ωt T ωt . In Equation (6), T is the period of the maneuver. The following relation is given between the gimbal angle vector x=[1, 2, 3, 4]T and the gimbal rate vector u, which define the system state and the control input, respectively. x u. (7) For the pyramid type CMG systems, CMG angular momentum h is sum of each CMG angular momentum vector hi. Fig. 2 Pyramid arrangement 199 4 h hi i 1 cos sin1 cos 2 cos sin 3 cos 4 h cos1 cos sin 2 cos 3 cos sin 4 , sin (sin 1 sin 2 sin 3 sin 4 ) (8) where h is the constant magnitude of the axial angular momentum of each CMG. The time derivative of Equation (8) is, h A x u. J int sin 2 cos cos1 A h sin1 cos cos 2 sin cos1 sin cos 2 cos cos 3 sin 4 sin 3 cos cos 4 . sin cos 3 sin cos 4 (10) This model is approximation of the real CMG dynamics which has been widely used in the studies on CMG systems. The control law for periodic maneuvering satellites is designed based on this dynamics in the following section. State 1 T u u ρ x - xopt T u. 2 (14) (9) In Equation (9), the matrix A has dimension of 3×4, 2.2 input variables that corresponds to the energy consumption, and second term (x-xopt)Tu represents the derivative of (xxopt)T(x-xopt); the square of the difference between current gimbal angles and optimal gimbal angles. The second term has the potential to stabilize current gimbal angles x to desired optimal gimbal angles at xopt. The criterion is integration of sum of two terms in one period. The integrant of the criterion Jint is, Feedback Control Law for Minimum J int subject to Au-S(t)=0 Applying the Lagrange multipliers can solve this optimization problem. Lagrangian, L is expressed using multiplier ∈R3 1 (15) L uT u x xopt T u λT Au S . 2 Periodic Maneuvering Satellites The control law for periodic maneuvering satellite is formulated as follows. Equation (2), (6), and (9) give periodic constraint for desired periodic maneuvers that has to be satisfied at all time. Au S t 0, It is possible to derive an optimal initial state and input with minimum criterion by numerical calculation. However, the gimbal rates between initial gimbal angles and optimal gimbal angles needs to be optimized, because both angles would be different in many cases. Hence, the local minimum solution is used to design the feedback control law to improve the energy consumption in every cycle. The local minimum solution solves the following optimal control problem to derive optimal input. Necessary conditions for minimum Jint are, (11) L L 0, 0. u λ (12) The CMG control law for periodic maneuvering satellites is obtained by the following expression, (16) where, t ωt Iωt ht . S t ht Iω n 1T J nT 1 T 2 u u ρ x - x opt T udt. S ρ E AT AAT us ρP x xopt , u AT AAT The matrix S follows the periodic condition, S(t+T)=S(t). This study aims to propose minimum energy control law for periodic maneuvering satellites. The criterion J is expressed as, 1 Ax x 1 opt (17) where, (13) P E AT AAT u s AT AA In Equation (13), n is the cycle number and is the weight. xopt indicates optimal gimbal trajectory for single cycle that achieves the minimum energy maneuver. The criterion contains two terms: the first term uTu is the square of the T 1 1 S, x opt uopt dt C 200 A, A A A T 1 T c c c (18) Sdt C . diag[10000,10000,10000][kgm2], pyramid skew angle , 45[deg], constant magnitude of the axial angular momentum h , 1000[kgm2/s], and weight , 0.005. These parameters are used throughout this paper. It is assumed that the satellite performs a 60-deg rotation and then returns back to its initial attitude in 60[s] around the rotation axis: r=[0.50.5 0.50.5 0]T. The maneuver is performed periodically. Let C in Equation (18) be set as [53.4 -53.4 53.4 -53.4]T [deg]. Fig. 3 shows the simulation results. Fig. 3a) and 3b) shows the time histories of four gimbal angles and four gimbal rates, respectively. Fig. 3c) indicates the criterion in each cycle. Fig. 3d) represents the unit quaternion of the attitude of the satellite. Note that the proposed control law achieves to stabilize the gimbal angles to the desired optimal initial gimbal angles [53.4 -53.4 53.4 -53.4]T [deg]. The gimbal rates are minimized accordingly. The commanded maneuver is confirmed by the periodic results of the quaternion that changes its value from q=[0, 0, 0, 1]T to q=[0.354, 0.354, 0, 0.866]T, and then back to q=[0, 0, 0, 1]T in one-cycle. In Equation (17), E is 4×4 unit matrix. The matrix P can be considered as projection matrix where P is idempotent PT=P and self-adjoint PPT=P. In Equation (18), optimal initial gimbal angles C is the constant value and is very important to achieve minimum-energy periodic maneuvers because of the following reason. Energy consumption for performing commanded maneuvers depends crucially on the initial gimbal angles. This optimal gimbal angles can be calculated numerically, where optimal angles are relative to various conditions: maneuver axis and angle, satellites moment of inertia, CMG pyramid skew angle, and so on. Ac in Equation (18) is almost the same form as A in Equation (10), but including the value of C. 2.3 Proof of Optimality In the preceding section, optimal gimbal angles are defined as Equation (18). This section proves the optimality of the set of optimal gimbal angles. The discussion stated in this section can be referred to the publication by Rao8). The solution of Au=S can be represented in the form u=GS where G is a generalized inverse of A that satisfies the condition, AGA=A, and general solution is u=GS+(IGA)z where z is arbitrary. Let G be a generalized inverse of A such that Gy is a minimum norm solution for all z and x, that is, the following inequality holds. z Gy Gy IGA z. GAx GAx IGA (19) It is necessary and sufficient for the condition (19) that the Gimbal angles [deg] 90 30 1 3 0 2 4 -30 -60 following equivalent conditions are satisfied; GAT I GA 0 or GAT GA. 60 -90 0 (20) Looking back at the control law shown in Equation (17), G generalized inverse of A, has the form AT(AAT)-1, which satisfies the necessary and sufficient condition (20). In CMG system, initial gimbal angles C plays a key role to decide the amount of energy consumption. Therefore the solution u=AcT(AcAcT)-1S including the optimal initial gimbal angle combination C is a minimum norm solution in the class of all solutions of Au=S(t). It can be concluded that the setting of desired optimal gimbal angles as Equation (18) has the optimality property for the system. 4 8 12 Cycle 16 20 24 a) Gimbal angles Gimbals Rate [deg/s] 3.0 2.4 Simulation Results This section shows simulation results with the CMG control law (17) for periodic maneuvering satellites. The simulation parameter values are as follows: inertia matrix I , u1 u3 1.5 u2 u4 0.0 -1.5 -3.0 0 4 8 Cycle b) Gimbal rates between 1st and 12th cycle 201 12 mission requirement referring to the index given by Equation (21). The results of Fig. 3 clearly indicate the effectiveness and utilization of the proposed control law for periodic maneuvering satellites. Criterion[1/s] 0.03 3. SINGULARITY PROBLEM 0.02 Singularities are major difficulty for using CMGs as attitude controller. This chapter is on singularity problem for using the proposed control law (17). This section provides simulation results to indicate the problem where singularity prevents gimbal angles from stabilizing at desired optimal gimbal angles. Discrete Lyapunov Stability Theory is used to analyze the singularity problem. 0.01 0 4 8 12 16 20 24 Cycle 3.1 Simulation Example of Singularity Problem Singularity problem is well known as major difficulty for using CMGs as attitude controller. Singularity is a gimbal angle combination that CMG system cannot produce torque about certain axis. To avoid the situation, singularity avoidances have been developed for many years including SR-inverse method, exact solution, and offline planning. Not only singularity avoidance but also singularity analysis and visualization9) have also been developed. The singularity problem can cause severe problems as shown in the following simulation. Fig. 4 shows simulation results of the case when maneuver axis r is [0.250.5, 0.250.5, 0.50.5]T. The satellite performs 60-deg maneuver and then returns to the initial attitude around the axis. Starting from the gimbal angles [60, -60, 60, -60]T [deg], let the desired optimal initial gimbal angle combination C be set as [13.1, -13.1, 13.1, 13.1]T [deg]. All other parameters are same as the previous simulation results. Fig. 4a and 4b show the time history of gimbal angles and gimbal rates, respectively. Fig. 4c and 4d show the criterion and the unit quaternion of the attitude of the satellite, respectively. From Fig. 4a, the results show that the gimbal angles stabilize at [43.5, -43.5, 27.0, -27.0]T [deg], which combination is apparently different from the desired optimal initial gimbal angles C. Criterion in onecycle on the desired optimal combination is about 0.051; on the contrary, the result of Fig. 4c, criterion cannot reach the minimum value. The gimbal angle combination [43.5, -43.5, 27.0, -27.0]T [deg] therefore can be considered as nonoptimal but stable gimbal angle combination. This undesirable stabilization of gimbal angles appears just before passing through a singularity [35.3, -35.3, 35.3, 35.3]T [deg] with zero-angular momentum that exists between initial and desired optimal gimbal angles. The singularity prevents gimbal angles from stabilizing at the c) Criterion 1.0 Quaternion 0.8 q1 q3 0.6 q2 q4 0.4 0.2 0.0 0 4 8 12 Cycle 16 20 24 d) Quaternion Fig. 3 Simulation results on r=[0.50.5 0.50.5 0]T Consequently, the satellite performs the commanded periodic maneuver with minimum energy consumption. In addition, the speed of stabilization of gimbal angles can be changed by the value of the weight in Equation (17). There is a method to determine the weight using the relationship between the weight and time constant . The relationship is described by, 1.0 (21) 1 exp 0.632 . The simulation results of Fig.3 assume that the time constant is 200[s]. The is determined from Equation (21) to be 0.005. Higher the value of is, faster the minimum energy periodic maneuver can be achieved, but the energy for the convergence increases as well. can be changed by 202 Discrete Lyapunov Stability guarantees the stable at steady state x0 if there is a positive definite function V(x0)≧ 0 for which V(x0)=0 and V(t+⊿t)-V(t)≦0 for some time step ⊿t＞0. Based on the theory, consider the Lyapunov function V, desired C. This is singularity problem for using the proposed control law, which problem is comparatively unique to ordinary singularity problem because of the following reason. Gimbal motion passing through a singularity is also a problem. This singularity problem needs to be deeply analyzed because minimum energy periodic maneuvers cannot be achieved. V 3.2 Numerical Stability Analysis It is difficult to discuss stability for complex non-linear system. Lyapunov Stability is one of the methods to analyze the system stability. In this study, Discrete Lyapunov Stability is used to analyze stability of desired optimal gimbal angles for the proposed control law (17). This section describes why the singularity problem occurs. 1 3 T P x x opt dt nT n 1T x x opt (22) T V is positive definite in all cycles and zero if and only if x=xopt because of the projection matrix P. 2 4 0.10 60 0.09 30 0.08 0 -30 -60 0.07 0.06 0.05 -90 0 4 8 12 Cycle 16 20 0.04 24 0 4 8 12 16 20 24 Cycle c) Criterion a) Gimbal angles 6 u1 u3 u2 u4 1.0 3 0.8 Quatenion Gimbals Rate [deg/s] P T P x x opt dt V n . nT Criterion[1/s] Gimbal angles [deg] 90 n 1T x x opt 0 -3 q1 q3 0.6 q2 q4 0.4 0.2 0.0 -6 0 4 8 12 0 Cycle b) Gimbal rates between 1st and 12th cycle 4 8 12 16 Cycle d) Quaternion Fig. 4 Simulation results: r=[0.250.5 0.250.5 0.50.5]T 203 20 24 V and V(n+1)-V(n) are numerically calculated to analyze the singularity problem. Fig. 5 shows gimbal angle combinations with zero-angular momentum. The horizontal axis shows the 1st gimbal angle and left and right vertical axis represent the 2nd and 3rd gimbal angle, respectively. There are seven gimbal angle combinations: [a, -a, a, -a]T, [a, a-180, a, a-180]T,[a, a+180, a, a+180]T, [a1, -a1, a2, a2]T, [-a2, a1, -a1, a2]T, [a2, -a2, a1, -a1]T, [-a1, a2, -a2, a1]T. The relation between a1 and a2 is, (23) Singularity exists at the cross points between each line in Fig. 5. In Fig. 5, the initial and optimal gimbal angles in Fig. 4 are also plotted. Fig. 5 shows important factor for control law to utilize null-motion to steer gimbal angles. The line indicates the trajectory of the gimbal motion where no output torque is generated to perform the transition to optimal gimbal angles. Lyapunov function V with the same parameters as the case of Fig. 4 is numerically calculated and shown in Fig. 6a and 6b. Fig. 6b is close-up view of V near singular point. Gimbal angles are steered in the direction that the value of V decreases, V(n+1)-V(n)＜0. Lyapunov Function V 150 125 100 75 50 60 40 25 20 00 40 Gimbal Angle: Singularity ― 3rd Gimbal angle - - 2nd Gimbal angle -180 20 60 0 1[deg] Gimbal Angle: 3[deg] tan 1 1 a 2 sin 1 sina1 tan 1 1 cos cos This numerical analysis for V and V(n+1)-V(n) can verify that the desired optimal initial gimbal angles [13.1, -13.1, 13.1, -13.1]T [deg] can be stable state, but also shows that the system can stabilize at the non-optimal gimbal angles at [43.5, -43.5, 27.0, -27.0]T [deg]. From Fig. 6b, there is significant decrease of Lyapunov function to the nonoptimal angles at the singularity. Gimbal angles are easy to steer toward [43.5, -43.5, 27.0, -27.0]T [deg] accordingly. Consequently, as shown in Fig. 4, the gimbal angles are stabilized at the non-optimal gimbal angles when approaching the singularity at [35.3, -35.3, 35.3, -35.3]T [deg]. Singularity problem, undesirable stabilization of gimbal angles is caused by the significant decrease of V at a singularity. a) Lyapunov function V 180 [a -a a -a] 0 0 [a1 -a1 a2 -a2] 90 180 -180 -90 -90 0 90 -180 180 30 20 10 030 32 34 36 1st Gimbal Ang le 38 40 [deg]: 40 38 36 34 32 30 3rd Gimbal Angle [deg]: 3 90 Lyapunov Function V -90 3rd Gimbal angle [deg]; a3 2nd Gimbal angle [deg]; a2 40 1 1st Gimbal angle [deg]; a1 b) Lyapunov function V around the cross point in Figure 6a Fig. 6 Numerical analysis of Lyapunov function Fig. 5 Gimbal angle combinations with zero-angular momentum 204 The significant decrease of V can be considered mainly due to the value of P(x-xopt), which is included in V. The P(x-xopt) in Equation (17) and (22) plays an important role in steering gimbal angles to optimal gimbal angles. Fig. 7a shows the 1st element of P(x-xopt) on gimbal angle combinations with zero-angular momentum. The other elements are similar to the 1st element. It can be confirmed by Fig. 7a that P(x-xopt) has sharp discontinuous drop when gimbal angles change its combination to [a1, -a1, a2, -a2]T at the singularity, which is induced by a difference between the value of P on [a1, - a1, a2, - a2]T and P on [a, -a, a, -a]T. W is 4×4 diagonal matrix. Substitute Equation (24) into the periodic constraint Au-S(t)=0, the modified control law can satisfy the constraint for all time. Fig. 7b shows the 1st element of PW(x-xopt) on the gimbal angle combinations with zero-angular momentum. The result of Fig. 7b is given from the same parameters presented in Figure 7a except for the W, diag[100, 100, 1, 1]. It is clear by comparing Fig. 7b to 7a that inserting proper W setting contributes to prevent the discontinuous drop of P(x-xopt), which leads to the singularity problem. To prevent the discontinuous drop, the effective W settings tend to be the form [100, 100, 1, 1], [100, 1, 100,1], 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 on [a, -a, a, -a]T P 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 [100, 1, 1, 100] … ; two diagonal elements in W are set large enough to pass a singularity. One of these effective W settings can be chosen from a singularity between initial and 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 on [a1, -a1, a2, -a2]T P 1 4 1 4 1 4 1 4 1 4 1 4 1 4 1 4 optimal gimbal angles, and non-optimal convergence gimbal angles. 1 .0 The sharp discontinuous drop of P(x-xopt) at a singularity lead to significant decrease of V and steers gimbal angles toward non-optimal gimbal angles. Next chapter proposes singularity robust control law that can solve the singularity problem based on the analysis in chapter 3. al An g] g ]: [d e 3 [d e 0 60 g le 40 1 .0 0 .6 Gi al mb PW(x-xo p )t 0 .8 3 rd 60 40 0 .4 0 .2 40 0 g] [d e g ]: 1 s t G im b a l A n g le [d e g ]: 1 [d e g ] 60 [d e 20 3 0 .0 0 g le (24) 20 An 20 a) P(x-xopt) 1 opt mb x x Gi us ρPW S ρ E W 1 AT AW 1 AT x xopt , 1 20 1 s t G im b a l A n g le [d e g ]: 1 [d e g ] 4.1 Singularity Robust Control Law The modified control law is proposed on the idea that a modification by inserting diagonal matrix W can increase the value of P(x-xopt) and steer gimbal angles toward desired optimal gimbal angles. 60 40 0 .4 0 .0 0 This chapter proposes modified version of the control law by inserting diagonal matrix to solve the singularity problem. Inserting diagonal matrix can eliminate the sharp discontinuous drop of P(x-xopt), which leads the system asymptotically stable at desired optimal gimbal angles. The modified control law results in improved performance, which is shown in simulation results. 0 .6 0 .2 4. SINGULARITY ROBUST CONTROL LAW u AT AAT 3 rd P(x-xo p )t 0 .8 where PW E W 1 AT AW 1 AT 1 b) PW(x-xopt) A. (25) 205 Fig. 7 P(x-xopt) and PW (x-xopt) about 1st gimbal angle 0, 1]T to q=[0.25, 0.25, 0.354, 0.866]T, and then back to q=[0, 0, 0, 1]T in one-cycle, which indicates that the modified control law with W can achieve the commanded periodic maneuver. Table 1 shows results for the case: rotation axis r, [1, 0, 0]T, and optimal initial gimbal angle C, [73, -73, 73, -73]T [deg]. Table 1 also clearly indicates that the modified law (24) can steer gimbal angles to the desired optimal angles; by contrast, the control law (17) fails to stabilize at the desired optimal state. It can be concluded through simulation results that the modified control law with proper diagonal matrix W has significant performance; it can pass through singularities on gimbal angle combinations with zero-angular momentum successfully and steer gimbal angles toward desired optimal gimbal angles. For the modified control law, let the Lyapunov function be redefined as, VW n 1T x x opt T PW T PW x x opt dt VW n. (26) nT Numerical analysis of the Lyapunov function VW (W=diag[100, 100, 1, 1]) is shown in Fig. 8 under the same condition of Figure 6. It is apparently different result from Fig. 6 that the non-optimal gimbal angles [43.5, -43.5, 27.0, -27.0]T [deg] is not stable and the desired optimal initial gimbal angles [13.1, -13.1, 13.1, -13.1]T [deg] becomes asymptotically stable state. 5 CONCLUSION 200 Periodic maneuvers can be utilized to widen the swath and observe multiple targets. This study proposes the minimum energy CMG control law for periodic maneuvering satellites. Simulation results verify its performance that gimbal angles can be stabilized at desired optimal gimbal angles during maneuver. However, in case there is a singularity between certain initial and desired optimal gimbal angles, there is possibility that the control law cannot stabilize at desired optimal gimbal angles. Discrete Lyapunov Stability technique is used to analyze the singularity problem. Numerical analysis of Lyapunov function clarifies that gimbal angles are easy to stabilize at non-optimal gimbal angles because of the following reason. Lyapunov function is decreased significantly at a singularity for non-optimal gimbal angles, which is mainly induced by sharp discontinuous drop of P(x-xopt) when gimbal angles change its combination with zero-angular momentum at a singularity. The proposed control law is modified by inserting diagonal matrix W that can eliminate the discontinuous drop of P(x-xopt). The modified control law makes the system asymptotically stable at desired optimal gimbal angles. Simulation results demonstrate the useful merits of the singularity robust control law that can stabilize at desired optimal gimbal angles even if a singularity appears in the stabilizing phase. As the result, the satellite achieves commanded periodic maneuvers with minimum energy consumption. The proposed CMG control law will enable spacecrafts, especially for next-generation Earth imaging satellites, to perform periodic attitude maneuvers much more effectively. Gimbal Angle: 3[de g] 100 60 40 50 20 0 0 20 Gimbal Angle: 1[d 40 Lyapunov Func tion V 150 0 60 eg] Fig. 8 Lyapunov Function VW in case W=[100, 100, 1, 1] 4.2 Performance Comparison using Simulation Results The performance of W is confirmed by comparing the simulation result from the case without W. Fig. 9 shows simulation results under the same parameter in Fig. 4 except for W, diag[100, 100, 1, 1]. Fig. 9a and 9b show timehistory of gimbal angles and gimbal rates of the modified control law, respectively. Fig. 9c and Fig. 9d show the comparison between criterions in each case and quaternion, respectively. Fig. 9e illustrates the stabilization of gimbal angles in two cases. Comparing Fig. 4 to 9a, it can be confirmed that the modified control law (24) can achieve to stabilize the gimbal angles at the desired optimal initial gimbal angles, [13.1, -13.1, 13.1, -13.1]T [deg]. The criterion in Fig. 9c, which implies amount of energy consumption, is obviously much less than that of Fig. 4. Fig. 9d shows the quaternion that changes its value from q=[0, 0, 206 1 3 60 0.10 2 4 30 0 -30 0.08 0.07 0.06 0.05 -60 0.04 -90 0 4 8 12 Cycle 16 20 0 24 4 8 12 16 20 24 Cycle a) Gimbal angles c) Comparison between criterion in Fig. 4 and Fig. 9a 6 u1 u3 u2 u4 1.0 3 0.8 Quatenion Gimbals Rate [deg/s] W=[100, 100, 1, 1] W=[1, 1, 1, 1] 0.09 Criterion[1/s] Gimbal angles [deg] 90 0 -3 q1 q3 0.6 q2 q4 0.4 0.2 0.0 -6 0 4 8 12 0 4 8 12 16 Cycle Cycle d) Quaternion b) Gimbal rates between 1st and 12th cycle W=diag[100, 100, 1, 1] W=diag[1, 1, 1, 1] [60, -60, 60, -60] [35.3, -35.3, 35.3, -35.3] [45, -45, 25, -25] [13.1, -13.1, 13.1, -13.1] e) Stabilization of gimbal angles in Fig. 4 and Fig. 9a Fig. 9 Simulation results with W, [100, 100, 1, 1] 207 20 24 Table 1 Simulation results with C, [73, -73, 73, -73]T [deg] and r, [1, 0, 0]T Takuya Ohmura He received his B.E. degrees from Yokohama National University in 2009. Currently, he is Initial gimbal angles [deg] Singularity [deg] [120,-120, 120,-120] [30,-30, 30,-30] Gimbal angles stabilize at a second-year master candidate at the same Control law (17) [deg] Modified control law (23) [deg] university. He received the “1st Place Student [90,-90, 90,-90] [97,-84,97,-84] [73,-73,73,-73] W=[100,1,100,1] Session Award” at 18th IFAC Symposium on [35,-35, 35,-35] [29,-29,42,-42] [73,-73,73,-73] W=[100,100,1,1] Automatic Control in Aerospace, Nara, Japan, 2010. Takehiro Higuchi (Member) REFERENCES He received the B.E., M.E., and Ph.D. degrees from Yokohama National 1） Philippe CAMPENON: PLEIADES AND SPOT 6: EARTH OBSERVATION IN HIGH University in 2000, 2002, and 2005, International Astronautical respectively. He has worked as assistant HIGH RESOLUTION, Proc. of 60th AND VARY professor at Yamaguchi University from Conference 2009 Daejeon. 2） Haruhisa Kurokawa: Survey of Theory and Steering Laws of 2005 to 2007, and has been working as Single-Gimbal Control Moment Gyros, Journal of Guidance, assistant professor at Yokohama National University from 2007. His research interests are optimal control Control, and Dynamics, 30-5, 1331/1340 (2007) and system design of aerospace systems. 3） N. S. Bedrossian, J. Paradiso, E. V. Bergmann and D. Rowell: Steering Law Design for Redundant Single-Gimbal Control Seiya Ueno (Member) Moment Gyroscope, Journal of Guidance, Control, and He received the B.E., M.E., and Ph.D. Dynamics, 13-6, 1083/1089(1990) degrees in aeronautical engineering from the 4） S. R. Vadali, H. S. Oh, and S. R. Walker: Preferred Gimbal university of Tokyo in 1980, 1982, and 1985, Angles for Single Gimbal Control Moment Gyros, Journal of respectively. He joined the Division of Guidance, Control, and Dynamics, 13-6, 1090/1095 (1990) Space Development, Toshiba corporation in 5） Bong Wie: Singularity Escape/Avoidance Steering Logic for 1985. In 1988, he moved to Yokohama Control Moment Gyro Systems, Journal of Guidance, Control, National University, where he has been a and Dynamics, 28-5, 948/956(2005) professor since 2002. His major research fields are applications of 6） Seiya Ueno: Optimal Trajectory of a Planar Space Crane as an optimal control in aerospace engineering. Adaptive Structure, Proc. of Third International Conference on Adaptive Structures, 320/334(1992) 7） Seiya Ueno, Takehiro Higuchi, and Takashi Ninomiya: Optimal Periodic Control for Manipulator of Variable Geometry Truss, The Japan Society of Mechanical Engineers, 12th Space Engineering Conference, 49/52(2001) 8） C. Radhakrishna Rao, and Sujit Kumar Mitra: GENERALIZED INVERSE OF A MATRIX AND ITS APPLICATIONS, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol.1, 601/620(1972) 9） Bong Wie: Singularity Analysis and Visualization for SingleGimbal Control Moment Gyro Systems, Journal of Guidance, Control, and Dynamics, 27-2, 271/282 (2004) 208