Global Trajectory Design for Position and Attitude Control of an Underactuated Satellite *

Transcription

1 Trans. Japan Soc. Aero. Space Sci. Vol. 59, No. 3, pp. 7 4, 26 Global Trajectory Design for Position and Attitude Control of an Underactuated Satellite * Yasuhiro YOSHIMURA, ) Takashi MATSUNO, 2) and Shinji HOKAMOTO 3) ) Department of Aerospace Engineering, Tokyo Metropolitan University, Hino, Tokyo 9 65, Japan 2) Department of Mechanical and Aerospace Engineering, Tottori University, Tottori, Tottori , Japan 3) Department of Aeronautics and Astronautics, Kyushu University, Fukuoka, Fukuoka , Japan Underactuated control offers fault-tolerance for satellite systems, which not only enables the position and attitude control of a satellite with fewer thrusters, but also can reduce the number of thrusters equipped on the satellite even when considering the need for backups. Due to having fewer thrusters, the coupling effect between the translational motion and rotational motion of the satellite cannot be avoided, and the coupled motion must be considered in control procedures. This paper presents a global trajectory design procedure required for the position and attitude control of an underactuated satellite. The satellite has four thrusters with constant thrust magnitudes on one plane of the satellite body. Then, an analytical solution for coupled motion between the rotation and translation of the satellite is obtained using three-step maneuvers of attitude control. The trajectory design based on the analytical solution is shown for the control of translational and rotational motion in three dimensions. Finally, a numerical simulation is performed to verify the effectiveness of the proposed design procedure. Key Words: Attitude Control, Position Control, Thrusters, Underactuated System, Constant Inputs Nomenclature c z : cos z CðÞ; SðÞ: Fresnel integrals f i : thrusters ði ¼ ; 2; 3; 4Þ F c : magnitude of thruster force F: external force vector in body-fixed frame J k : moment of inertia ðk ¼ x; y; zþ m: satellite mass k : angular velocity ðk ¼ x; y; zþ : w parameter phase QðÞ: skew symmetric matrix R bi : direction cosine matrix from inertial frame to body-fixed frame R ib : direction cosine matrix from body-fixed frame to inertial frame s z : sin z k : moment of inertia ratio ðk ¼ x; y; zþ t: input duration T j : control torque ðj ¼ x; zþ V: translational velocity vector in inertial frame w ;w 2 ;z: wz-parameters X: position vector in inertial frame Subscripts : initial value of single spin motion x; y; z: body-fixed coordinates X; Y; Z: inertial coordinates Superscripts i: i-th on-interval 26 The Japan Society for Aeronautical and Space Sciences + Received June 24; final revision received 29 December 25; accepted for publication 8 January 26. Corresponding author, yyoshi@tmu.ac.jp n: number of thrust firings : positive directional control torque : negative directional control torque. Introduction Thrusters can be used to control the position and attitude of satellite systems. In practice, the force directions of the thrusters are fixed with respect to the satellite body, and a satellite is equipped with more than thrusters to control its position and attitude. This number of thrusters is clearly more than the minimum necessary. However, the minimum number to control both the satellite position and attitude has not been specified. Moreover, no control methods for the translational and rotational motion of an underactuated satellite have been developed yet. The position and attitude control of an underactuated satellite is useful for reducing the number of thrusters equipped on satellites even when considering the need for backups. Furthermore, such control techniques enable continuing satellite operations even when some of the thrusters have failed, which enhances the faulttolerance of satellite systems. The attitude control of underactuated satellites has been extensively studied. Some researchers 5) have shown that torques about two principal axes of a satellite can control the angular velocities and orientation angles about three axes because the rotational motion of a rigid body forms first-order nonintegrable constraints (i.e., first-order nonholonomic constraints). In these studies, the control torques are assumed to be available in both positive and negative directions using control moment gyros (CMGs) or pairs of gas-jet thrusters, and the assumption of the bidirectional inputs allows the use of some excellent results. 6,7) 7

2 Trans. Japan Soc. Aero. Space Sci., Vol. 59, No. 3, 26 This study presents a global trajectory design procedure for the position and attitude control of a satellite with fewer thrusters. When the thrusters are fixed with respect to the satellite body, the translational equations form second-order nonintegrable constraints. Moreover, each thrust direction is restricted in one direction due to the thruster mechanisms. Some studies have dealt with systems under second-order nonholonomic constraints 8 ) that have bidirectional control inputs. The position and attitude control problem of a satellite using a sufficient number of thrusters,2) is also discussed. Their techniques, however, are not applicable to the system discussed in this paper due to the input constraints caused by four thrusters and their input directions. To tackle these severe constraints, in this paper an analytical solution for coupled motion between the rotation and translation of the satellite is obtained using three-step maneuvers of attitude control. The trajectory design based on the analytical solution is then shown and helpful for the control of translational and rotational motion in three dimensions. The designed trajectory indicates the control method for ideal thruster conditions (i.e., there is neither misalignment nor unmodeled nonlinear dynamics of thrusters). For realistic conditions, the trajectory designed in this paper shows a clue for feedback control. This paper is organized as follows. In Section 2, a thruster configuration and the equations of motion of a satellite are shown. Section 3 first derives an attitude control technique that consists of three-step maneuvers, because the governing equations of rotational motion are more complicated than those of translational motion. On the basis of the attitude control algorithm and Fresnel integrals, the analytical solution for coupled motion between the rotation and translation of the satellite is derived. Using the analytical solution, a trajectory design procedure to simultaneously control the position and attitude is presented. Finally, a numerical simulation is performed to demonstrate the validity of the proposed trajectory design in Section 4, and Section 5 provides the conclusions. 2. Problem Formulation In this paper, a Cartesian frame fx; Y; Zg is considered as an inertial reference system. The body-fixed frame fx b ;y b ;z b g is assumed to be coincident with the principal axes of a satellite and the frame origin is placed at the satellite mass center. 2.. Thruster configuration The minimum number of thrusters required to control satellite position and attitude has not been specified due to theoretical and mathematical complexities. Determining the minimum number of thrusters and their configuration with regard to controllability is difficult because the theories developed by Sussmann 3) and Goodwine and Burdick 4) are not applicable to systems with constant and unilateral inputs. In the present study, four thrusters are placed parallel to a satellite principal axis y b for simplicity, as shown in Fig.. All thrusters are assumed to generate the same magnitude Fig.. Thruster configuration. of constant force F c and have the same length of moment arms about the x b and z b axes. Although this thruster configuration may not be the minimum number, it should enable the position and attitude control of the satellite by using its nonholonomic constraints. The control torques about two axes can be generated by the following combinations of the thrusters. Tx : f ¼ f 2 ¼ ; f 3 ¼ f 4 ¼ F c ðþ Tx : f ¼ f 2 ¼ F c ; f 3 ¼ f 4 ¼ ð2þ Tz : f ¼ f 4 ¼ F c ; f 2 ¼ f 3 ¼ ð3þ Tz : f ¼ f 4 ¼ ; f 2 ¼ f 3 ¼ F c ð4þ Due to the unilateral constraint on the thrust forces, the thrusters cannot attenuate translational velocity without changing the attitude of the satellite. Thus, simultaneous control of the satellite rotation and translation is needed. It should be noted that although the thruster forces are assumed to be along the y b axis, the proposed method is applicable to the other axes without a loss of generality by redefining the body-fixed frame Rotational equations of motion This study assumes only two control torques around a satellite x b and z b axes for attitude control. The dynamic equations of the underactuated spacecraft are written as _ x ¼ x y z T x _ y ¼ y z x _ z ¼ z x y T z J z The variables k are the moment of inertia ratios: x ¼ J y J z J x ð8þ y ¼ J z J x J y ð9þ z ¼ J x J y J z ðþ where y 6¼ (i.e., J z 6¼ J x ). Otherwise, Eq. (6) indicates that the rotational velocity around the y b axis becomes uncontrollable. The kinematic relation between the direction cosine matrix (DCM) and angular velocity satisfies the following equation. 5) _Rbi ¼ QðÞRbi ðþ where QðÞ expresses the skew symmetric matrix consisting J x ð5þ ð6þ ð7þ 8

3 of the angular velocity vector : 2 3 z y 6 QðÞ ¼4 z 7 x 5 ð2þ y x The kinematics of satellite attitude are represented as rotations from the inertial frame to the body-fixed frame. In this study, the satellite attitude is expressed with wz-parameters, which have been proposed by Tsiotras and Longuski. 3) The wz-parameters use two successive rotations to represent the Trans. Japan Soc. Aero. Space Sci., Vol. 59, No. 3, 26 three-dimensional attitude, whereas Euler angles and quaternions take three rotations and one rotation, respectively. The first rotational angle is represented by z parameter, and the second rotational angle and axis are expressed by parameters w and w 2. Using the wz-parameters for the attitude representation makes it possible to foresee a proper control procedure and control the satellite attitude to a desired target within a few maneuvers. 6) The DCM using the wz-parameters is described as 2 w 2 w2 2 cz 2w w 2 s z w 2 3 w2 2 sz 2w w 2 c z 2w 2 R bi ¼ w 2 2w w 2 c z w 2 w2 w2 2 sz 2w w 2 s z w 2 6 w2 2 cz 2w ð3þ 2 2w 2 c z 2w s z 2w 2 s z 2w c z w 2 w2 2 The kinematic equations of a satellite are written as follows: _w ¼ z w 2 y w w 2 x 2 w2 w2 2 _w 2 ¼ z w x w w 2 y 2 w2 2 w2 _z ¼ z x w 2 y w ð4þ ð5þ ð6þ Note that parameter z does not appear in the kinematic equations because it denotes the first rotational angle of the satellite attitude. 3) 2.3. Translational equations of motion The DCM is an orthonormal matrix and satisfies R ib ¼ R T bi. The translational equation in the Cartesian frame is written as m _V ¼ R ib F ð7þ The kinematic equation of the translational motion is described by _X ¼ V ð8þ When thrusters are arranged parallel to the y b axis (i.e., F x ¼ F z ¼ ), Eq. (7) becomes F y m _V X ¼ w 2 2w w 2 cos z w 2 w2 w2 2 sin z 2 ð9þ F y m _V Y ¼ w 2 2w w 2 sin z w 2 w2 w2 2 cos z 2 ð2þ 2F y m _V Z ¼ w 2 w ð2þ w Control Procedure This section presents a trajectory design procedure for a satellite s position and attitude control. In a previous study of attitude stabilization using constant inputs, Kojima 7) has shown that two constant torques around two principal axes of a satellite can stabilize the satellite angular velocity to zero on the basis of the manifolds discussed by Rafael and Bong. 8) The present study thus assumes that a satellite has no initial angular velocity. If the satellite has non-zero initial angular velocities, the controller 7) is applied to converge the angular velocities to zero beforehand. 3.. Attitude control This study uses the wz-parameters to represent the satellite attitude. Although Euler angles or quaternions can be used, the wz-parameters are previously shown to enable control of the satellite attitude angles using a three-step maneuver. 6) Since the inertial frame can be defined arbitrarily, without loss of generality, the target state is set to zero attitude angles in the inertial frame. The proposed attitude control method consists of three steps, which are summarized in Table. Note that, for constant control inputs, analytical solutions can be derived to obtain the control procedure for each maneuver, and thus the attitude parameters are sequentially controlled to zero. For instance, after w 2 is converged to zero in Maneuver, the next maneuver controls w while w 2 is kept invariant (i.e., _w 2 ¼ ). Each maneuver is explained in the following. In Maneuver, a positive control torque about the z b axis Maneuver Table. Target state ðw ;w 2 ;zþ Attitude control maneuvers. Input timing ð; ; Þ d ¼ T z ðt Þ 2 z 2J t z 2 ð; ; Þ w d ¼ tan T x ðt Þ 2 x 4J x 2 t 3 ð; ; Þ z d ¼ T z ðt Þ 2 z 2J t z 9

4 Trans. Japan Soc. Aero. Space Sci., Vol. 59, No. 3, 26 is applied during an arbitrary finite time t to generate a single spin motion around the z b axis, where the superscript + denotes a parameter when a positive directional control torque is applied. Since the single spin motion makes the coupling effect of the angular velocity zero, the dynamic equations are described as follows: _ x ¼ _ y ¼ ð22þ J z _ z ¼ Tz ð23þ Equation (23) is integrable, and the analytical solution is obtained as z ¼ T z J z t z ð24þ Hereafter, the subscript means the initial value when a single spin motion is generated or attenuated for each maneuver. Since x ¼ y ¼ in the first maneuver, the kinematic equations, Eqs. (4) (6), are expressed as follows: _w ¼ z w 2 ð25þ _w 2 ¼ z w ð26þ _z ¼ z ð27þ The analytical solutions of w and w 2 are described as follows: w ¼ A cos ð28þ w 2 ¼ A sin ð29þ where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ¼ w 2 ; w2 2; ð3þ ¼ arctanðw 2 =w Þ ð3þ The analytical solutions in Eqs. (28) and (29) also mean that w and w 2 cannot be controlled to zero simultaneously. Thus, w 2 is controlled to zero in the first maneuver by applying the control input at a proper timing defined in the following. Because the motion of parameter w 2 is expressed with the phase angle as shown in Eq. (29), should be controlled to zero to achieve w 2 ¼. The time derivative of is obtained from Eqs. (25), (26), and (3): _w 2 w _w w 2 _ ¼ 2 w 2 =w w 2 ð32þ ¼ z Substitution of Eq. (23) into Eq. (32) yields the analytical solution of d ¼ T z 2J z t ð Þ 2 z t ð34þ At the end of this maneuver, the state becomes ð; ; Þ where the is arbitrary values of w and z. In Maneuver 2, a control torque is applied to generate a single spin motion about the x b axis. Since y ¼ z ¼ is kept during this maneuver and the attitude parameter w 2 was controlled to zero in the previous maneuver, the time derivatives of w 2 and z are expressed from Eqs. (5) and (6) as From Eq. (4), _w 2 ¼ _z ¼ _w ¼ x 2 w2 ð35þ ð36þ Equations (35) and (36) show that w can be controlled independently, and Eq. (36) is integrated as follows: dw w 2 ¼ x 2 dt ¼ 2 T x J x t x dt ) arctanðw Þ ¼ T x t 2 x 4J x 2 t arctan w ; ð37þ Similarly, this analytical solution presents the input timing to control w to an arbitrary value. Thus, Maneuver 2 can make the state ð; ; Þ. Finally, the attitude parameter z is controlled using a torque about the z b axis in Maneuver 3. Then, since _w ¼ _w 2 ¼, the equation around the z b axis is written as _z ¼ z ð38þ Equation (38) is integrable, and z can be easily controlled to a target state. Although the above discussion is for a case when a positive control torque is applied to generate a single spin motion, a similar process can be obtained for a negative control torque Analytical solutions for translational motion In this subsection, the analytical solutions of the satellite translational motion during Maneuvers 2 and 3 are derived. The corresponding translational equations are described from Eqs. (9) (2) as follows: Maneuver 2 ðtþ ¼ T z 2J z t 2 z t ð33þ w m _V 2 X ¼F y w 2 sin z ð39þ This analytical solution provides the desired input timing d to control and z to zero at the same time. Since jtz j¼jt z j is assumed, the input duration t to despin the single spin motion is determined as t ¼ t. The desired input timing to converge the angular rate and attitude angle to zero is thus specified as follows. Maneuver 3 w m _V 2 Y ¼ F y w 2 cos z m _V Z ¼ F y 2w w 2 m _V X ¼F y sin z ð4þ ð4þ ð42þ

5 Trans. Japan Soc. Aero. Space Sci., Vol. 59, No. 3, 26 m _V Y ¼ F y cos z m _V Z ¼ ð43þ ð44þ CðxÞ :¼ Z x cos 2 t2 dt ð55þ In Maneuver 3, Eq. (44) indicates that the translational velocity along the Z axis is uncontrollable. Thus the satellite position along the Z axis must be controlled to the target value before Maneuver 3 is implemented. As shown below, the analytical solutions for Maneuver 3 are firstly found, and then the analytic solutions for Maneuver 2 are derived. As shown in Eq. (38), the orientation angle for a single spin can be described analytically for Maneuver 3. Thus, the analytical solutions of the translational equations are derived to rewrite Eqs. (42) and (43) as follows: m _V X ¼F y sin m _V Y ¼ F y cos T z 2J z ðt Þ 2 z t z T z 2J z ðt Þ 2 z t z Equation (45) can be transformed into m _V X ¼F y sin 2 ð Þ 2 B where :¼ T z J z 2 B :¼ z z 2 :¼ t z Similarly, Eq. (46) is rewritten as m _V Y ¼ F y cos 2 ð Þ 2 B Furthermore, Eqs. (47) and (5) are transformed into _V X ¼ F y m sin 2 ð Þ 2 cos B cos 2 ð Þ 2 sin B _V Y ¼ F y m cos 2 ð Þ 2 cos B sin 2 ð Þ 2 sin B ð45þ ð46þ ð47þ ð48þ ð49þ ð5þ ð5þ ð52þ ð53þ Equations (52) and (53) cannot be integrated nor expressed with elementary functions. However, Fresnel integrals make it possible to obtain the analytical solutions for translational motion by integrating Eqs. (52) and (53). Normalized Fresnel integrals are defined as SðxÞ :¼ Z x sin 2 t2 dt ð54þ Then, the integrations of Eqs. (52) and (53) are described as follows: V X ¼ F y m ðs cos B C sin B ÞV X V Y ¼ F y m ðs sin B C cos B ÞV Y where Z t z S :¼ sin z 2 ð Þ 2 d ¼ S t z S C Z t z :¼ cos z 2 ð Þ 2 d ¼ C t z C z z ð56þ ð57þ ð58þ ð59þ Note that the integral interval is changed from t to z = t z = due to the replacement of t with in Eq. (5). To obtain an analytical solution for the satellite position, Eqs. (56) and (57) need to be integrated. Since Fresnel integrals are further integrable as Z x SðxÞdx ¼ xsðxþ cos 2 x2 ð6þ Z x CðxÞdx ¼ xcðxþ sin 2 x2 ð6þ the integrations of Eqs. (56) and (57) provide the following analytical solutions for the satellite position: ffi X ¼ F y cos B S2 m ðt z Þ S t F y sin B C m 2 ðt Þ C z t F y m S 2 ðþ cos B C 2 ðþ sin B VX t X Y ¼ F y cos B C m F y sin B S m 2 ðt Þ C 2 ðt Þ S s ffiffiffiffiffiffi z z t t ð62þ

6 Trans. Japan Soc. Aero. Space Sci., Vol. 59, No. 3, 26 F y m C 2 ðþ cos B S2 ðþ sin B VY t Y ð63þ where S2 ðt Þ¼ 2 t z t z S 2 A 2 C 2 ðt Þ¼ 2 t z A t z C t z t z ð64þ ð65þ The analytical solutions of Maneuver 2 can be obtained in a similar manner. The translational motion along the Z axis should be controlled to a target value in Maneuver 2, and Eq. (4) is transformed into w m _V Z ¼ 2F y pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi w 2 w 2 ð66þ ¼ F y sinð2 arctan w Þ From Eq. (37), Eq. (66) is rewritten as m _V Z ¼ F y sin T x t 2 x 2J t 2 arctan w ; ð67þ x Because this equation has the same form as Eq. (45), the analytical solution along the Z axis in Maneuver 2 can be obtained in the same manner Control of translational velocity This subsection presents a control method to drive the satellite translational velocity and attitude angle to zero in Maneuver 3. As shown in Eq. (67), the analytical solution in Maneuver 2 becomes the same form as that in Maneuver 3 and the control technique described in the following discussion is applicable to both Maneuvers 2 and 3. The translational velocity in Eqs. (56) and (57) is affected by not only the satellite attitude angle, but also the angular velocity. For one thruster firing on -interval, the attitude angle at which the thrust force should be applied is uniquely specified according to the angular velocity. That is, the translational velocity and attitude angle cannot be controlled to the target state at the same time. However, applying several on-intervals to reduce the rotation, we can vary the final translational velocity after the despin maneuver. When n on-intervals are applied for the thruster firings, the following relation is needed to despin the rotational motion: t ¼ Xn ðt Þ i ð68þ Equations (56) and (57) are rewritten in matrix form as i¼ " # " #" # " V X ¼ F y cos B sin B S V # X m sin B cos B V Y ) V ¼ F y m R BV V C V Y ð69þ The matrix R B forms a rotation matrix and indicates that the direction of V depicts a circle whose center is V on the V X V Y plane (Fig. 2). The time profile of the translational velocity V can vary with different firing timings because the matrix R B includes the initial attitude angle z i for each on-interval. Furthermore, since V can be changed by varying the input interval t, the velocity vector V can be changed to any value in the inner area of the circle. That is, it indicates the satellite velocity can be controlled arbitrarily. As mentioned above, the attitude angle of the n-th thruster firing z n is unique according to the angular velocity n z due to the constant inputs. The angular rate n z is specified by z n and the (n )-th interval t n as n z ¼ n z t n. Thus, to cancel the translational velocity after the n-th despin maneuver, the following conditions must be satisfied by two proper variables t n and z n V n X V n Y Fig. 2. Graphic interpretation of V. : VX n ðzn ;t n ÞVX n ðzn ;t n Þ¼ ð7þ VY n ðz n ;t n ÞVY n ðzn ;t n Þ¼ ð7þ The solutions satisfying the above two conditions can be obtained numerically using the analytical solutions of the translational velocity. Note that Eqs. (7) and (7) have solutions because the initial velocities, VX n and VY n, can be made relatively small before the (n )-th maneuver using the discrete input intervals t i ði ¼ ;...;n2þ. Thus, the satellite translational velocity and attitude can be converged to zero simultaneously Position control When the spacecraft has a negative translational velocity V Y < with zero attitude angle during Maneuver 3, the satellite s position and translational velocity along the Y axis can be easily controlled using Eqs. (42) and (43). The translational motion of the satellite can be converged to the origin once the satellite reaches the following states: ðx ¼ V X ¼ Z ¼ V Z ¼ ; V Y < ; Y>Þ. 2

7 Trans. Japan Soc. Aero. Space Sci., Vol. 59, No. 3, 26 Fig. 3. Proposed control procedure in Maneuver 3. Table 2. Simulation parameters and initial condition. Simulation parameters Satellite mass, m 5. [kg] Constant thrust force, F c. [N] Moment of inertia ðj x ;J y ;J z Þ (25., 3., 35.) [kgm 2 ] Moment arm. [m] Initial condition Angular rate ð x ; y ; z Þ (.,.,.) [rad/s] ZYX Euler angles 2., 3., 45. [deg] Translational velocity ðv X ;V Y ;V Z Þ (.,.,.) [m/s] Position ðx; Y; ZÞ (¹2., ¹2., ¹2.) [m] Figure 3 describes the position and attitude control procedures in Maneuver 3. When the satellite has the state (X >; Y ; V X < ; V Y < ), the input timings to control the attitude angles and translational velocity in the X direction are calculated. Then, Eq. (62) provides the position increments X for such maneuvers. Thus, summing those increments can specify the start position of the thrust firings. Consequently, maneuvering the satellite to the Y axis with the target attitude angles can control and stop the satellite at the origin. The proposed control procedure is summarized as follows. Maneuver : The angular rate z and attitude parameter w 2 are simultaneously controlled to zero with a single spin motion around the z b axis. The input timing is uniquely determined by Eq. (34). Maneuver 2: After a single spin motion around the x b axis is generated, the satellite translational velocity and position along the Z axis are controlled to zero with discrete thruster firings. The angular rate x and attitude parameter w are simultaneously converged to the target state in the maneuver. The thruster firing timings are determined by the analytical solutions obtained from Eq. (67). Maneuver 3: The satellite is first driven to a preferred state that satisfies ðx >; Y ; V X < ; V Y < Þ by a controlled single rotation around the z b axis. Then, through drift motion, the satellite translational velocity and position along the X axis are controlled onto the Y axis with discrete inputs satisfying Eq. (7). The satellite moves along the Y axis and stops at the origin of the inertial frame. Fig. 4. Time history of the angular velocities. Fig. 5. Time history of the attitude angles. 4. Numerical Simulation A numerical simulation is performed to demonstrate the validity of the proposed control algorithm. As an example, the simulation parameters and initial conditions are summarized in Table 2. The initial attitude angles are expressed with ZYX Euler angles for a better understanding. The number of thruster firings n to despin rotation are set to two and one for Maneuvers 2 and 3, respectively. Although a despin maneuver is generally completed with multiple thrust firings, the despin maneuver in this simulation uses the solution of Eq. (7) with one thruster firing. Figures 4 and 5 show the time histories of the angular velocity and attitude angles, respectively. Since the inertial frame is defined to be coincident with the target attitude Fig. 6. Time history of the translational velocities. 3

8 Trans. Japan Soc. Aero. Space Sci., Vol. 59, No. 3, 26 This study, from a theoretical point of view, has dealt with a global trajectory design for the position and attitude control of an underactuated satellite using four thrusters with a constant force magnitude. The thrusters are fixed relative to the satellite body, which is an equivalent thruster configuration of the minimum number of thrusters for attitude control. To tackle this quite difficult problem, the following procedures have been adopted. First, by a technique using wz-parameters, the attitude control method has been derived to simplify the coupled motion between the rotation and translation of the satellite. Then, the analytical solutions of the translational motion were obtained using Fresnel integrals. The global trajectory design for the translational and rotational motion control with discrete inputs has been shown on the basis of the analytical solutions derived. The control procedure consists of three-step maneuvers and each input timing has been designed with the analytical solutions. Numerical simulations have verified the effectiveness of the proposed control algorithm. For real satellite systems, from a practical viewpoint, controllers are required to be robust against modeling errors or external disturbances. Since the global trajectory designed with the proposed procedure is based on open-loop control, a feedback controller should be superimposed on the designed trajectory. In such cases, motion variables which cannot be directly controlled with only thrusters must be controlled first, then thrusters should control other variables by feedback control. However, the development of such a feedback control method is out of the scope of this paper. References and every state parameter become zero at the target, the satellite attitude is successfully controlled. Figures 6 and 7 show the time histories of the satellite translational velocity and position. In Maneuver 3, the satellite is initially controlled to the Y axis, (i.e., V X ¼ X ¼ ). The satellite position and velocity along the Y axis are then successfully converged to the origin. 5. Conclusions Fig. 7. Time history of the position. ) Crouch, P.: Spacecraft Attitude Control and Stabilization: Applications of Geometric Control Theory to Rigid Body Models, IEEE Trans. Automatic Control, 29 (984), pp ) Krishnan, H., McClamroch, N. H., and Reyhanoglu, M.: Attitude Stabilization of a Rigid Spacecraft Using Two Momentum Wheel Actuators, J. Guid. Control Dynam., 8 (995), pp ) Tsiotras, P. and Longuski, J. M.: A New Parameterization of the Attitude Kinematics, J. Astronaut. Sci., 43 (995), pp ) Morin, P. and Samson, C.: Time-varying Exponential Stabilization of a Rigid Spacecraft with Two Control Torques, IEEE Trans. Automatic Control, 42 (997), pp ) Aicardi, M., Cannata, G., and Casalino, G.: Attitude Feedback Control: Unconstrained and Nonholonomic Constrained Cases, J. Guid. Control Dynam., 23 (2), pp ) Murray, R. M. and Sastry, S. S.: Nonholonomic Motion Planning: Steering Using Sinusoids, IEEE Trans. Automatic Control, 38 (993), pp ) Luo, J. and Tsiotras, P.: Exponentially Convergent Control Laws for Nonholonomic Systems in Power Form, Systems Control Lett., 35 (998), pp ) Wichlund, K. Y. and Sordalen, O. J.: Control of Vehicles with Second- Order Nonholonomic Constraints Underactuated Vehicles, Proceedings of the Third Eur. Control Conference, 995, pp ) He, G. and Geng, Z.: The Nonholonomic Redundancy of Second-order Nonholonomic Mechanical Systems, Robotics Autonomous Systems, 56 (28), pp ) Oriolo, G. and Nakamura, Y.: Control of Mechanical Systems with Second-order Nonholonomic Constraints: Underactuated Manipulators, Proceedings of the 3th IEEE Conference on Decision and Control, 99, pp ) Curti, F., Romano, M., and Bevilacqua, R.: Lyapunov-Based Thrusters Selection for Spacecraft Control: Analysis and Experimentation, J. Guid. Control Dynam., 33 (2), pp ) Terui, F.: Position and Attitude Control of a Spacecraft by Sliding Mode Control, Proceedings of the American Control Conference, 998, pp ) Sussmann, H. J.: A General Theorem on Local Controllability, SIAM J. Control Optimization, 25 (987), pp ) Goodwine, B. and Burdick, J.: Controllability with Unilateral Control Inputs, Proceedings of the 35th IEEE Decision and Control, 996, pp ) Shuster, M. D.: A Survey of Attitude Representations, J. Astronaut. Sci., 4 (993), pp ) Yoshimura, Y., Matsuno, T., and Hokamoto, S.: 3-D Attitude Control of an Underactuated Satellite with Constant Inputs, Proceedings of the 28th International Symposium on Space Technology and Science, 2, 2-d-6. 7) Kojima, H.: Stabilization of Angular Velocity of Asymmetrical Rigid Body Using Two Constant Torques, J. Guid. Control Dynam., 3 (27), pp ) Rafael, L. and Bong, W.: New Results for an Asymmetric Rigid Body with Constant Body-Fixed Torques, J. Guid. Control Dynam., 2 (997), pp H. Fujii Associate Editor 4