- Email: [email protected]
Translational Motion Control for Asymmetric Flexible Satellite and Its Vibration Suppression Effect
Translational Motion Control for Asymmetric Flexible Satellite and Its Vibration Suppression Effect Tsutomu Nakamura ∗ ∗
The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, Japan. (e-mail: [email protected])
Abstract: As a case of asymmetric satellite with flexible appendages, it is known that there exists coupled vibration between rotational motion and translational motion. Therefore, when an asymmetric satellite rotates, elastic vibration excited by control input of rotational motion may be transmitted to translational motion. Reversely, suppression of such an excited vibration of translational motion would help realizing attitude maneuver with little vibration. From the reason mentioned above, in this paper, we propose translational motion control system using proof-mass actuators and co-located accelerometers for asymmetric satellite. Then, translational motion control law with consideration of actuator control loop was studied. The proposed translational motion control system was verified through numerical simulations. Keywords: Attitude Control, Flexible Structure, Actuators 1. INTRODUCTION In recent years, satellite development goes toward enlargement due to its advanced and complex mission. However, the carrying ability of the launching rocket limits the satellite weight, and thus, elastic behavior of a satellite becomes non-negligible. In addition, it is on the increase to install huge flexible appendages such as a large deployable antenna shown in Figs. 1 and 2. Therefore, interference between attitude control bandwidth and mechanical vibration bandwidth should be considered in attitude control system (ACS) design of such a flexible satellite. Typical approach to this problem to date is to lower the ACS bandwidth sufficiently enough under the principle of band separation. A radio telescope satellite ASTRO-G (Fig. 1) is a large flexible satellite which has a high-speed maneuvering mission for Radio Astronomical Observatory (Tsuboi (2008); Nakamura et al (2008)). The elasticity of this satellite mainly originates in 8 m diameter large deployable antenna shown in Fig. 1. Radio Astronomical Observatory mission imposes this satellite to observe two celestial bodies alternately in one minute cycle. Antenna re-orientation needed for this observation is considered to be realized via attitude maneuver of whole satellite, which result in the demand for high-speed attitude maneuver. Typical example of the high-speed attitude maneuver is to maneuver 3 deg in 15 s with attitude error less than 0.001 deg at the end of the maneuver. As mentioned above, ASTRO-G is a large flexible satellite with high-speed and high-precision maneuvering demand. Thus, it is difficult to achieve the mission demand with conventional ACS design. By the way, ASTRO-G has a particular property: the whole body is geometrically asymmetric as shown in Fig. 1. It is known that the asymmetric property makes dynamic
Fig. 1. Configuration of a large flexible satellite “ASTROG.” ASTRO-G is one of geometrically asymmetric satellites. characteristics different from ‘symmetric satellite’ such as Engineering Test Satellite-VIII (ETS-VIII) shown in Fig. 2. The difference exists on coupling dynamics between rotational motion and translational motion. To understand the difference, consider that there exists elastic vibration coupling with translational motion on symmetric satellite and asymmetric satellite, as illustrated on Fig. 3 (a) and (b), respectively. In case of symmetric satellite, the vibration occurring on translational motion does not affect the attitude dynamics as shown in Fig. 3 (a), while in case of asymmetric satellite, the vibration appears not only translational motion but also rotational motion (Fig. 3 (b)). In this way, the asymmetric property of a satellite makes translational motion and rotational motion coupled with each other. Thus, flexible modes excited via high-speed maneuvering such as 3 deg/15 s maneuver of ASTROG will appear in translational motion. Herein, if the translational vibration can be eliminated via controlling translational motion assuming some kind of translational actuators, improved performance of vibration suppression
stroke length proof-mass
Base
Structure
Flexible Appendage
Fig. 4. Proof-mass actuator attached to a structure with one flexible mode
Fig. 2. Configuration of ETS-VIII: An example of symmetric satellite SAP
Revolving 2
SAP
Body
Body
1
2
Generally, attitude angle and its rate are observable. Thus the observation variables are represented as follows:
SAP
1 2
2
2 Translational motion
Translational motion
(a) Symmetric Satellite
(b) Asymmetric Satellite
R3×3 the mass matrix, I ∈ R3×3 the moment of inertia, ζ the modal damping ratio , σ the natural angular frequency, and δ0 and δ1 the zeroth- and first-order coupling matrices of the flexible appendage, respectively.
[ ]T y = θT θ˙T .
(4)
2.2 Modeling with Additional Actuators and Sensors
Fig. 3. Difference of dynamics between (a) symmetric satellite and (b) asymmetric satellite when translational vibration occurs. can be expected such as high-speed maneuvering with no or little excitation of flexible modes. From the motivation mentioned above, in this paper, we propose translational motion control system for vibration suppression. The remainder of this paper is organized as follows. Section 2 describes modeling of flexible satellite containing translational motion. Section 3 details in proposed translational motion control law. The translational motion control law was derived from velocity feedback control with consideration of accelerometer bias problem and actuator feedback control loop characteristics. Section 4 is the numerical example of proposed method. Attitude controller, translational motion controller and proof-mass actuator feedback controller were designed for the numerical model of ASTRO-G. Then, numerical simulations of high-speed maneuvering were carried out to show the vibration suppression effect of proposed translational motion control system. This section also includes feasibility study. 2. MODELING
To design translational motion control system, proof-mass actuator (Zimmerman et al (1988), Wie (1992)) was selected for the force-producing actuator. Figure 4 shows a proof-mass actuator attached to a structure with one flexible mode. Proof-mass actuator is an internal force actuator which is attached to the structure and moves its (proof-)mass to generate force F on the structure. Unlike reaction jet which is an external force actuator, proof-mass actuator can provide force with high-accuracy for persistent duration. However, because proof-mass actuator is an internal actuator, it has a limit that the displacement of the proof-mass can not exceed the stroke length. Otherwise the proof-mass runs into the stops and will damage the structure. This is known to be stroke saturation (Lindner (1994)) and should be considered in control design. To observe translational motion, accelerometers were assumed to be installed on the satellite body. Moreover, the accelerometers were assumed to be co-located with proofmass actuators because the co-location property makes the control easy. Assuming 3-axes proof-mass actuators which yield force F ∈ R3 to the satellite body, (1) is replaced by Mx ¨ + δ0 η¨ = F,
2.1 General Modeling including Translational Motion
(5)
while the proof-mass reacts with the reaction force, Considering translational motion, equation of motion (EOM) of a flexible satellite is represented as hybrid equations consisting of attitude, translational motion and elastic dynamics (Kasai et al (2009)): Mx ¨ + δ0 η¨ = 0, I θ¨ + δ1 η¨ = T, δ0T x ¨
+
δ1T θ¨ +
2
η¨ + 2ζσ η˙ + σ η = 0,
(1) and
(2) (3)
where x ∈ R3 is the translational displacement vector, θ ∈ R3 the attitude angle vector, η ∈ Rn the constrained modal coordinate vector, T ∈ R3 the torque input, M ∈
Ma x¨a = −F,
(6)
here Ma is the mass matrix of the proof-mass actuators and xa is the displacement of the proof-mass. Then, the EOM has represented as hybrid equations with (5), (6), (2) and (3). The displacement xa of the proof-mass is assumed to be observable. Then, the observation variables of (4) are extended as [ ]T y = θT θ˙T x ¨T xTa .
(7)
−20
Gain [dB]
• Flexible modes appear in wide range of frequency, sharing same eigen-frequency. • All flexible modes are inphase with rigid mode. Thus, phase characteristics are ranged from -180 to 0 deg for any frequency.
y axis
−40
z axis
−60 −80 −100
x axis
−120
Because of this similarity, the translational motion control design can be achieved by adapting time-tested attitude control design theory.
−140 −160
Phase [deg]
0
In attitude control design theory, Direct Velocity FeedBack (DVFB) control by Balas (1979) is known to be useful for flexible spacecraft. Mass-spring-damper system with DVFB control law is represented as
−45 −90 −135 −180 −1 10
0
1
10
2
10
10
Frequency [rad/s]
Fig. 5. Bode diagram of attitude dynamics1 −20 x axis
Gain [dB]
−40
y axis
−60 −80
3.2 Integration of Accelerometer Output
−100 −120
As seen in DVFB control, passive stabilization of velocity feedback is an effective technique to suppress the vibration. Indeed, if velocity feedback was used for ACS, all flexible modes are phase stabilized and increase in damping coefficient can be expected. Because accelerometers are assumed in this research, the accelerometer signal should have time integration to generate velocity signal. However, integration of acceleration signal may bring velocity bias problem which result in saturation of an internal actuator.
z axis
−140
Phase [deg]
0 −45 −90 −135 −180 −1 10
Mx ¨ + Dx˙ + Kx = −KD x, ˙ (8) where M represents mass matrix, D damper matrix, K stiffness matrix, and KD velocity gain, respectively. The negative velocity feedback in DVFB control can be considered to be of most effective way for vibration suppression when actuators and sensors are co-located. Then, authors propose the translational controller based on velocity feedback.
0
1
10
10
2
10
Vibration control experiments called FLEX was carried out on ETS-VI in order to demonstrate the effectiveness of Fig. 6. Bode diagram of translational dynamics1 . This advanced control theory applied for large flexible satellite Bode diagram was calculated with the transfer func- (NASDA (1996)). In the FLEX experiments, there was a vibration control experiment of non-colocated feedback tion from force F to translational displacement x. using accelerometers on flexible appendages. In this exper3. TRANSLATIONAL MOTION CONTROL DESIGN iment, the problem of integration of accelerometer output mentioned above was avoided using low-pass filter, which 3.1 Similarity between Rotational Motion and Translational they call low-pass type integration represented as follows: Motion 1 F (s) = . (9) As shown in (2) and (5), both the rotational motion and s + ωint the translational motion are represented as spring-mass Quasi-integration of acceleration using (9) is thought to be system. In addition, if translational actuators and sensors an effective way for eliminating integration bias, and thus are co-located as mentioned in Sec. 2.2, the characteristics used in this research. of translational motion and rotational motion are similar to each other. Figures 5 and 6 show the Bode diagrams 1 3.3 Proof-Mass Control Loop of attitude and translational motion, respectively (Bode diagram of translational motion shown in Fig. 6 was drawn As mentioned in Sec. 2.2, the stroke saturation of a proofwith the transfer function from force F to translational mass actuator should be carefully considered (Lindner displacement x for ease of understanding the similarity). (1994)) when designing proof-mass actuator control sysComparing Figs. 5 and 6, following similarities can be tem. In Zimmerman et al (1988) or Lindner (1994), actuafound: tor displacement and velocity are fed back to yield center• 40 dB/dec characteristics on whole frequency range ing force so that the actuator would not run into its stops. Figure 7 shows translational motion control block diagram because of double pole on complex plane origin. 1 All of the Bode diagrams in this paper were calculated with a with the actuator feedback loop. In Fig. 7, translational motion control of the satellite is designed with velocity numerical model of ASTRO-G under review. Note that this model is not the final model. feedback gain KD and low-pass type integral in (9), while Frequency [rad/s]
Actuator Control Loop
40
Gain [dB]
20 0 −20 −40 −60
Vibration Suppression Loop
90
actuator control loop is designed with PD control. The total force f applied to the satellite body is calculated by adding vibration control force fv and actuator centering force fa . Figure 7 shows that translational motion control loop includes the actuator control loop dynamics and suggests that vibration control force fv is not directly applied to the satellite body. The transfer function Ta (s) from vibration control force fv to force f applied to the satellite can be written as Ma s2 Ta (s) = , (10) 2 Ma s + KDa s + KP a where KP a and KDa is the P and D gain of the actuator control loop, respectively. Thus, the characteristics of the actuator feedback loop viewed from the satellite translational control system is a second order high-pass filter. In designing translational motion control system, it is rather easy to deal with second order high-pass filter than to deal with actuator mass Ma and proof-mass actuator gain KP a and KDa directly. Therefore, we designed the translational motion control system with following controller KD Ktrs (s) = H(s), (11) s + ωint where H(s) is the high-pass filter with cut-off frequency ωH : s2 . (12) (s + ωH )2 The proof-mass control loop gains are calculated after designing vibration control loop. The relationship among cut-off frequency ωH , proof-mass actuator mass and actuator control gain can be obtained by letting H(s) in (12) equivalent with Ta (s) in (10), and represented as (13) and (14). H(s) =
2 KP a = Ma ωH
(13)
KDa = 2Ma ωH
(14)
4. NUMERICAL SIMULATION Numerical simulations were carried out using numerical model of ASTRO-G shown in Fig. 1. In this section, attitude controller and translational controller were designed with this numerical model, with some guidelines for the controller design. Then, numerical simulations of 1 deg attitude maneuver around x-axis were carried out.
Phase [deg]
45
Fig. 7. Block diagram of the translational motion control: vibration loop and proof-mass actuator control loop
0 −45 −90 −135 −180 −1 10
0
1
10
10
2
10
Frequency [rad/s]
Fig. 8. Bode diagram of attitude loop transfer function (x-axis) 4.1 Attitude Control System Design The attitude controller was designed with PD control. Because the purpose of the numerical simulations are to present the vibration suppression effect of the translational motion control system, the simulations deal with attitude maneuver around x-axis where 4 dominant flexible modes appears. For the sake of simplicity, only x-axis attitude controller was designed. The PD control law are designed under following principles: 1) all flexible modes are designed to be phase stabilized, and 2) distinguishing highgain flexible modes are designed so that their peak gain become over 0 dB for active damping. After iterative design and simulations, PD control law of (15) are designed. Krotx = 1000 +
3000ωkd s s + ωkd
(15)
Here ωkd was designed to be 30 rad/s. Figure 8 shows the Bode diagram of loop transfer function of attitude x-axis.
4.2 Translational Motion Control System Design Translational x-, y- and z-axis controllers were designed. Figure 9 shows Bode diagram of transfer function from translational force to translational acceleration. Controller of each axis were designed without considering interference with other translational axes. The translational controllers were designed based on guidelines shown below: (1) The controllers were designed to let all flexible modes satisfy desirable phase condition, i.e., phase of flexible modes from anti-resonance frequency to resonance frequency are designed to be in the range of -90 deg to 90 deg, as possible. (2) The controllers were designed to let peak gain of predominant flexible modes over 0 dB in order that their damping coefficients become large. (3) To avoid deteriorated transient response of the proofmass actuator, the rigid mode gain under the lowest eigen frequency is set to be under 0 dB.
−20
1.5
z axis
Gain [dB]
−30 −40
1
x axis
−50 0.5
−60 −70 y axis Angle [deg]
−80 −90
Phase [deg]
180
0 0 1.015
10
20
30
40
50
60
70
80
90
100
w/ translational motion control
1.01 1.005
135 1
90
0.995
45
0.99
0 −1 10
1
0
10
0.985 0
2
10
10
w/o translational motion control
10
20
30
40
Frequency [rad/s]
Fig. 9. Bode diagram of transfer function from force to acceleration
50 60 Time [sec]
70
80
90
100
Fig. 11. Simulation results of 1 deg attitude maneuver: time response of attitude angle
40 0.01
Gain [dB]
20 0
0
z axis
−20
−0.005
x axis Acceleration [g]
−40 y axis
−60 180 Phase [deg]
x axis
0.005
90
−0.01 0.1 y axis
0.05 0 −0.05 −0.1 0.1
z axis
0.05
0
0
−90 −2 10
−0.05 −1
10
0
10 Frequency [rad/s]
1
10
2
10
Fig. 10. Bode diagram of loop transfer function of translational motion control system Through iterated trial-and-error of designing controller on Bode diagram and numerical simulations, the translational controllers were finally designed as bellow:
−0.1 0
5
10
15 Time [sec]
20
25
30
Fig. 12. Simulation result of 1 deg attitude maneuver: time response of acceleration control system. Simulation contents are to carry out 1 deg attitude maneuver with step reference.
800 s2 , (16) s + 0.5 (s + 0.5)2 3000 s2 Ktrsy = , and (17) s + 2.0 (s + 2.0)2 1500 s2 Ktrsz = , (18) s + 0.8 (s + 0.8)2 Figure 10 shows the loop transfer function of translational motion system using (16), (17) and (18).
Figure 11 shows time response of attitude on x-axis. From Fig. 11, attitude maneuvering converges 1 deg in about 30 sec. In addition, in the case of using translational motion control indicated as red broken line in Fig. 11, residual vibration are sufficiently suppressed at the end of the maneuver, while residual vibration exsists in the case of without translational motion control. Thus, the proposed translational motion control can effectively suppress the excited vibration and shown to be effective.
The mass of the proof-mass actuator was set to be 2 kg in each axis. Then, feedback gain of the proof-mass actuator KP a and KDa are calculated with (13) and (14).
Figure 12 shows the acceleration of the satellite body in the case with translational motion control. From Fig. 12, though the maximum acceleration are found to be 0.1 g on y- and z-axis on 1 sec, approximately 0.01 g acceleration are observed after 1.1 sec.
Ktrsx =
4.3 Numerical Simulation Results Numerical simulations were carried out with attitude controller in Sec. 4.1 and translational controllers and proofmass actuator controllers designed in Sec. 4.2. Numerical simulations are on 2 cases: with and without translational
Finally, Fig. 13 shows the time response of the displacement of 2 kg proof-mass actuator. From Fig. 13, the maximum amplitude of the actuator displacement is found to be about 0.6 m in z-axis, which can be considered to be feasible moving range.
Thus, there is a tradeoff between vibration suppression performance and actuator transient response.
0.2
Actuator Displacement [m]
x axis
Feasibility study was carried out with the results of numerical simulations. However, simulations in this paper were based on feedback controller and step reference where elastic modes tend to be excited. Assuming future highspeed maneuvering mission, feedforward control or restto-rest reference for anti-vibration should be used in the control system, which realize high-speed maneuvering with little excitation of vibration. In such cases, accelerometer sensitivity will be the critical term for the feasibility of proposed system.
0
y axis
-0.2
z axis
-0.4
-0.6
REFERENCES 0
10
20
30
40 50 Time [sec]
60
70
80
90
100
Fig. 13. Simulation result of 1 deg attitude maneuver: time response of actuator with 2 kg proof-mass 5. CONCLUSION In this paper, we proposed translational motion control system to provide additional vibration suppression performance on asymmetric flexible satellite. This proposal utilizes translational 3 degree-of-freedom (DOF) dynamics which have been ignored in past satellite ACS design. The 3 DOF translational dynamics is coupled with attitude dynamics in the case of geometrically asymmetric satellite. The effectiveness of proposed method was demonstrated through numerical simulations which compare the residual vibrations in two cases: with and without proposed translational control system. The simulation results show that translational motion control system can suppress residual vibration significantly at the end of the maneuver. The vibration suppression ability can be improved with a high gain translational motion controller which weight the phase condition and gain condition at eigen frequency of fleixble modes (see first and second guidelines of Sec. 4.2). However, such a high gain controller will end in failure because of huge amplitude of proof-mass actuator transient with which the third guideline of Sec. 4.2 closely concerns.
Tsuboi, M. (2008). VSOP2/ASTRO-G project, J. of Physics: Conference Series, Vol. 131, No. 1, p. 012048. Nakamura, T. et al. (2008). Agile and Robust Attitude Control System for VSOP-2 Satellite “ASTRO-G,” The 10th International Workshop on Advanced Motion Control, TRENTO, 26–28, March. Kasai, T. et al. (2009). On-Orbit System Identification Experiments of the Engineering Test Satellite-VIII, Transaction on Space Technology Japan, Vol. 7, Issue 26, pp. 79–84. Zimmerman, D., Horner, G., and Inman, D. (1988).Microprocessor Controlled Force Actuator, J. Guid., Control, and Dynm., Vol. 11, No. 3, pp. 230–236. Wie, B. (1992).Experimental Demonstration of a Classical Approach to Flexible Structure Control, J. of Guid., Control, and Dynm., Vol. 15, No. 6, pp. 1327–1333. Lindner D., Zvonar, G., and Borojevie, D. (1994). Performance and Control of Proof-Mass Actuators Accounting for Stroke Saturation, J. of Guidance, Control, and Dynamics, Vol. 17, No. 5., pp. 1103–1108. Balas, M. (1979). Direct Velocity Feedback Control of Large Space Structures, J. Guid. and Control, Vol. 2, No. 3, pp. 252–253. NASDA (1996). On-orbit flexible structure control experiment of Engineering Test Satellite-6 (ETS-6), NASDA Special Publication, (in Japanese).
The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo, Japan. (e-mail: [email protected])
Abstract: As a case of asymmetric satellite with flexible appendages, it is known that there exists coupled vibration between rotational motion and translational motion. Therefore, when an asymmetric satellite rotates, elastic vibration excited by control input of rotational motion may be transmitted to translational motion. Reversely, suppression of such an excited vibration of translational motion would help realizing attitude maneuver with little vibration. From the reason mentioned above, in this paper, we propose translational motion control system using proof-mass actuators and co-located accelerometers for asymmetric satellite. Then, translational motion control law with consideration of actuator control loop was studied. The proposed translational motion control system was verified through numerical simulations. Keywords: Attitude Control, Flexible Structure, Actuators 1. INTRODUCTION In recent years, satellite development goes toward enlargement due to its advanced and complex mission. However, the carrying ability of the launching rocket limits the satellite weight, and thus, elastic behavior of a satellite becomes non-negligible. In addition, it is on the increase to install huge flexible appendages such as a large deployable antenna shown in Figs. 1 and 2. Therefore, interference between attitude control bandwidth and mechanical vibration bandwidth should be considered in attitude control system (ACS) design of such a flexible satellite. Typical approach to this problem to date is to lower the ACS bandwidth sufficiently enough under the principle of band separation. A radio telescope satellite ASTRO-G (Fig. 1) is a large flexible satellite which has a high-speed maneuvering mission for Radio Astronomical Observatory (Tsuboi (2008); Nakamura et al (2008)). The elasticity of this satellite mainly originates in 8 m diameter large deployable antenna shown in Fig. 1. Radio Astronomical Observatory mission imposes this satellite to observe two celestial bodies alternately in one minute cycle. Antenna re-orientation needed for this observation is considered to be realized via attitude maneuver of whole satellite, which result in the demand for high-speed attitude maneuver. Typical example of the high-speed attitude maneuver is to maneuver 3 deg in 15 s with attitude error less than 0.001 deg at the end of the maneuver. As mentioned above, ASTRO-G is a large flexible satellite with high-speed and high-precision maneuvering demand. Thus, it is difficult to achieve the mission demand with conventional ACS design. By the way, ASTRO-G has a particular property: the whole body is geometrically asymmetric as shown in Fig. 1. It is known that the asymmetric property makes dynamic
Fig. 1. Configuration of a large flexible satellite “ASTROG.” ASTRO-G is one of geometrically asymmetric satellites. characteristics different from ‘symmetric satellite’ such as Engineering Test Satellite-VIII (ETS-VIII) shown in Fig. 2. The difference exists on coupling dynamics between rotational motion and translational motion. To understand the difference, consider that there exists elastic vibration coupling with translational motion on symmetric satellite and asymmetric satellite, as illustrated on Fig. 3 (a) and (b), respectively. In case of symmetric satellite, the vibration occurring on translational motion does not affect the attitude dynamics as shown in Fig. 3 (a), while in case of asymmetric satellite, the vibration appears not only translational motion but also rotational motion (Fig. 3 (b)). In this way, the asymmetric property of a satellite makes translational motion and rotational motion coupled with each other. Thus, flexible modes excited via high-speed maneuvering such as 3 deg/15 s maneuver of ASTROG will appear in translational motion. Herein, if the translational vibration can be eliminated via controlling translational motion assuming some kind of translational actuators, improved performance of vibration suppression
stroke length proof-mass
Base
Structure
Flexible Appendage
Fig. 4. Proof-mass actuator attached to a structure with one flexible mode
Fig. 2. Configuration of ETS-VIII: An example of symmetric satellite SAP
Revolving 2
SAP
Body
Body
1
2
Generally, attitude angle and its rate are observable. Thus the observation variables are represented as follows:
SAP
1 2
2
2 Translational motion
Translational motion
(a) Symmetric Satellite
(b) Asymmetric Satellite
R3×3 the mass matrix, I ∈ R3×3 the moment of inertia, ζ the modal damping ratio , σ the natural angular frequency, and δ0 and δ1 the zeroth- and first-order coupling matrices of the flexible appendage, respectively.
[ ]T y = θT θ˙T .
(4)
2.2 Modeling with Additional Actuators and Sensors
Fig. 3. Difference of dynamics between (a) symmetric satellite and (b) asymmetric satellite when translational vibration occurs. can be expected such as high-speed maneuvering with no or little excitation of flexible modes. From the motivation mentioned above, in this paper, we propose translational motion control system for vibration suppression. The remainder of this paper is organized as follows. Section 2 describes modeling of flexible satellite containing translational motion. Section 3 details in proposed translational motion control law. The translational motion control law was derived from velocity feedback control with consideration of accelerometer bias problem and actuator feedback control loop characteristics. Section 4 is the numerical example of proposed method. Attitude controller, translational motion controller and proof-mass actuator feedback controller were designed for the numerical model of ASTRO-G. Then, numerical simulations of high-speed maneuvering were carried out to show the vibration suppression effect of proposed translational motion control system. This section also includes feasibility study. 2. MODELING
To design translational motion control system, proof-mass actuator (Zimmerman et al (1988), Wie (1992)) was selected for the force-producing actuator. Figure 4 shows a proof-mass actuator attached to a structure with one flexible mode. Proof-mass actuator is an internal force actuator which is attached to the structure and moves its (proof-)mass to generate force F on the structure. Unlike reaction jet which is an external force actuator, proof-mass actuator can provide force with high-accuracy for persistent duration. However, because proof-mass actuator is an internal actuator, it has a limit that the displacement of the proof-mass can not exceed the stroke length. Otherwise the proof-mass runs into the stops and will damage the structure. This is known to be stroke saturation (Lindner (1994)) and should be considered in control design. To observe translational motion, accelerometers were assumed to be installed on the satellite body. Moreover, the accelerometers were assumed to be co-located with proofmass actuators because the co-location property makes the control easy. Assuming 3-axes proof-mass actuators which yield force F ∈ R3 to the satellite body, (1) is replaced by Mx ¨ + δ0 η¨ = F,
2.1 General Modeling including Translational Motion
(5)
while the proof-mass reacts with the reaction force, Considering translational motion, equation of motion (EOM) of a flexible satellite is represented as hybrid equations consisting of attitude, translational motion and elastic dynamics (Kasai et al (2009)): Mx ¨ + δ0 η¨ = 0, I θ¨ + δ1 η¨ = T, δ0T x ¨
+
δ1T θ¨ +
2
η¨ + 2ζσ η˙ + σ η = 0,
(1) and
(2) (3)
where x ∈ R3 is the translational displacement vector, θ ∈ R3 the attitude angle vector, η ∈ Rn the constrained modal coordinate vector, T ∈ R3 the torque input, M ∈
Ma x¨a = −F,
(6)
here Ma is the mass matrix of the proof-mass actuators and xa is the displacement of the proof-mass. Then, the EOM has represented as hybrid equations with (5), (6), (2) and (3). The displacement xa of the proof-mass is assumed to be observable. Then, the observation variables of (4) are extended as [ ]T y = θT θ˙T x ¨T xTa .
(7)
−20
Gain [dB]
• Flexible modes appear in wide range of frequency, sharing same eigen-frequency. • All flexible modes are inphase with rigid mode. Thus, phase characteristics are ranged from -180 to 0 deg for any frequency.
y axis
−40
z axis
−60 −80 −100
x axis
−120
Because of this similarity, the translational motion control design can be achieved by adapting time-tested attitude control design theory.
−140 −160
Phase [deg]
0
In attitude control design theory, Direct Velocity FeedBack (DVFB) control by Balas (1979) is known to be useful for flexible spacecraft. Mass-spring-damper system with DVFB control law is represented as
−45 −90 −135 −180 −1 10
0
1
10
2
10
10
Frequency [rad/s]
Fig. 5. Bode diagram of attitude dynamics1 −20 x axis
Gain [dB]
−40
y axis
−60 −80
3.2 Integration of Accelerometer Output
−100 −120
As seen in DVFB control, passive stabilization of velocity feedback is an effective technique to suppress the vibration. Indeed, if velocity feedback was used for ACS, all flexible modes are phase stabilized and increase in damping coefficient can be expected. Because accelerometers are assumed in this research, the accelerometer signal should have time integration to generate velocity signal. However, integration of acceleration signal may bring velocity bias problem which result in saturation of an internal actuator.
z axis
−140
Phase [deg]
0 −45 −90 −135 −180 −1 10
Mx ¨ + Dx˙ + Kx = −KD x, ˙ (8) where M represents mass matrix, D damper matrix, K stiffness matrix, and KD velocity gain, respectively. The negative velocity feedback in DVFB control can be considered to be of most effective way for vibration suppression when actuators and sensors are co-located. Then, authors propose the translational controller based on velocity feedback.
0
1
10
10
2
10
Vibration control experiments called FLEX was carried out on ETS-VI in order to demonstrate the effectiveness of Fig. 6. Bode diagram of translational dynamics1 . This advanced control theory applied for large flexible satellite Bode diagram was calculated with the transfer func- (NASDA (1996)). In the FLEX experiments, there was a vibration control experiment of non-colocated feedback tion from force F to translational displacement x. using accelerometers on flexible appendages. In this exper3. TRANSLATIONAL MOTION CONTROL DESIGN iment, the problem of integration of accelerometer output mentioned above was avoided using low-pass filter, which 3.1 Similarity between Rotational Motion and Translational they call low-pass type integration represented as follows: Motion 1 F (s) = . (9) As shown in (2) and (5), both the rotational motion and s + ωint the translational motion are represented as spring-mass Quasi-integration of acceleration using (9) is thought to be system. In addition, if translational actuators and sensors an effective way for eliminating integration bias, and thus are co-located as mentioned in Sec. 2.2, the characteristics used in this research. of translational motion and rotational motion are similar to each other. Figures 5 and 6 show the Bode diagrams 1 3.3 Proof-Mass Control Loop of attitude and translational motion, respectively (Bode diagram of translational motion shown in Fig. 6 was drawn As mentioned in Sec. 2.2, the stroke saturation of a proofwith the transfer function from force F to translational mass actuator should be carefully considered (Lindner displacement x for ease of understanding the similarity). (1994)) when designing proof-mass actuator control sysComparing Figs. 5 and 6, following similarities can be tem. In Zimmerman et al (1988) or Lindner (1994), actuafound: tor displacement and velocity are fed back to yield center• 40 dB/dec characteristics on whole frequency range ing force so that the actuator would not run into its stops. Figure 7 shows translational motion control block diagram because of double pole on complex plane origin. 1 All of the Bode diagrams in this paper were calculated with a with the actuator feedback loop. In Fig. 7, translational motion control of the satellite is designed with velocity numerical model of ASTRO-G under review. Note that this model is not the final model. feedback gain KD and low-pass type integral in (9), while Frequency [rad/s]
Actuator Control Loop
40
Gain [dB]
20 0 −20 −40 −60
Vibration Suppression Loop
90
actuator control loop is designed with PD control. The total force f applied to the satellite body is calculated by adding vibration control force fv and actuator centering force fa . Figure 7 shows that translational motion control loop includes the actuator control loop dynamics and suggests that vibration control force fv is not directly applied to the satellite body. The transfer function Ta (s) from vibration control force fv to force f applied to the satellite can be written as Ma s2 Ta (s) = , (10) 2 Ma s + KDa s + KP a where KP a and KDa is the P and D gain of the actuator control loop, respectively. Thus, the characteristics of the actuator feedback loop viewed from the satellite translational control system is a second order high-pass filter. In designing translational motion control system, it is rather easy to deal with second order high-pass filter than to deal with actuator mass Ma and proof-mass actuator gain KP a and KDa directly. Therefore, we designed the translational motion control system with following controller KD Ktrs (s) = H(s), (11) s + ωint where H(s) is the high-pass filter with cut-off frequency ωH : s2 . (12) (s + ωH )2 The proof-mass control loop gains are calculated after designing vibration control loop. The relationship among cut-off frequency ωH , proof-mass actuator mass and actuator control gain can be obtained by letting H(s) in (12) equivalent with Ta (s) in (10), and represented as (13) and (14). H(s) =
2 KP a = Ma ωH
(13)
KDa = 2Ma ωH
(14)
4. NUMERICAL SIMULATION Numerical simulations were carried out using numerical model of ASTRO-G shown in Fig. 1. In this section, attitude controller and translational controller were designed with this numerical model, with some guidelines for the controller design. Then, numerical simulations of 1 deg attitude maneuver around x-axis were carried out.
Phase [deg]
45
Fig. 7. Block diagram of the translational motion control: vibration loop and proof-mass actuator control loop
0 −45 −90 −135 −180 −1 10
0
1
10
10
2
10
Frequency [rad/s]
Fig. 8. Bode diagram of attitude loop transfer function (x-axis) 4.1 Attitude Control System Design The attitude controller was designed with PD control. Because the purpose of the numerical simulations are to present the vibration suppression effect of the translational motion control system, the simulations deal with attitude maneuver around x-axis where 4 dominant flexible modes appears. For the sake of simplicity, only x-axis attitude controller was designed. The PD control law are designed under following principles: 1) all flexible modes are designed to be phase stabilized, and 2) distinguishing highgain flexible modes are designed so that their peak gain become over 0 dB for active damping. After iterative design and simulations, PD control law of (15) are designed. Krotx = 1000 +
3000ωkd s s + ωkd
(15)
Here ωkd was designed to be 30 rad/s. Figure 8 shows the Bode diagram of loop transfer function of attitude x-axis.
4.2 Translational Motion Control System Design Translational x-, y- and z-axis controllers were designed. Figure 9 shows Bode diagram of transfer function from translational force to translational acceleration. Controller of each axis were designed without considering interference with other translational axes. The translational controllers were designed based on guidelines shown below: (1) The controllers were designed to let all flexible modes satisfy desirable phase condition, i.e., phase of flexible modes from anti-resonance frequency to resonance frequency are designed to be in the range of -90 deg to 90 deg, as possible. (2) The controllers were designed to let peak gain of predominant flexible modes over 0 dB in order that their damping coefficients become large. (3) To avoid deteriorated transient response of the proofmass actuator, the rigid mode gain under the lowest eigen frequency is set to be under 0 dB.
−20
1.5
z axis
Gain [dB]
−30 −40
1
x axis
−50 0.5
−60 −70 y axis Angle [deg]
−80 −90
Phase [deg]
180
0 0 1.015
10
20
30
40
50
60
70
80
90
100
w/ translational motion control
1.01 1.005
135 1
90
0.995
45
0.99
0 −1 10
1
0
10
0.985 0
2
10
10
w/o translational motion control
10
20
30
40
Frequency [rad/s]
Fig. 9. Bode diagram of transfer function from force to acceleration
50 60 Time [sec]
70
80
90
100
Fig. 11. Simulation results of 1 deg attitude maneuver: time response of attitude angle
40 0.01
Gain [dB]
20 0
0
z axis
−20
−0.005
x axis Acceleration [g]
−40 y axis
−60 180 Phase [deg]
x axis
0.005
90
−0.01 0.1 y axis
0.05 0 −0.05 −0.1 0.1
z axis
0.05
0
0
−90 −2 10
−0.05 −1
10
0
10 Frequency [rad/s]
1
10
2
10
Fig. 10. Bode diagram of loop transfer function of translational motion control system Through iterated trial-and-error of designing controller on Bode diagram and numerical simulations, the translational controllers were finally designed as bellow:
−0.1 0
5
10
15 Time [sec]
20
25
30
Fig. 12. Simulation result of 1 deg attitude maneuver: time response of acceleration control system. Simulation contents are to carry out 1 deg attitude maneuver with step reference.
800 s2 , (16) s + 0.5 (s + 0.5)2 3000 s2 Ktrsy = , and (17) s + 2.0 (s + 2.0)2 1500 s2 Ktrsz = , (18) s + 0.8 (s + 0.8)2 Figure 10 shows the loop transfer function of translational motion system using (16), (17) and (18).
Figure 11 shows time response of attitude on x-axis. From Fig. 11, attitude maneuvering converges 1 deg in about 30 sec. In addition, in the case of using translational motion control indicated as red broken line in Fig. 11, residual vibration are sufficiently suppressed at the end of the maneuver, while residual vibration exsists in the case of without translational motion control. Thus, the proposed translational motion control can effectively suppress the excited vibration and shown to be effective.
The mass of the proof-mass actuator was set to be 2 kg in each axis. Then, feedback gain of the proof-mass actuator KP a and KDa are calculated with (13) and (14).
Figure 12 shows the acceleration of the satellite body in the case with translational motion control. From Fig. 12, though the maximum acceleration are found to be 0.1 g on y- and z-axis on 1 sec, approximately 0.01 g acceleration are observed after 1.1 sec.
Ktrsx =
4.3 Numerical Simulation Results Numerical simulations were carried out with attitude controller in Sec. 4.1 and translational controllers and proofmass actuator controllers designed in Sec. 4.2. Numerical simulations are on 2 cases: with and without translational
Finally, Fig. 13 shows the time response of the displacement of 2 kg proof-mass actuator. From Fig. 13, the maximum amplitude of the actuator displacement is found to be about 0.6 m in z-axis, which can be considered to be feasible moving range.
Thus, there is a tradeoff between vibration suppression performance and actuator transient response.
0.2
Actuator Displacement [m]
x axis
Feasibility study was carried out with the results of numerical simulations. However, simulations in this paper were based on feedback controller and step reference where elastic modes tend to be excited. Assuming future highspeed maneuvering mission, feedforward control or restto-rest reference for anti-vibration should be used in the control system, which realize high-speed maneuvering with little excitation of vibration. In such cases, accelerometer sensitivity will be the critical term for the feasibility of proposed system.
0
y axis
-0.2
z axis
-0.4
-0.6
REFERENCES 0
10
20
30
40 50 Time [sec]
60
70
80
90
100
Fig. 13. Simulation result of 1 deg attitude maneuver: time response of actuator with 2 kg proof-mass 5. CONCLUSION In this paper, we proposed translational motion control system to provide additional vibration suppression performance on asymmetric flexible satellite. This proposal utilizes translational 3 degree-of-freedom (DOF) dynamics which have been ignored in past satellite ACS design. The 3 DOF translational dynamics is coupled with attitude dynamics in the case of geometrically asymmetric satellite. The effectiveness of proposed method was demonstrated through numerical simulations which compare the residual vibrations in two cases: with and without proposed translational control system. The simulation results show that translational motion control system can suppress residual vibration significantly at the end of the maneuver. The vibration suppression ability can be improved with a high gain translational motion controller which weight the phase condition and gain condition at eigen frequency of fleixble modes (see first and second guidelines of Sec. 4.2). However, such a high gain controller will end in failure because of huge amplitude of proof-mass actuator transient with which the third guideline of Sec. 4.2 closely concerns.
Tsuboi, M. (2008). VSOP2/ASTRO-G project, J. of Physics: Conference Series, Vol. 131, No. 1, p. 012048. Nakamura, T. et al. (2008). Agile and Robust Attitude Control System for VSOP-2 Satellite “ASTRO-G,” The 10th International Workshop on Advanced Motion Control, TRENTO, 26–28, March. Kasai, T. et al. (2009). On-Orbit System Identification Experiments of the Engineering Test Satellite-VIII, Transaction on Space Technology Japan, Vol. 7, Issue 26, pp. 79–84. Zimmerman, D., Horner, G., and Inman, D. (1988).Microprocessor Controlled Force Actuator, J. Guid., Control, and Dynm., Vol. 11, No. 3, pp. 230–236. Wie, B. (1992).Experimental Demonstration of a Classical Approach to Flexible Structure Control, J. of Guid., Control, and Dynm., Vol. 15, No. 6, pp. 1327–1333. Lindner D., Zvonar, G., and Borojevie, D. (1994). Performance and Control of Proof-Mass Actuators Accounting for Stroke Saturation, J. of Guidance, Control, and Dynamics, Vol. 17, No. 5., pp. 1103–1108. Balas, M. (1979). Direct Velocity Feedback Control of Large Space Structures, J. Guid. and Control, Vol. 2, No. 3, pp. 252–253. NASDA (1996). On-orbit flexible structure control experiment of Engineering Test Satellite-6 (ETS-6), NASDA Special Publication, (in Japanese).