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Thrust vector control of upper stage with a gimbaled thruster during orbit transfer
Author’s Accepted Manuscript Thrust Vector Control of Upper Stage with a Gimbaled Thruster during Orbit Transfer Zhaohui Wang, Yinghong Jia, Lei Jin, Jiajia Duan
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S0094-5765(16)30526-4 http://dx.doi.org/10.1016/j.actaastro.2016.06.002 AA5848
To appear in: Acta Astronautica Received date: 7 February 2015 Revised date: 22 May 2016 Accepted date: 1 June 2016 Cite this article as: Zhaohui Wang, Yinghong Jia, Lei Jin and Jiajia Duan, Thrust Vector Control of Upper Stage with a Gimbaled Thruster during Orbit Transfer, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2016.06.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Thrust Vector Control of Upper Stage with a Gimbaled Thruster during Orbit Transfer Zhaohui Wang1, Yinghong Jia1*, Lei Jin1, Jiajia Duan2 (1 Beihang University, School of Astronautics, Beijing, 100191, P.R. China; 2 Shanghai Institute of Aerospace Control Technology，Shanghai, 201109，China) Abstract: In launching Multi-Satellite with One-Vehicle, the main thruster provided by the upper stage is mounted on a two-axis gimbal. During orbit transfer, the thrust vector of this gimbaled thruster (GT) should theoretically pass through the mass center of the upper stage and align with the command direction to provide orbit transfer impetus. However, it is hard to be implemented from the viewpoint of the engineering mission. The deviations of the thrust vector from the command direction would result in large velocity errors. Moreover, the deviations of the thrust vector from the upper stage mass center would produce large disturbance torques. This paper discusses the thrust vector control (TVC) of the upper stage during its orbit transfer. Firstly, the accurate nonlinear coupled kinematic and dynamic equations of the upper stage body, the two-axis gimbal and the GT are derived by taking the upper stage as a multi-body system. Then, a thrust vector control system consisting of the special attitude control of the upper stage and the gimbal rotation of the gimbaled thruster is proposed. The special attitude control defined by the desired attitude that draws the thrust vector to align with the command direction when the gimbal control makes the thrust vector passes through the upper stage mass center. Finally, the validity of the proposed method is verified through numerical simulations. Keyword: Upper stage, Thrust vector control, Orbit transfer
*Corresponding author. Email address: [email protected] 1
I. Introduction
THRUST vector control has been considered as a crucial area of research in the launch systems, because of the extremely advantageous ability to control the thrust vector of any propulsion system. For the launch systems
[1-5]
, the direction of thrust vector of the propulsion
system should align with the command direction. The TVC of the launch systems can be obtained by gimbaling the engine via servo actuation system. The deviations of the thrust vector in the launch systems lead to large velocity and positioning errors, along with extra requirements of orbit control. For the upper stage, it should transfer to the other orbit when it completes the deploy mission in one orbit. During orbit transfer, the thrust vector should theoretically align with the command direction and pass through the mass center of upper stage. The thrust vector misalignments of the upper stage also produce disturbance torques and make the upper stage rotate. Furthermore, it results in long orbit transfer time and extra requirements of attitude/orbit control. TVC has become a significant technology for the launch systems to control their trajectories and accomplish high maneuverability. The requirements for the TVC of space shuttles were presented in Penchuk’s work [1]. The TVC for space shuttles obtained by Omni-axis vectoring of nozzle was studied in reference [2]. The controller can maintain the vehicle trajectory and the direction of the thrust vector by independently controlling the gimbal angles of the engine. The model, estimation, implementation and control techniques of the TVC-electromechanical actuators for the rocket engines were investigated in references [3-5]. Taking fuel slosh dynamics into consideration, Jaime [6] researched on the TVC for the rocket engine. However, three more independent torques were needed in the control system. The TVC for a liquid-bipropellant upper stage spacecraft was presented by Wie [7]. The controller can keep the pointing of the thrust
2
vector and the stability of the spacecraft. In researches [8, 9], the TVC for a solar-sail spacecraft was proposed. The sail angle with respect to the sunlight was utilized to produce a net torque for the attitude control and thrust vectoring. For the satellite systems [10, 11], the TVC was used to correct the misalignments produced by the thruster firing and the shift of the mass center for a satellite due to fuel consumption, solar panels deployment, onboard experiments, etc. For the upper stage, the selection of a TVC system actuation technology and subsystem technology was discussed in Garrison’s research [12]. Most achievements above were concentrated on Linear Actuators with small engine deflection defined in Suchitra’s study [13], where Linear Actuators are most powered by pneumatic or hydraulic actuators, and Rotary Actuators are motor powered by electromechanical actuators. The Linear actuators were used for small engine deflection ( 4 ), whereas for higher gimballing of the engine; preference goes to Rotary Actuators. In this paper, Rotary Actuators are utilized to solve the higher gimbaling case. Disturbance torques produced by the thrust vector misalignments, eight reaction control subsystem (RCS) thrusters, together with the main thruster, are utilized in the TVC in order to eliminate the effects of the thrust vector misalignments. The thruster configuration presented in this paper was defined as the optimal one in Hwang’s investigation [14]. The steering law for the RCS thrusters is obtained by pulse-width and pulse-frequency modulator (PWPFM) which is first utilized in the attitude controller by Bong W. [15] and has a first order filter compensates the Schmitt trigger output in the feedback path. The advantage of the PWPFM is the static parameters that are independent with respect to the parameters of the spacecraft. The accurate nonlinear model of the upper stage is derived by taking the upper stage as a multi-body system which consists of the upper stage body, the gimbaled thruster actuator and the gimbaled thruster. The superiority of this context is the thrust vector control algorithm which completes by the
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special attitude control of the upper stage and the gimbal rotation of the gimbaled thruster. To achieve this goal, the algorithm that determines the desired attitude for the special attitude control and the gimbal control method that based on the special attitude control is proposed. Due to the property of the rotary actuator, the rotation motion of the gimbal and the GT is strongly coupled with the attitude motion of the upper stage body. The control scheme takes the nonlinear coupling effect into consideration because the controller based on the linearization model is inappropriate in this situation. The remainder of this paper is outlined as follows. First, the comprehensive mathematical formulations are proposed considering the coupling of the orbit and attitude motion, the upper stage’s motion and GT’s rotation. Next, the desired attitude of the upper stage is obtained using the newly derived algorithms to guarantee the direction of the GT’s thrust vector aligning with the command direction. In the same way, the desired gimbal angles of the GT are got to make sure the GT’s thrust vector passing through the mass center of the upper stage. Then, a TVC system including the upper stage’s attitude control and the GT’s gimbal control is designed, ensuring the thrust vector of the GT not only passes through the mass center of the upper stage, but also aligns with the command direction. Finally, the numerical simulations are performed, followed by analysis and conclusions. II. Mathematical Models The system discussed in this paper consists of an upper stage body, a two-axis gimbal and a GT. The kinematic and dynamic equations of the system are derived in this section. The configuration of the system is shown in Fig.1.
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RCS Thruster RCS Thruster Upper Stage yb
Gimbaled Thruster
xb
Upper Stage
Fm
ob
Gimbaled Thruster
RCS Thruster RCS Thruster
RCS Thruster
R Z
RCS Thruster
RCS Thruster
Y X
Fig. 1 Configuration of upper stage A. Coordinate frames To clearly represent the dynamics of the system, the coordinate frames are defined as follows. An Earth centered inertial frame is denoted by f I , with its origin located at the center of the Earth, the z I axis pointing towards the celestial North Pole, the xI axis pointing towards the vernal equinox and the y I axis completes the right-handed reference frame. The orbit reference frame ( f o ), has its origin located at the mass center of the upper stage, with the zo axis pointing towards the Earth center, the xo axis pointing towards the direction of tangential velocity and the
yo axis completes the right-handed reference frame. The principal body-fixed coordinate frame of the upper stage is denoted by f b with its origin located at the mass center of the upper stage. Attitude orientation of the body fixed frame f b with respect to frame f o is represented by ( ZXY ) set of Euler angles: (pitch) about z axis, (roll) about x axis and (yaw) about y axis. The body-fixed reference frame of the gimbal is denoted by f k , and the gimbal angle of f k with respect to f b is represented by Euler angle: (roll) about x axis. The body-fixed reference 5
frame of the GT is denoted by f m , and the gimbal angle of frame f m with respect to frame f k is represented by Euler angle: (yaw) about y axis. The coordinate frames are shown in Fig. 2. zo
yk ym z
Upper Stage
yb
xb ob
fb
zb m
zk
f k / f m ok om
R
xo
Z
fo O
yb yo
Gimbaled Thruster
fI
Fm
xk xm
xb
fb
Y
X
Fig. 2 Definition of coordinate frames B. Thrusts and Torques The thrust vector of the GT is denoted by Fm , which is demonstrated in Fig. 2 and defined as
Fm Fm I m 0J m 0 K m Fm cos sin Io Fm sin J o Fm cos cos K o
(1)
where Ii , J i , K i are unit vectors along the axes of frame f i ( i o, b, k , m ), is the elevation angle, the azimuth angle, and Fm the constant thrust level of the GT. Jo Fm
Oo
Io
Ko
Fig. 3 Gimbaled Thruster azimuth and elevation
6
To calculate the thrusts and torques provided by the GT, two consecutive gimbal rotations are utilized. First, the two-axis gimbal rotates with respect to the upper stage body about the xk axis,
is the gimbal angle. Second, the GT rotates with respect to the two-axis gimbal about the intermediate axis y ' , is the gimbal angle. Then, Fm expressed in frame f b is
Fm Fm cos Ib Fm sin sin Jb Fm sin cos Kb
(2)
rm rmx Ib rmy Jb rmz Kb is the vector presenting the placement from the upper stage’s mass
center to the action point of the GT. Then, the torques provided by GT is
Tm rm Fm
(3)
In this paper, we assume that all thrusters can provide constant thrusts. The torque provided by the RCS thrusters is expressed as
Tx 4Fg rg , Tx 4Fg rg
(4)
where Tx , and Tx represents the positive and negative torque along xb axis respectively, Fg the constant thrust level of the RCS thrusters, and rg the moment arm.
C. Kinematic Equations Based on the definition of the attitude orientation for the upper stage, the kinematic equation gives
rx cos rz sin σ ry tan rx sin rz cos A ω Abo ωo ( sin cos ) / cos rx rz
(5)
where σ is the attitude of the upper stage, A the transformation matrix from T
ωr to σ , ωr rx ry
T
rz the angular velocity of the upper stage body with respect to
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frame f o , ω x y
T
z the angular velocity of the upper stage body with respect to
frame f I , A ji the transformation matrix from frame f i to frame f j ( i, j o, b, k , m ) and ωo the orbit angular velocity. The rotating equation for the gimbal with respect to the upper stage body is
ωk Γ k
(6)
where ωk is the angular velocity of the gimbal with respect to the upper stage body expressed in frame f k . Γ k [1 0 0]T is determined by the configuration of the system. The rotating equation of the GT with respect to the gimbal gives ωm Γ m
(7)
where ωm is the angular velocity of the GT with respect to the gimbal expressed in frame f m .
Γ m [0 1 0]T is determined by the configuration of the system. D. Dynamic Equations The dynamic equations including both translational and rotational motion of the system are derived based on Kane equation [16]. The general formula of Kane equation is defined as Fi G Fi A 0
(8)
where Fi G , Fi A are the generalized inertia force and generalized active force of the system with respect to the ith order generalized velocity. First, the dynamic equation for the translational motion of the upper stage is
mvb Sbt ω Abk Skt Γ k Abm Sm Γ m Fb,n Fb Fw where
Fb,n mωvb ωSb*ω Abk Ωk Sk* Ωk Abm Ωm Sm Ωm Abk Skt Akb ωAbk Ωk Abm Sm Amk Ωk Akm Ωm 8
(9)
where m mb mk mm , mb , mk , mm are the masses of the upper stage, the two-axis gimbal and the GT respectively, vb is the velocity of the upper stage defined in frame f b , Sbt the static moment of the system expressed in frame f b , S kt the static moment of the gimbal and the GT expressed in frame f k , S k the static moment of the gimbal expressed in frame f k , S m the static moment of the GT expressed in frame f m , Sb the static moment of the upper stage’s augment expressed in frame f b , S k the static moment of the gimbal’s augment expressed in frame f k ,
Ωk the angular velocity of the gimbal, Ωm the angular velocity of the GT, Fb the thrust and Fw the gravity that act on the system. The dynamic equation for the rotational motion of the upper stage is
SbtT vb Ibt ω Abk I kb Γ k Abm I mb Γ m Tb,n Tb
(10)
where
Tb,n Sbt ωvb I ω Abk Ib k Ωk Abm Ib m Ωm Abk Ikb Akb ωAbk Ωk Abm Imb Amk Ωk Akm Ωm where I bt is the inertia matrix of the system expressed in frame f b , I kb the coupling inertia matrix of the gimbal, I mb the coupling inertia matrix of the GT, I the quasi inertia matrix of the system, I b k the quasi inertia matrix of the gimbal, I b m the quasi inertia matrix of the GT and Tb the torque which acts on the system. The dynamic equation of the gimbal rotates with respect to the upper stage body is
Γ kT SktT Akbvb ( I kb )T Akb ω I kt Γ k Akm I mk Γ m Tk ,n Tk where
Tk ,n Skt Akb ωvb SktT Akb ωrb,k ω Akm Ik m Ωm Ikt Akb ωAbk Ωk I k Ωk Akm Imk Amk Ωk Akm Ωm
9
(11)
where I m denote the inertia matrix of the GT, I mk the coupling inertia matrix of the GT, I Ωk the quasi inertia matrix of the gimbal, I Ωk m the quasi inertia matrix of the GT, and Tk the rotational control torque which acts on the gimbal. The dynamic equation of GT rotates with respect to the gimbal is
Γ mT SmT Amb vb ( I mb )T Amb ω ( I mk )T Amk Γ k I m Γ m Tm,n Tm
(12)
where
Tm,n Sm Amb ωvb SmT Amb ωrmω Ωm Im Ωm ( Imk )T Amb ωAbk Ωk Im Amk Ωk Akm Ωm where Tm is the rotational control torque that acts on the GT. Obviously，Eq. (9) is coupled with Eqs. (10), (11) , and (12). For TVC, Eqs. (10), (11) and (12) are independent. Substituting Eqs. (5) and (9) into Eqs. (10), (11) and (12), the independent dynamic equations can be rewritten as
At x T Tn where x σ T
T
(13)
T
, T TbT Tk Tm . The elements of the matrix At and the nonlinear
thrust Tn are given as
1 T 1 I bt Sbt Sbt A m 1 1 At Γ kT ( I kb )T Akb SktT Akb Sbt A m T b T 1 T 1 Γ m ( I m ) Amb Sm Amb Sbt A m
1 T b Abk I k Sbt Abk Skt Γ k m 1 Γ kT I kt SktT Skt Γ k m 1 Γ mT ( I mk )T Amk SmT Amk Skt Γ k m
10
1 T b Abm I m Sbt Abm Sm Γ m m 1 Γ kT Akm I mk SktT Akm Sm Γ m m 1 Γ mT I m SmT Sm Γ m m
1 I bt SbtT Sbt m SbtT Tb,n T b T 1 T T 1 T T 1 1 Tn Γ k Tk ,n Γ k Skt Akb Fb Fw Fb ,n Γ k ( I k ) Akb Skt Akb Sbt A A A σ Abo ωo m T T m T Γ mTm,n Γ m Sm Amb T b T 1 T Γ m ( I m ) Amb Sm Amb Sbt m
III. Control Law This section discusses the TVC system including the attitude control of the upper stage and the gimbal rotation control of the GT. First, the desired gimbal angles d , d , which ensure the thrust vector of the GT passes through the mass center of the upper stage, are given. Then, the desired attitude of the upper stage denoted by σ d is calculated; guaranteeing the direction of the thrust vector of the GT denoted by , aligns with its command direction d , d . The methods and analysis tools of variable structure control (VSC) are utilized in this paper due to their robustness to nonlinear model errors. A. Desired States Assume that the thrust vector of the GT passes through the mass center of the upper stage so that the desired gimbal angles d , d are defined as follows
Fm cos d / rmx Fm sin d sin d / rmy Fm sin d cos d / rmz
(14)
The thrust vector of the GT also can be defined by two Euler angles , which indicate the rotation of GT with respect to frame f o . The first rotation is about the yaw ( zo ) axis, and the rotate angle. The second rotation is about the intermediate pitch axis ( y ' ), and the rotate angle. The transformation matrix Amo and thrust vector Fm in terms of , are
11
cos cos Amo sin sin cos
cos sin cos sin sin
sin 0 cos
(15)
Fm Fm cos cos I o Fm cos sin J o Fm sin K o
(16)
The rotation angles , can be calculated by the azimuth angle and the elevation angle
,
as
arctan tan sin arcsin cos cos
(17)
The transformation matrix Amo can also be defined by σ, , as
Amo Amb Abo
(18)
where
cos Amb 0 sin
sin sin cos cos sin
cos cos sin sin sin Abo cos sin sin cos cos sin sin
sin cos sin cos cos
cos sin sin sin cos cos cos sin sin cos sin cos
(19)
sin cos sin cos cos
(20)
First, the command direction d , d of the thrust vector for the GT is determined by the guidance system. Then, the desired rotation angles d ,d , the desired transformation matrix d d Amo , the desired gimbal angles d , d , and the desired transformation matrix Amb can be
calculated by Eqs. (17), (15), (14) and (19), respectively. Finally, the desired transformation matrix Abod is obtained as d Abod Amb Amod 1
12
(21)
Thus, the desired attitude σ d can be obtained from Eqs. (20) and (21). B. Design of Control Law For a TVC system, the attitude control signals are the current desired gimbal angles cd , cd providing the attitude control torque about yb axis and zb axis, and the on-off thrust of the RCS thrusters providing the attitude control torque about xb axis. The GT’s gimbal rotation control signal is the control torque provided by the electrical motor. The methods and analysis tools of VSC are utilized in this paper due to their robustness to nonlinear model errors. If the desired system states are defined as xd σd
cd , a
cd
T
sliding surface is defined as
S e k1e where e x xd is the error of system states, k1
55
(22) is the weighing matrix for e .
Substituting Eq. (13) into the differentiating Eq. (22), yield
S e k1e x xd k1 x xd At1 T Tn xd k1 x xd
(23)
The approaching law is chosen as
S KS ε sgn S where K
55
, ε
55
(24)
, and sgn is the sign function defined as
1 if x 0 sgn x 0 if x 0 1 if x 0 Substituting Eqs. (24) and (13) into Eq. (23), we obtain
T Tn At xd Kk1 x xd ε sgn x xd k1 x xd K k1 x xd
13
(25)
To determine the current desired gimbal angles of the GT cd , cd , substituting Eq. (25) into Eq. (3), yield
rmz Fm cos cd rmx Fm sin cd cos cd T 2 rmy Fm cos cd rmx Fm sin cd sin cd T 3
(26)
where T 2 , T 3 represent the second and third element of control torque (Eq. (25)), respectively. The current desired gimbal angles cd , cd can be figured out by Eq. (26). Because of the coupling, the GT would also produce torques about xb axis, which should be counteracted by the RCS thrusters. Then, the torques that the RCS thrusters should provide is Tbx T 1 rmz Fm sin cd sin cd rmy Fm sin cd cos cd
(27)
The control torque about xb provided by the RCS thrusters is defined by Eq. (4). The steering law of the RCS thruster is obtained by the PWPFM method [15,17,18]. In order to verify the stability of the controller, set the Lyapunov Function as V S T S / 2 . Substituting Eqs. (25) and (24) into the differentiation of the Lyapunov Function, yield
V S T S S T K T S sgn S T εT S K j S 2j j S j 0 5
5
j 1
j 1
5
5
j 1
j 1
(28)
Since V S T S / 2 0 , and V K j S 2j j S j 0 , V 0 exists only when S 0 . From the LaSalle invariance principle, we conclude that the controller designed previously is global asymptotic stable. The paper, in order to avoid of chattering at the sliding surface boundary, utilizes the saturation function to instead of the sign function in the controller, which is commonly applied in engineering.
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IV. Numerical simulations In this section, the control strategy of the TVC for upper stage with a gimbaled thruster designed previously is simulated numerically. A. Simulation Parameters The parameters are mb 3600kg , mk 10kg , mm 30kg . Fm 5000N , Fg 25N .
rmx
rmy
rmz 1.107 0.0025 0.0022 m , rg 1m . rmc 0.1 0 0 m is the T
T
T
placement vector of the mass center for the GT expressed in frame f m . The inertia matrix of upper stage ( I1 ), the gimbal ( I 2 ) and the GT ( I 3 ) are expressed as follows.
0 0 4800 4 6.4 0.5 0 0.75 0 2 2 I1 4 6000 2.6 kg m , I 2 0 0.7 0 kg m , I 3 0 1.3 0 kg m 2 0 0 0.6 0 1.4 6.4 2.6 5800 0 The command direction of the thrust vector is d
d 0 T
T
0 ; The initial altitude of
the upper stage orbit is 1120km ; The initial attitude of the GT is 5 T
initial gimbal angles is
0 0 T
T
53.5174
T
T
5 ; The
0.1551 , The initial gimbal angular rate is
T
/s .
The parameters of the designed controller are
k1 diag 1.5 1.5 1.5 2 2 , K diag 0.25 0.25 0.25 0.25 0.25 ε diag 0.0001 0.0001 0.0001 0.0001 0.0001
B. Simulation Results and Analysis The simulation results are demonstrated in Fig. (4)~Fig. (8).
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As a part of the thrust vector control, the simulation results of the special attitude control are shown as follows. The attitude of the upper stage and its error in the desired attitude is shown in Fig. 4, which illustrates that the upper stage’s attitude σ converges towards the desired attitude
σ d . It is shown that the attitude control of upper stage during orbit transfer is available. Fig. 5a) implies the time history of the elevation angle and the azimuth angle of the thrust vector, and Fig. 5b) verifies the angles converge towards the command angles determined by the guidance system. The other part of thrust vector control is the gimbal control of the GT. The time histories of the GT’s attitude and gimbal angles demonstrated in Fig. 6 indicate that the gimbal rotation of the GT converges towards a stabilized state. Fig.4~Fig.6 reveals the thrust vector control of upper stage is complete by the special attitude control of upper stage and the gimbal control of the GT. Fig. 7 demonstrates the time histories of the attitude control torques acting on the upper stage, showing that the torques about zb and yb axis are converge. For the torque about xb provided by the RCS thrusters, the roll attitude approaches the corresponding limit loops, whose radius is a near-zero constant. Fig. 8 shows the time histories of the gimbal rotation control torques provided by electrical motors and indicates that the control torques are converge. Fig.6~Fig.8 depicts the GT’s thrust vector passing through the mass center of the upper stage is satisfied by the gimbal control of the GT. In addition to the definition of the special attitude control process of the upper stage, the thrust vector of GT aligning with the command direction is accomplished displays in Fig.5 when the special attitude control is achieved shown in Fig.4. In summary, Fig. 4 to Fig.8 indicate that controlling the attitude of the upper stage and the gimbal rotation of the GT, the thrust vector of the GT not only converges towards the command direction determined by the guidance system, but also passes through the upper stage mass center, avoiding the error of velocity vector and the disturbance torques.
16
40
20
Attitude error of upper stage/()
Attitude of upper stage/()
60
0
-20
0
10
20
30
40
50
10
8
6 4 2 0 -2
0
10
20
t/s
30
40
50
t/s
a)
b)
100
80
60 40 20 0 -20
0
10
20
30
40
50
Azimuth and elevation error of GT/()
Azimuth and elevation of GT/()
Fig. 4 Attitude (a) and attitude error (b) of upper stage 10
5 0 -5
-10
0
10
20
t/s
30
40
50
t/s
a)
b) Fig. 5 Angles (a) and angles error (b) of GT’s thrust vector
8
20 Gimbal angles of GT/()
Attitude of GT/()
6 4 2 0 -2
0
10
20
30
40
-20 -40
-60
50
0
0
t/s
20
40 t/s
a)
b)
17
60
Gimbal control torques of GT/(Nm)
Fig. 6 Attitude (a) and gimbal angles (b) of GT
Attitude control torques/(Nm)
200 T bx T by
100
T bz 0 -100
-200
0
10
20
30
40
50
3 Tg
2
Tm 1 0 -1 -2 -3
0
10
20
30
40
50
t/s
t/s
Fig. 7 Attitude control torques
Fig. 8 Gimbal control torques
V. Conclusion The disturbance torques, which is produced by the thrust vector misalignments and used to control the attitude of the upper stage, is designed positively by controlling the gimbal angle. The convergence of the attitude controller ensures the controller itself to eliminate the effects of the thrust vector misalignments, and the desired attitude of the upper stage is designed to keep the thrust vector direction of the GT same with the command’s after the stabilization of the controller. The RCS thrusters in roll direction were employed to provide the control torque about the roll axis. The gimbal rotations of the two-axis gimbal and the GT were controlled by the electrical motor. A thrust vector control procedure including the upper stage attitude control and the GT’s gimbal control was proposed. The control strategy guarantees that the thrust vector of the GT not only aligns with the command direction but also passes through the upper stage’s mass center. That is to say, the thrust vector of the upper stage accomplishes the goal to provide the orbit transfer impetus effectively and do not produce disturbance torques. Numerical simulations verified the effectiveness of the proposed control algorithm. The TVC of upper stage presented in this paper will decrease the disturbance torques which act on the upper stage,
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increase the accuracy of orbit transfer and decrease the consumption of fuel in the practical launch mission. The problem discussed in this paper was realized in recent launch missions. Our method and results in this paper can be utilized in the TVC of the upper stage during orbit transfer in the mission of One-vehicle with Multi-satellite in the future. This paper only concentrates on the TVC of the upper stage. Detailed relations between the TVC and the orbit transfer control are of great significance and worthy of further research. Acknowledgments As a part of One-Vehicle with Multi-Satellite Project, this work is supported by Shanghai Institute of Aerospace Control Technology and National Natural Science Foundation (NO. 11272027). References 1
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for Spacecraft Thrust Vector Control,” Journal of Guidance, Control, and Dynamics, Vol. 28, NO. 1, 2005, pp. 185-188. 11
L. Felicetti, M. Sabatini, A. Pisculli, et al , “Adaptive Thrust Vector Control during On-
Orbit Servicing,” AIAA Space 2014 Conference and Exposition, San Diego, CA, 2014, pp. 1-18. 12
Michael Garrison, Mark Davis, Scott Steffan, “Human-Rated Upper Stage Thrust Vector
Control System Architecture Selection,” 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Cincinnati, OH, 2007, PP. 1-20. 13
Suchitra P, Priya C Kurian, Rini Jones S B, “Optimal Controller based Servo System Design
for the Thrust Vector Control of Liquid Propellant Engine of three Stage Launch Vehicle,” International Conference on Magnetics, Machines & Drives, 2014, pp. 1-6
20
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Tae Won Hwang, Chang-Su Park, Min-Jea Tahk, Hyochoong Bang, “Upper Stage Launch
Vehicle Servo Controller Design Considering Optimal Thruster Configuration,” AIAA Guidance, Navigation, and Control Conference and Exhibit, Austin, Texas, 2003, pp. 1-8 15
Tobin C. Anthony, Bong Wie, “Pulse-Modulated Control Synthesis for a Flexible
Spacecraft,” Journal of Guidance, Control, and Dynamics, Vol. 13, NO. 6, 1989, pp. 1014-1022. 16
Zhaohui Wang, Lei Jin, Yinghong Jia. “Attitude Control Technology of Upper Stage during
Orbital Transfer,” China Space Science and Technology, NO. 5, 2014, PP. 1-9. 17
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Highlights: 1) The rotary actuator is used due to the higher gimballing ability. 2) Coupled mathematical model of the upper stage as a multi-body system. 3) A control system consisting of the special attitude control and gimbal rotation. 4) The desired attitude computing method and the gimbal steering law.
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PII: DOI: Reference:
S0094-5765(16)30526-4 http://dx.doi.org/10.1016/j.actaastro.2016.06.002 AA5848
To appear in: Acta Astronautica Received date: 7 February 2015 Revised date: 22 May 2016 Accepted date: 1 June 2016 Cite this article as: Zhaohui Wang, Yinghong Jia, Lei Jin and Jiajia Duan, Thrust Vector Control of Upper Stage with a Gimbaled Thruster during Orbit Transfer, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2016.06.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Thrust Vector Control of Upper Stage with a Gimbaled Thruster during Orbit Transfer Zhaohui Wang1, Yinghong Jia1*, Lei Jin1, Jiajia Duan2 (1 Beihang University, School of Astronautics, Beijing, 100191, P.R. China; 2 Shanghai Institute of Aerospace Control Technology，Shanghai, 201109，China) Abstract: In launching Multi-Satellite with One-Vehicle, the main thruster provided by the upper stage is mounted on a two-axis gimbal. During orbit transfer, the thrust vector of this gimbaled thruster (GT) should theoretically pass through the mass center of the upper stage and align with the command direction to provide orbit transfer impetus. However, it is hard to be implemented from the viewpoint of the engineering mission. The deviations of the thrust vector from the command direction would result in large velocity errors. Moreover, the deviations of the thrust vector from the upper stage mass center would produce large disturbance torques. This paper discusses the thrust vector control (TVC) of the upper stage during its orbit transfer. Firstly, the accurate nonlinear coupled kinematic and dynamic equations of the upper stage body, the two-axis gimbal and the GT are derived by taking the upper stage as a multi-body system. Then, a thrust vector control system consisting of the special attitude control of the upper stage and the gimbal rotation of the gimbaled thruster is proposed. The special attitude control defined by the desired attitude that draws the thrust vector to align with the command direction when the gimbal control makes the thrust vector passes through the upper stage mass center. Finally, the validity of the proposed method is verified through numerical simulations. Keyword: Upper stage, Thrust vector control, Orbit transfer
*Corresponding author. Email address: [email protected] 1
I. Introduction
THRUST vector control has been considered as a crucial area of research in the launch systems, because of the extremely advantageous ability to control the thrust vector of any propulsion system. For the launch systems
[1-5]
, the direction of thrust vector of the propulsion
system should align with the command direction. The TVC of the launch systems can be obtained by gimbaling the engine via servo actuation system. The deviations of the thrust vector in the launch systems lead to large velocity and positioning errors, along with extra requirements of orbit control. For the upper stage, it should transfer to the other orbit when it completes the deploy mission in one orbit. During orbit transfer, the thrust vector should theoretically align with the command direction and pass through the mass center of upper stage. The thrust vector misalignments of the upper stage also produce disturbance torques and make the upper stage rotate. Furthermore, it results in long orbit transfer time and extra requirements of attitude/orbit control. TVC has become a significant technology for the launch systems to control their trajectories and accomplish high maneuverability. The requirements for the TVC of space shuttles were presented in Penchuk’s work [1]. The TVC for space shuttles obtained by Omni-axis vectoring of nozzle was studied in reference [2]. The controller can maintain the vehicle trajectory and the direction of the thrust vector by independently controlling the gimbal angles of the engine. The model, estimation, implementation and control techniques of the TVC-electromechanical actuators for the rocket engines were investigated in references [3-5]. Taking fuel slosh dynamics into consideration, Jaime [6] researched on the TVC for the rocket engine. However, three more independent torques were needed in the control system. The TVC for a liquid-bipropellant upper stage spacecraft was presented by Wie [7]. The controller can keep the pointing of the thrust
2
vector and the stability of the spacecraft. In researches [8, 9], the TVC for a solar-sail spacecraft was proposed. The sail angle with respect to the sunlight was utilized to produce a net torque for the attitude control and thrust vectoring. For the satellite systems [10, 11], the TVC was used to correct the misalignments produced by the thruster firing and the shift of the mass center for a satellite due to fuel consumption, solar panels deployment, onboard experiments, etc. For the upper stage, the selection of a TVC system actuation technology and subsystem technology was discussed in Garrison’s research [12]. Most achievements above were concentrated on Linear Actuators with small engine deflection defined in Suchitra’s study [13], where Linear Actuators are most powered by pneumatic or hydraulic actuators, and Rotary Actuators are motor powered by electromechanical actuators. The Linear actuators were used for small engine deflection ( 4 ), whereas for higher gimballing of the engine; preference goes to Rotary Actuators. In this paper, Rotary Actuators are utilized to solve the higher gimbaling case. Disturbance torques produced by the thrust vector misalignments, eight reaction control subsystem (RCS) thrusters, together with the main thruster, are utilized in the TVC in order to eliminate the effects of the thrust vector misalignments. The thruster configuration presented in this paper was defined as the optimal one in Hwang’s investigation [14]. The steering law for the RCS thrusters is obtained by pulse-width and pulse-frequency modulator (PWPFM) which is first utilized in the attitude controller by Bong W. [15] and has a first order filter compensates the Schmitt trigger output in the feedback path. The advantage of the PWPFM is the static parameters that are independent with respect to the parameters of the spacecraft. The accurate nonlinear model of the upper stage is derived by taking the upper stage as a multi-body system which consists of the upper stage body, the gimbaled thruster actuator and the gimbaled thruster. The superiority of this context is the thrust vector control algorithm which completes by the
3
special attitude control of the upper stage and the gimbal rotation of the gimbaled thruster. To achieve this goal, the algorithm that determines the desired attitude for the special attitude control and the gimbal control method that based on the special attitude control is proposed. Due to the property of the rotary actuator, the rotation motion of the gimbal and the GT is strongly coupled with the attitude motion of the upper stage body. The control scheme takes the nonlinear coupling effect into consideration because the controller based on the linearization model is inappropriate in this situation. The remainder of this paper is outlined as follows. First, the comprehensive mathematical formulations are proposed considering the coupling of the orbit and attitude motion, the upper stage’s motion and GT’s rotation. Next, the desired attitude of the upper stage is obtained using the newly derived algorithms to guarantee the direction of the GT’s thrust vector aligning with the command direction. In the same way, the desired gimbal angles of the GT are got to make sure the GT’s thrust vector passing through the mass center of the upper stage. Then, a TVC system including the upper stage’s attitude control and the GT’s gimbal control is designed, ensuring the thrust vector of the GT not only passes through the mass center of the upper stage, but also aligns with the command direction. Finally, the numerical simulations are performed, followed by analysis and conclusions. II. Mathematical Models The system discussed in this paper consists of an upper stage body, a two-axis gimbal and a GT. The kinematic and dynamic equations of the system are derived in this section. The configuration of the system is shown in Fig.1.
4
RCS Thruster RCS Thruster Upper Stage yb
Gimbaled Thruster
xb
Upper Stage
Fm
ob
Gimbaled Thruster
RCS Thruster RCS Thruster
RCS Thruster
R Z
RCS Thruster
RCS Thruster
Y X
Fig. 1 Configuration of upper stage A. Coordinate frames To clearly represent the dynamics of the system, the coordinate frames are defined as follows. An Earth centered inertial frame is denoted by f I , with its origin located at the center of the Earth, the z I axis pointing towards the celestial North Pole, the xI axis pointing towards the vernal equinox and the y I axis completes the right-handed reference frame. The orbit reference frame ( f o ), has its origin located at the mass center of the upper stage, with the zo axis pointing towards the Earth center, the xo axis pointing towards the direction of tangential velocity and the
yo axis completes the right-handed reference frame. The principal body-fixed coordinate frame of the upper stage is denoted by f b with its origin located at the mass center of the upper stage. Attitude orientation of the body fixed frame f b with respect to frame f o is represented by ( ZXY ) set of Euler angles: (pitch) about z axis, (roll) about x axis and (yaw) about y axis. The body-fixed reference frame of the gimbal is denoted by f k , and the gimbal angle of f k with respect to f b is represented by Euler angle: (roll) about x axis. The body-fixed reference 5
frame of the GT is denoted by f m , and the gimbal angle of frame f m with respect to frame f k is represented by Euler angle: (yaw) about y axis. The coordinate frames are shown in Fig. 2. zo
yk ym z
Upper Stage
yb
xb ob
fb
zb m
zk
f k / f m ok om
R
xo
Z
fo O
yb yo
Gimbaled Thruster
fI
Fm
xk xm
xb
fb
Y
X
Fig. 2 Definition of coordinate frames B. Thrusts and Torques The thrust vector of the GT is denoted by Fm , which is demonstrated in Fig. 2 and defined as
Fm Fm I m 0J m 0 K m Fm cos sin Io Fm sin J o Fm cos cos K o
(1)
where Ii , J i , K i are unit vectors along the axes of frame f i ( i o, b, k , m ), is the elevation angle, the azimuth angle, and Fm the constant thrust level of the GT. Jo Fm
Oo
Io
Ko
Fig. 3 Gimbaled Thruster azimuth and elevation
6
To calculate the thrusts and torques provided by the GT, two consecutive gimbal rotations are utilized. First, the two-axis gimbal rotates with respect to the upper stage body about the xk axis,
is the gimbal angle. Second, the GT rotates with respect to the two-axis gimbal about the intermediate axis y ' , is the gimbal angle. Then, Fm expressed in frame f b is
Fm Fm cos Ib Fm sin sin Jb Fm sin cos Kb
(2)
rm rmx Ib rmy Jb rmz Kb is the vector presenting the placement from the upper stage’s mass
center to the action point of the GT. Then, the torques provided by GT is
Tm rm Fm
(3)
In this paper, we assume that all thrusters can provide constant thrusts. The torque provided by the RCS thrusters is expressed as
Tx 4Fg rg , Tx 4Fg rg
(4)
where Tx , and Tx represents the positive and negative torque along xb axis respectively, Fg the constant thrust level of the RCS thrusters, and rg the moment arm.
C. Kinematic Equations Based on the definition of the attitude orientation for the upper stage, the kinematic equation gives
rx cos rz sin σ ry tan rx sin rz cos A ω Abo ωo ( sin cos ) / cos rx rz
(5)
where σ is the attitude of the upper stage, A the transformation matrix from T
ωr to σ , ωr rx ry
T
rz the angular velocity of the upper stage body with respect to
7
frame f o , ω x y
T
z the angular velocity of the upper stage body with respect to
frame f I , A ji the transformation matrix from frame f i to frame f j ( i, j o, b, k , m ) and ωo the orbit angular velocity. The rotating equation for the gimbal with respect to the upper stage body is
ωk Γ k
(6)
where ωk is the angular velocity of the gimbal with respect to the upper stage body expressed in frame f k . Γ k [1 0 0]T is determined by the configuration of the system. The rotating equation of the GT with respect to the gimbal gives ωm Γ m
(7)
where ωm is the angular velocity of the GT with respect to the gimbal expressed in frame f m .
Γ m [0 1 0]T is determined by the configuration of the system. D. Dynamic Equations The dynamic equations including both translational and rotational motion of the system are derived based on Kane equation [16]. The general formula of Kane equation is defined as Fi G Fi A 0
(8)
where Fi G , Fi A are the generalized inertia force and generalized active force of the system with respect to the ith order generalized velocity. First, the dynamic equation for the translational motion of the upper stage is
mvb Sbt ω Abk Skt Γ k Abm Sm Γ m Fb,n Fb Fw where
Fb,n mωvb ωSb*ω Abk Ωk Sk* Ωk Abm Ωm Sm Ωm Abk Skt Akb ωAbk Ωk Abm Sm Amk Ωk Akm Ωm 8
(9)
where m mb mk mm , mb , mk , mm are the masses of the upper stage, the two-axis gimbal and the GT respectively, vb is the velocity of the upper stage defined in frame f b , Sbt the static moment of the system expressed in frame f b , S kt the static moment of the gimbal and the GT expressed in frame f k , S k the static moment of the gimbal expressed in frame f k , S m the static moment of the GT expressed in frame f m , Sb the static moment of the upper stage’s augment expressed in frame f b , S k the static moment of the gimbal’s augment expressed in frame f k ,
Ωk the angular velocity of the gimbal, Ωm the angular velocity of the GT, Fb the thrust and Fw the gravity that act on the system. The dynamic equation for the rotational motion of the upper stage is
SbtT vb Ibt ω Abk I kb Γ k Abm I mb Γ m Tb,n Tb
(10)
where
Tb,n Sbt ωvb I ω Abk Ib k Ωk Abm Ib m Ωm Abk Ikb Akb ωAbk Ωk Abm Imb Amk Ωk Akm Ωm where I bt is the inertia matrix of the system expressed in frame f b , I kb the coupling inertia matrix of the gimbal, I mb the coupling inertia matrix of the GT, I the quasi inertia matrix of the system, I b k the quasi inertia matrix of the gimbal, I b m the quasi inertia matrix of the GT and Tb the torque which acts on the system. The dynamic equation of the gimbal rotates with respect to the upper stage body is
Γ kT SktT Akbvb ( I kb )T Akb ω I kt Γ k Akm I mk Γ m Tk ,n Tk where
Tk ,n Skt Akb ωvb SktT Akb ωrb,k ω Akm Ik m Ωm Ikt Akb ωAbk Ωk I k Ωk Akm Imk Amk Ωk Akm Ωm
9
(11)
where I m denote the inertia matrix of the GT, I mk the coupling inertia matrix of the GT, I Ωk the quasi inertia matrix of the gimbal, I Ωk m the quasi inertia matrix of the GT, and Tk the rotational control torque which acts on the gimbal. The dynamic equation of GT rotates with respect to the gimbal is
Γ mT SmT Amb vb ( I mb )T Amb ω ( I mk )T Amk Γ k I m Γ m Tm,n Tm
(12)
where
Tm,n Sm Amb ωvb SmT Amb ωrmω Ωm Im Ωm ( Imk )T Amb ωAbk Ωk Im Amk Ωk Akm Ωm where Tm is the rotational control torque that acts on the GT. Obviously，Eq. (9) is coupled with Eqs. (10), (11) , and (12). For TVC, Eqs. (10), (11) and (12) are independent. Substituting Eqs. (5) and (9) into Eqs. (10), (11) and (12), the independent dynamic equations can be rewritten as
At x T Tn where x σ T
T
(13)
T
, T TbT Tk Tm . The elements of the matrix At and the nonlinear
thrust Tn are given as
1 T 1 I bt Sbt Sbt A m 1 1 At Γ kT ( I kb )T Akb SktT Akb Sbt A m T b T 1 T 1 Γ m ( I m ) Amb Sm Amb Sbt A m
1 T b Abk I k Sbt Abk Skt Γ k m 1 Γ kT I kt SktT Skt Γ k m 1 Γ mT ( I mk )T Amk SmT Amk Skt Γ k m
10
1 T b Abm I m Sbt Abm Sm Γ m m 1 Γ kT Akm I mk SktT Akm Sm Γ m m 1 Γ mT I m SmT Sm Γ m m
1 I bt SbtT Sbt m SbtT Tb,n T b T 1 T T 1 T T 1 1 Tn Γ k Tk ,n Γ k Skt Akb Fb Fw Fb ,n Γ k ( I k ) Akb Skt Akb Sbt A A A σ Abo ωo m T T m T Γ mTm,n Γ m Sm Amb T b T 1 T Γ m ( I m ) Amb Sm Amb Sbt m
III. Control Law This section discusses the TVC system including the attitude control of the upper stage and the gimbal rotation control of the GT. First, the desired gimbal angles d , d , which ensure the thrust vector of the GT passes through the mass center of the upper stage, are given. Then, the desired attitude of the upper stage denoted by σ d is calculated; guaranteeing the direction of the thrust vector of the GT denoted by , aligns with its command direction d , d . The methods and analysis tools of variable structure control (VSC) are utilized in this paper due to their robustness to nonlinear model errors. A. Desired States Assume that the thrust vector of the GT passes through the mass center of the upper stage so that the desired gimbal angles d , d are defined as follows
Fm cos d / rmx Fm sin d sin d / rmy Fm sin d cos d / rmz
(14)
The thrust vector of the GT also can be defined by two Euler angles , which indicate the rotation of GT with respect to frame f o . The first rotation is about the yaw ( zo ) axis, and the rotate angle. The second rotation is about the intermediate pitch axis ( y ' ), and the rotate angle. The transformation matrix Amo and thrust vector Fm in terms of , are
11
cos cos Amo sin sin cos
cos sin cos sin sin
sin 0 cos
(15)
Fm Fm cos cos I o Fm cos sin J o Fm sin K o
(16)
The rotation angles , can be calculated by the azimuth angle and the elevation angle
,
as
arctan tan sin arcsin cos cos
(17)
The transformation matrix Amo can also be defined by σ, , as
Amo Amb Abo
(18)
where
cos Amb 0 sin
sin sin cos cos sin
cos cos sin sin sin Abo cos sin sin cos cos sin sin
sin cos sin cos cos
cos sin sin sin cos cos cos sin sin cos sin cos
(19)
sin cos sin cos cos
(20)
First, the command direction d , d of the thrust vector for the GT is determined by the guidance system. Then, the desired rotation angles d ,d , the desired transformation matrix d d Amo , the desired gimbal angles d , d , and the desired transformation matrix Amb can be
calculated by Eqs. (17), (15), (14) and (19), respectively. Finally, the desired transformation matrix Abod is obtained as d Abod Amb Amod 1
12
(21)
Thus, the desired attitude σ d can be obtained from Eqs. (20) and (21). B. Design of Control Law For a TVC system, the attitude control signals are the current desired gimbal angles cd , cd providing the attitude control torque about yb axis and zb axis, and the on-off thrust of the RCS thrusters providing the attitude control torque about xb axis. The GT’s gimbal rotation control signal is the control torque provided by the electrical motor. The methods and analysis tools of VSC are utilized in this paper due to their robustness to nonlinear model errors. If the desired system states are defined as xd σd
cd , a
cd
T
sliding surface is defined as
S e k1e where e x xd is the error of system states, k1
55
(22) is the weighing matrix for e .
Substituting Eq. (13) into the differentiating Eq. (22), yield
S e k1e x xd k1 x xd At1 T Tn xd k1 x xd
(23)
The approaching law is chosen as
S KS ε sgn S where K
55
, ε
55
(24)
, and sgn is the sign function defined as
1 if x 0 sgn x 0 if x 0 1 if x 0 Substituting Eqs. (24) and (13) into Eq. (23), we obtain
T Tn At xd Kk1 x xd ε sgn x xd k1 x xd K k1 x xd
13
(25)
To determine the current desired gimbal angles of the GT cd , cd , substituting Eq. (25) into Eq. (3), yield
rmz Fm cos cd rmx Fm sin cd cos cd T 2 rmy Fm cos cd rmx Fm sin cd sin cd T 3
(26)
where T 2 , T 3 represent the second and third element of control torque (Eq. (25)), respectively. The current desired gimbal angles cd , cd can be figured out by Eq. (26). Because of the coupling, the GT would also produce torques about xb axis, which should be counteracted by the RCS thrusters. Then, the torques that the RCS thrusters should provide is Tbx T 1 rmz Fm sin cd sin cd rmy Fm sin cd cos cd
(27)
The control torque about xb provided by the RCS thrusters is defined by Eq. (4). The steering law of the RCS thruster is obtained by the PWPFM method [15,17,18]. In order to verify the stability of the controller, set the Lyapunov Function as V S T S / 2 . Substituting Eqs. (25) and (24) into the differentiation of the Lyapunov Function, yield
V S T S S T K T S sgn S T εT S K j S 2j j S j 0 5
5
j 1
j 1
5
5
j 1
j 1
(28)
Since V S T S / 2 0 , and V K j S 2j j S j 0 , V 0 exists only when S 0 . From the LaSalle invariance principle, we conclude that the controller designed previously is global asymptotic stable. The paper, in order to avoid of chattering at the sliding surface boundary, utilizes the saturation function to instead of the sign function in the controller, which is commonly applied in engineering.
14
IV. Numerical simulations In this section, the control strategy of the TVC for upper stage with a gimbaled thruster designed previously is simulated numerically. A. Simulation Parameters The parameters are mb 3600kg , mk 10kg , mm 30kg . Fm 5000N , Fg 25N .
rmx
rmy
rmz 1.107 0.0025 0.0022 m , rg 1m . rmc 0.1 0 0 m is the T
T
T
placement vector of the mass center for the GT expressed in frame f m . The inertia matrix of upper stage ( I1 ), the gimbal ( I 2 ) and the GT ( I 3 ) are expressed as follows.
0 0 4800 4 6.4 0.5 0 0.75 0 2 2 I1 4 6000 2.6 kg m , I 2 0 0.7 0 kg m , I 3 0 1.3 0 kg m 2 0 0 0.6 0 1.4 6.4 2.6 5800 0 The command direction of the thrust vector is d
d 0 T
T
0 ; The initial altitude of
the upper stage orbit is 1120km ; The initial attitude of the GT is 5 T
initial gimbal angles is
0 0 T
T
53.5174
T
T
5 ; The
0.1551 , The initial gimbal angular rate is
T
/s .
The parameters of the designed controller are
k1 diag 1.5 1.5 1.5 2 2 , K diag 0.25 0.25 0.25 0.25 0.25 ε diag 0.0001 0.0001 0.0001 0.0001 0.0001
B. Simulation Results and Analysis The simulation results are demonstrated in Fig. (4)~Fig. (8).
15
As a part of the thrust vector control, the simulation results of the special attitude control are shown as follows. The attitude of the upper stage and its error in the desired attitude is shown in Fig. 4, which illustrates that the upper stage’s attitude σ converges towards the desired attitude
σ d . It is shown that the attitude control of upper stage during orbit transfer is available. Fig. 5a) implies the time history of the elevation angle and the azimuth angle of the thrust vector, and Fig. 5b) verifies the angles converge towards the command angles determined by the guidance system. The other part of thrust vector control is the gimbal control of the GT. The time histories of the GT’s attitude and gimbal angles demonstrated in Fig. 6 indicate that the gimbal rotation of the GT converges towards a stabilized state. Fig.4~Fig.6 reveals the thrust vector control of upper stage is complete by the special attitude control of upper stage and the gimbal control of the GT. Fig. 7 demonstrates the time histories of the attitude control torques acting on the upper stage, showing that the torques about zb and yb axis are converge. For the torque about xb provided by the RCS thrusters, the roll attitude approaches the corresponding limit loops, whose radius is a near-zero constant. Fig. 8 shows the time histories of the gimbal rotation control torques provided by electrical motors and indicates that the control torques are converge. Fig.6~Fig.8 depicts the GT’s thrust vector passing through the mass center of the upper stage is satisfied by the gimbal control of the GT. In addition to the definition of the special attitude control process of the upper stage, the thrust vector of GT aligning with the command direction is accomplished displays in Fig.5 when the special attitude control is achieved shown in Fig.4. In summary, Fig. 4 to Fig.8 indicate that controlling the attitude of the upper stage and the gimbal rotation of the GT, the thrust vector of the GT not only converges towards the command direction determined by the guidance system, but also passes through the upper stage mass center, avoiding the error of velocity vector and the disturbance torques.
16
40
20
Attitude error of upper stage/()
Attitude of upper stage/()
60
0
-20
0
10
20
30
40
50
10
8
6 4 2 0 -2
0
10
20
t/s
30
40
50
t/s
a)
b)
100
80
60 40 20 0 -20
0
10
20
30
40
50
Azimuth and elevation error of GT/()
Azimuth and elevation of GT/()
Fig. 4 Attitude (a) and attitude error (b) of upper stage 10
5 0 -5
-10
0
10
20
t/s
30
40
50
t/s
a)
b) Fig. 5 Angles (a) and angles error (b) of GT’s thrust vector
8
20 Gimbal angles of GT/()
Attitude of GT/()
6 4 2 0 -2
0
10
20
30
40
-20 -40
-60
50
0
0
t/s
20
40 t/s
a)
b)
17
60
Gimbal control torques of GT/(Nm)
Fig. 6 Attitude (a) and gimbal angles (b) of GT
Attitude control torques/(Nm)
200 T bx T by
100
T bz 0 -100
-200
0
10
20
30
40
50
3 Tg
2
Tm 1 0 -1 -2 -3
0
10
20
30
40
50
t/s
t/s
Fig. 7 Attitude control torques
Fig. 8 Gimbal control torques
V. Conclusion The disturbance torques, which is produced by the thrust vector misalignments and used to control the attitude of the upper stage, is designed positively by controlling the gimbal angle. The convergence of the attitude controller ensures the controller itself to eliminate the effects of the thrust vector misalignments, and the desired attitude of the upper stage is designed to keep the thrust vector direction of the GT same with the command’s after the stabilization of the controller. The RCS thrusters in roll direction were employed to provide the control torque about the roll axis. The gimbal rotations of the two-axis gimbal and the GT were controlled by the electrical motor. A thrust vector control procedure including the upper stage attitude control and the GT’s gimbal control was proposed. The control strategy guarantees that the thrust vector of the GT not only aligns with the command direction but also passes through the upper stage’s mass center. That is to say, the thrust vector of the upper stage accomplishes the goal to provide the orbit transfer impetus effectively and do not produce disturbance torques. Numerical simulations verified the effectiveness of the proposed control algorithm. The TVC of upper stage presented in this paper will decrease the disturbance torques which act on the upper stage,
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increase the accuracy of orbit transfer and decrease the consumption of fuel in the practical launch mission. The problem discussed in this paper was realized in recent launch missions. Our method and results in this paper can be utilized in the TVC of the upper stage during orbit transfer in the mission of One-vehicle with Multi-satellite in the future. This paper only concentrates on the TVC of the upper stage. Detailed relations between the TVC and the orbit transfer control are of great significance and worthy of further research. Acknowledgments As a part of One-Vehicle with Multi-Satellite Project, this work is supported by Shanghai Institute of Aerospace Control Technology and National Natural Science Foundation (NO. 11272027). References 1
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Highlights: 1) The rotary actuator is used due to the higher gimballing ability. 2) Coupled mathematical model of the upper stage as a multi-body system. 3) A control system consisting of the special attitude control and gimbal rotation. 4) The desired attitude computing method and the gimbal steering law.
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