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Thrust distribution for attitude control in a variable thrust propulsion system with four ACS nozzles
Accepted Manuscript Thrust Distribution for Attitude Control in a Variable Thrust Propulsion System with four ACS Nozzles Yeerang Lim, Wonsuk Lee, Hyochoong Bang, Hosung Lee PII: DOI: Reference:
S0273-1177(17)30015-7 http://dx.doi.org/10.1016/j.asr.2017.01.002 JASR 13041
To appear in:
Advances in Space Research
Received Date: Revised Date: Accepted Date:
3 May 2016 19 December 2016 3 January 2017
Please cite this article as: Lim, Y., Lee, W., Bang, H., Lee, H., Thrust Distribution for Attitude Control in a Variable Thrust Propulsion System with four ACS Nozzles, Advances in Space Research (2017), doi: http://dx.doi.org/ 10.1016/j.asr.2017.01.002
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Thrust Distribution for Attitude Control in a Variable Thrust Propulsion System with four ACS Nozzles Yeerang Lim1 Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Republic of Korea Wonsuk Lee2 Agency for Defense Development, Daejeon, 305-600, Republic of Korea Hyochoong Bang3 Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Republic of Korea and Hosung Lee4 Agency for Defense Development, Daejeon, 305-600, Republic of Korea
A thrust distribution approach is proposed in this paper for a variable thrust solid propulsion system with an attitude control system (ACS) that uses a reduced number of nozzles for a three-axis attitude maneuver. Although a conventional variable thrust solid propulsion system needs six ACS nozzles, this paper proposes a thrust system with four ACS nozzles to reduce the complexity and mass of the system. The performance of the new system was analyzed with numerical simulations, and the results show that the performance of the system with four ACS nozzles was similar to the original system while the mass of the whole system was simultaneously reduced. Moreover, a feasibility analysis was performed to determine whether a thrust system with three ACS nozzles is possible.
Nomenclature Cf
=
thrust coefficient
At
=
throat area, m 2
FAi , FDi
=
Thrust of the ith attitude/divert control nozzle, N
Fsys
=
total system thrust generated under chamber pressure control, N
FDsum
=
Available total thrust after substituting the attitude control system thrusts, N
1
Ph.D. Candidate, Aerospace Engineering Department, [email protected]. Ph.D., Senior Researcher. 3 Professor, Aerospace Engineering Department, Senior Member AIAA. 4 Senior Researcher. 2
Fy , Fz
=
y/z axis forces, N
F , F , F
=
roll, pitch, and yaw axis forces, N
=
installation angle of the ACS nozzles, deg
=
Tilt angle of the ACS nozzles, deg
=
roll, pitch, and yaw angles, deg
, ,
1. Introduction
V
ARIABLE thrust propulsion systems with solid propellant have been widely adopted for various missions. In
addition to the advantages of solid propulsion systems which include light weight and simple design, a fast response and the availability of precise thrust control are the additional advantages of thrust variability. These additional advantages can be achieved by controlling the nozzle throat area using a pintle actuator (Beardsley and Shipley, 1976; Morris et al., 1995). Based on these characteristics, a variable thrust solid propulsion system (VTSPS) could be a viable solution for systems that require agile translational and rotational maneuvers. To control the attitude of a VTSPS, the system should have multiple nozzles to generate torque with respect to all body axes. Moreover, multiple nozzle systems are advantageous for a simple method to control the position of the thrust system. Because all the nozzles should share a combustion chamber, thrust distribution should be done under controlled chamber pressure conditions. In other words, all the pintles with respect to each nozzle should be controlled to change the throat area, while the throat area of the whole system has to satisfy the constraint of maintaining the pressure. Therefore, the pressure control and thrust distribution problems affect one other. Lee et al. (2013) introduced an integrated algorithm generating variable thrust based on pressure stabilization with maximum thrust efficiency. The research was based on a conventional VTSPS shown in Fig. 1, which has four divert control system (DCS) nozzles and six attitude control system (ACS) nozzles (Joner and Quinquis, 2006). Attitude maneuvers with respect to the yaw axis can be decoupled using a thrust system with six ACS nozzles, while roll/pitch maneuvers are coupled. By extension of this thrust system with six ACS nozzles, this main objective of this study is to propose a VTSPS with four ACS nozzles to reduce system weight. Morris et al. (1995) has designed similar system with reduced ACS nozzles, however, any details about maneuver efficiency or thrust distributions are derived. A thrust distribution algorithm and with pressure control strategy is established for the thrust system with only four ACS nozzles.
The control of the propulsion system with a reduced number of ACS nozzles consists of two parts. First, chamber pressure control should be considered because the resulting thrust is a function of the chamber pressure. To resolve the problem associated with the characteristics of the combustion chamber being nonlinear and time-varying, Bergmans and Di Salvo (2003) studied an adaptive matching technique. Lee et al. (2013) introduced an adaptive controller based on a feedback linearization (Choi and Lim, 2003; Marino and Tomei, 1997; Diao and Passino, 2004) technique, and analyzed the stability of the proposed algorithm. Because the number of nozzles are not related to the chamber pressure dynamics, this control strategy could be used in a similar way. After the chamber pressure is stabilized, the desired thrust calculations and thrust assignment problems are derived for the multiple nozzles sharing the combustion chamber. The thrust sum of the nozzles is limited because of the chamber pressure control. A thrust distribution strategy that minimizes the pintle movement is introduced to maintain the pressure. ACS thrust distribution is separated from DCS thrust distribution and the pressure maintenance loop. ACS thrust assignment is performed before the DCS thrust assignment because it is more beneficial from the perspective of attitude control to separate the ACS thrusts from the pressure control, which adds complexity to the system. Since all the attitude maneuvers are coupled to each other because of the reduced number of ACS nozzles, pseudothrust algorithms are introduced to solve the problem. Using the four ACS nozzle arrangement suggested in this paper, attitude control performance will be maintained as well as the system mass can be decreased. This means the agile performance of the system can be increased without any attitude control degradation. Favorable and adverse nozzle placement are derived respectively, and the minimum number of the ACS nozzle is also analyzed. Thrusts to produce yaw moment will cause roll/pitch moments also with reduced configuration, with is originally separated with six ACS nozzles. A proper thrust distribution method with the coupled nozzles is described and verified with simulation studies.
2. Variable Thrust Solid Propulsion System A conventional variable thrust solid propulsion system includes four DCS nozzles D1 , D2 , D3 , D4 and six ACS nozzles A1 , A2 , A3 , A4 , A5 , A6 . Because the x -axis represents the agile direction of the whole propulsion system, the DCS nozzles are dedicated to the translational motion for y, z -axes, respectively. Because the nozzles share a
single chamber, all the desired thrusts for the nozzles should satisfy the throat area constraint. The sum of the throat area of all the nozzles should be equal to the desired throat area to control the pressure of the thrust chamber.
Fig. 1 Variable thrust solid propulsion system and DCS/ACS nozzle positions
Six ACS nozzles are in charge of attitude maneuvers: roll (about the body fixed x axis), pitch (about the body fixed y axis), and yaw (about the body fixed z axis). A yaw maneuver is independent of the other two attitude maneuvers and can be achieved with the nozzle A1 or A4 . Because the other four ACS nozzles are coupled, a roll maneuver can be achieved with the ( A2 , A5 ) or ( A3 , A6 ) pair based on the sign of the desired thrust, while ( A2 , A3 ) or ( A5 , A6 ) pair could be engaged for the pitch maneuver. The final thrust inputs, which are assigned to
each ACS nozzles, are the sum of these desired thrusts regarding each attitude maneuver. This will be described in detail in section III.
FAi Fi Fi Fi ,
i 1, 2,3, 4,5,6
(1)
Because DCS nozzles are in charge of controlling the chamber pressure, the divert thrusts are assessed after the ACS thrust vectors. The sum of the divert thrusts is the difference of the available thrust, which is obtained from the throat area, to maintain the pressure and the sum of the desired ACS thrust. Divert nozzles ( D1 , D2 ) are dedicated to y-directional thrust, and ( D3 , D4 ) are for z-directional maneuvers. 6
4
j 1
i 1
Fsys FAj FDsum FDi Fy FD1 FD2
(2)
Fz FD3 FD4 with
0 FDi FDsum
The whole control loop for the variable thrust propulsion system is summarized in Figure 2. From the pressure and state (position and attitude) errors, the desired chamber pressure and the control thrusts are constructed first. Thrust distribution is also performed in this step. Additionally, the desired throat area, which is the actual command for the actuator, is estimated to distribute thrust command properly for the desired pressure control. Finally, the pintle actuator should move based on the throat area command, and the actual pressure and state errors are fed back to the error state computation module.
Fig. 2 Nozzle input mechanism
3. Desired Attitude Thrust Calculation and Distribution The sum of available thrust of the whole system can be calculated from the chamber pressure such that
F C f At Pc
(3)
where C f represents a thrust coefficient defined as 1 1 Pe 2 2 2 1 Pe Pa Ae Cf 1 ( ) 1 1 Pc Pc At
(4)
Because the thrust of each nozzle is not a feedback variable, the thrust command should be calculated based on the position and attitude errors. A PD (Proportional plus Derivative) controller was employed in this research. In addition, a thrust distribution algorithm should be constructed because there are more than two nozzles. The whole thrust distribution algorithm is presented in Figure 3. The resultant FDi s and FA j s should satisfy the throat area constraint to maintain the pressure at a desired level.
Fig. 3 Thrust algorithm block diagram
Traditionally, a variable thrust propulsion system consists four DCS nozzles and six ACS nozzles in many previous studies. In this study, the number of ACS nozzles is reduced from six to four to minimize the system mass
shown in Figure 4. Because the roll maneuver should be available, tilt angle is combined with a new design parameter: the installation angle . The new proposed thrust distribution algorithm is based on the parameters and .
Fig. 4 Rear view of the variable solid propulsion system with 4 ACSs
3.1 Attitude Control System (ACS) Thrust Command The desired ACS thrusts are simply derived with the PD controllers as follows: F
1 (k (cmd ) k d ) d
F
1 (k ( cmd ) k d ) d
F
1 (k ( cmd ) k d ) d
(5)
where k , k , k are the proportional control gains, and k d , k d , k d are the derivative control gains (refer to the Appendix A for stability analysis). This desired thrust vector should be allocated to all the ACS nozzles properly. In the case of six ACS nozzles shown in Figure 1, because nozzle A1 and A4 are dedicated to the yaw maneuver only, the desired yaw thrust F is assigned to A1 or A4 based on the sign of F , while F and F are assigned to A2 , A3 , A5 , and A6 ..
Table 1 Pseudothrust algorithm for attitude thrust distribution – 6 ACSs (Lee et al., 2013) Allocation Roll if F 0,
Pitch
Yaw if F 0,
if F 0,
F2 F / 2
F1 F
F2 F / 2
F5 F / 2
F3 F / 2 if F 0,
if F 0, F3 F / 2
F5 F / 2
F6 F / 2
F6 F / 2
if F 0, F4 F
Distribution
FAi Fi Fi Fi , i 1, 2,3, 4,5,6
Thrust assignment logic should be different for the four ACS nozzle case shown in Figure 4. A proper thrust distribution strategy for the four ACS nozzles is presented in Table 2. Because there are four nozzles only, unlike Figure 1, now pitch, yaw, and roll maneuvers are all coupled. Table 2 Pseudothrust algorithm for attitude thrust distribution – 4 ACSs Allocation if F 0, F2 , F3
if F 0,
if F 0, F
F1 , F2
2sin F 2sin
F3 , F4
F2 , F4
F 2 cos( )
if F 0,
if F 0,
if F 0, F1 , F4
F 2sin( ) F 2sin( )
F1 , F3
F 2 cos( )
Distribution
FAi Fi Fi Fi , i 1, 2,3, 4
Based on the thrust distribution logic in Table 2, the resultant torque inputs for each attitude maneuver can be derived as follows: r ( FA2 FA3 ) sin M r ( FA1 FA4 ) sin d ( FA1 FA2 ) sin( ) M d ( FA3 FA4 ) sin( ) d ( FA1 FA3 ) cos( ) M d ( FA2 FA4 ) cos( )
for M 0 for M 0 for M 0 for M 0 for M 0 for M 0
(6)
where d denotes the distance of the ACS nozzles from the center of mass, and r represents the radius of the chamber intersection, which is the moment arm of the ACS nozzles. Note that the performance of each attitude maneuver is determined by the installation angle and the tilt angle
. Maneuver performance for all the axes should be equalized far as possible for arbitrary maneuver. Under the assumption that the thrust limit of each ACS nozzle is equal, should be equal to 45deg as follows to drive the pitch and yaw performance to be equal.
d (2 Fmax )sin( ) d (2 Fmax ) cos( )
(7)
( ) 45deg
Once is restricted to 45deg , tilt angle is a key parameter to equalize the roll performance with the other two maneuvers. The tilt angle equ that maximize the roll maneuver performance can be derived by setting the roll moment M same as the pitch moment M or the yaw moment M . Then equ can be derived as a function of the moment arm d and r . Fmax r sin equ Fmax d sin( equ )
equ
Fmax d
d d sin 1 sin( equ ) sin 1 r r 2
2
(8)
where the thrust for each axis is assumed as the maximum thrust Fmax . Based on Eq. (8), should be greater than 45deg , because d is usually longer than r for the thrust propulsion system exterior. However, the boundary of the tilt angle is 0 (45deg ) because should be positive, and the sum of the installation angle and the tilt angle is constrained to 45deg . Therefore, there is no which actually allows M to be equal to M . Under such a restriction, should be decided as maximum among the boundary, which is 45deg , to maximize the roll maneuver as much as possible. The maximum minimizes the difference between M and M , while for the maximum roll performance is equal to 0. This will be analyzed later with numerical simulations.
3.2 Divert Control System (DCS) Thrust Command Divert thrust can be constructed after the ACS thrust distribution is completed. The available thrust is FDsum as described in Eq. (1), which is the remaining thrust after ACS thrust distribution. There are two DCS thrust distribution approaches. One is to distribute the y-axis thrust and z-axis thrust separately with a performance limit for each maneuver which is FDsum / 2 . This approach is not efficient because the available thrust cannot be focused on a single axis maneuver. The second approach is to maximize the DCS thrusts within the FDsum limit. This limit can be expressed as Fy Fz FDsum , and the solution to this problem corresponds to the intersection of two lines in Figure 5, where is the thrust direction in the y-z plane.
Fig. 5 Graphical representation for the maximum force problem
The solution Fy and Fz are summarized in Table 3 as a function of the thrust direction . Table 3 Maximum force available based on saturation Quadrant Force
Fy
Fz
1 FDsum
2 FDsum
3 FDsum
4 FDsum
1 tan FDsum tan 1 tan
1 tan FDsum tan 1 tan
1 tan FDsum tan 1 tan
1 tan FDsum tan 1 tan
Once the desired translational thrusts are calculated, then these thrust should be allocated to each DCS nozzle efficiently. If the minimum pintle movement is treated as the most efficient way to distribute the divert thrust, the problem can be expressed as an optimization problem to minimize the cost function as follows: 4
f min ( FDi Feq )
2
i 1
FD1 FD2 Fy where
FD3 FD4 Fz
(9)
FD1 FD2 FD3 FD4 FDsum Feq FDsum 4
Because of three equality constraints, the problem can be written as a function of one thrust variable. If FD4 is chosen as a variable, the solution to this optimization problem can be readily derived from f FD4 0 . 1 ( FDsum Fy Fz ) FD4 2 1 FD2 ( FDsum Fy Fz ) FD4 2 FD3 Fz FD4 FD1
FD4
(10)
1 ( FDsum 2 Fz ) 4
The optimally distributed divert thrust commands are applied to translational motion control.
4. Pressure Control Algorithm 4.1 Chamber Pressure Dynamics Chamber pressure dynamics should be derived before describing the pressure control algorithm. Figure 6 is the schematic diagram of the combustion chamber with a pintle actuator. The dynamics can be derived from the law of conservation of mass as Eq. (11), which means the mass flow rate inside the chamber should be equal to the sum of the mass produced by the combustion and the exhausted mass from the nozzle. m p , me , c and Vc represent the produced mass by combustion, exhausted mass, gas density and the free volume, respectively, in the chamber (Young et al., 2003; Sutton and Biblarz, 2001).
mp
d cVc me dt
(11)
Fig. 6 General variable thrust solid propulsion system
From the mass flow rate in Eq. (11) and the ideal gas assumption, the chamber pressure dynamics can be derived as Eq. (12), where R, Tc and k denote the gas constant, gas temperature and heat capacity ratio, respectively. Ab and p represent the propellant properties, cross-sectional area, and density. r is the burning ratio of the chamber, n which satisfies r aPc . At represents the throat area of the nozzle, which is a main control parameter in this study.
Pc
RTc Ab r p Vc
k 1
RT A P c t c Vc
k RTc
2 k 1 Pc Vc Vc k 1
(12)
4.2 Chamber Pressure Control Algorithm Because the chamber contains hot gas which may not be ideal, there must be several uncertainties in the free volume estimation and the invariant system estimation. Therefore, a robust pressure control strategy under these uncertainties are required. An adaptive control approach was selected in this study to deal with this real-world problem. The system can be expressed as follows:
x ( k (t ) ( x, t )) ( k (t ) ( x, t ))u yx
(13)
where k (t ), k (t ) are the known dynamics, and ( x, t ), ( x, t ) represent the nonlinear time-varying dynamics of the unknown uncertainties. The above formula can be applied to the chamber pressure model shown below:
k
RTc Ab p
Vc
Pc Vc Vc 1
2 1 RTc 1
RT P k c c Vc
(14)
where the control input of the pressure model is At . The control input can be derived as a combination of the control inputs from feedback linearization and the sliding mode controller such that
u uc us
(15)
The control algorithm can be achieved by the so-called Model Reference Adaptive Controller (MRAC) approach as follows:
uc
(( k (t ) ˆ ( x, t )) (t )) (t ) ˆ ( x, t )
(16)
k
with the following update laws for the adaptive controller,
ˆ
e
1
ue , ˆ c
2
(17)
where 1 and 2 correspond to the Lyapunov function gain (Lee et al., 2013; Diao and Passino, 2004). Furthermore, (t ) in Eq. (16) can be derived from the error dynamics shown below. In this formulation, ym means a reference model, and k is a positive parameter, whose value is decided from the error dynamics.
(t ) ym (t ) ke(t ) e(t ) y (t ) ym (t )
(18)
The control input from the sliding mode controller is described below. The sliding mode control is adopted to eliminate system estimation errors and enhance the control performance (Lee et al., 2013) under uncertainties: us
W
0
e sat( )
where sat( x) function is decided based on the norm of the parameter x such that
(19)
x sat( x) sgn( x)
if if
x 1 x 1
(20)
This control law makes chamber pressure error as uniformly asymptotically stable (refer to the Appendix B for stability analysis).
5. Numerical Simulation Numerical simulation was performed to confirm the proposed idea for the new nozzle orientation design. Position and attitude of the thrust system are integrated into the following dynamic equations: F Ma
T Iω ω Iω
(21)
where F, a, M denote the external force, acceleration and the system mass, whereas T, ω, I represent the external torque, angular velocity, and the moment of inertia of the system, respectively. The simulation conditions are provided in Table 4. The control gains for the PD controllers are applied as follows: k y , kz 400 , k yd , k zd 70 , k 70 , k , k 400 , k 400 , k d 50 , and k d , k d 350 , respectively, for fast dynamic responses.
Table 4 Simulation parameters Property Propellant density p , kg/m3
Value 1,558
Property Atmospheric pressure Pa , Pa
Burning rate exponent n Burning rate coefficient a, in/sec
0.5672
Vehicle mass, kg
0.0025
Moment of inertia I xx , kg m
Burning area
0.0292
Moment of inertia I yy , kg m2
10.4
Moment of inertia I zz , kg m Pintle actuator time constant , sec
10.4
Heat capacity ratio
Value 101,325 80
2
2
1.18
1.6
Gas constant Gas temperature Tc , K
407.35 1,453
Adaptive controller variables
k 30, 1 5 103 , 2 5 1011 , W 1.46 107 , 0 12 1011
System diameter d , m System intersection diameter r, m
0.01 0.5 0.2
The pintle actuator dynamics is assumed to be a first-order lag system as below, where sm 0.05 and 0.01 . Ym ( s)
1
sm s 1
R( s )
(22)
Atd ( s)
1 A (s) s 1 t
(23)
To evaluate the system performance in terms of a numerical measure, a score measure is established as a function of the velocity of the attitude maneuver and the system weight to account for the savings in weight due to the reduced number of nozzles. Attcmd represents the attitude maneuver command, and C1 is a constant for nondimensionalization parameter being equal to 10deg in this study. The weight(%) is the percent ratio between the thrust propulsion system and the thrust propulsion system with six ACS nozzles. This value should be less than unity with four ACS nozzles. Furthermore, t is the time to reach a specific level in Attcmd , which corresponds to 80% of the attitude command in this study. There is no perfect score, only higher score means better performance. Theoretically the score can be infinite if the convergence time is zero. score
Attcmd (deg) t / C1 weight(%)
(24)
5.1 Roll Maneuver Maximization First, the numerical simulation is performed to confirm the suggested tilt angle orientation for the maximum roll maneuver. The attitude maneuver simulation is conducted with the same 5deg roll, pitch and yaw commands. Table 5 ACS simulation scenario – roll maneuver maximization Time (sec) 0.0
Attitude command (deg) ( , , ) (0,0,0)
1.0
( , , ) (5,0,0)
1.5
( , , ) (5,5,0)
2.0
( , , ) (5,5,5)
(1) Six ACS nozzles – for comparison The first simulation case is to compare the performance of the attitude maneuver. Figure 8 shows that FDsum is decreased to maintain the pressure under the attitude maneuver. Figure 7 indicates that the pressure control results are stable. Due to the limitation of the ACS nozzle thrust and the moment arm length, the actual F , F , F do not reach the attitude thrust command in Figure 10. This tendency is also observed in Figure 11.
300
Sum (At
thrust
)
0.5 0
(N)
Pcntr
200 100
Dsum
Input(cm2)
U 1
F
1.5
0
0
0.5
1
1.5
2
2.5
3
Divert input(cm2)
Fy(N)
D
Sum (At result) D
0.5 0
0
0.5
1
1.5
2
2.5
2
2.5
3
495
CMD result
0
0.5
1
1.5 Time(sec)
2
2.5
0.5
1
Fz(N)
3
2.5
3
1.5 Time(sec)
2
2.5
3
Fig. 8 Fy , Fz thrust history (6 ACSs) 0 F -20
F
F (N)
Err(Deg)
2
CMD result
0
0
0
0.5
1
1.5
2
2.5
ACS ACS
CMD result
-40
3 -60
0 CMD Result
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5 Time(sec)
2
2.5
3
60
-5 0.5
1
1.5
2
2.5
3
40
0
F (N)
Phi(Deg)
1.5
-200
5
0
20
5 0 0 0
0.5
1
1.5
2
2.5
3
0 0
0.5
1
1.5 Time(sec)
2
2.5
3
Fig. 9 Attitude control history (6 ACSs)
40
Nozzle2(N)
F CMD A
F result A
20
0
1
2
Nozzle4(N)
20
0
1
2
20
1
2
40
3
F CMD A
F result A
20 0
3
0
0
1
0
1
2
3
2
3
Time(sec) Nozzle6(N)
40 20 0
0
1
2
3
40 20 0
Time(sec)
Fig. 11 ACS thrust history (6 ACSs)
20 0
Fig. 10 F , F , F thrust history (6 ACSs)
40
0
3
40
0
40
5
0
60 F (N)
Theta(Deg)
1
0
Fig. 7 Chamber pressure history (6 ACSs)
Psi(Deg)
0.5
200
P P CMD
500
Nozzle1(N)
1.5
0
3
505
Nozzle3(N)
1
-200
510
Nozzle5(N)
0.5
200
Sum (At CMD) 1
515
Press(Psi)
0
1.5
The resultant performance indicators are shown below. Because only A1 , A4 are only working for the yaw maneuver compared to the roll/pitch maneuver performed by ( A2 , A3, A5 , A6 ) pair, the yaw performance is slower than that of the roll/pitch maneuver. The roll maneuver is faster than that of the pitch rotation because of the difference in the moment of inertia of the vehicle. Table 6 Performance indicator results – 6 ACSs Maneuver Roll ( ) Pitch ( ) Yaw ( )
Indicator values (non-dimensional) 12.20 10.20 6.02
(2) Four ACS nozzles – case 1 ( 40deg, 5deg, 45deg ) The same attitude maneuver simulation is performed by 4 ACS nozzles with a 40deg installation angle and
5deg tilt angle. Since there are two less ACS nozzles, total mass will be reduced more than 5% of the wet mass. The tilt angle is set to be much smaller than suggested to verify the degradation in roll performance. Because the tilt angle is small, the performance of the roll maneuver is quite poor compared to the other two indicators as expected. The reason why the pitch/yaw performance is similar to the six ACS nozzle case is because the ACS
0 F
-20
F
0
F (N)
5
0
0.5
1
1.5 Time(sec)
2
2.5
3
-40 -60
0 CMD Result
-5 0.5
1
1.5
2
2.5
ACS ACS
CMD result
0
0.5
1
1.5 Time(sec)
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
60 3
40
5 0 0
0.5
1
1.5
2
2.5
3
60 40
5 0 0
20 0
F (N)
Psi(Deg)
Theta(Deg)
0
F (N)
Phi(Deg)
Err(Deg)
nozzle orientation satisfies the 45deg constraint.
0.5
1
1.5
2
2.5
3
Fig. 12 Attitude control history (4 ACSs – case 1)
20 0
Fig. 13 F , F , F thrust history (4 ACSs – case 1)
F CMD A
F result
Nozzle2(N)
50
Nozzle3(N)
0
50
Nozzle4(N)
Nozzle1(N)
50
50
0
0
A
0
0.5
1
1.5 Time(sec)
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
F CMD A
F result 0
A
0
0.5
1
1.5
2
2.5
3
Fig. 14 ACS thrust history (4 ACSs – case 1) Table 7 Performance indicator results – 4 ACSs (case 1) Maneuver Roll ( ) Pitch ( ) Yaw ( )
Indicator values (non-dimensional) 4.51 8.37 8.37
(3) Four ACS nozzles – case 2 ( 0deg, 45deg, 45deg ) Numerical simulation by 4 ACSs is performed with different installation and tilt angles. The installation angle is zero and 45deg , which corresponds to a maximum for the roll maneuver maximization. The resultant performance indicators are shown below. Because 45deg same as case (2), the performance of the pitch/yaw maneuver is analogous to the previous case. Additionally, because 45deg , a maximum value, the performance of the roll maneuver is improved.
F
-20
F
F (N)
0
0
0.5
1
1.5 Time(sec)
2
2.5
3
-40 -60
0 CMD Result
-5 0.5
1
1.5
2
2.5
ACS ACS
CMD result
0
0.5
1
1.5 Time(sec)
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
60 3
40
5
0.5
1
1.5
2
2.5
3
60
0 0
0.5
1
1.5
2
2.5
20 0
3
Fig. 16 F , F , F thrust history (4 ACSs – case 2)
50 F CMD A
F result
Nozzle2(N)
0
50
Nozzle3(N)
Nozzle1(N)
40
5
Fig. 15 Attitude control history (4 ACSs – case 2)
50
Nozzle4(N)
20 0
0 0
F (N)
Psi(Deg)
Theta(Deg)
0
F (N)
Err(Deg) Phi(Deg)
0
5
0
0
A
0
0.5
1
1.5 Time(sec)
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
50 F CMD A
F result 0
A
0
0.5
1
1.5
2
2.5
3
Fig. 17 ACS thrust history (4 ACSs – case 2)
Table 8 Performance indicator results – 4 ACSs (case 2) Maneuver Roll ( ) Pitch ( ) Yaw ( )
Indicator values (non-dimensional) 15.06 8.37 8.37
(4) Four ACS nozzles – case 3 ( 15deg, 5deg, 20deg ) Numerical simulation by 4 ACSs is performed with different installation and tilt angles, 15deg and
5deg , which do not satisfy the 45deg condition this time. Simulation results and the corresponding performance indicators are presented in the following figures and Table 9. Since 20deg , the pitch moment arm sin( ) is smaller than the proposed one, 45deg . Therefore, the
pitch performance is slower than that of the yaw maneuver. On the other hand, the yaw performance improves compared to case (2) or (3) and even case (1). A design approach could be proposed from this result: if the yaw performance requirement is more stringent than that of the pitch performance, should be less than 45deg to make the yaw moment arm cos( ) greater than that of the pitch moment arm sin( ) . In contrast, if the pitch performance is a higher priority, it should be greater than 45deg to extend the pitch moment arm and shorten
0 F
-20
F
0
F (N)
5
0
0.5
1
1.5 Time(sec)
2
2.5
3
-40 -60
0 CMD Result
-5 0.5
1
1.5
2
2.5
ACS
CMD result
0
0.5
1
1.5 Time(sec)
2
2.5
3
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
40
5
0.5
1
1.5
2
2.5
3
60
Nozzle1(N)
40
5 0 0
0.5
1
1.5
2
2.5
20 0
3
Fig. 18 Attitude control history (4 ACSs – case 3)
Fig. 19 F , F , F thrust history (4 ACSs – case 4)
50 F CMD A
F result 0
A
0
0.5
1
1.5 Time(sec)
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
50
Nozzle3(N)
0
50
Nozzle4(N)
Nozzle2(N)
20 0
0 0
F (N)
Psi(Deg)
Theta(Deg)
0
ACS
60 F (N)
Phi(Deg)
Err(Deg)
the yaw moment arm.
50
0
F CMD A
F result 0
A
0
0.5
1
1.5
2
2.5
3
Fig. 20 ACS thrust history (4 ACSs – case 4)
Table 9 Performance indicator results – 4 ACSs (case 3) Maneuver Pitch ( ) Yaw ( )
Indicator value (non-dimensional) 4.51 10.34
6. Feasibility Analysis of Three ACS Thrusters In this section, the feasibility of attitude control with only three ACS nozzles is analyzed. The most general nozzle configuration is shown in Figure 21. All three nozzles have a different installation angle i and tilt angle i .
Fig. 22 Rear view of VTSPS, with 3 ACSs (symmetric)
Fig. 21 Rear view of VTSPS, with 3 ACSs (nonsymmetric)
If the desired thrust vector is F f of the ACS nozzles FA1
FA2
Fq
T
Fy , the relationship between the desired thrust vector and the thrust
T
FA3 can be derived as
FA1 F sin 1 sin 2 sin 3 FA1 F sin(1 1 ) sin( 2 2 ) sin( 3 3 ) FA2 C FA2 F cos(1 1 ) cos( 2 2 ) cos( 3 3 ) F A3 FA3
(25)
Therefore, the existence of C1 determines the feasibility of the three ACS nozzles. The determinant of C is shown below.
det(C) sin 1 sin( 2 3 2 3 ) +sin 2 sin(1 3 1 3 ) sin 3 sin(1 2 1 2 )
(26)
The determinant is generally not equal to zero, which indicates that three ACS nozzles are feasible. However, because the nozzle orientation configuration is not symmetric, the performance of each attitude maneuver will be different due to the sign of the desired thrust sets. If the nozzle configuration is symmetric, the general case can be described in Figure 22. In this case, the relationship between F
F
F and FA1 T
FA2
T
FA3 can be simplified as
FA1 F 0 sin sin FA1 F 1 sin( ) sin( ) FA2 C FA2 F 0 cos( ) cos( ) F A3 FA3
(27)
with det(C) sin cos( ) sin cos( ) 0
zero determinant. Consequently, Eq. (27) tells us that the symmetric configuration of three ACS nozzles is not feasible for the actual system.
7. Conclusion A variable thrust solid propulsion system (VTSPS) with multiple nozzles was introduced in this paper. Basic control logics for the desired thrust calculation and pressure control were covered. An adaptive control approach was used to control the chamber pressure under uncertainties such as a hot gas environment and free volume estimation error. The number of ACS nozzles was reduced from six to four, which enabled a reduction in mass for the whole system. This mass decrement enables more efficient agile maneuver. Nozzle installation angle and tilt angle were introduced to analyze the effects of the nozzle orientation to prevent performance degradation due to the reduced number of nozzles. The optimized nozzle orientation was investigated based on the simple moment arm equation, and the equations were verified through numerical simulations. The best configuration condition is
45deg , which equalizes the performance of the pitch and yaw maneuver. Under this condition, individually, and have no effect on the pitch/yaw performance. However, the angle is a key parameter if the roll maneuver is important.
Although the desired throat area for each nozzle increases due to the reduced number of nozzles, the simulation study showed that there is no significant performance degradation. All of the results matched fairly well with our expectations. Furthermore, the feasibility of three ACS nozzles was analytically analyzed, and the results showed that the minimum number of ACS nozzles to guarantee desired performances of attitude maneuvers should be at least four.
Appendix A: Stability of PD Controller To prove the stability of PD controller for attitude control (in section 3.1), attitude dynamics with angular velocity
p q r . T
I I u
(A1)
With diagonal inertia matrix and small Euler angle assumption, equation (A1) can be approximated as a function of T
the Euler angle derivative (Sidi, 1997). I u
(A2)
If the control input is set as PD controller in section 3.1, with positive control gains K p and K d u K p (t ) Kd (t )
(A3)
Lyapunov function can be defined as equation (A4). V
1 T 1 K p T I 0 2 2
(A4)
Then Lyapunov derivative can be derived as below, which has negative value or zero. It will be zero when the Euler angle derivatives are conserved to zero.
V T K p T I T [ K p I] T [ K p u ] T [ K p K p K d ] K d 0 T
(A5)
Appendix B: Stability of Adaptive Controller To prove the stability of the adaptive controller for chamber pressure maintenance (in section 4.2), Lyapunov function should be defined as follows (Lee et al., 2013), V
1 2 1 1 e 1 2 2 2 2 2 2
(B1)
where error states ˆ (t ) and ˆ (t ) is defined as (B2), with the equation (13) and (18).
(t ) ˆ (t ) (t ) (t ) ˆ (t ) (t )
(B2)
Then the derivative of the Lyapunov function can be derived as: V ee 1 2
(B3)
e y ym [( k ) ( k )u ( ke)] ke [ ( k ) ( k )u] ke [ ( ) ( ˆ )u ( ˆ )u ]
(B4)
The derivative of the error state e ,
k
k
could be compressed as equation (B5) with the feedback linearization control input defined in equation (16). e ke u
(B5)
Then Lyapunov derivative in equation (B3) can be written as follows:
V ke2 e ue 1 (ˆ ) 2 (ˆ )
(B6)
This will be equivalent with equation (B7), with the update law in equation (17). V ke2 ( 1 2 )
(B7)
If magnitude of , , and , are assumed bounded, the time-varying effect can be bounded also as equation (B8), ( 1 2 ) W
(B8)
and Lyapunov derivative (B7) will be negative in the error bound e W k . V ke2 W
(B9)
To compensate the estimation error completely in this adaptive controller, sliding mode controller is also adopted in equation (15). With the additional control input u s , the derivative of the error state in equation (B4) and the Lyapunov function derivative in equation (B6) can be modified as below. e ke [ ( k ) ( k )uc ( k )us ] ke [ ( k ) ( k ˆ )uc ( ˆ )uc ( k )us ] ke uc ( k )us
(B10)
V ke2 e uc e (k )us e 1 (ˆ ) 2 (ˆ )
If the following supplement is assumed,
e ,
e
(B11)
the time-varying effect in equation (B8) can be refreshed as follow.
( 1 2 ) W e
(B13)
This new bound makes the Lyapunov derivative also bounded, and finally has negative value, V ke2 W e ( k )
W
0
sgn(e) ke2
(B14)
when the lower bound of the uncertainty is set as 0 .
k (t ) ( x, t ) 0
(B15)
Acknowledgments This work was supported by the Defense Acquisition Program Administration and Agency for Defense Development under the contract UD110093CD. The authors truly appreciate their financial support.
References Beardsley, J. D., Shipley, J. D., 1976. System for Controlling the Nozzle Throat Area of a Rocket Motor. U.S. Patent 3,948,042. Bergmans, J. L., Di Salvo, R., 2003. Solid Rocket Motor Control: Theoretical Motivation and Experimental Demonstration. 39th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit. Choi, H., Lim, J., 2003. Feedback Linearisation of Time-Varying Nonlinear Systems via Time-Varying Diffeomorphism. IEEE Procedings: Control Theory and Applications, 150 (3), 279-284. Diao, Y., Passino, K. M., 2004. Stable Adaptive Control of Feedback Linearizable Time-Varying Non-Linear Systems with Application to Fault-Tolerant Engine Control. International Journal of Control, 77 (17), 1463-1480. Joner, S., Quinquis, I., 2006. Control of an Exoatmospheric Kill Vehicle with a Solid Propulsion Attitude Control System. AIAA Guidance, Navigation, and Control Conference and Exhibit. Lee W., Eun Y., Bang H., et al., 2013. Efficient Thrust Distribution with Adaptive Pressure Control for Multi-Nozzle Solid Propulsion System. AIAA Journal of Propulsion and Power, 29 (6), 1410-1419. Marino, R., Tomei, P., 1997. Adaptive Output Feedback Tracking for Nonlinear Systems with Time-Varying Parameters. Proceedings of the 36th IEEE Conference on Decision and Control, 3, 2483-2488. Morris, J. W., Calson, R. W., Peterson, K. L., et al., 1995. Multiple Pintle Nozzle Propulsion Control System. U.S. Patent 5,456,425. Sidi, M., 1997. Spacecraft Dynamics and Control: A Practical Engineering Approach. Cambridge University Press. Sutton, G.P., Biblarz, O., 2001. Rocket Propulsion Elements, Wiley, New York, 52-74. Young, G., Bruck, H. A., Gowrisankaran, S., 2014. Modeling of Rocket Motor Ballistics for Functionally Graded Propellants. Proceedings of the 39th JANNAF Combustion Meeting.
S0273-1177(17)30015-7 http://dx.doi.org/10.1016/j.asr.2017.01.002 JASR 13041
To appear in:
Advances in Space Research
Received Date: Revised Date: Accepted Date:
3 May 2016 19 December 2016 3 January 2017
Please cite this article as: Lim, Y., Lee, W., Bang, H., Lee, H., Thrust Distribution for Attitude Control in a Variable Thrust Propulsion System with four ACS Nozzles, Advances in Space Research (2017), doi: http://dx.doi.org/ 10.1016/j.asr.2017.01.002
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Thrust Distribution for Attitude Control in a Variable Thrust Propulsion System with four ACS Nozzles Yeerang Lim1 Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Republic of Korea Wonsuk Lee2 Agency for Defense Development, Daejeon, 305-600, Republic of Korea Hyochoong Bang3 Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Republic of Korea and Hosung Lee4 Agency for Defense Development, Daejeon, 305-600, Republic of Korea
A thrust distribution approach is proposed in this paper for a variable thrust solid propulsion system with an attitude control system (ACS) that uses a reduced number of nozzles for a three-axis attitude maneuver. Although a conventional variable thrust solid propulsion system needs six ACS nozzles, this paper proposes a thrust system with four ACS nozzles to reduce the complexity and mass of the system. The performance of the new system was analyzed with numerical simulations, and the results show that the performance of the system with four ACS nozzles was similar to the original system while the mass of the whole system was simultaneously reduced. Moreover, a feasibility analysis was performed to determine whether a thrust system with three ACS nozzles is possible.
Nomenclature Cf
=
thrust coefficient
At
=
throat area, m 2
FAi , FDi
=
Thrust of the ith attitude/divert control nozzle, N
Fsys
=
total system thrust generated under chamber pressure control, N
FDsum
=
Available total thrust after substituting the attitude control system thrusts, N
1
Ph.D. Candidate, Aerospace Engineering Department, [email protected]. Ph.D., Senior Researcher. 3 Professor, Aerospace Engineering Department, Senior Member AIAA. 4 Senior Researcher. 2
Fy , Fz
=
y/z axis forces, N
F , F , F
=
roll, pitch, and yaw axis forces, N
=
installation angle of the ACS nozzles, deg
=
Tilt angle of the ACS nozzles, deg
=
roll, pitch, and yaw angles, deg
, ,
1. Introduction
V
ARIABLE thrust propulsion systems with solid propellant have been widely adopted for various missions. In
addition to the advantages of solid propulsion systems which include light weight and simple design, a fast response and the availability of precise thrust control are the additional advantages of thrust variability. These additional advantages can be achieved by controlling the nozzle throat area using a pintle actuator (Beardsley and Shipley, 1976; Morris et al., 1995). Based on these characteristics, a variable thrust solid propulsion system (VTSPS) could be a viable solution for systems that require agile translational and rotational maneuvers. To control the attitude of a VTSPS, the system should have multiple nozzles to generate torque with respect to all body axes. Moreover, multiple nozzle systems are advantageous for a simple method to control the position of the thrust system. Because all the nozzles should share a combustion chamber, thrust distribution should be done under controlled chamber pressure conditions. In other words, all the pintles with respect to each nozzle should be controlled to change the throat area, while the throat area of the whole system has to satisfy the constraint of maintaining the pressure. Therefore, the pressure control and thrust distribution problems affect one other. Lee et al. (2013) introduced an integrated algorithm generating variable thrust based on pressure stabilization with maximum thrust efficiency. The research was based on a conventional VTSPS shown in Fig. 1, which has four divert control system (DCS) nozzles and six attitude control system (ACS) nozzles (Joner and Quinquis, 2006). Attitude maneuvers with respect to the yaw axis can be decoupled using a thrust system with six ACS nozzles, while roll/pitch maneuvers are coupled. By extension of this thrust system with six ACS nozzles, this main objective of this study is to propose a VTSPS with four ACS nozzles to reduce system weight. Morris et al. (1995) has designed similar system with reduced ACS nozzles, however, any details about maneuver efficiency or thrust distributions are derived. A thrust distribution algorithm and with pressure control strategy is established for the thrust system with only four ACS nozzles.
The control of the propulsion system with a reduced number of ACS nozzles consists of two parts. First, chamber pressure control should be considered because the resulting thrust is a function of the chamber pressure. To resolve the problem associated with the characteristics of the combustion chamber being nonlinear and time-varying, Bergmans and Di Salvo (2003) studied an adaptive matching technique. Lee et al. (2013) introduced an adaptive controller based on a feedback linearization (Choi and Lim, 2003; Marino and Tomei, 1997; Diao and Passino, 2004) technique, and analyzed the stability of the proposed algorithm. Because the number of nozzles are not related to the chamber pressure dynamics, this control strategy could be used in a similar way. After the chamber pressure is stabilized, the desired thrust calculations and thrust assignment problems are derived for the multiple nozzles sharing the combustion chamber. The thrust sum of the nozzles is limited because of the chamber pressure control. A thrust distribution strategy that minimizes the pintle movement is introduced to maintain the pressure. ACS thrust distribution is separated from DCS thrust distribution and the pressure maintenance loop. ACS thrust assignment is performed before the DCS thrust assignment because it is more beneficial from the perspective of attitude control to separate the ACS thrusts from the pressure control, which adds complexity to the system. Since all the attitude maneuvers are coupled to each other because of the reduced number of ACS nozzles, pseudothrust algorithms are introduced to solve the problem. Using the four ACS nozzle arrangement suggested in this paper, attitude control performance will be maintained as well as the system mass can be decreased. This means the agile performance of the system can be increased without any attitude control degradation. Favorable and adverse nozzle placement are derived respectively, and the minimum number of the ACS nozzle is also analyzed. Thrusts to produce yaw moment will cause roll/pitch moments also with reduced configuration, with is originally separated with six ACS nozzles. A proper thrust distribution method with the coupled nozzles is described and verified with simulation studies.
2. Variable Thrust Solid Propulsion System A conventional variable thrust solid propulsion system includes four DCS nozzles D1 , D2 , D3 , D4 and six ACS nozzles A1 , A2 , A3 , A4 , A5 , A6 . Because the x -axis represents the agile direction of the whole propulsion system, the DCS nozzles are dedicated to the translational motion for y, z -axes, respectively. Because the nozzles share a
single chamber, all the desired thrusts for the nozzles should satisfy the throat area constraint. The sum of the throat area of all the nozzles should be equal to the desired throat area to control the pressure of the thrust chamber.
Fig. 1 Variable thrust solid propulsion system and DCS/ACS nozzle positions
Six ACS nozzles are in charge of attitude maneuvers: roll (about the body fixed x axis), pitch (about the body fixed y axis), and yaw (about the body fixed z axis). A yaw maneuver is independent of the other two attitude maneuvers and can be achieved with the nozzle A1 or A4 . Because the other four ACS nozzles are coupled, a roll maneuver can be achieved with the ( A2 , A5 ) or ( A3 , A6 ) pair based on the sign of the desired thrust, while ( A2 , A3 ) or ( A5 , A6 ) pair could be engaged for the pitch maneuver. The final thrust inputs, which are assigned to
each ACS nozzles, are the sum of these desired thrusts regarding each attitude maneuver. This will be described in detail in section III.
FAi Fi Fi Fi ,
i 1, 2,3, 4,5,6
(1)
Because DCS nozzles are in charge of controlling the chamber pressure, the divert thrusts are assessed after the ACS thrust vectors. The sum of the divert thrusts is the difference of the available thrust, which is obtained from the throat area, to maintain the pressure and the sum of the desired ACS thrust. Divert nozzles ( D1 , D2 ) are dedicated to y-directional thrust, and ( D3 , D4 ) are for z-directional maneuvers. 6
4
j 1
i 1
Fsys FAj FDsum FDi Fy FD1 FD2
(2)
Fz FD3 FD4 with
0 FDi FDsum
The whole control loop for the variable thrust propulsion system is summarized in Figure 2. From the pressure and state (position and attitude) errors, the desired chamber pressure and the control thrusts are constructed first. Thrust distribution is also performed in this step. Additionally, the desired throat area, which is the actual command for the actuator, is estimated to distribute thrust command properly for the desired pressure control. Finally, the pintle actuator should move based on the throat area command, and the actual pressure and state errors are fed back to the error state computation module.
Fig. 2 Nozzle input mechanism
3. Desired Attitude Thrust Calculation and Distribution The sum of available thrust of the whole system can be calculated from the chamber pressure such that
F C f At Pc
(3)
where C f represents a thrust coefficient defined as 1 1 Pe 2 2 2 1 Pe Pa Ae Cf 1 ( ) 1 1 Pc Pc At
(4)
Because the thrust of each nozzle is not a feedback variable, the thrust command should be calculated based on the position and attitude errors. A PD (Proportional plus Derivative) controller was employed in this research. In addition, a thrust distribution algorithm should be constructed because there are more than two nozzles. The whole thrust distribution algorithm is presented in Figure 3. The resultant FDi s and FA j s should satisfy the throat area constraint to maintain the pressure at a desired level.
Fig. 3 Thrust algorithm block diagram
Traditionally, a variable thrust propulsion system consists four DCS nozzles and six ACS nozzles in many previous studies. In this study, the number of ACS nozzles is reduced from six to four to minimize the system mass
shown in Figure 4. Because the roll maneuver should be available, tilt angle is combined with a new design parameter: the installation angle . The new proposed thrust distribution algorithm is based on the parameters and .
Fig. 4 Rear view of the variable solid propulsion system with 4 ACSs
3.1 Attitude Control System (ACS) Thrust Command The desired ACS thrusts are simply derived with the PD controllers as follows: F
1 (k (cmd ) k d ) d
F
1 (k ( cmd ) k d ) d
F
1 (k ( cmd ) k d ) d
(5)
where k , k , k are the proportional control gains, and k d , k d , k d are the derivative control gains (refer to the Appendix A for stability analysis). This desired thrust vector should be allocated to all the ACS nozzles properly. In the case of six ACS nozzles shown in Figure 1, because nozzle A1 and A4 are dedicated to the yaw maneuver only, the desired yaw thrust F is assigned to A1 or A4 based on the sign of F , while F and F are assigned to A2 , A3 , A5 , and A6 ..
Table 1 Pseudothrust algorithm for attitude thrust distribution – 6 ACSs (Lee et al., 2013) Allocation Roll if F 0,
Pitch
Yaw if F 0,
if F 0,
F2 F / 2
F1 F
F2 F / 2
F5 F / 2
F3 F / 2 if F 0,
if F 0, F3 F / 2
F5 F / 2
F6 F / 2
F6 F / 2
if F 0, F4 F
Distribution
FAi Fi Fi Fi , i 1, 2,3, 4,5,6
Thrust assignment logic should be different for the four ACS nozzle case shown in Figure 4. A proper thrust distribution strategy for the four ACS nozzles is presented in Table 2. Because there are four nozzles only, unlike Figure 1, now pitch, yaw, and roll maneuvers are all coupled. Table 2 Pseudothrust algorithm for attitude thrust distribution – 4 ACSs Allocation if F 0, F2 , F3
if F 0,
if F 0, F
F1 , F2
2sin F 2sin
F3 , F4
F2 , F4
F 2 cos( )
if F 0,
if F 0,
if F 0, F1 , F4
F 2sin( ) F 2sin( )
F1 , F3
F 2 cos( )
Distribution
FAi Fi Fi Fi , i 1, 2,3, 4
Based on the thrust distribution logic in Table 2, the resultant torque inputs for each attitude maneuver can be derived as follows: r ( FA2 FA3 ) sin M r ( FA1 FA4 ) sin d ( FA1 FA2 ) sin( ) M d ( FA3 FA4 ) sin( ) d ( FA1 FA3 ) cos( ) M d ( FA2 FA4 ) cos( )
for M 0 for M 0 for M 0 for M 0 for M 0 for M 0
(6)
where d denotes the distance of the ACS nozzles from the center of mass, and r represents the radius of the chamber intersection, which is the moment arm of the ACS nozzles. Note that the performance of each attitude maneuver is determined by the installation angle and the tilt angle
. Maneuver performance for all the axes should be equalized far as possible for arbitrary maneuver. Under the assumption that the thrust limit of each ACS nozzle is equal, should be equal to 45deg as follows to drive the pitch and yaw performance to be equal.
d (2 Fmax )sin( ) d (2 Fmax ) cos( )
(7)
( ) 45deg
Once is restricted to 45deg , tilt angle is a key parameter to equalize the roll performance with the other two maneuvers. The tilt angle equ that maximize the roll maneuver performance can be derived by setting the roll moment M same as the pitch moment M or the yaw moment M . Then equ can be derived as a function of the moment arm d and r . Fmax r sin equ Fmax d sin( equ )
equ
Fmax d
d d sin 1 sin( equ ) sin 1 r r 2
2
(8)
where the thrust for each axis is assumed as the maximum thrust Fmax . Based on Eq. (8), should be greater than 45deg , because d is usually longer than r for the thrust propulsion system exterior. However, the boundary of the tilt angle is 0 (45deg ) because should be positive, and the sum of the installation angle and the tilt angle is constrained to 45deg . Therefore, there is no which actually allows M to be equal to M . Under such a restriction, should be decided as maximum among the boundary, which is 45deg , to maximize the roll maneuver as much as possible. The maximum minimizes the difference between M and M , while for the maximum roll performance is equal to 0. This will be analyzed later with numerical simulations.
3.2 Divert Control System (DCS) Thrust Command Divert thrust can be constructed after the ACS thrust distribution is completed. The available thrust is FDsum as described in Eq. (1), which is the remaining thrust after ACS thrust distribution. There are two DCS thrust distribution approaches. One is to distribute the y-axis thrust and z-axis thrust separately with a performance limit for each maneuver which is FDsum / 2 . This approach is not efficient because the available thrust cannot be focused on a single axis maneuver. The second approach is to maximize the DCS thrusts within the FDsum limit. This limit can be expressed as Fy Fz FDsum , and the solution to this problem corresponds to the intersection of two lines in Figure 5, where is the thrust direction in the y-z plane.
Fig. 5 Graphical representation for the maximum force problem
The solution Fy and Fz are summarized in Table 3 as a function of the thrust direction . Table 3 Maximum force available based on saturation Quadrant Force
Fy
Fz
1 FDsum
2 FDsum
3 FDsum
4 FDsum
1 tan FDsum tan 1 tan
1 tan FDsum tan 1 tan
1 tan FDsum tan 1 tan
1 tan FDsum tan 1 tan
Once the desired translational thrusts are calculated, then these thrust should be allocated to each DCS nozzle efficiently. If the minimum pintle movement is treated as the most efficient way to distribute the divert thrust, the problem can be expressed as an optimization problem to minimize the cost function as follows: 4
f min ( FDi Feq )
2
i 1
FD1 FD2 Fy where
FD3 FD4 Fz
(9)
FD1 FD2 FD3 FD4 FDsum Feq FDsum 4
Because of three equality constraints, the problem can be written as a function of one thrust variable. If FD4 is chosen as a variable, the solution to this optimization problem can be readily derived from f FD4 0 . 1 ( FDsum Fy Fz ) FD4 2 1 FD2 ( FDsum Fy Fz ) FD4 2 FD3 Fz FD4 FD1
FD4
(10)
1 ( FDsum 2 Fz ) 4
The optimally distributed divert thrust commands are applied to translational motion control.
4. Pressure Control Algorithm 4.1 Chamber Pressure Dynamics Chamber pressure dynamics should be derived before describing the pressure control algorithm. Figure 6 is the schematic diagram of the combustion chamber with a pintle actuator. The dynamics can be derived from the law of conservation of mass as Eq. (11), which means the mass flow rate inside the chamber should be equal to the sum of the mass produced by the combustion and the exhausted mass from the nozzle. m p , me , c and Vc represent the produced mass by combustion, exhausted mass, gas density and the free volume, respectively, in the chamber (Young et al., 2003; Sutton and Biblarz, 2001).
mp
d cVc me dt
(11)
Fig. 6 General variable thrust solid propulsion system
From the mass flow rate in Eq. (11) and the ideal gas assumption, the chamber pressure dynamics can be derived as Eq. (12), where R, Tc and k denote the gas constant, gas temperature and heat capacity ratio, respectively. Ab and p represent the propellant properties, cross-sectional area, and density. r is the burning ratio of the chamber, n which satisfies r aPc . At represents the throat area of the nozzle, which is a main control parameter in this study.
Pc
RTc Ab r p Vc
k 1
RT A P c t c Vc
k RTc
2 k 1 Pc Vc Vc k 1
(12)
4.2 Chamber Pressure Control Algorithm Because the chamber contains hot gas which may not be ideal, there must be several uncertainties in the free volume estimation and the invariant system estimation. Therefore, a robust pressure control strategy under these uncertainties are required. An adaptive control approach was selected in this study to deal with this real-world problem. The system can be expressed as follows:
x ( k (t ) ( x, t )) ( k (t ) ( x, t ))u yx
(13)
where k (t ), k (t ) are the known dynamics, and ( x, t ), ( x, t ) represent the nonlinear time-varying dynamics of the unknown uncertainties. The above formula can be applied to the chamber pressure model shown below:
k
RTc Ab p
Vc
Pc Vc Vc 1
2 1 RTc 1
RT P k c c Vc
(14)
where the control input of the pressure model is At . The control input can be derived as a combination of the control inputs from feedback linearization and the sliding mode controller such that
u uc us
(15)
The control algorithm can be achieved by the so-called Model Reference Adaptive Controller (MRAC) approach as follows:
uc
(( k (t ) ˆ ( x, t )) (t )) (t ) ˆ ( x, t )
(16)
k
with the following update laws for the adaptive controller,
ˆ
e
1
ue , ˆ c
2
(17)
where 1 and 2 correspond to the Lyapunov function gain (Lee et al., 2013; Diao and Passino, 2004). Furthermore, (t ) in Eq. (16) can be derived from the error dynamics shown below. In this formulation, ym means a reference model, and k is a positive parameter, whose value is decided from the error dynamics.
(t ) ym (t ) ke(t ) e(t ) y (t ) ym (t )
(18)
The control input from the sliding mode controller is described below. The sliding mode control is adopted to eliminate system estimation errors and enhance the control performance (Lee et al., 2013) under uncertainties: us
W
0
e sat( )
where sat( x) function is decided based on the norm of the parameter x such that
(19)
x sat( x) sgn( x)
if if
x 1 x 1
(20)
This control law makes chamber pressure error as uniformly asymptotically stable (refer to the Appendix B for stability analysis).
5. Numerical Simulation Numerical simulation was performed to confirm the proposed idea for the new nozzle orientation design. Position and attitude of the thrust system are integrated into the following dynamic equations: F Ma
T Iω ω Iω
(21)
where F, a, M denote the external force, acceleration and the system mass, whereas T, ω, I represent the external torque, angular velocity, and the moment of inertia of the system, respectively. The simulation conditions are provided in Table 4. The control gains for the PD controllers are applied as follows: k y , kz 400 , k yd , k zd 70 , k 70 , k , k 400 , k 400 , k d 50 , and k d , k d 350 , respectively, for fast dynamic responses.
Table 4 Simulation parameters Property Propellant density p , kg/m3
Value 1,558
Property Atmospheric pressure Pa , Pa
Burning rate exponent n Burning rate coefficient a, in/sec
0.5672
Vehicle mass, kg
0.0025
Moment of inertia I xx , kg m
Burning area
0.0292
Moment of inertia I yy , kg m2
10.4
Moment of inertia I zz , kg m Pintle actuator time constant , sec
10.4
Heat capacity ratio
Value 101,325 80
2
2
1.18
1.6
Gas constant Gas temperature Tc , K
407.35 1,453
Adaptive controller variables
k 30, 1 5 103 , 2 5 1011 , W 1.46 107 , 0 12 1011
System diameter d , m System intersection diameter r, m
0.01 0.5 0.2
The pintle actuator dynamics is assumed to be a first-order lag system as below, where sm 0.05 and 0.01 . Ym ( s)
1
sm s 1
R( s )
(22)
Atd ( s)
1 A (s) s 1 t
(23)
To evaluate the system performance in terms of a numerical measure, a score measure is established as a function of the velocity of the attitude maneuver and the system weight to account for the savings in weight due to the reduced number of nozzles. Attcmd represents the attitude maneuver command, and C1 is a constant for nondimensionalization parameter being equal to 10deg in this study. The weight(%) is the percent ratio between the thrust propulsion system and the thrust propulsion system with six ACS nozzles. This value should be less than unity with four ACS nozzles. Furthermore, t is the time to reach a specific level in Attcmd , which corresponds to 80% of the attitude command in this study. There is no perfect score, only higher score means better performance. Theoretically the score can be infinite if the convergence time is zero. score
Attcmd (deg) t / C1 weight(%)
(24)
5.1 Roll Maneuver Maximization First, the numerical simulation is performed to confirm the suggested tilt angle orientation for the maximum roll maneuver. The attitude maneuver simulation is conducted with the same 5deg roll, pitch and yaw commands. Table 5 ACS simulation scenario – roll maneuver maximization Time (sec) 0.0
Attitude command (deg) ( , , ) (0,0,0)
1.0
( , , ) (5,0,0)
1.5
( , , ) (5,5,0)
2.0
( , , ) (5,5,5)
(1) Six ACS nozzles – for comparison The first simulation case is to compare the performance of the attitude maneuver. Figure 8 shows that FDsum is decreased to maintain the pressure under the attitude maneuver. Figure 7 indicates that the pressure control results are stable. Due to the limitation of the ACS nozzle thrust and the moment arm length, the actual F , F , F do not reach the attitude thrust command in Figure 10. This tendency is also observed in Figure 11.
300
Sum (At
thrust
)
0.5 0
(N)
Pcntr
200 100
Dsum
Input(cm2)
U 1
F
1.5
0
0
0.5
1
1.5
2
2.5
3
Divert input(cm2)
Fy(N)
D
Sum (At result) D
0.5 0
0
0.5
1
1.5
2
2.5
2
2.5
3
495
CMD result
0
0.5
1
1.5 Time(sec)
2
2.5
0.5
1
Fz(N)
3
2.5
3
1.5 Time(sec)
2
2.5
3
Fig. 8 Fy , Fz thrust history (6 ACSs) 0 F -20
F
F (N)
Err(Deg)
2
CMD result
0
0
0
0.5
1
1.5
2
2.5
ACS ACS
CMD result
-40
3 -60
0 CMD Result
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5 Time(sec)
2
2.5
3
60
-5 0.5
1
1.5
2
2.5
3
40
0
F (N)
Phi(Deg)
1.5
-200
5
0
20
5 0 0 0
0.5
1
1.5
2
2.5
3
0 0
0.5
1
1.5 Time(sec)
2
2.5
3
Fig. 9 Attitude control history (6 ACSs)
40
Nozzle2(N)
F CMD A
F result A
20
0
1
2
Nozzle4(N)
20
0
1
2
20
1
2
40
3
F CMD A
F result A
20 0
3
0
0
1
0
1
2
3
2
3
Time(sec) Nozzle6(N)
40 20 0
0
1
2
3
40 20 0
Time(sec)
Fig. 11 ACS thrust history (6 ACSs)
20 0
Fig. 10 F , F , F thrust history (6 ACSs)
40
0
3
40
0
40
5
0
60 F (N)
Theta(Deg)
1
0
Fig. 7 Chamber pressure history (6 ACSs)
Psi(Deg)
0.5
200
P P CMD
500
Nozzle1(N)
1.5
0
3
505
Nozzle3(N)
1
-200
510
Nozzle5(N)
0.5
200
Sum (At CMD) 1
515
Press(Psi)
0
1.5
The resultant performance indicators are shown below. Because only A1 , A4 are only working for the yaw maneuver compared to the roll/pitch maneuver performed by ( A2 , A3, A5 , A6 ) pair, the yaw performance is slower than that of the roll/pitch maneuver. The roll maneuver is faster than that of the pitch rotation because of the difference in the moment of inertia of the vehicle. Table 6 Performance indicator results – 6 ACSs Maneuver Roll ( ) Pitch ( ) Yaw ( )
Indicator values (non-dimensional) 12.20 10.20 6.02
(2) Four ACS nozzles – case 1 ( 40deg, 5deg, 45deg ) The same attitude maneuver simulation is performed by 4 ACS nozzles with a 40deg installation angle and
5deg tilt angle. Since there are two less ACS nozzles, total mass will be reduced more than 5% of the wet mass. The tilt angle is set to be much smaller than suggested to verify the degradation in roll performance. Because the tilt angle is small, the performance of the roll maneuver is quite poor compared to the other two indicators as expected. The reason why the pitch/yaw performance is similar to the six ACS nozzle case is because the ACS
0 F
-20
F
0
F (N)
5
0
0.5
1
1.5 Time(sec)
2
2.5
3
-40 -60
0 CMD Result
-5 0.5
1
1.5
2
2.5
ACS ACS
CMD result
0
0.5
1
1.5 Time(sec)
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
60 3
40
5 0 0
0.5
1
1.5
2
2.5
3
60 40
5 0 0
20 0
F (N)
Psi(Deg)
Theta(Deg)
0
F (N)
Phi(Deg)
Err(Deg)
nozzle orientation satisfies the 45deg constraint.
0.5
1
1.5
2
2.5
3
Fig. 12 Attitude control history (4 ACSs – case 1)
20 0
Fig. 13 F , F , F thrust history (4 ACSs – case 1)
F CMD A
F result
Nozzle2(N)
50
Nozzle3(N)
0
50
Nozzle4(N)
Nozzle1(N)
50
50
0
0
A
0
0.5
1
1.5 Time(sec)
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
F CMD A
F result 0
A
0
0.5
1
1.5
2
2.5
3
Fig. 14 ACS thrust history (4 ACSs – case 1) Table 7 Performance indicator results – 4 ACSs (case 1) Maneuver Roll ( ) Pitch ( ) Yaw ( )
Indicator values (non-dimensional) 4.51 8.37 8.37
(3) Four ACS nozzles – case 2 ( 0deg, 45deg, 45deg ) Numerical simulation by 4 ACSs is performed with different installation and tilt angles. The installation angle is zero and 45deg , which corresponds to a maximum for the roll maneuver maximization. The resultant performance indicators are shown below. Because 45deg same as case (2), the performance of the pitch/yaw maneuver is analogous to the previous case. Additionally, because 45deg , a maximum value, the performance of the roll maneuver is improved.
F
-20
F
F (N)
0
0
0.5
1
1.5 Time(sec)
2
2.5
3
-40 -60
0 CMD Result
-5 0.5
1
1.5
2
2.5
ACS ACS
CMD result
0
0.5
1
1.5 Time(sec)
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
60 3
40
5
0.5
1
1.5
2
2.5
3
60
0 0
0.5
1
1.5
2
2.5
20 0
3
Fig. 16 F , F , F thrust history (4 ACSs – case 2)
50 F CMD A
F result
Nozzle2(N)
0
50
Nozzle3(N)
Nozzle1(N)
40
5
Fig. 15 Attitude control history (4 ACSs – case 2)
50
Nozzle4(N)
20 0
0 0
F (N)
Psi(Deg)
Theta(Deg)
0
F (N)
Err(Deg) Phi(Deg)
0
5
0
0
A
0
0.5
1
1.5 Time(sec)
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
50 F CMD A
F result 0
A
0
0.5
1
1.5
2
2.5
3
Fig. 17 ACS thrust history (4 ACSs – case 2)
Table 8 Performance indicator results – 4 ACSs (case 2) Maneuver Roll ( ) Pitch ( ) Yaw ( )
Indicator values (non-dimensional) 15.06 8.37 8.37
(4) Four ACS nozzles – case 3 ( 15deg, 5deg, 20deg ) Numerical simulation by 4 ACSs is performed with different installation and tilt angles, 15deg and
5deg , which do not satisfy the 45deg condition this time. Simulation results and the corresponding performance indicators are presented in the following figures and Table 9. Since 20deg , the pitch moment arm sin( ) is smaller than the proposed one, 45deg . Therefore, the
pitch performance is slower than that of the yaw maneuver. On the other hand, the yaw performance improves compared to case (2) or (3) and even case (1). A design approach could be proposed from this result: if the yaw performance requirement is more stringent than that of the pitch performance, should be less than 45deg to make the yaw moment arm cos( ) greater than that of the pitch moment arm sin( ) . In contrast, if the pitch performance is a higher priority, it should be greater than 45deg to extend the pitch moment arm and shorten
0 F
-20
F
0
F (N)
5
0
0.5
1
1.5 Time(sec)
2
2.5
3
-40 -60
0 CMD Result
-5 0.5
1
1.5
2
2.5
ACS
CMD result
0
0.5
1
1.5 Time(sec)
2
2.5
3
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
40
5
0.5
1
1.5
2
2.5
3
60
Nozzle1(N)
40
5 0 0
0.5
1
1.5
2
2.5
20 0
3
Fig. 18 Attitude control history (4 ACSs – case 3)
Fig. 19 F , F , F thrust history (4 ACSs – case 4)
50 F CMD A
F result 0
A
0
0.5
1
1.5 Time(sec)
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
50
Nozzle3(N)
0
50
Nozzle4(N)
Nozzle2(N)
20 0
0 0
F (N)
Psi(Deg)
Theta(Deg)
0
ACS
60 F (N)
Phi(Deg)
Err(Deg)
the yaw moment arm.
50
0
F CMD A
F result 0
A
0
0.5
1
1.5
2
2.5
3
Fig. 20 ACS thrust history (4 ACSs – case 4)
Table 9 Performance indicator results – 4 ACSs (case 3) Maneuver Pitch ( ) Yaw ( )
Indicator value (non-dimensional) 4.51 10.34
6. Feasibility Analysis of Three ACS Thrusters In this section, the feasibility of attitude control with only three ACS nozzles is analyzed. The most general nozzle configuration is shown in Figure 21. All three nozzles have a different installation angle i and tilt angle i .
Fig. 22 Rear view of VTSPS, with 3 ACSs (symmetric)
Fig. 21 Rear view of VTSPS, with 3 ACSs (nonsymmetric)
If the desired thrust vector is F f of the ACS nozzles FA1
FA2
Fq
T
Fy , the relationship between the desired thrust vector and the thrust
T
FA3 can be derived as
FA1 F sin 1 sin 2 sin 3 FA1 F sin(1 1 ) sin( 2 2 ) sin( 3 3 ) FA2 C FA2 F cos(1 1 ) cos( 2 2 ) cos( 3 3 ) F A3 FA3
(25)
Therefore, the existence of C1 determines the feasibility of the three ACS nozzles. The determinant of C is shown below.
det(C) sin 1 sin( 2 3 2 3 ) +sin 2 sin(1 3 1 3 ) sin 3 sin(1 2 1 2 )
(26)
The determinant is generally not equal to zero, which indicates that three ACS nozzles are feasible. However, because the nozzle orientation configuration is not symmetric, the performance of each attitude maneuver will be different due to the sign of the desired thrust sets. If the nozzle configuration is symmetric, the general case can be described in Figure 22. In this case, the relationship between F
F
F and FA1 T
FA2
T
FA3 can be simplified as
FA1 F 0 sin sin FA1 F 1 sin( ) sin( ) FA2 C FA2 F 0 cos( ) cos( ) F A3 FA3
(27)
with det(C) sin cos( ) sin cos( ) 0
zero determinant. Consequently, Eq. (27) tells us that the symmetric configuration of three ACS nozzles is not feasible for the actual system.
7. Conclusion A variable thrust solid propulsion system (VTSPS) with multiple nozzles was introduced in this paper. Basic control logics for the desired thrust calculation and pressure control were covered. An adaptive control approach was used to control the chamber pressure under uncertainties such as a hot gas environment and free volume estimation error. The number of ACS nozzles was reduced from six to four, which enabled a reduction in mass for the whole system. This mass decrement enables more efficient agile maneuver. Nozzle installation angle and tilt angle were introduced to analyze the effects of the nozzle orientation to prevent performance degradation due to the reduced number of nozzles. The optimized nozzle orientation was investigated based on the simple moment arm equation, and the equations were verified through numerical simulations. The best configuration condition is
45deg , which equalizes the performance of the pitch and yaw maneuver. Under this condition, individually, and have no effect on the pitch/yaw performance. However, the angle is a key parameter if the roll maneuver is important.
Although the desired throat area for each nozzle increases due to the reduced number of nozzles, the simulation study showed that there is no significant performance degradation. All of the results matched fairly well with our expectations. Furthermore, the feasibility of three ACS nozzles was analytically analyzed, and the results showed that the minimum number of ACS nozzles to guarantee desired performances of attitude maneuvers should be at least four.
Appendix A: Stability of PD Controller To prove the stability of PD controller for attitude control (in section 3.1), attitude dynamics with angular velocity
p q r . T
I I u
(A1)
With diagonal inertia matrix and small Euler angle assumption, equation (A1) can be approximated as a function of T
the Euler angle derivative (Sidi, 1997). I u
(A2)
If the control input is set as PD controller in section 3.1, with positive control gains K p and K d u K p (t ) Kd (t )
(A3)
Lyapunov function can be defined as equation (A4). V
1 T 1 K p T I 0 2 2
(A4)
Then Lyapunov derivative can be derived as below, which has negative value or zero. It will be zero when the Euler angle derivatives are conserved to zero.
V T K p T I T [ K p I] T [ K p u ] T [ K p K p K d ] K d 0 T
(A5)
Appendix B: Stability of Adaptive Controller To prove the stability of the adaptive controller for chamber pressure maintenance (in section 4.2), Lyapunov function should be defined as follows (Lee et al., 2013), V
1 2 1 1 e 1 2 2 2 2 2 2
(B1)
where error states ˆ (t ) and ˆ (t ) is defined as (B2), with the equation (13) and (18).
(t ) ˆ (t ) (t ) (t ) ˆ (t ) (t )
(B2)
Then the derivative of the Lyapunov function can be derived as: V ee 1 2
(B3)
e y ym [( k ) ( k )u ( ke)] ke [ ( k ) ( k )u] ke [ ( ) ( ˆ )u ( ˆ )u ]
(B4)
The derivative of the error state e ,
k
k
could be compressed as equation (B5) with the feedback linearization control input defined in equation (16). e ke u
(B5)
Then Lyapunov derivative in equation (B3) can be written as follows:
V ke2 e ue 1 (ˆ ) 2 (ˆ )
(B6)
This will be equivalent with equation (B7), with the update law in equation (17). V ke2 ( 1 2 )
(B7)
If magnitude of , , and , are assumed bounded, the time-varying effect can be bounded also as equation (B8), ( 1 2 ) W
(B8)
and Lyapunov derivative (B7) will be negative in the error bound e W k . V ke2 W
(B9)
To compensate the estimation error completely in this adaptive controller, sliding mode controller is also adopted in equation (15). With the additional control input u s , the derivative of the error state in equation (B4) and the Lyapunov function derivative in equation (B6) can be modified as below. e ke [ ( k ) ( k )uc ( k )us ] ke [ ( k ) ( k ˆ )uc ( ˆ )uc ( k )us ] ke uc ( k )us
(B10)
V ke2 e uc e (k )us e 1 (ˆ ) 2 (ˆ )
If the following supplement is assumed,
e ,
e
(B11)
the time-varying effect in equation (B8) can be refreshed as follow.
( 1 2 ) W e
(B13)
This new bound makes the Lyapunov derivative also bounded, and finally has negative value, V ke2 W e ( k )
W
0
sgn(e) ke2
(B14)
when the lower bound of the uncertainty is set as 0 .
k (t ) ( x, t ) 0
(B15)
Acknowledgments This work was supported by the Defense Acquisition Program Administration and Agency for Defense Development under the contract UD110093CD. The authors truly appreciate their financial support.
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