- Email: [email protected]
Design of a robust control law for the Vega launcher ballistic phase
Acta Astronautica 71 (2012) 92–98
Contents lists available at SciVerse ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Design of a robust control law for the Vega launcher ballistic phase$ Monica Valli a,n, Miche le R. Lavagna a, Thomas Panozzo b a b
Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via la Masa 34, 20156 Milano, Italy Arianespace, Boulevard de l’Europe BP 177, 91006 Evry-Courcouronnes, France
a r t i c l e in f o
abstract
Article history: Received 3 February 2011 Received in revised form 5 August 2011 Accepted 9 August 2011 Available online 31 August 2011
This work presents the design of a robust control law, and the related control system architecture, for the Vega launcher ballistic phase, taking into account the complete six degrees of freedom dynamics. To gain robustness a non-linear control approach has been preferred: more specifically the Lyapunov’s second stability theorem has been exploited, being a very powerful tool to guarantee asymptotic stability of the controlled dynamics. The dynamics of Vega’s actuators has also been taken into account. The system performance has been checked and analyzed by numerical simulations run on real mission data for different operational and configuration scenarios, and the effectiveness of the synthesized control highlighted: in particular scenarios including a wide range of composite’s inertial configurations performing various typologies of maneuvers have been run. The robustness of the controlled dynamics has been validated by 100 cases Monte Carlo analysis campaign: the containment of the dispersion for the controlled variables – say the composite roll, yaw and pitch angles – confirmed the wide validity and generality of the proposed control law. This paper will show the theoretical approach and discuss the obtained results. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Vega Ballistic phase Lyapunov RACS
1. Introduction Over the last years worldwide space agencies focused on small satellite missions requiring lower investments and shorter development time. Small satellites are increasingly being considered a suitable alternative to traditional satellites: new standardized satellite platforms have been developed in Europe, with mass, cost and manufacturing time considerably reduced. Moreover science-based, Earth observation as well as telecommunication missions are, more often, based on multiple small missions instead of relying on a single large satellite. To guarantee access to space and commercial success in this new scenario, Europe solved the need of appropriate
$
This paper was presented during the 61st IAC in Prague. Corresponding author. Tel.: þ 39 0223998428. E-mail addresses: [email protected] (M. Valli), [email protected] (M.R. Lavagna), [email protected] (T. Panozzo). n
0094-5765/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2011.08.002
launch services in the light payload class by developing the Vega small launcher [1,2]. The ballistic phase is one of the most challenging flight phases for a launcher: even a minimal error during maneuvers execution can seriously compromise the payload release and lead the mission to a failure. In this phase, the dedicated control system must be able to manage any maneuver set in the flight program respecting accuracy, especially regarding the composite (i.e. launch vehicle last stage, adapter and payloads) pointing direction before separation. Immediately after Vega injection into the required orbit for payload separation, the launcher last stage (AVUM) main engine is cut off and the composite starts following its orbital path. It is at this point that the ballistic phase begins and the roll and attitude control system (RACS) is activated by the on-board computer to initiate the orbital sequence. The RACS includes both the piloting algorithm and the dedicated actuation system used to control the composite during the ballistic phase.
M. Valli et al. / Acta Astronautica 71 (2012) 92–98
The piloting algorithm calculates the control action that must be applied to the system in order to properly execute the flight program and works out sets of commands to manage the actuators. The core of this algorithm is represented by control laws designed taking into account both launch vehicle upper composite configuration and main ballistic phase technical requirements. The actuation system is based on reaction thrusters: two identical clusters of three thrusters each are mounted in diametrically opposite positions on the external skin of the AVUM. Due to the thrusters specific position, any torque around the launcher roll, pitch and yaw axis is possible as well as longitudinal thrust. The whole sequence of ballistic phase maneuvers is optimized for every Vega mission, taking into account payload requirements for separation and launcher-spacecraft constraints. The paper is organized as follows. The main requirements for the RACS are discussed in Section 2 where some considerations concerning critical issues of the Vega upper stage configuration are also presented. In Section 3, the RACS piloting algorithm design is established. In order to verify the efficiency of the proposed technique, some ballistic phase simulations and a Monte Carlo analysis have been carried out as presented in Section 4. 2. RACS algorithm requirements and challenges The RACS shall be designed to perform a wide range of maneuvers:
Slew maneuvers from an arbitrary state to a final
orientation through three-axis maneuvers, including large-angle maneuvers in limited times. Spin up and spin down maneuvers. Waiting phases. Three-axis controlled boost maneuvers to control the composite’s position in space and its distance from the released bodies.
All these maneuvers that specify the ballistic phase are illustrated in Fig. 1, where a typical double launch RACS sequence is presented. Clearly this sequence becomes more complex in case of multiple launches. The capability to perform a large set of maneuvers represents the basic requirement for the RACS, but other important requirements must be specified. Since the composite needs to be controlled both before and after separation, the RACS shall be able to manage a very wide range of geometrical and
93
inertial configurations. This requirement represents a real challenge as it is possible to see looking at the following data. Considering the case of a single launch of a 1250 kg satellite, through separation the composite pitch lever arm decreases by 75%, the principal roll inertia by 60% and the principal transversal inertia even by 90%. Moreover, being the moments of inertia of the AVUM very small – an order of magnitude of 320 kgm2 – it follows that the inertial and geometrical configuration of the composite is almost completely defined by the payload configuration (and not by the launcher upper stage) and hence can strongly change not only through separation but also from mission to mission. These considerations clearly specify that the RACS shall be robust enough to face a wide range of system parameters’ modifications. Another important feature to be considered is related to the propulsion system. The actuation system is based on hydrazine thrusters. One tank operated in blow-down mode feeds a catalytic bed through flow control valves [3]. In blow-down mode the tank is loaded with propellant and locked up with a specified gas mass that results in diminishing pressure during operations and hence in reducing the available thrust. The RACS shall therefore take into account that the thrust level does not remain constant during the ballistic phase but decreases with each thrusters opening. The initial pressure of the tanks is set before launch to 26 bar, corresponding to a thrust level of 215 N for each thruster at the beginning of use. Since the RACS is also activated during the P80, Z23, Z9 [1] flight phases in order to control the launcher roll angle, the available thrust level at the beginning of the ballistic phase will be smaller. In this work it has been assumed to be 150 N. This thrust level – and more in detail its coupling with the Vega launcher upper stage inertial configuration – represents a very critical feature for the RACS design. A 150 N thrust level is in fact enormous if compared to the small moments of inertia of the body that must be controlled. In Table 1 some data concerning the Vega
Table 1 Comparison between Ariane 4 and Vega configurations before payload separation during a single launch mission. Property
Ariane 4
Vega
Ratio (%)
Principal roll inertia (kg m2) Principal transversal inertia (kg m2) Thrust level (N)
4687 85870 49
978 3860 150
20.9 4.5 306
Fig. 1. Typical single launch RACS sequence, courtesy of Arianespace.
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M. Valli et al. / Acta Astronautica 71 (2012) 92–98
upper composite configuration are compared with the data of another launcher, i.e. Ariane 4 (the data presented for Ariane 4 and Vega both refer to single launch missions characterized by similar features). On Ariane 4 a thrust level of 49 N was available in order to manage a considerable inertial configuration while Vega has three times the thrust level available on Ariane 4 in order to manage an inertial configuration till 20 times smaller. After separation, the RACS working conditions become even more critical since there is a further decrease of the moments of inertia while the thrust level practically remains unchanged (Table 2). The RACS control laws shall then be designed taking into account all these critical issues searching for good robustness properties. 3. RACS piloting algorithm design In this work the attitude of the composite has been described using quaternions. So, the attitude of the composite in space is described by a quaternion representing the angle between the body reference frame (geometrical or principal) and the inertial reference frame (the one defined by the inertial platform output 3 s before the first stage ignition). Then the Euler’s equations and the quaternions kinematic differential equations allow describing the composite attitude dynamical evolution during the whole ballistic phase [4,5].
the stability of the system is guaranteed, the attention is focused on which action must be applied to the system in order to guarantee the asymptotic stability [6,7]. This is the main idea of the RACS control law design presented hereafter. 3.2. Control law design Let us consider the dynamical system: x_ ¼ f ðx,uÞ
where x and u denote the state vector and the external input to the system. If we refer to the equilibrium condition (1) as X 2 W, with W Rn , Lyapunov’s second stability theorem says that if it is possible to find a scalar function VðxÞ 2 C 1 : W-R such that:
VðX Þ ¼ 0; VðxÞ 4 0 8x 2 W, xaX ; dVðxÞ=dt o0 8x 2 W, xaX , dVðxÞ=dt ¼ 0 if x ¼ X then the equilibrium point X is asymptotically stable [8]. So, according to Lyapunov’s second stability theorem, the control problem can be expressed as the pursuit of a scalar function V(x) that satisfies the previous constraints and from which drawing the control law after the definition of simple analytical conditions. The function (2) has been selected as the best Lyapunov’s candidate function VðxÞ ¼ kL ðtÞT L ðtÞ
3.1. Choice of the control technique Since the RACS shall be able to perform variable-angle slew maneuvers, it is not possible to linearize the dynamics equations around one equilibrium condition. This results in a considerable coupling between the dynamics around the three body axes and in a strong nonlinearity of the equations that describe the dynamical evolution of the system. This consideration, together with the requirement for the control system to be strongly adaptable to any flight program variation and robust enough to face any change in the system physical configuration, led to the decision to work in the non-linear control techniques field. A control law able to guarantee the asymptotic stability of the controlled system (independently of the composite inertial and geometrical configuration or the typology of the maneuvers) has been designed referring to Lyapunov’s second stability method. Lyapunov’s theory is commonly used to study the stability properties of already designed non-linear systems. However, it can be used in the design process of a control system if instead of wondering under which conditions Table 2 Comparison between Vega’s composite main properties before and after payload separation. Property 2
Principal roll inertia (kg m ) Principal transversal inertia (kg m2) Thrust level (N)
Before
After
Ratio (%)
978 3860 150
389 376 147
40 9.7 98
ð1Þ
ð2Þ
where k is an arbitrary positive constant and L denotes the error quaternion, i.e. the angle between the composite current attitude and the target attitude at time t. This error quaternion can be easily calculated combining the rotations from the inertial reference frame to the body reference frame and from the inertial reference frame to the target reference frame [4,9–12]. The chosen Lyapunov’s function is then derived and set equal to the arbitrary negative defined function (3) to obtain (4): f ¼ L ðtÞT L ðtÞ
ð3Þ
_ ðtÞT þ L ðtÞT LðtÞ ¼ 0 ½2kL
ð4Þ
Clearly, Eq. (4) must be true for every LðtÞ. Rearranging the constants into a single variable t1 , it is possible to write a first order differential equation that describes the dynamical evolution of the attitude error:
t1 L_ ðtÞ þ L ðtÞ ¼ 0
ð5Þ
A very interesting result can be found studying Eq. (5) in conjunction with the quaternion kinematics differential equations: ( v_ ¼ 12 q4 o 12o 4v ð6Þ q_ 4 ¼ 12o v where u and q4 denote the vectorial and the scalar part of the quaternion error. More in detail, it is clear that if Eq. (5) must be satisfied also t1 v_ ðtÞ þvðtÞ ¼ 0 must be valid, from which it follows v_ ¼ ð1=t1 Þv. It is easy to demonstrate that in order to guarantee this last result
M. Valli et al. / Acta Astronautica 71 (2012) 92–98
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(and therefore to guarantee the exponential decay of the attitude error to zero) a constraint on the composite angular velocity must be satisfied, i.e. the composite angular rate must be equal to a target angular rate at each time step. This target angular rate O D is given by
OD ¼
2 u
ð7Þ
t1 q4
In fact, replacing this expression in the first equation of (6) we obtain (8) which is exactly the condition that must be satisfied: 1 2 v 1 2 v 1 ð8Þ v_ ¼ q4 4v ¼ v 2 2 t1 q4 t1 q4 t1 Since the system is provided with on–off thrusters it is impossible to operate a punctual control of the composite angular rate and exactly guarantee o ðtÞ ¼ O D at each time step. Hence it is necessary to refer again to Lyapunov’s second stability theorem in order to guarantee the convergence of the composite angular rate to the target angular rate. For this purpose the following function is chosen: V1 ðxÞ ¼ C½o ðtÞO D ðtÞT ½o ðtÞO D ðtÞ
ð9Þ
and, following the same procedure presented above, a differential equation that rules the dynamical evolution of the angular rate is obtained:
t2 o_ ðtÞ þ o ðtÞ ¼ O D ðtÞ
ð10Þ
Finally, considering Eq. (10) and Euler’s equations of motion, it is possible to write the nominal control torque that must be applied to the system in order to achieve the required maneuver: M ¼I
O D ðtÞo ðtÞ
t2
þ o ðtÞ4I o ðtÞ
ð11Þ
where I and M respectively denote the composite moments of inertia matrix and the torques applied to the system. Clearly (11) are theoretical torques. In order to calculate the real torques it is necessary to refer and take into account the dynamics of the actuation system. 3.3. Transforming the torque command into thrusters activation time Since the RACS reaction thrusters are activated in a pulsing mode only, it is necessary to find a dedicated algorithm that can transform the torque command into correctly timed activation of the relevant thrusters. A simple algorithm suitable for this purpose is the one that uses the pulse width modulation principle (PWM) [4]. Thresholds of the thrusters ontimes, that would introduce an additional non-linearity in the model, have been neglected. The algorithm naturally depends on the physical set-up of the thrusters. Referring to the Vega RACS thrusters configuration presented in Fig. 2, it is easy to see that thrusters (2,4) and (1,3) respectively provide the negative and positive control torque around the roll axis (X-axis); the main component of thrusters 5 and 6 respectively provide the negative and the positive control torque around the yaw axis (Y-axis) while thrusters (1,4) and (2,3) respectively provide the negative and the positive control torque around the pitch axis (Z-axis).
Fig. 2. RACS reaction thrusters geometry [1].
Once that the geometrical configuration is well known it is possible to write the acceleration around the body axes as follows: 8 x > < Gx ¼ ðh1 þ h3 h2 h4 ÞG0 y Gy ¼ ðh6 h5 ÞG0 ð12Þ > : G ¼ ðh þh h h ÞGz z 2 3 1 4 0 where hi is 1 or 0 depending on whether the i-th thruster is on or off and Gj0 denotes the nominal acceleration that the i-th thruster, characterized by the nominal thrust Fi, can produce around the j-axis of inertia Ij exploiting the lever arm bij, as F0j ¼
X Fi bij i
ð13Þ
Ij
The required nominal acceleration GD can be calculated from the nominal control torques provided by the RACS control algorithm with a sampling time Tsam. Obviously, the average acceleration provided by the actuators during a sampling time depends upon the time t that the thrusters remain opened with respect to Tsam. Defining the variables ðwx , wy , wz Þ as presented in the first part of Eq. (14) we obtain a relation that can be used to solve the problem and calculate the thrusters activation time: 8 Gx > > > wx ¼ ðh1 þ h3 h2 h4 Þt ¼ Dx Tsam > > G0 > > > > < GyD wy ¼ ðh6 h5 Þt ¼ y Tsam ð14Þ G0 > > > > z > > GD > > > wz ¼ ðh2 þh3 h1 h4 Þt ¼ z Tsam : G 0
In this work only the procedure to calculate the thrusters activation time for the thrusters 3 and 4 will be presented and can be summarized as 8 wx þ wz w þ wz > > 4 0 ) t3 ¼ x , t4 ¼ 0 < if 2 2 ð15Þ wx þ wz w þ wz > > o 0 ) t4 ¼ x , t3 ¼ 0 : if 2 2 3.4. Three-axis controlled boost phase On Vega the RACS thrusters must also provide longitudinal thrust to control the composite’s position in
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M. Valli et al. / Acta Astronautica 71 (2012) 92–98
space. Hence the RACS algorithm shall also be able to perform a three-axis controlled boost maneuver. In order to give a complete description of this maneuver two variables must be set: the DV that must be given to the composite and the maneuver time. First of all the RACS algorithm calculates the attitude error with respect to the boost direction and verifies whether the pointing accuracy is respected. Depending on the accuracy test response, the piloting algorithm considers the following different cases:
If the RACS thrusters must provide a longitudinal
thrust but there is also the need for three-axis control, the piloting algorithm first of all calculates the control torque required to recover the pointing accuracy and the corresponding thrusters activation time; secondly it joints this information with the need for the thrusters 5 and 6 to remain opened as much as possible in order to give the composite the required DV. Thus any torque around the Y direction is compensated by firing thruster 5 or thruster 6 (depending on the direction of the required control torque) for a shorter time than Tsam. If the RACS thrusters do not have to provide longitudinal thrust (since the required DV has been obtained before the end of the maneuver time), but there is the need for three-axis control, the piloting algorithm carries out a normal three-axis control phase.
4. Simulations and results The RACS piloting algorithm has been implemented and tested with success for all the maneuvers presented in Section 2. Case studies with no particular constraints over the attitude (e.g. no constraints over the sun direction in the spacecraft reference frame) have been considered. Simulation results are presented hereafter in terms of Euler angles (321 sequence). Fig. 3 shows an example of slew maneuver, followed by a waiting phase, before payload separation. The maneuver convergence time is very good. After only 26 s from injection the composite has already reached the target attitude and is pointing the correct direction for separation. Once the convergence is achieved, the control system carries out a waiting phase, working to maintain the pointing direction
(a threshold of 70.51 for the transversal angle; 70.751 for the roll angle; 70.31/s for the transversal rate and 70.81/s for the roll rate have been imposed) [3,13,14]. Some important considerations can be done concerning the composite angular rates. More in detail, it has been noticed that rather high angular rates can be reached during the simulation of some slew maneuvers. This feature is strongly related to Vega’s configuration and in particular to the coupling between Vega’s upper stage inertial configuration and the RACS thrust level. The RACS high thrust level applied to a composite characterized by such small inertia naturally inclines towards rapid attitude changes and so towards high angular rates. If the resulting angular rates are not compatible with the payload specifications or the mission constraints, it is possible to vary them acting on the RACS algorithm control gains that can be conveniently changed according to the flight phase. In particular, it is possible to reduce the angular rates by increasing the time constant t1 (Eq. (5)). The RACS piloting algorithm also succeeds in controlling the composite after payload separation. In Fig. 4 a slew maneuver followed by a waiting phase after payload separation is presented. For this phase a threshold of 70.11 for the transversal angle, 71.01 for the roll angle, 70.51/s for the transversal rate and 71.01/s for the roll rate have been imposed. The control system robustness has been tested with a Monte Carlo analysis made up by 100 cases and considering dispersions on the AVUM, the payload and other simulation parameters such as the thrust level, thrust coefficients, thrusters orientation, perturbing torques, etc. The Monte Carlo analysis showed that the control system is very robust before payload separation while after separation its performance is strongly influenced by the typology of the maneuver that must be carried out (another demonstration that the coupling between Vega small inertias and RACS high thrust level represents a very critical issue). Complete ballistic phase simulations have also been carried out. In Fig. 5 the composite attitude evolution during a complete ballistic phase simulation is presented. In this example, the composite performs a first slew maneuver before payload separation to achieve the correct pointing direction for separation, a waiting phase that coincides with payload separation, a second slew maneuver after payload separation, a spin up maneuver
Fig. 3. Composite attitude evolution during a slew maneuver before payload separation.
M. Valli et al. / Acta Astronautica 71 (2012) 92–98
97
Fig. 4. Composite attitude evolution during a slew maneuver after payload separation.
Fig. 5. Composite attitude evolution during a complete ballistic phase.
Fig. 6. Upper stage and payload orbits osculatory apogees.
and a final waiting phase of the spinning composite before passivation. In Fig. 6 it is possible to see the effect of the three-axis controlled boost phase. A safety distance between the composite and the released payload is guaranteed with a difference of 21 km between the two orbits osculatory apogees. 5. Conclusions The design of a dedicated robust control law for the Vega launcher ballistic phase has been presented. The control system design is based on a non-linear control approach and Lyapunov’s second stability theorem has been exploited. Both launch vehicle upper composite configuration and main ballistic phase technical requirements have been taken into account. It has been
demonstrated that the designed RACS piloting algorithm can perform all the typical ballistic phase maneuvers. The control system can manage a wide range of geometrical and inertial configuration of the controlled composite including multiple launches. In addition, some critical issues related to the Vega launcher configuration have been highlighted. The principal criticality is represented by the coupling between Vega upper composite small inertias and RACS high thrust level. It has been demonstrated that, if neglected, this feature can affect the system’s response. Nevertheless, an appropriate tuning of the control system gains according to the flight phase can be studied as a solution to keep the composite attitude dynamical evolution under control. The robustness of the system has been validated by a Monte Carlo campaign. References [1] E. Perez, Vega user’s manual, Arianespace, 2004. [2] A.G. Accettura, M. Balduccini, A. Rinalducci, The baseline performance capabilities and the envisaged future service options of the Vega launch system, in: 54th International Astronautical Congress of the International Astronautical Federation, Bremen, Germany, 2003. [3] I. Cruciani, C. Roux, F. Carducci, P. Bellomi, R. Fabrizi, Performance model of RACS, ELV, Rev.1, 2007. [4] M.J. Sidi, Spacecraft Dynamics and Control: A Practical Engineering Approach, Cambridge University Press, 1997. [5] D.A. Vallado, Fundamentals of Astrodynamics and Applications, second ed., Space Technology Library, 2004. [6] Notice d’utilisation du programme de simulation SCAR, Rev. 1, CNES, 1984. [7] S. Chavy, F. Fantar, Design of attitude control system for injection of spinned payloads, Acta Astronautica 25 (4) (1991) 185–197.
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[8] A.M. Lyapunov, The general problem of the stability motion, International Journal of Control 55 (1992) 531–534. [9] J.R. Wertz, Spacecraft Attitude Determination and Control, Kluwer Academic Publishers, 1978. [10] V.A. Chobotov, Spacecraft Attitude Dynamics and Control, Krieger Publishing Company, 1991. [11] B. Wie, Space Vehicles Dynamics and Control, AIAA Education Series Edition, 1998.
[12] M.J. Sidi, Design of Robust Control Systems, Krieger Publishing Company, 2001. [13] F. Battie, C. Corba, P. Bini, C. Dumaz, F. Carducci, P. Bellomi, A. Liberati, R. Fabrizi, Input data to Vega GNC analysis, ELV, Rev. 1, 2006. [14] I. Cruciani, F. Carducci, P. Bini, P. Bellomi, R. Fabrizi, Vegamath functional file, ELV, Rev. 1, 2006.
Contents lists available at SciVerse ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Design of a robust control law for the Vega launcher ballistic phase$ Monica Valli a,n, Miche le R. Lavagna a, Thomas Panozzo b a b
Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via la Masa 34, 20156 Milano, Italy Arianespace, Boulevard de l’Europe BP 177, 91006 Evry-Courcouronnes, France
a r t i c l e in f o
abstract
Article history: Received 3 February 2011 Received in revised form 5 August 2011 Accepted 9 August 2011 Available online 31 August 2011
This work presents the design of a robust control law, and the related control system architecture, for the Vega launcher ballistic phase, taking into account the complete six degrees of freedom dynamics. To gain robustness a non-linear control approach has been preferred: more specifically the Lyapunov’s second stability theorem has been exploited, being a very powerful tool to guarantee asymptotic stability of the controlled dynamics. The dynamics of Vega’s actuators has also been taken into account. The system performance has been checked and analyzed by numerical simulations run on real mission data for different operational and configuration scenarios, and the effectiveness of the synthesized control highlighted: in particular scenarios including a wide range of composite’s inertial configurations performing various typologies of maneuvers have been run. The robustness of the controlled dynamics has been validated by 100 cases Monte Carlo analysis campaign: the containment of the dispersion for the controlled variables – say the composite roll, yaw and pitch angles – confirmed the wide validity and generality of the proposed control law. This paper will show the theoretical approach and discuss the obtained results. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Vega Ballistic phase Lyapunov RACS
1. Introduction Over the last years worldwide space agencies focused on small satellite missions requiring lower investments and shorter development time. Small satellites are increasingly being considered a suitable alternative to traditional satellites: new standardized satellite platforms have been developed in Europe, with mass, cost and manufacturing time considerably reduced. Moreover science-based, Earth observation as well as telecommunication missions are, more often, based on multiple small missions instead of relying on a single large satellite. To guarantee access to space and commercial success in this new scenario, Europe solved the need of appropriate
$
This paper was presented during the 61st IAC in Prague. Corresponding author. Tel.: þ 39 0223998428. E-mail addresses: [email protected] (M. Valli), [email protected] (M.R. Lavagna), [email protected] (T. Panozzo). n
0094-5765/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2011.08.002
launch services in the light payload class by developing the Vega small launcher [1,2]. The ballistic phase is one of the most challenging flight phases for a launcher: even a minimal error during maneuvers execution can seriously compromise the payload release and lead the mission to a failure. In this phase, the dedicated control system must be able to manage any maneuver set in the flight program respecting accuracy, especially regarding the composite (i.e. launch vehicle last stage, adapter and payloads) pointing direction before separation. Immediately after Vega injection into the required orbit for payload separation, the launcher last stage (AVUM) main engine is cut off and the composite starts following its orbital path. It is at this point that the ballistic phase begins and the roll and attitude control system (RACS) is activated by the on-board computer to initiate the orbital sequence. The RACS includes both the piloting algorithm and the dedicated actuation system used to control the composite during the ballistic phase.
M. Valli et al. / Acta Astronautica 71 (2012) 92–98
The piloting algorithm calculates the control action that must be applied to the system in order to properly execute the flight program and works out sets of commands to manage the actuators. The core of this algorithm is represented by control laws designed taking into account both launch vehicle upper composite configuration and main ballistic phase technical requirements. The actuation system is based on reaction thrusters: two identical clusters of three thrusters each are mounted in diametrically opposite positions on the external skin of the AVUM. Due to the thrusters specific position, any torque around the launcher roll, pitch and yaw axis is possible as well as longitudinal thrust. The whole sequence of ballistic phase maneuvers is optimized for every Vega mission, taking into account payload requirements for separation and launcher-spacecraft constraints. The paper is organized as follows. The main requirements for the RACS are discussed in Section 2 where some considerations concerning critical issues of the Vega upper stage configuration are also presented. In Section 3, the RACS piloting algorithm design is established. In order to verify the efficiency of the proposed technique, some ballistic phase simulations and a Monte Carlo analysis have been carried out as presented in Section 4. 2. RACS algorithm requirements and challenges The RACS shall be designed to perform a wide range of maneuvers:
Slew maneuvers from an arbitrary state to a final
orientation through three-axis maneuvers, including large-angle maneuvers in limited times. Spin up and spin down maneuvers. Waiting phases. Three-axis controlled boost maneuvers to control the composite’s position in space and its distance from the released bodies.
All these maneuvers that specify the ballistic phase are illustrated in Fig. 1, where a typical double launch RACS sequence is presented. Clearly this sequence becomes more complex in case of multiple launches. The capability to perform a large set of maneuvers represents the basic requirement for the RACS, but other important requirements must be specified. Since the composite needs to be controlled both before and after separation, the RACS shall be able to manage a very wide range of geometrical and
93
inertial configurations. This requirement represents a real challenge as it is possible to see looking at the following data. Considering the case of a single launch of a 1250 kg satellite, through separation the composite pitch lever arm decreases by 75%, the principal roll inertia by 60% and the principal transversal inertia even by 90%. Moreover, being the moments of inertia of the AVUM very small – an order of magnitude of 320 kgm2 – it follows that the inertial and geometrical configuration of the composite is almost completely defined by the payload configuration (and not by the launcher upper stage) and hence can strongly change not only through separation but also from mission to mission. These considerations clearly specify that the RACS shall be robust enough to face a wide range of system parameters’ modifications. Another important feature to be considered is related to the propulsion system. The actuation system is based on hydrazine thrusters. One tank operated in blow-down mode feeds a catalytic bed through flow control valves [3]. In blow-down mode the tank is loaded with propellant and locked up with a specified gas mass that results in diminishing pressure during operations and hence in reducing the available thrust. The RACS shall therefore take into account that the thrust level does not remain constant during the ballistic phase but decreases with each thrusters opening. The initial pressure of the tanks is set before launch to 26 bar, corresponding to a thrust level of 215 N for each thruster at the beginning of use. Since the RACS is also activated during the P80, Z23, Z9 [1] flight phases in order to control the launcher roll angle, the available thrust level at the beginning of the ballistic phase will be smaller. In this work it has been assumed to be 150 N. This thrust level – and more in detail its coupling with the Vega launcher upper stage inertial configuration – represents a very critical feature for the RACS design. A 150 N thrust level is in fact enormous if compared to the small moments of inertia of the body that must be controlled. In Table 1 some data concerning the Vega
Table 1 Comparison between Ariane 4 and Vega configurations before payload separation during a single launch mission. Property
Ariane 4
Vega
Ratio (%)
Principal roll inertia (kg m2) Principal transversal inertia (kg m2) Thrust level (N)
4687 85870 49
978 3860 150
20.9 4.5 306
Fig. 1. Typical single launch RACS sequence, courtesy of Arianespace.
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upper composite configuration are compared with the data of another launcher, i.e. Ariane 4 (the data presented for Ariane 4 and Vega both refer to single launch missions characterized by similar features). On Ariane 4 a thrust level of 49 N was available in order to manage a considerable inertial configuration while Vega has three times the thrust level available on Ariane 4 in order to manage an inertial configuration till 20 times smaller. After separation, the RACS working conditions become even more critical since there is a further decrease of the moments of inertia while the thrust level practically remains unchanged (Table 2). The RACS control laws shall then be designed taking into account all these critical issues searching for good robustness properties. 3. RACS piloting algorithm design In this work the attitude of the composite has been described using quaternions. So, the attitude of the composite in space is described by a quaternion representing the angle between the body reference frame (geometrical or principal) and the inertial reference frame (the one defined by the inertial platform output 3 s before the first stage ignition). Then the Euler’s equations and the quaternions kinematic differential equations allow describing the composite attitude dynamical evolution during the whole ballistic phase [4,5].
the stability of the system is guaranteed, the attention is focused on which action must be applied to the system in order to guarantee the asymptotic stability [6,7]. This is the main idea of the RACS control law design presented hereafter. 3.2. Control law design Let us consider the dynamical system: x_ ¼ f ðx,uÞ
where x and u denote the state vector and the external input to the system. If we refer to the equilibrium condition (1) as X 2 W, with W Rn , Lyapunov’s second stability theorem says that if it is possible to find a scalar function VðxÞ 2 C 1 : W-R such that:
VðX Þ ¼ 0; VðxÞ 4 0 8x 2 W, xaX ; dVðxÞ=dt o0 8x 2 W, xaX , dVðxÞ=dt ¼ 0 if x ¼ X then the equilibrium point X is asymptotically stable [8]. So, according to Lyapunov’s second stability theorem, the control problem can be expressed as the pursuit of a scalar function V(x) that satisfies the previous constraints and from which drawing the control law after the definition of simple analytical conditions. The function (2) has been selected as the best Lyapunov’s candidate function VðxÞ ¼ kL ðtÞT L ðtÞ
3.1. Choice of the control technique Since the RACS shall be able to perform variable-angle slew maneuvers, it is not possible to linearize the dynamics equations around one equilibrium condition. This results in a considerable coupling between the dynamics around the three body axes and in a strong nonlinearity of the equations that describe the dynamical evolution of the system. This consideration, together with the requirement for the control system to be strongly adaptable to any flight program variation and robust enough to face any change in the system physical configuration, led to the decision to work in the non-linear control techniques field. A control law able to guarantee the asymptotic stability of the controlled system (independently of the composite inertial and geometrical configuration or the typology of the maneuvers) has been designed referring to Lyapunov’s second stability method. Lyapunov’s theory is commonly used to study the stability properties of already designed non-linear systems. However, it can be used in the design process of a control system if instead of wondering under which conditions Table 2 Comparison between Vega’s composite main properties before and after payload separation. Property 2
Principal roll inertia (kg m ) Principal transversal inertia (kg m2) Thrust level (N)
Before
After
Ratio (%)
978 3860 150
389 376 147
40 9.7 98
ð1Þ
ð2Þ
where k is an arbitrary positive constant and L denotes the error quaternion, i.e. the angle between the composite current attitude and the target attitude at time t. This error quaternion can be easily calculated combining the rotations from the inertial reference frame to the body reference frame and from the inertial reference frame to the target reference frame [4,9–12]. The chosen Lyapunov’s function is then derived and set equal to the arbitrary negative defined function (3) to obtain (4): f ¼ L ðtÞT L ðtÞ
ð3Þ
_ ðtÞT þ L ðtÞT LðtÞ ¼ 0 ½2kL
ð4Þ
Clearly, Eq. (4) must be true for every LðtÞ. Rearranging the constants into a single variable t1 , it is possible to write a first order differential equation that describes the dynamical evolution of the attitude error:
t1 L_ ðtÞ þ L ðtÞ ¼ 0
ð5Þ
A very interesting result can be found studying Eq. (5) in conjunction with the quaternion kinematics differential equations: ( v_ ¼ 12 q4 o 12o 4v ð6Þ q_ 4 ¼ 12o v where u and q4 denote the vectorial and the scalar part of the quaternion error. More in detail, it is clear that if Eq. (5) must be satisfied also t1 v_ ðtÞ þvðtÞ ¼ 0 must be valid, from which it follows v_ ¼ ð1=t1 Þv. It is easy to demonstrate that in order to guarantee this last result
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(and therefore to guarantee the exponential decay of the attitude error to zero) a constraint on the composite angular velocity must be satisfied, i.e. the composite angular rate must be equal to a target angular rate at each time step. This target angular rate O D is given by
OD ¼
2 u
ð7Þ
t1 q4
In fact, replacing this expression in the first equation of (6) we obtain (8) which is exactly the condition that must be satisfied: 1 2 v 1 2 v 1 ð8Þ v_ ¼ q4 4v ¼ v 2 2 t1 q4 t1 q4 t1 Since the system is provided with on–off thrusters it is impossible to operate a punctual control of the composite angular rate and exactly guarantee o ðtÞ ¼ O D at each time step. Hence it is necessary to refer again to Lyapunov’s second stability theorem in order to guarantee the convergence of the composite angular rate to the target angular rate. For this purpose the following function is chosen: V1 ðxÞ ¼ C½o ðtÞO D ðtÞT ½o ðtÞO D ðtÞ
ð9Þ
and, following the same procedure presented above, a differential equation that rules the dynamical evolution of the angular rate is obtained:
t2 o_ ðtÞ þ o ðtÞ ¼ O D ðtÞ
ð10Þ
Finally, considering Eq. (10) and Euler’s equations of motion, it is possible to write the nominal control torque that must be applied to the system in order to achieve the required maneuver: M ¼I
O D ðtÞo ðtÞ
t2
þ o ðtÞ4I o ðtÞ
ð11Þ
where I and M respectively denote the composite moments of inertia matrix and the torques applied to the system. Clearly (11) are theoretical torques. In order to calculate the real torques it is necessary to refer and take into account the dynamics of the actuation system. 3.3. Transforming the torque command into thrusters activation time Since the RACS reaction thrusters are activated in a pulsing mode only, it is necessary to find a dedicated algorithm that can transform the torque command into correctly timed activation of the relevant thrusters. A simple algorithm suitable for this purpose is the one that uses the pulse width modulation principle (PWM) [4]. Thresholds of the thrusters ontimes, that would introduce an additional non-linearity in the model, have been neglected. The algorithm naturally depends on the physical set-up of the thrusters. Referring to the Vega RACS thrusters configuration presented in Fig. 2, it is easy to see that thrusters (2,4) and (1,3) respectively provide the negative and positive control torque around the roll axis (X-axis); the main component of thrusters 5 and 6 respectively provide the negative and the positive control torque around the yaw axis (Y-axis) while thrusters (1,4) and (2,3) respectively provide the negative and the positive control torque around the pitch axis (Z-axis).
Fig. 2. RACS reaction thrusters geometry [1].
Once that the geometrical configuration is well known it is possible to write the acceleration around the body axes as follows: 8 x > < Gx ¼ ðh1 þ h3 h2 h4 ÞG0 y Gy ¼ ðh6 h5 ÞG0 ð12Þ > : G ¼ ðh þh h h ÞGz z 2 3 1 4 0 where hi is 1 or 0 depending on whether the i-th thruster is on or off and Gj0 denotes the nominal acceleration that the i-th thruster, characterized by the nominal thrust Fi, can produce around the j-axis of inertia Ij exploiting the lever arm bij, as F0j ¼
X Fi bij i
ð13Þ
Ij
The required nominal acceleration GD can be calculated from the nominal control torques provided by the RACS control algorithm with a sampling time Tsam. Obviously, the average acceleration provided by the actuators during a sampling time depends upon the time t that the thrusters remain opened with respect to Tsam. Defining the variables ðwx , wy , wz Þ as presented in the first part of Eq. (14) we obtain a relation that can be used to solve the problem and calculate the thrusters activation time: 8 Gx > > > wx ¼ ðh1 þ h3 h2 h4 Þt ¼ Dx Tsam > > G0 > > > > < GyD wy ¼ ðh6 h5 Þt ¼ y Tsam ð14Þ G0 > > > > z > > GD > > > wz ¼ ðh2 þh3 h1 h4 Þt ¼ z Tsam : G 0
In this work only the procedure to calculate the thrusters activation time for the thrusters 3 and 4 will be presented and can be summarized as 8 wx þ wz w þ wz > > 4 0 ) t3 ¼ x , t4 ¼ 0 < if 2 2 ð15Þ wx þ wz w þ wz > > o 0 ) t4 ¼ x , t3 ¼ 0 : if 2 2 3.4. Three-axis controlled boost phase On Vega the RACS thrusters must also provide longitudinal thrust to control the composite’s position in
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space. Hence the RACS algorithm shall also be able to perform a three-axis controlled boost maneuver. In order to give a complete description of this maneuver two variables must be set: the DV that must be given to the composite and the maneuver time. First of all the RACS algorithm calculates the attitude error with respect to the boost direction and verifies whether the pointing accuracy is respected. Depending on the accuracy test response, the piloting algorithm considers the following different cases:
If the RACS thrusters must provide a longitudinal
thrust but there is also the need for three-axis control, the piloting algorithm first of all calculates the control torque required to recover the pointing accuracy and the corresponding thrusters activation time; secondly it joints this information with the need for the thrusters 5 and 6 to remain opened as much as possible in order to give the composite the required DV. Thus any torque around the Y direction is compensated by firing thruster 5 or thruster 6 (depending on the direction of the required control torque) for a shorter time than Tsam. If the RACS thrusters do not have to provide longitudinal thrust (since the required DV has been obtained before the end of the maneuver time), but there is the need for three-axis control, the piloting algorithm carries out a normal three-axis control phase.
4. Simulations and results The RACS piloting algorithm has been implemented and tested with success for all the maneuvers presented in Section 2. Case studies with no particular constraints over the attitude (e.g. no constraints over the sun direction in the spacecraft reference frame) have been considered. Simulation results are presented hereafter in terms of Euler angles (321 sequence). Fig. 3 shows an example of slew maneuver, followed by a waiting phase, before payload separation. The maneuver convergence time is very good. After only 26 s from injection the composite has already reached the target attitude and is pointing the correct direction for separation. Once the convergence is achieved, the control system carries out a waiting phase, working to maintain the pointing direction
(a threshold of 70.51 for the transversal angle; 70.751 for the roll angle; 70.31/s for the transversal rate and 70.81/s for the roll rate have been imposed) [3,13,14]. Some important considerations can be done concerning the composite angular rates. More in detail, it has been noticed that rather high angular rates can be reached during the simulation of some slew maneuvers. This feature is strongly related to Vega’s configuration and in particular to the coupling between Vega’s upper stage inertial configuration and the RACS thrust level. The RACS high thrust level applied to a composite characterized by such small inertia naturally inclines towards rapid attitude changes and so towards high angular rates. If the resulting angular rates are not compatible with the payload specifications or the mission constraints, it is possible to vary them acting on the RACS algorithm control gains that can be conveniently changed according to the flight phase. In particular, it is possible to reduce the angular rates by increasing the time constant t1 (Eq. (5)). The RACS piloting algorithm also succeeds in controlling the composite after payload separation. In Fig. 4 a slew maneuver followed by a waiting phase after payload separation is presented. For this phase a threshold of 70.11 for the transversal angle, 71.01 for the roll angle, 70.51/s for the transversal rate and 71.01/s for the roll rate have been imposed. The control system robustness has been tested with a Monte Carlo analysis made up by 100 cases and considering dispersions on the AVUM, the payload and other simulation parameters such as the thrust level, thrust coefficients, thrusters orientation, perturbing torques, etc. The Monte Carlo analysis showed that the control system is very robust before payload separation while after separation its performance is strongly influenced by the typology of the maneuver that must be carried out (another demonstration that the coupling between Vega small inertias and RACS high thrust level represents a very critical issue). Complete ballistic phase simulations have also been carried out. In Fig. 5 the composite attitude evolution during a complete ballistic phase simulation is presented. In this example, the composite performs a first slew maneuver before payload separation to achieve the correct pointing direction for separation, a waiting phase that coincides with payload separation, a second slew maneuver after payload separation, a spin up maneuver
Fig. 3. Composite attitude evolution during a slew maneuver before payload separation.
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Fig. 4. Composite attitude evolution during a slew maneuver after payload separation.
Fig. 5. Composite attitude evolution during a complete ballistic phase.
Fig. 6. Upper stage and payload orbits osculatory apogees.
and a final waiting phase of the spinning composite before passivation. In Fig. 6 it is possible to see the effect of the three-axis controlled boost phase. A safety distance between the composite and the released payload is guaranteed with a difference of 21 km between the two orbits osculatory apogees. 5. Conclusions The design of a dedicated robust control law for the Vega launcher ballistic phase has been presented. The control system design is based on a non-linear control approach and Lyapunov’s second stability theorem has been exploited. Both launch vehicle upper composite configuration and main ballistic phase technical requirements have been taken into account. It has been
demonstrated that the designed RACS piloting algorithm can perform all the typical ballistic phase maneuvers. The control system can manage a wide range of geometrical and inertial configuration of the controlled composite including multiple launches. In addition, some critical issues related to the Vega launcher configuration have been highlighted. The principal criticality is represented by the coupling between Vega upper composite small inertias and RACS high thrust level. It has been demonstrated that, if neglected, this feature can affect the system’s response. Nevertheless, an appropriate tuning of the control system gains according to the flight phase can be studied as a solution to keep the composite attitude dynamical evolution under control. The robustness of the system has been validated by a Monte Carlo campaign. References [1] E. Perez, Vega user’s manual, Arianespace, 2004. [2] A.G. Accettura, M. Balduccini, A. Rinalducci, The baseline performance capabilities and the envisaged future service options of the Vega launch system, in: 54th International Astronautical Congress of the International Astronautical Federation, Bremen, Germany, 2003. [3] I. Cruciani, C. Roux, F. Carducci, P. Bellomi, R. Fabrizi, Performance model of RACS, ELV, Rev.1, 2007. [4] M.J. Sidi, Spacecraft Dynamics and Control: A Practical Engineering Approach, Cambridge University Press, 1997. [5] D.A. Vallado, Fundamentals of Astrodynamics and Applications, second ed., Space Technology Library, 2004. [6] Notice d’utilisation du programme de simulation SCAR, Rev. 1, CNES, 1984. [7] S. Chavy, F. Fantar, Design of attitude control system for injection of spinned payloads, Acta Astronautica 25 (4) (1991) 185–197.
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[8] A.M. Lyapunov, The general problem of the stability motion, International Journal of Control 55 (1992) 531–534. [9] J.R. Wertz, Spacecraft Attitude Determination and Control, Kluwer Academic Publishers, 1978. [10] V.A. Chobotov, Spacecraft Attitude Dynamics and Control, Krieger Publishing Company, 1991. [11] B. Wie, Space Vehicles Dynamics and Control, AIAA Education Series Edition, 1998.
[12] M.J. Sidi, Design of Robust Control Systems, Krieger Publishing Company, 2001. [13] F. Battie, C. Corba, P. Bini, C. Dumaz, F. Carducci, P. Bellomi, A. Liberati, R. Fabrizi, Input data to Vega GNC analysis, ELV, Rev. 1, 2006. [14] I. Cruciani, F. Carducci, P. Bini, P. Bellomi, R. Fabrizi, Vegamath functional file, ELV, Rev. 1, 2006.