MODELING AND PERFORMANCE EVALUATION OF MULTISTAGE LAUNCH VEHICLES THROUGH FIREWORK ALGORITHM M. Pallone∗, M. Pontaniy, P. Teofilattoz Scuola di Ingegneria Aerospaziale, University of Rome ”La Sapienza”, Rome, Italy ABSTRACT and applied a hybrid analytic/numerical approach, based on homotopy and starting with the generation of the optimal so- Multistage launch vehicles of reduced size, such as Super lution in a vacuum. They adopted the approximate linear Strypi or Sword, are currently investigated for the purpose gravity model, and the same did Lu and Pan [13] and Lu of providing launch opportunities for microsatellites. A sim- et al. [14], who applied a multiple-shooting method to op- ple open-loop guidance strategy is proposed in this research timizing exoatmospheric trajectories composed of two pow- and applied to the Scout rocket, a micro-launcher used in the ered phases separated by a coast arc. Weigel and Well [15] past. Aerodynamics and propulsion are modeled with high used a similar indirect, multiple-shooting approach to ana- fidelity through interpolation of available data. Unlike the lyze and optimize the ascent trajectories of two launch vehi- original Scout, the terminal optimal ascent path is determined cles with splash-down constraints. Miele [2] developed and for the upper stage, using a firework algorithm in conjunction applied the indirect multiple-subarc gradient restoration algo- with the Euler-Lagrange equations and the Pontryagin min- rithm to optimizing the two-dimensional ascending trajectory imum principle. Firework algorithms represent a recently- of a three-stage rocket in the presence of dynamical and con- introduced heuristic technique inspired by the firework explo- trol constraints. The previously cited works [2, 10, 12, 13, sions in the night sky. 14, 15] make use of indirect algorithms and require a con- Index Terms— Multistage launch vehicles, trajectory op- siderable deal of effort for deriving the analytical conditions timization, firework algorithm optimization for optimality and for the subsequent programming and de- bugging. Furthermore, these methods can suffer from a slow 1. INTRODUCTION rate of convergence and an excessive dependence on the start- ing guess. This difficulty has been occasionally circumvented Currently, microsatellites can be launched according to the through homotopy [10, 12, 16], but this adds further com- time and orbital requirements of a main payload. The limited plexity to the solution process. Other papers deal with direct costs of microsatellites and their capability to be produced numerical techniques applied to multistage rocket trajectory and ready for use in short time make them particularly suit- optimization. Roh and Kim [17] used a collocation method able to face an emergency (responsive space), therefore small for optimizing the performance of a four-stage rocket, whose launch vehicles dedicated to microsatellites would be very two-dimensional trajectory was assumed to be composed of useful. On the other hand, in order to reduce the launcher three thrust phases and a coast arc of specified duration. Col- size without increasing too much the launch cost per kg of location was also employed by Jamilnia and Naghash [11], payload it is necessary to simplify the launch system as much with the additional task of determining the optimal staging, as possible, including the guidance algorithms. and by Martinon et al. [16], for the purpose of validating the In general, the numerical solution of aerospace trajectory numexrical results attained through indirect shooting. This optimization problems is not trivial and has been pursued with latter paper refers to the Ariane V launch vehicle and is specif- different approaches in the past. Indirect methods, such as the ically devoted to investigating singular arcs. Direct methods gradient-restoration algorithm [1, 2] and the shooting method convert the optimal control problem into a nonlinear program- [3], or direct techniques, such as direct collocation [4, 5], di- ming problem involving a large number of unknown param- rect transcription [6, 7], and differential inclusion [8, 9], to eters to optimize. The disadvantage is in the need of using name a few. However, only a relatively small number of pub- specialized nlp solvers, such as SNOPT [20]. Recently, dif- lications are concerned with trajectory optimization of mul- ferent methodologies appeared that do not belong to the cat- tistage launch vehicles [1, 2, 10, 11, 12, 13, 14, 15, 16, 17, egory of either the indirect nor the direct techniques. These 18, 19]. Calise et al. [10] and Gath and Calise [12] proposed techniques are usually referred to as heuristics and have been sporadically applied to optimizing ascent trajectories of mul- ∗ corresponding author: [email protected] tistage rockets. Bayley et al. [18] used a genetic algorithm for [email protected] zpaolo.teofi[email protected] the purpose of minimizing the overall rocket mass in the con- text of a high fidelity model of the entire vehicle. Three- and into low orbit. It has specified structural, propulsive and aero- four-stage rockets were considered, with ascending trajecto- dynamic characteristics and it is represented in Figure 1. ries composed (respectively) of three or four powered arcs. Lastly, Qazi et al. [19] integrated neural networks, particle swarm optimization, and sequential quadratic programming for the simultaneous conceptual design and trajectory opti- mization of a new multistage launch vehicle. The work that follows is concerned with a novel ap- proach, which is intended to supply a fast performance evalu- ation for multistage rockets with given characteristics, under some simplifying assumptions. The technique described in this work is applied to a four-stage rocket, whose two- dimensional trajectory is composed of the following thrust phases and coast arcs: 1. first stage propulsion 2. second stage propulsion 3. third stage propulsion Fig. 1: Scout rocket geometry 4. coast arc (after the third stage separation) 5. fourth stage thrust phase To simplify the analysis at hand, the mass distribution of the launch vehicle can be described in terms of masses of In general, the inclusion of a coast arc (between two pow- subrockets: subrocket 1 is the entire rocket with all the four ered phases) leads to substantial propellant savings and this stages, subrocket 2 is the launch vehicle after the separation of circumstance justifies the trajectory structure assumed in this the first stage, subrocket 3 is the launch vehicle after the sep- research. Usually the coast duration increases as the injec- aration of the first two stages and subrocket 4 is represented tion altitude increases, as remarked by Lu et al. [14]. In (i) by the last stage only. Let m0 denote the initial mass of the order to simplify the open loop guidance law employed for (i) subrocket i, this mass is composed of a structural mass m , the first three stages, the aerodynamic angle of attack is as- S m(i) m(i) sumed constant for each stage. In the last stage thrust phase a propellant mass P and a payload mass U : the problem of minimizing the propellant is solved defining (i) (i) (i) (i) a Hamiltonian function which is minimized through the Pon- m0 = mS + mP + mU (1) tryagin minimum principle. The optimization algorithm used (i) in this work is the firework algorithm [21], a novel swarm in- For the first three subrockets mU (i=1,2,3) coincides with the (i) (i+1) telligence algorithm inspired by the explosions of fireworks initial mass of the subsequent subrocket (i.e. mU = m0 ). in a night sky. The concept that underlies this method is rel- With regard to the fourth subrocket, its propellant mass is to atively simple: a firework explodes in the search space of the be minimized. The mass distribution for the Scout rocket is unknown parameters, with amplitude and number of sparks in the Table 1. For the last stage the initial mass is 514 kg. determined dynamically. The succeeding iterations preserve Minimizing the propellant mass is equivalent to maximizing the best sparks. The method that is being presented requires the payload mass. a reduced deal of effort in programming, debugging, and test- ing the algorithmic codes, as existing routines are used, in Table 1: Mass distribution for the first three subrockets conjunction with analytical developments and a simple im- plementation of firework algorithm. Hence, the methodology (i) (i) (i) (i) treated in this paper is intended to: (i) yield a reasonable solu- i m0 mS mP mU tion for performance evaluation of multistage rockets and (ii) 1 21643 kg 1736 kg 12810 kg 6897 kg represent a technique for generating a suitable first-attempt 2 6897 kg 915 kg 3749 kg 2033 kg guess trajectory to be employed by more refined algorithms 3 2033 kg 346 kg 1173 kg 514 kg tailored to optimizing the overall trajectory. 2. ROCKET MODELING 2.1. Propulsive thrust The four stage rocket that is being considered is the Scout The propulsive characteristic of the launch vehicle can be de- launcher, which is a rocket designed to place small satellites scribed in terms of thrust magnitude T (j) and specific impulse Table 2: Specific impulse for the four stages where the coefficients CL and CD depend on the Mach num- ber M and the aerodynamic angle of attack α. The aerody- j 1 2 3 4 namic surfaces used in the computation are the cross surfaces and are listed in the Table 3. The C and C coefficient Isp(j) 260 s 288 s 284 s 270 s L D Table 3: Aerodynamic surfaces for the first three subrockets (j) ISP , with superscript j referring to the stage number. In Ta- ble 2 the specific impulses for the four stages are listed. j 1 2 3 While the specific impulse is considered time-independent S(i) 1.026 m2 0.487 m2 0.458 m2 for all the four stages, the thrust is obtained through a linear interpolation of the experimental thrust data which are given are obtained using a four degree polynomial interpolation to at discrete times.