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Figure 6. The trajectory of Design ID 20245 and reference points. point represents a solution in multi-objective optimization where no further improvement in one objective can occur without compromising another. This concept enables decision- makers to explore trade-offs and identify balanced solutions that maximize multiple ob- jectives simultaneously, offering a comprehensive and efficient approach to complex op- timization problems. As Design ID 20245 points to a Pareto optimal solution, the results of Design ID 20245 in this study were compared with the original Minotaur-I launch ve- hicle. Figure 6 provides the comparison of trajectory, velocity, and flight path during the flight for the optimization results of Design ID 20245 for the hybrid last stage, along with the reference points of the solid last stage, for the Minotaur-I launch vehicle [56].

Figure 6 The trajectory of Design ID 20245 and reference points. point represents a solution in multi-objective optimization where no further improvement in one objective can occur without compromising another. This concept enables decision- makers to explore trade-offs and identify balanced solutions that maximize multiple ob- jectives simultaneously, offering a comprehensive and efficient approach to complex op- timization problems. As Design ID 20245 points to a Pareto optimal solution, the results of Design ID 20245 in this study were compared with the original Minotaur-I launch ve- hicle. Figure 6 provides the comparison of trajectory, velocity, and flight path during the flight for the optimization results of Design ID 20245 for the hybrid last stage, along with the reference points of the solid last stage, for the Minotaur-I launch vehicle [56].

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Table 1. Comparison of advantages and disadvantages of solid, liquid, and hybrid rocket motors.
Table 1. Comparison of advantages and disadvantages of solid, liquid, and hybrid rocket motors.
According to Table 1, solid iability due to minimal moving age and fixed thrust profiles as restart capabilities for intricate propellant combinations. Nonet rocket propulsion offers advantages in simplicity and re- parts, featuring stable propellants ideal for extended stor- seen in boosters. However, a drawback is their lack of throttle control, impeding precise adjustments, and once ignited, they are not easily shut down. This contrasts with liquid propulsion, which provides precise throttle control and missions like orbital maneuvers. Liquid propellants en- hance fuel efficiency and performance with high specific impulse allowing for a variety of heless, the complexity of liquid motors leads to increased manufacturing and maintenance costs, and volatile propellants demand cautious han- dling due to potential degradation. On another point, hybrid propulsion amalgamates solid safety with liquid control, excelling in throttle adaptability and potentially higher efficiency, and some hybrid propellants are environmentally friendlier.
According to Table 1, solid iability due to minimal moving age and fixed thrust profiles as restart capabilities for intricate propellant combinations. Nonet rocket propulsion offers advantages in simplicity and re- parts, featuring stable propellants ideal for extended stor- seen in boosters. However, a drawback is their lack of throttle control, impeding precise adjustments, and once ignited, they are not easily shut down. This contrasts with liquid propulsion, which provides precise throttle control and missions like orbital maneuvers. Liquid propellants en- hance fuel efficiency and performance with high specific impulse allowing for a variety of heless, the complexity of liquid motors leads to increased manufacturing and maintenance costs, and volatile propellants demand cautious han- dling due to potential degradation. On another point, hybrid propulsion amalgamates solid safety with liquid control, excelling in throttle adaptability and potentially higher efficiency, and some hybrid propellants are environmentally friendlier.
oO Oo tics. There vehicles and all subsystems, making it crucial to obtain the most optimal design o Ss Minotaur- 8 Mass, moment of inertia, and center of gravity are fundamental parameters in flight simulation within the structural model. The mass value of the launch vehicle has a deci- sive impact on acceleration, velocity, and maneuverability. The moment of inertia deter- mines the response to angular motions, influencing maneuverability. The center of gravity represents the balance point of the launch vehicle and affects stability if not in an ap priate position. Accurate modeling and simulation of these parameters are crucial to un- derstanding how the launch vehicle will behave in real-world conditions. Careful trac tructural eometry, re oO f these parameters in flight simulation is critical for safe and effective flight operati In the design phase, accurately determining these parameters plays a significant ro ptimizing the launch vehicle’s performance and achieving the desired flight characteris- fore, the vehicle structure serves as the mechanical interface between launch f the f the configuration to meet mission requirements. For the structural models o launch vehicle, CATIA V5R19 product design software was utilized [24]. f the fina in the optimization process. pro- king ons. ein The stage characteristics, and dimensions utilized in both the structural and aero- ynamic models are depicted in Figure 1. It is important to highlight that the dimensions stage were updated and optimized using a hybrid rocket motor and included Figure 1. The representation of the Minotaur-I launch vehicle geometry used in structural and aer- odynamic subsystems, obtained from DATCOM.
oO Oo tics. There vehicles and all subsystems, making it crucial to obtain the most optimal design o Ss Minotaur- 8 Mass, moment of inertia, and center of gravity are fundamental parameters in flight simulation within the structural model. The mass value of the launch vehicle has a deci- sive impact on acceleration, velocity, and maneuverability. The moment of inertia deter- mines the response to angular motions, influencing maneuverability. The center of gravity represents the balance point of the launch vehicle and affects stability if not in an ap priate position. Accurate modeling and simulation of these parameters are crucial to un- derstanding how the launch vehicle will behave in real-world conditions. Careful trac tructural eometry, re oO f these parameters in flight simulation is critical for safe and effective flight operati In the design phase, accurately determining these parameters plays a significant ro ptimizing the launch vehicle’s performance and achieving the desired flight characteris- fore, the vehicle structure serves as the mechanical interface between launch f the f the configuration to meet mission requirements. For the structural models o launch vehicle, CATIA V5R19 product design software was utilized [24]. f the fina in the optimization process. pro- king ons. ein The stage characteristics, and dimensions utilized in both the structural and aero- ynamic models are depicted in Figure 1. It is important to highlight that the dimensions stage were updated and optimized using a hybrid rocket motor and included Figure 1. The representation of the Minotaur-I launch vehicle geometry used in structural and aer- odynamic subsystems, obtained from DATCOM.
Figure 2. Circular fuel grain port.
Figure 2. Circular fuel grain port.
Mathematically, the optimization problem can be defined as shown in Equation (8). The objective is to determine the optimal values for the propulsion characteristics of the hybrid stage and the TVC angles of the first and last stages. These values should maximize the total impulse-to-weight ratio (/;/wo) while adhering to I,, constraints and ensuring the target orbit parameters are met, as stated in Equations (9)-(12). The objectives are:
Mathematically, the optimization problem can be defined as shown in Equation (8). The objective is to determine the optimal values for the propulsion characteristics of the hybrid stage and the TVC angles of the first and last stages. These values should maximize the total impulse-to-weight ratio (/;/wo) while adhering to I,, constraints and ensuring the target orbit parameters are met, as stated in Equations (9)-(12). The objectives are:
Figure 3. Flowchart of coupled analysis and multi-objective MDO approach. Table 2. Minotaur-I launch vehicle characteristics.
Figure 3. Flowchart of coupled analysis and multi-objective MDO approach. Table 2. Minotaur-I launch vehicle characteristics.
Figure 4. Workflow of the multi-objective multidisciplinary optimization process.
Figure 4. Workflow of the multi-objective multidisciplinary optimization process.
Table 3. The couplings between subsystems of launch vehicles. The optimization process was executed on a workstation equipped with an Intel Xeon Silver 4216 CPU operating at 2.10 GHz, 64 cores, and 128 GB RAM. The process runs for approximately 8 h and 11 min. To expedite the process, 30 models were created and run simultaneously in parallel. The initial design parameters were selected using the de- sign of the experiment methodology. Throughout the optimization process, the design parameters were constrained to scan a wide range. However, due to the simultaneous fulfillment of targeted trajectory parameters being a highly challenging criterion, it has been observed that as the iteration steps progress during optimization, the algorithm con- verges until the global optimum solution is reached, and the optimization process is ter- minated. A total of 30,000 iterations were performed, resulting in 29,688 unfeasible de- signs and 310 errors. An unfeasible solution indicates that a specific point in the design space cannot be physically realized. In other words, it denotes situations where con- straints related to that design configuration are not satisfied or conflict. Errors signify a potential issue with simulation or analysis tools. This could be due to various reasons such as an error in input parameters, model configuration, or missing/corrupted data files. However, since only 1.03% of the entire solution space is affected, it does not carry critical importance in the optimization solution here. Moreover, two feasible points were ob- tained for the multi-objective multidisciplinary problem. The reasons for finding a few feasible points lie in the complexity of multi-stage launch vehicles and hybrid-propellant motor optimizations. Additionally, the scarcity of feasible points indicates that solutions
Table 3. The couplings between subsystems of launch vehicles. The optimization process was executed on a workstation equipped with an Intel Xeon Silver 4216 CPU operating at 2.10 GHz, 64 cores, and 128 GB RAM. The process runs for approximately 8 h and 11 min. To expedite the process, 30 models were created and run simultaneously in parallel. The initial design parameters were selected using the de- sign of the experiment methodology. Throughout the optimization process, the design parameters were constrained to scan a wide range. However, due to the simultaneous fulfillment of targeted trajectory parameters being a highly challenging criterion, it has been observed that as the iteration steps progress during optimization, the algorithm con- verges until the global optimum solution is reached, and the optimization process is ter- minated. A total of 30,000 iterations were performed, resulting in 29,688 unfeasible de- signs and 310 errors. An unfeasible solution indicates that a specific point in the design space cannot be physically realized. In other words, it denotes situations where con- straints related to that design configuration are not satisfied or conflict. Errors signify a potential issue with simulation or analysis tools. This could be due to various reasons such as an error in input parameters, model configuration, or missing/corrupted data files. However, since only 1.03% of the entire solution space is affected, it does not carry critical importance in the optimization solution here. Moreover, two feasible points were ob- tained for the multi-objective multidisciplinary problem. The reasons for finding a few feasible points lie in the complexity of multi-stage launch vehicles and hybrid-propellant motor optimizations. Additionally, the scarcity of feasible points indicates that solutions
Figure 5. Broken constraints and distribution of the feasible designs for y, h, V. Upon successful completion of the optimization process, feasible designs and the cor responding broken constraints are acquired. These results are presented in Figure 5 where each bubble with a red foreground marker represents a feasible solution. Due t the complexity of the multi-objective multidisciplinary problem, the plot specificalh showcases the minimum flight path and minimum V error constraints. Furthermore, Tabl 4 provides the properties of the feasible points displayed in Figure 5. The broken con straint pie chart illustrates the percentage values at which constraints are broken for al design points obtained throughout the entire optimization process. This enables an as sessment of which constraint is more challenging.
Figure 5. Broken constraints and distribution of the feasible designs for y, h, V. Upon successful completion of the optimization process, feasible designs and the cor responding broken constraints are acquired. These results are presented in Figure 5 where each bubble with a red foreground marker represents a feasible solution. Due t the complexity of the multi-objective multidisciplinary problem, the plot specificalh showcases the minimum flight path and minimum V error constraints. Furthermore, Tabl 4 provides the properties of the feasible points displayed in Figure 5. The broken con straint pie chart illustrates the percentage values at which constraints are broken for al design points obtained throughout the entire optimization process. This enables an as sessment of which constraint is more challenging.
Table 4. The properties of feasible solutions. Observing the broken constraints in Figure 5, it can be noted that the y, h, and V constraints are broken to a similar extent in the broken constraint percentages, while the Is) constraints exhibit smaller deviations compared to other constraints. In this problem, it is indicated that all constraints, namely y, h, and V, are equally important for achieving the required orbital parameters to place the payload into the targeted orbit. If any of these constraints are not met, the payload cannot be placed into the targeted orbit. This high- lights the challenge of simultaneously satisfying all these constraints. On the other hand, aiming for a higher specific impulse than the solid-propelled original stage (287 s) is less restrictive relative to other constraints, and limits on design points with lower specific impulse have been broken. Therefore, while it can be stated that performance improve- ment is achieved with an I,, value greater than 287 s, it can be explicitly expressed that the mission is unsuccessful when any of the orbital parameters cannot be fulfilled. As a result, a few feasible points are obtained in the optimization solution. A Pareto optimal
Table 4. The properties of feasible solutions. Observing the broken constraints in Figure 5, it can be noted that the y, h, and V constraints are broken to a similar extent in the broken constraint percentages, while the Is) constraints exhibit smaller deviations compared to other constraints. In this problem, it is indicated that all constraints, namely y, h, and V, are equally important for achieving the required orbital parameters to place the payload into the targeted orbit. If any of these constraints are not met, the payload cannot be placed into the targeted orbit. This high- lights the challenge of simultaneously satisfying all these constraints. On the other hand, aiming for a higher specific impulse than the solid-propelled original stage (287 s) is less restrictive relative to other constraints, and limits on design points with lower specific impulse have been broken. Therefore, while it can be stated that performance improve- ment is achieved with an I,, value greater than 287 s, it can be explicitly expressed that the mission is unsuccessful when any of the orbital parameters cannot be fulfilled. As a result, a few feasible points are obtained in the optimization solution. A Pareto optimal
Table 5. Trajectory comparison of optimization results and reference points. Table 6. Propulsion comparison of optimization results and reference points.
Table 5. Trajectory comparison of optimization results and reference points. Table 6. Propulsion comparison of optimization results and reference points.
4. Comparison of the Comprehensive Modeling
4. Comparison of the Comprehensive Modeling
Figure 7. The simulation results of the 3DOF model and 6DOF model for the hybrid last stage.
Figure 7. The simulation results of the 3DOF model and 6DOF model for the hybrid last stage.
Figure 8. The simulation results of the 3DOF model and 6DOF model for the original Minotaur-I launch vehicle.
Figure 8. The simulation results of the 3DOF model and 6DOF model for the original Minotaur-I launch vehicle.
Table 7. The comparison summary of 3DOF and 6DOF results for the hybrid last stage.
Table 7. The comparison summary of 3DOF and 6DOF results for the hybrid last stage.
Table 8. The comparison summary of 3DOF and 6DOF results for the solid last stage. When Table 7 is examined, in the 3DOF model, complex angular velocity calculations, such as those found in the 6DOF model, are not present. Additionally, effects like moment of inertia, center of mass, and moment changes are not included, causing the gap between the two models to widen as the simulation progresses. Firstly, when we examine the ve- locity (V) parameter, it is observed that the (DOF model reaches a speed of 7425 m/s, while the 3DOF model reaches 6567 m/s. The target speed is set at 7482.2 m/s, and in this case, it is observed that the (DOF model deviates by 0.76%, whereas the 3DOF model deviates by 11.55%. Focusing on another parameter, the flight path angle (y), the 6DOF model achieves —0.0074 degrees, while the 3DOF model reaches 7.2708 degrees. The target angle of attack is set at 0 degrees, and in this situation, it is observed that the 6DOF model is closer to this target, whereas t he 3DOF model exhibits a significant deviation. Lastly, when we concentrate on the altitude (i) parameter, the (DOF model reaches an altitude of 747.6 km, while the 3DOF model reaches 807.6 km. The target altitude value is set at 741.2 km, and in this case, it is observed that the 6DOF model deviates by 0.86%, whereas the 3DOF model deviates by 8.02%. This analysis provides a detailed insight into the performance differences between the two models.
Table 8. The comparison summary of 3DOF and 6DOF results for the solid last stage. When Table 7 is examined, in the 3DOF model, complex angular velocity calculations, such as those found in the 6DOF model, are not present. Additionally, effects like moment of inertia, center of mass, and moment changes are not included, causing the gap between the two models to widen as the simulation progresses. Firstly, when we examine the ve- locity (V) parameter, it is observed that the (DOF model reaches a speed of 7425 m/s, while the 3DOF model reaches 6567 m/s. The target speed is set at 7482.2 m/s, and in this case, it is observed that the (DOF model deviates by 0.76%, whereas the 3DOF model deviates by 11.55%. Focusing on another parameter, the flight path angle (y), the 6DOF model achieves —0.0074 degrees, while the 3DOF model reaches 7.2708 degrees. The target angle of attack is set at 0 degrees, and in this situation, it is observed that the 6DOF model is closer to this target, whereas t he 3DOF model exhibits a significant deviation. Lastly, when we concentrate on the altitude (i) parameter, the (DOF model reaches an altitude of 747.6 km, while the 3DOF model reaches 807.6 km. The target altitude value is set at 741.2 km, and in this case, it is observed that the 6DOF model deviates by 0.86%, whereas the 3DOF model deviates by 8.02%. This analysis provides a detailed insight into the performance differences between the two models.
Table 9 provides a comprehensive comparison of modeling results for the hybrid las tage, considering different scenarios involving variations in center of gravity, moment c nertia, and constant aerodynamics. The table includes results for the (DOF model, com aring them to the target values, and showcases the percentage differences for each spe ific case. Looking at the V parameter, the 6DOF model’s initial result is 7425 m/s, whil he target value is set at 7482.2 m/s. When the model is simulated without a change i enter of gravity, the velocity drops to 6603.8 m/s, resulting in an 11.06% deviation fron he target. Similarly, without inertia changes, the velocity decreases to 6499.4 m/s, show ng a 12.47% deviation. In the case of constant aerodynamics, the velocity further de reases to 6195.5 m/s, with a 16.56% deviation. The combined effect of all three cases re ults in a significant deviation, with the velocity decreasing to 4613.5 m/s, exhibiting 7.87% difference from the target orbit velocity. For the y parameter, the initial 6DOF re ult is -0.0074 degrees, aiming to align with a target of 0 degrees. Without a change in CC increases to 21.8 degrees, deviating from the target. Similarly, without a moment c nertia change, y becomes —8.54 degrees, showing a deviation. In the case of constant ae1 dynamics, y increases even more to 27.41 degrees, with a substantial deviation. Th ombined effect of all three cases results in y of 80.03 degrees, indicating a significar leviation from the target. Examining the h parameter, the initial 6DOF result is 747.6 kr vith a target set at 741.2 km. Without a change in CG, the altitude increases to 1249.4 kr esulting in a 68.6% deviation from the target. Without inertia changes, the altitude de reases to 896.02 km, showing a 20.9% deviation. In the case of constant aerodynamic: he altitude further increases to 1546.61 km, with a substantial 108.7% deviation. The com ined effect of all three cases results in a significant altitude increase to 2417.6 km, exhib ting a 226.2% difference from the target. Table 9. The comparison summary of comprehensive modeling results for the hybrid last-stage ve- hicle.
Table 9 provides a comprehensive comparison of modeling results for the hybrid las tage, considering different scenarios involving variations in center of gravity, moment c nertia, and constant aerodynamics. The table includes results for the (DOF model, com aring them to the target values, and showcases the percentage differences for each spe ific case. Looking at the V parameter, the 6DOF model’s initial result is 7425 m/s, whil he target value is set at 7482.2 m/s. When the model is simulated without a change i enter of gravity, the velocity drops to 6603.8 m/s, resulting in an 11.06% deviation fron he target. Similarly, without inertia changes, the velocity decreases to 6499.4 m/s, show ng a 12.47% deviation. In the case of constant aerodynamics, the velocity further de reases to 6195.5 m/s, with a 16.56% deviation. The combined effect of all three cases re ults in a significant deviation, with the velocity decreasing to 4613.5 m/s, exhibiting 7.87% difference from the target orbit velocity. For the y parameter, the initial 6DOF re ult is -0.0074 degrees, aiming to align with a target of 0 degrees. Without a change in CC increases to 21.8 degrees, deviating from the target. Similarly, without a moment c nertia change, y becomes —8.54 degrees, showing a deviation. In the case of constant ae1 dynamics, y increases even more to 27.41 degrees, with a substantial deviation. Th ombined effect of all three cases results in y of 80.03 degrees, indicating a significar leviation from the target. Examining the h parameter, the initial 6DOF result is 747.6 kr vith a target set at 741.2 km. Without a change in CG, the altitude increases to 1249.4 kr esulting in a 68.6% deviation from the target. Without inertia changes, the altitude de reases to 896.02 km, showing a 20.9% deviation. In the case of constant aerodynamic: he altitude further increases to 1546.61 km, with a substantial 108.7% deviation. The com ined effect of all three cases results in a significant altitude increase to 2417.6 km, exhib ting a 226.2% difference from the target. Table 9. The comparison summary of comprehensive modeling results for the hybrid last-stage ve- hicle.
Table 10. The comparison summary of comprehensive modeling results for the original Minotaur-I launch vehicle.
Table 10. The comparison summary of comprehensive modeling results for the original Minotaur-I launch vehicle.
Related topics:
Multidisciplinary Design OptimizationHybrid Rocket MotorsSatellite Launch Vehicles
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