The Development of Autonomous Dynamic Trajectory Optimization Control of Launch Vehicles

SONG Zhengyu

China Academy of Launch Vehicle Technology,Beijing 100076

Abstract:This paper first introduces the technical requirements for autonomous flight,with a brief review of the International Academy of Astronautics (IAA) study group,“autonomous dynamic trajectory optimization control of launch vehicle”.Two research scenarios,ascent rescue and powered descent,are compared from the viewpoint of optimal control.On this basis,the technologies on the autonomous trajectory planning and control under the thrust-drop failures in the ascending phase,and the autonomous guidance method during the powered landing for the recovery of the rockets are discussed respectively.For the ascending problem,the characteristics of different solutions,including the iterative guidance method (IGM)-based residual carrying capacity evaluation,the state-triggered indices (STI),the joint planning with the payload's performance,and the multiple graded optimization (MGO),are analyzed for comparison.For the landing problem,the challenges such as the feasible region reduction caused by high thrust weight ratio (HTWR)and the disturbance adaptability brought by the limited feasible region,are studied in detail,as well as the onboard planning demonstration flight in China are introduced.Finally,the foundations supporting the above methods are summarized,which play an important role in promoting the flight autonomy.

Key words:launch vehicle,dynamic optimization,onboard planning,rescue orbit,powered descent

1 INTRODUCTION

Autonomous dynamic trajectory planning is proposed with the increasing demand for the autonomous flight of launch vehicles.For a long time,it was taken that the launch vehicle flies automatically based on the prescribed scenario.The demand for autonomous flight planning is mainly due to the following two scenarios not foreseen in the traditional design.One is for the failure of thrust in the ascending flight,where the time of the failure and the magnitude of the thrust reduction cannot be predicted,hence leading to many mission failures.If online trajectory planning was available,the possibility of the payload reaching a parking orbit or even reaching its target orbit,would be increased.The second occurs in the powered descent (PD)phase before the rocket returns to Earth vertically.The switch conditions between the aerodynamic deceleration and the PD phases are difficult to be predicted in advance,so offline flight trajectory planning wouldn't make any sense,hence online planning is absolutely necessary.

The autonomous dynamic trajectory planning method can plan the flight trajectory or guidance commands online during the real-time motion of the rocket,and meet the complicated process and terminal constraints in-flight.Considering the whole flight process,this method has strong adaptability and robustness to fulfill the mission with strict constraints during strong environmental uncertainty.The research in this area is still at its initial stage,and few achievements have been released (especially for the thrust drop failure while ascending).However the study has attracted great attention worldwide,leading to the work of adaptive guidance and mission planning[1],adaptive optimal guidance[2],end-to-end dynamic optimization[3],autonomous guidance[4],and computational guidance[5].

With the above background,some scholars from China and other countries proposed to set up a study group which focuses on autonomous dynamic trajectory optimal control of launch vehicles[6,7].This paper introduces the latest progress on this study.

2 REVIEW OF THE STUDY GROUP

A data-driven artificial intelligence (AI) algorithm may be the first choice to improve the autonomy of the launch vehicle.However,an AI algorithm,based on neural networks (NN),requires a large number of training samples,which are related to specific flight scenarios.Here the demand on online planning in-flight mainly comes from two kinds of uncertainties:the unpredictability of the failure in the ascending phase and the uncertainty of the initial state in the PD phase.It's difficult to generate training samples of uncertain situations,which weakens the NN's performance.

The flight of a rocket can be expressed by a mathematical model in theory,so we can use indirect or direct methods to solve the optimal control problems.However,the indirect method is limited by the treatment of reducing the order of the dynamic equations and simplifying the model and constraints; while the direct method,although widely used for offline planning with many outstanding nonlinear programming (NLP) solvers,still faces huge technical obstacles for an online application,such as real-time performance and convergence.

However,thanks to the rapid development of computer technologies in the past 30 years,model-based online optimization methods have emerged in engineering practice[8],and would have a broad application if their performance could be further improved.

At the spring meeting of the International Academy of Astronautics (IAA) in April 2019,Chinese scholars proposed a study group (SG) on the autonomous dynamic trajectory optimal control of launch vehicles,which was approved by the Science Action Committee of IAA.The SG members come from China,Germany,France and Russia.They admitted that the loss of many missions may be saved by advanced guidance and control technology.The mission reliability could be improved,or any negative impact reduced,if onboard re-planning could be conducted.In addition,the powered landing of a rocket is affected by the wind disturbance or atmospheric uncertainty,so pinpoint soft landing would benefit from autonomous trajectory planning.

From the view of the optimal control,the features of autonomous trajectory planning in the ascent and landing scenarios are slightly different,see comparison in Table 1.

From Table 1,it can be seen that the terminal constraints of the ascent flight are functions of the state variables with stronger nonlinear characteristics.It is difficult for the CVX approach to solve the rescue problem,but the numerical computation based on ACM could be adopted.At the same time,a reasonable initial guess is important to meet real-time needs.For the landing task,the terminal constraints are the state variables themselves,here the CVX or SCP approach is usually adopted.

In the case that the prescribed target orbit is un-reachable,the question about how to define the optimal rescue orbit needs to be discussed first,this can be seen from the studies of different methods in Section 3.So,the IAA study group proposed the following technical routes,see Figure 1.

Autonomous trajectory planning is not only the key technology for engine-out/thrust-drop rescue and powered landing recovery,but also widely used for pinpoint landing in an atmospheric environment (such as Mars),large-diverting of vehicles on planets,surface cruise around gravity-free celestial bodies,and even orbit transfer control,etc.It has become the fundamental prerequisite for greater scientific returns in deep space exploration.

Table 1 Comparison of the autonomous trajectory planning between ascent rescue and powered descent scenarios

Figure 1 Technical routes for SG 3.32

3 AUTONOMOUS GUIDANCE METHOD FOR ASCENT FLIGHT

When the propulsion system fails,but no explosion or other catastrophic events occur,the resultant could be divided into four categories:the mission is still saved,such as the launch of GPS 2F-3 satellite by Delta 4 on October 4,2012[9]; the mission is partially affected,such as the loss of the second or rideshare payload[10]; the payload is returned although the mission fails,which occurred for example in the era of the Space Shuttle[11]; and the most serious consequence,the complete loss of the mission and payload.

In December 2010,a new version of the Proton rocket equipped with the DM-03 upper stage launched three GLONASS satellites from Baikonur.Due to the increased amount of fuel in the upper stage,the rocket couldn't insert the upper stage into the target orbit,leading to the failing of all satellites.This failure was not caused by a reduction or loss of thrust,but the consequence was the same as that.It should be pointed out that the upper stage had sufficient propellant to make up for the lack of the velocity and altitude.At the 2016 IAF conference,Russian scholars proposed adaptive optimal guidance[2]to deal with this type of failure.Through online re-planning by the upper stage,the mission may still be accomplished[2],demonstrating the advantages of end-to-end optimization.In February 2019,the Soyuz-2-1b/Fregat-M suffered a similar situation when launching the EgyptSat-A satellite from Baikonur,where a premature cutoff of the 3rd stage engines caused by the amount of propellant loaded switched between fuel and oxidizer.The altitude inserted by the Soyuz-2-1b rocket was 57 km lower than the prescribed value,but fortunately,the flight computer aboard the upper stage,Fregat-M,had detected the abnormal shutdown and determined that it had enough propellant for compensating for the deficiency in velocity and altitude.It autonomously commanded its main propulsion system to burn longer than planned.The payload was finally inserted into an orbit close to the target orbit with the exception of the longitude of the ascending node,hence this was a successful rescue[12].

For the cases that the rocket could still enter into the planned orbit when failure occurs,they usually occur when the rocket has a margin of carrying capacity for the mission,so only the flight trajectory to the target orbit needs to be re-planned.However,when the capacity loss is large enough that the target orbit is un-reachable,flying to it could cause payload falling,which has happened many times.Under these situations,it is necessary to find an optimal orbit match with the remaining carrying capacity while planning the flight trajectory to it.The above two situations can be jointly optimized as the problem was first discussed in the reference [13].

Very few literatures were published regarding the study of the optimal rescue orbit,but the emergent engine-out adaptability has gained increased focus.For example,NASA requires the latest SLS Block-1B launch vehicle to ensure the safety of the crew in the event of a core engine failure,and select new targets or abort tasks when the original target orbit is not reachable,but the response strategy and the time to switch to the alternative orbit are determined by offline simulation[14]rather than onboard real-time optimization.

The followings are some online autonomous rescue methods for discussion.

3.1 Residual Carrying Capacity Evaluation Based on IGM Integral

The iterative guidance algorithm (IGM) can solve the flight trajectory on-line,but it is difficult to directly evaluate whether the target trajectory is reachable.In theory,if the time-to-go (tgo),calculated by IGM,is less than the time when the propellant is exhausted,the target orbit could be reached.However,in order to obtain an analytical solution,the gravity is approximately treated in IGM,causing errors intgoevaluation when the flight trajectory is a large arc.The error has no effect on normal flight,because when approaching the target,the shorter the arc,the smaller the deviation,i.e.,the accuracy oftgoprediction is constantly improved.

However,when the failure occurs earlier than the scheduled injection time,it would misjudge if the target orbit can be reached according totgo.For this reason,a three-degree-offreedom integration based on a forward-step IGM command is conducted onboard,then the propellant required to reach the target or the lowest acceptable safe orbit is calculated,and compared with the remaining propellant.If there's not enough propellant for the safe orbit,the rescue should be given up.If the target orbit is achievable,IGM should be used to re-plan the flight trajectory.Otherwise,it should find an optimal rescue orbit and plan the flight trajectory[15].

The optimality of the rescue orbit is defined by the weighted sum of the absolute deviations of the orbital elements,as shown in the following:

The adaptive collocation method (ACM) is used to solve the problem,where the initial guess plays a key role for ACM to quickly converge to the solution.

3.2 State-triggered Indices

Due to the strong nonlinear relationship between the five orbital elements,it is difficult to determine the weights in Eq.(1).So,a state-triggered indices (STI) strategy was proposed,as shown in Figure 2[16].

The first priority when a failure occurs is to ensure the safety of the payload and avoid crashing to the ground,so the STI method looks for the maximum circular orbit in the orbital plane formed at the failure point (the objective is set asJ1).If the height of the circular orbit is less than a safety threshold,the rescue would be given up; if the height exceeds the perigee height of the target orbit,it indicates that there is extra capacity to adjust the orbital plane deviation (so the objective is set asJ2); if the solution ofJ2is coplanar with the target orbit and there's still residual propellant left,the orbital shape is further adjusted (the objective is set asJ3).Each objective is triggered only when a certain state is satisfied and replaces the previous one.At the same time,the solution of the former objective could be used as the initial guess of the current one,accelerating the computation.For a more detailed discussion,see reference [16].

The thrust will not stay fixed after a propulsion system fails,so the STI method is called in every guidance cycle,so as to suit the thrust fluctuations.

3.3 End-to-end Planning

The end-to-end (ETE) optimization method is to find a rescue orbit considering payload capacity,and minimum propellant consumption for payload orbital transfer.The consumption is related to the orbital elements of the rescue and target orbits.

Using the simulation model of reference [16],Figure 3 shows the different effects for one case in STI and ETE planning,whereJ1in the lower part of Figure 3 refers to theJ1objective in Section 3.2.

It's assumed that the thrust reduces to 77.3% of the original at 48.9 s (the flight time of the second stage),when the orbit height is 149.3 km.The target orbit is an elliptical orbit of 200 km × 300 km.The solution of STI is a 160 km circular orbit,which is in the orbital plane formed at the time of failure; the fuel consumption for payload orbit transfer is 1051.5 kg.The solution of ETE is an elliptical orbit of 99.1 km × 300 km,which is coplanar with the target orbit; the payload could transfer orbit at the height of 200.3 km,and the fuel consumption would be 610 kg,less than that of STI method.

The STI method aims to leave the payload in an orbit,while the ETE method regards the payload as the last stage of the rocket.The above example is the case found during off-line simulation,while the study of online optimization is still in progress.

3.4 Multiple Graded Optimization

With the increase of constellation missions,multiple satellites sharing one rocket has become a common launch mode.However,once the launch fails,all the payloads onboard may be lost,resulting in considerable losses.

If the satellites could be handled in groups,i.e.,launching as many satellites as possible into their target orbit,meanwhile sending the rest into a parking orbit to wait for rescue,it can avoid total loss.Hence some rideshare payloads should be separated in advance,and we call this strategy multiple graded optimization (MGO).MGO can be widely adopted for all rideshare missions,where the main payload has the priority to the target orbit if a failure occurs.MGO involves the evaluation of the maximum carrying capacity to the target orbit under the current state,and takes into account the multiple burns and the relevant degraded orbits.This study is also under way.

In addition,if the fault does not occur in the last stage,or the last stage has the ability of multiple burns,the solving for the onboard rescue problem will be more challenging.

Figure 2 The block diagram of STI based joint dynamic optimization strategy

Figure 3 The comparison between ETE and STI optimization

4 GUIDANCE METHOD FOR POWERED DESCENT

The vertical landing by retro-propulsion has been proved to be feasible by Falcon 9,so the other rockets are also seeking their own recovery possibilities.However,the boosters of the rockets from SpaceX or Blue Origin companies are configured with multiple engines,or engine cluster.During landing,only one or a few engines are re-started,having the equivalent effect of deep throttling.At the same time,the engines can also be throttled,extending the range of thrust regulation and greatly reducing the difficulty of guidance control.

The engine cluster solution was considered to be defective in reliability design at one time,where the complex composition of the propulsion system and low structural efficiency were the concerns.It's a common choice for launch vehicles in-service today to adopt a few but larger thrust engines operating in parallel.However,even if the throttling ratio of the engine is the same as that of the Falcon 9's engines,these rockets in-service cannot have the same throttling effect,resulting in a thrust far greater than the gravity in the PD phase.In addition it is also very hard to further adjust the throttling ratio of high thrust engines,leading to the challenges to the guidance control.If this problem cannot be solved,the launchers in-service may not be recovered just through engine throttling alone,and the cost and time spent would be tremendous to design a reusable rocket with an new engine cluster.

The guidance method for powered descent is also one of the key technologies for deep space exploration missions.In reference [17],a unified mathematical model for guidance control is constructed for landing on Earth,the moon,and Mars,where the differences between these missions are:

1) For lunar landing,the aerodynamic forces are negligible;the mass-flow-rate of the thruster is also very limited,so the lander mass could be regarded as constant.An analytical or tracking algorithm is qualified for the mission,and the online planning method can also be adopted when confronting obstacle avoidance or optimal landing site selection.

2) For a pinpoint soft landing on Mars,the initial conditions of the PD phase cannot be determined in advance due to the limited control accuracy during aerodynamic deceleration in the previous phase,so online trajectory planning is needed.The aerodynamic drag during the PD phase could be ignored because the velocity has decreased,so the tracking method is effective in the following control process.

If the requirement of landing accuracy is not so high,the offline planning and onboard tracking method can still be adopted,resulting in landing position deviations in kilometers.

3) Vertical landing on Earth is most challenging for guidance control.Aerodynamic drag,the nonlinear relationship between the atmospheric density and the altitude,the limited engine throttling range,and the large mass-flow-rate of the rocket engines,make onboard planning difficult,and iterative optimization is needed throughout the process.

The problems of vertical landing with a limited engine throttling range,i.e.,under high thrust-to-weight ratio (HTWR) condition are discussed as follows.

4.1 HTWR and Physically Feasible Region

Due to the influence of the atmosphere and restricted by the control measures in the decelerating phase,the initial conditions of the PD phase have a large dispersion,so online trajectory planning is required.The physically feasible region refers to the state space represented by the velocity and position.When the state of the rocket is in the feasible region,a pinpoint soft landing can be realized by adjusting the thrust magnitude and direction,i.e.,a solution can be found under this condition; and the thrust regulation range of the rocket determines the range of the feasible region.The vehicle's state during the landing process must be in this region,as the smaller the feasible region,the more difficult the guidance control.So it is very important to analyze the feasible region and deviation adaptability affected by HTWR conditions.

For a HTWR rocket,it always decelerates once the engines start,and shuts down when the position,velocity and attitude meet the terminal constraints simultaneously.An off-line algorithm solving the physically feasible region is proposed in reference [18],and using the algorithm,the following scenario is discussed:with a maximum thrust of 1200 kN,and an engine specific impulse of 300 s; assuming an initial mass entering the PD phase is 39.6 t,with a structural weight of 32 t,and theSrefis 8.8 m2.The feasible regions under differentκ,the engine throttling depth,are shown in Figure 4.

It can be seen from Figure 4 that the narrower the thrust adjustment range,the smaller the feasible range,which means that the switch conditions of the aerodynamic deceleration and the powered descent phases,and the deviations of the velocity and position during the landing,should be well controlled.This leads to the discussion in the next section.

Figure 4 The physically feasible region under different throttling depths

4.2 Maximizing Disturbance Adaptability

With the a fuel-optimal objective,the guidance control sequence usually presents a “bang-bang” feature,i.e.,the engines mainly work in the maximum or minimum state.Taking Figure 5 for example to analyze the adaptability to the disturbances.

Assuming that the required thrust in the axial directionTy=Tmax.If the disturbance momentMdoccurs at this time,such as the wind interference,the engine must swing at an angle to generate a control momentMcto compensate it,

whereαrepresents the angle between the thrust and the body axis,andLis the distance from the engine to the center of mass.

Figure 5 The compensation for disturbances

Even ifT'=Tmax(the thrust cannot be further increased),the thrust in the axial direction is less than the required thrust,Tmax,

The thrust deviation caused by disturbance compensation affects the motion of the rocket and eventually affects the landing accuracy.So,the fuel-optimal bang-bang control command is not preferable for rocket landing when disturbances and uncertainties exist.In case of HTWR,the range of the feasible region is so small that it would not be able find a solution if the rocket deviates slightly to exceed the feasible region,resulting in violating the terminal constraints.

In reference [18],a new objective is proposed to maximize the disturbance (deviation) adaptation range.It adopts a trajectory propelled by a middle thrust within the regulation range as the reference,which is planned off-line,and redefines the SOCP constraint to handle the deviations.The simulation shows that this method is more adaptive,but also consumes more propellant.

4.3 GNC Demonstration

The guidance control plays an important role in the flight of the launch vehicle,which leads to a conservative trend when choosing a new guidance method for the rocket.For example,an improved PEG guidance method[14,19,20]is selected for NASA's SLS after detailed comparison,and the optimal guidance algorithm was put on the shelf again due to the relatively lower maturity,although it also has a long research history[21].It's also very rare to use the numerical method in-flight,which was only reported in the SpaceX rocket recovery mission[22].The CVX based algorithm,G-FOLD[4,23],sponsored by NASA and aiming at pinpoint soft landing on Mars,has been tested intensively on the demonstrator,Xombie,but there's still no report on actual future application.

It has been widely accepted to verify an algorithm in an embedded environment on GNC demonstrators,such as in the EAGLE of DLR,FROG of CNES,CALLISTO driven jointly by CNES,DLR,and JAXA.Beijing Aerospace Automatic Control Institute also tested an onboard trajectory optimization algorithm on a demonstrator[17],shown in Figure 6.

The demonstrator was designed with reference to the slenderness and thrust-to-weight ratios of LM-8 core booster,equipped with two small UAV engines and a composite shell.All avionics were COTS products.

Based on the ECOS solver,the switch conditions between different phases,and the cost function which guides the demonstrator to the top of the landing site,the algorithm has been successfully verified on several flights.The switch condition is fitting in the longitudinal direction,shown as the dark yellow curve in Figure 7.The intersection point of the height-velocity curve and the switch condition represents the state when the engines are re-started.

The landing accuracy was also evaluated by Monte Carlo simulations,shown in Figure 8.

Figure 8 Monte Carlo simulation

The real-time performance,convergence of the autonomous guidance method with the embedded device,the applicability of CVX under HTWR,the feasibility of the switch condition,and the delay tolerance for the time spent on the calculation,were assessed in the flights.When comparing rocket recovery,the mass of the demonstrator was almost constant due to the lower mass-flow-rate; the disturbances were weak which are usually proportional to the amplitude of the thrust,so the impact of the attitude control on the guidance accuracy was not fully simulated; the thrust regulation was faster than that of the rocket engine,so the delay tolerance by the throttling of the rocket engines was not checked.These differences would be tested later by other vertical-takeoff-vertical-landing vehicles equipped with rocket engines.

5 CONCLUSION

Autonomous trajectory optimization control was introduced in this paper.It is an on-line dynamic system optimization process based on an effective model.It is difficult to obtain analytical solutions for these problems when considering the complicated process and terminal constraints,except for that when the control accuracy is relaxed so the model and constraints could be tailored.However,the deviation brought about by the approximation and simplification approach could not meet higher accuracy requirements,so we adopted the numerical method,which is supported by the development of the following technologies:

1) Automatic tuners for internal procedures for the optimization,including numerical integration,array gradient calculation,and intelligent decision algorithms to guide and redirect the complex optimization iterative process.

2) Autonomous and smart initial guess generator for onboard mission design.Utilising a reasonable initial value,the numerical method can present the same calculation efficiency as the analytical method.So the smart initial guess becomes a cornerstone,and the analytical solution of a simplified model can also be taken as the initial value.

3) Powerful mission design and optimization software.

4) New technologies which support real-time online operations in new flight regimes.

With the dynamic trajectory optimization method,flight control could be flexible and adaptive,exhibiting a certain degree of autonomy.