Launch Vehicle Reconfigurable Guidance Method Based on Online Trajectory Optimization

XU Guoqiang

Beijing Aerospace Automatic Institute,Beijing 100854

Abstract:In order to improve the safety and reliability of a launch vehicle,we present a reconfigurable guidance method based on online trajectory optimization for thrust decrease at the ascending stage of the launch vehicle in this paper.We used a multiple shooting method to convert the optimal control problem into a nonlinear programming (NLP)problem.Finally,we used the interior point method to solve the NLP problem.The simulation results show that the method can effectively adapt to thrust decreases by 5% and 10%.This reconfigurable guidance method based on online planning is proposed to realize launch missions under the condition of power descent,thus verifying the feasibility of the method.

Key words:trajectory optimization,multiple shooting method,optimal control,NLP,launch vehicle safety

1 INTRODUCTION

In 2002,the United States proposed the Space Launch Initiative (SLI) to study advanced guidance and control technology to improve the reliability,safety and reduce the cost of the second generation launch vehicle.When faults are encountered in the flight process,they will have a great impact on the control performance,requiring a reconfigurable ability of inner control loop.For an autonomous system,the guidance system must also have an adaptive capability,using control reconfiguration and an online trajectory generation system to mitigate faults,enabling the vehicle to recover from non-fatal failures.

The typical adaptive guidance and reconfigurable control system mainly includes three parts:control reconfiguration,guidance reconfiguration and trajectory reconfiguration.The task of control reconfiguration is to keep the attitude stable and restore the control performance as much as possible in case of failure.In the past few years,great achievements have been made in control reconfiguration,leading the development in autonomous guidance and control.The main task of guidance reconfiguration is to track the command of the reference trajectory and restore the flight stability margin to a certain extent in the face of reduced control performance.More representative research can be found in [4].Trajectory reconstruction is to replan the trajectory in order to meet the ideal terminal conditions of the mission when the inner loop control system reconstruction cannot meet the requirements in the case of a large fault.In terms of trajectory reconstruction,the more mature method is the Optimal-Path-To-Go (OPTG) method,which is based on variational theory and has been successfully applied to many guidance and control systems,including UAVs and reusable launch vehicles.

A combination of adaptive guidance and trajectory reconstruction algorithm was tested on the X-40A reentry vehicle using online simulation demonstration technology.This experiment was a flight demonstration for the first fault tolerant autonomous landing system,which shows that the X-40A can adapt to multiple control interface failures when equipped with this system.

For the research on various adaptive guidance and control technologies,researchers tested the algorithm according to their own research objectives,as oppose to collating their objectives.Hence,a series of test results for different advanced guidance and control technologies for X-33 were recorded in MSFC,and the algorithm is implemented in non-real time with high confidence simulation for X-33.The tests included power failure,control interface failure,wind interference,changing aerodynamic characteristics and mass characteristics.In addition to guidance methods,a large number of advanced control methods were tested in this project,including an adaptive reconfigurable control law based on dynamic inversion,with a linear programming method based on control allocation.

Generally speaking,there are direct method and indirect method in solving the optimal control problem (OCP).The direct methodtransforms the nonlinear control problem into the OCP,which is to solve the nonlinear programming (NLP)problem by the discretization method.The direct method can easily deal with any dynamic constraints,but it is expensive and difficult to obtain a convergent solution.The indirect methodtransforms the OCP into a boundary value problem,which is solved by Pontryaguin maximum principle (PMP) explicitly or implicitly.The indirect method can obtain the convergent solution quickly,but it is highly sensitive to the initial value.In addition,in recent years,some scholars have studied the hybrid method,which combines the direct method with the indirect method.Firstly,using the direct method to establish a good initial guess,and then using the indirect method to achieve accurate convergence.

In this paper,an online trajectory optimization algorithm is used to achieve reconfigurable control.The NLP method is used to solve the OCP when the thrust decreases.Firstly,the OCP is summarized,and a dynamic model is derived.Then several construction methods of NLP problem are given.The multiple shooting method is used to convert it into a nonlinear programming problem,which uses the inter-point method to solve the problem.Finally,this is applied to the simulation results of a two-stage launch vehicle for verification.The results show that online optimization can ensure the realization of the final reconfigurable goal.

2 OPTIMAL CONTROL PROBLEMS

The OCP can be described as establishing a type of optimal control function for a given controlled system.The system can transfer from one state to another desired by the designer under its action,and achieve a certain performance index relative to the system optimal.For dynamic optimization problems,we must define the cost function (also known as the objective function),dynamic constraints,mixed state control path constraints,terminal constraints (including initial and terminal boundary constraints) and other additional equality or inequality conditions related to state variables and control variables.

Minimizing the objective function:

Dynamic constrain:

Path constrain:

Boundary constrain:

States and control constrain:

For the optimization process of a multi-stage launch vehicle,the above OCP needs to be expanded into a multi-stage optimization problem.Generally speaking,the standard of segmentation is the shutdown time of engines at all levels.According to different shutdown conditions,the multi-stage OCP can be written as:

The connection conditions between the stages are

L

(

x

)(

t

)

,u

)(

t

)

,t;q

)).

It is mainly used to solve the discontinuity of state variables,including the discontinuity of engine separation time,mass change and other parameters.

3 DYNAMIC MODEL

The general motion equation of the ascending stage launch vehicle in the inertial frame is as follows:

where

r

is the inertial position,

v

is the velocity vector,g is the acceleration of gravity,

t

is the acceleration of gravity,

T

is the amplitude of the vacuum thrust,

t

is the current thrust,

A

and

N

are the axial and normal aerodynamic forces,u is the thrust vector direction,

m

(

t

) is the mass at the current moment,

Isp

and

ɡ

are the specific impulse and gravity.

The gravity model is J2:

The terminal conditions include position amplitude,flight path angle and corresponding orbit elements:

Flight path constraints generally include dynamic pressure,product of dynamic pressure angle of attack,product of dynamic pressure sideslip angle,as well as ensuring flight altitude and control vector unity of the rocket.

The initial condition is determined by the coordinates of the launch point and the initial conditions of the rocket.

The terminal conditions ensure that the speed and position meet the conditions of entering orbit.

The above parameters are semi major axis,eccentricity,orbit inclination,longitude of ascending node and amplitude of near-Earth point.

4 NLP PROBLEMS

At present,the most commonly used methods to transform the OCP into NLP include the shooting method and collocation method.The shooting method includes the single shooting method and multiple shooting method.

4.1 Single Shooting Method

NLP decision variables:

Cost function:

4.2 Multiple Shooting Method

The direct multiple shooting method is a numerical method to solve the boundary value problem,and its algorithm structure is shown in Figure 1.This method divides the time interval of seeking solutions into several smaller time intervals,solves the initial value problem in each smaller time interval,and applies additional matching conditions to form a solution over the whole-time interval.Compared with the single target method,this method has significant improvement in nonlinear distribution and numerical stability.

Figure 1 Multiple shooting method

4.3 NLP Solutions

At present,constrained optimization methods based on feasible direction search are widely used in NLP processing,including the inner point trust region line search algorithm and active set trust region line search algorithm.Among them,the representative is the sequential quadratic programming (SQP)algorithm and interior-point iteration (IPI) algorithm,which were matured in the late 1980s.In this paper,the interior point method is used to solve NLP problem.

The interior point method solves the constrained optimization problem by the penalty function method.It uses a sequence of unconstrained problems,which is called the sequence unconstrained minimization technique.By minimizing the following functions:

5 SIMULATION RESULTS

Given a two-stage launch vehicle as the model object,the online trajectory optimization algorithm based on the multiple shooting method was used to verify the feasibility of the reconfigurable guidance algorithm when the thrust decreases by 5%and 10%.

The simulation results under standard conditions are shown in Figures 2-5.

The simulation results of 5% thrust decrease are shown in Figures 6-9.

Figure 2 Altitude result

Figure 3 Velocity result

Figure 4 Mass result

Figure 5 Thrust result

Figure 6 Altitude result

Figure 7 Velocity result

Figure 8 Mass result

Figure 9 Thrust result

Figure 10 Altitude result

Figure 11 Velocity result

Figure 12 Mass result

Figure 13 Thrust result

The simulation results of 10% thrust decrease are shown in Figures 10-13.

It can be seen from the simulation results that when the thrust decreases by 5%,the algorithm based on online trajectory optimization can meet the target conditions;when the thrust decreases by 10%,the flight time increases through online planning,and finally the flight target can be achieved.

6 CONCLUSION

For launch vehicles,the most important mission characteristic is reliability.Among all the launch failures in history,the power system failures account for the majority,so how to reconstruct the guidance and control system to achieve the success of a mission is a challenge.The proposed method can meet the requirements of on-board computing ability and the target conditions with the thrust decreases by 10%,which is valuable for engineering application.