Optimal feedback based control for Mars entry trajectory tracking

HUANG Yuechen,LI Haiyang,*,and SHEN Hongxin

1.College of Aerospace Science and Engineering,National University of Defense Technology,Changsha 410073,China;

2.State Key Laboratory of Astronautic Dynamics,Xi’an Satellite Control Center,Xi’an 710043,China

1.Introduction

Entry trajectory control is an onboard process in which the appropriate commands are generated to lead the vehicles to the specified final states. The entry trajectory control showed a great success in the Apollo program. The Apollo entry control law uses the nominal guidancemethod which is also called as the reference trajectory tracking control. In the framework of tracking control, as mentioned by [1],an optimization tool is usually employed to obtain an optimal reference trajectory. The theory of proportional integral derivative (PID) control or optimal control is then used to design the control law to track the optimal trajectory online. The other type of entry control methods is the predictor-corrector control, which predicts the future states in the entry process and adjusts the control command according to the range from the target point online. The accuracy of predictor-corrector control depends on the precision of the build-in model. Comparing with the predictor corrector control, the reference trajectory tracking control has not only relatively low dependence on the model, but also lower requirement on on-board computation. In fact,since the Apollo age, the reference trajectory tracking controlhas been well developed and applied in the Mars mission [2]. However, the high uncertainties of Mars atmosphere as well as the aerodynamic performance of entry vehicle still pose great challenge to the design of the entry trajectory control law [3]. Moreover, the thin Mars atmospheric density results in the difficulties of entry aerodynamic deceleration and trajectory correction control. The situation is worse for the low lift-to-drag ratio vehicles.Therefore, it is necessary to develop a robust and accurate entry control law.

To improve the performance of tracking control, a variety of approaches has been developed. A typical feedback linearization (FBL) based control law was proposed by Coate and successfully adopted by the Apollo entry capsule [4]. The robustness of the FBL control law was then enhanced by Talole through adding a sliding mode observer into the drag tracking [5]. The control ability of the drag tracking control law was further enhanced by Benito by introducing the idea of nonlinear prediction. Besides,the problem of control saturation was settled by introducing the information of drag and its rate [6]. Another special kind of feedback control is the sliding mode control [7,8].In practice, the high tracking accuracy of sliding modecontrol in entry trajectory control has been verified by Furfaro [9]. Ross et al [10,11] formulated a pseudo spectral theory considering the possibility of real-time optimal control. And Bollino [12] introduced this method into the design of the entry trajectory control law and developed an optimal nonlinear feedback control law with the use of standard 3-degree of the freedom dynamical model.

As a classical method,feedback control has its advantages of low requirement for control system and is easy to implement.However,the outstanding drawback of the feedback control is its weak robustness,which limits its application.In this paper,an entry trajectory control law based on optimal feedback(OFB)technique is proposed.Unlike previous work,the feedback gains are generated by the theory of optimal control and the dynamic equations are not to be linearized.Thus,the robustness of the control law is improved.

The proposed control law is divided into longitudinal control and lateral control.The longitudinal control law,in which feedback gains are generated by means of the optimal control theory,generates the amplitude of the bank angle.The reversal of bank angle depends on the lateral control law,in which a corridor with respect to the cross-range is designed.The accuracy and robustness of the proposed closed-loop OFB based control law in tracking the reference trajectory is verified via 500 deviation simulations,in which modeling errors and external disturbances are considered.In addition,the FBL-based control law and a numerical predictor-corrector(NPC)control law are introduced to make comparison with the proposed method.

This paper is organized as follows.In Section 2,the entry problem is formulated.In Section 3,the control scheme,including longitudinal control based on optimal feedback and the lateral control strategy,is described.In Section4,simulation results and discussions are presented.In Section 5,the conclusions are given.

2.Problem formulation

The attitude motion of the entry vehicle is neglected in this paper.The 3-degree of freedom(DOF)entry motion equations describing the vehicle entering the rotating,spherical Mars can be found in[13]and are given by

where the derivations of six state variables,including the radial distance r,longitude θ,latitude ϕ,velocity V, flight path angle γ and heading angle ψ,are with respect to time.σ denotes the bank angle,ω denotes the rotation rate of Mars,and g denotes the Mars surface gravitational acceleration.The lift and drag accelerations are the functions with respect to the velocity and atmospheric density ρ and are given by

The terms CLand CDdenote lift and drag coefficient,respectively.Srefand m denotes the reference area and mass,respectively.The atmospheric density can be approximated by the following exponential function,which can be found in[14].

where β(h)represents a polynomial of altitude.

3.Trajectory tracking control scheme

The entry trajectory control can be artificially decoupled into longitudinal and lateral channels.Considering that the bank angle is the sole designable variable for entering vehicles,the amplitude of the bank angle is obtained by the longitudinal control law and the bank angle sign or the timing of the bank reversals is determined by the lateral control strategy.In this section,control methods in the two separated channels are presented respectively.

3.1 Longitudinal control law based on optimal feedback

In the longitudinal plane,the above 3-DOF equations in(1)–(6)can be reduced to the following 2-dimensional equations with the assumption of a non-rotating Mars as established in[15].

with

where Cσdenotes the rate of the bank angle,R0denotes the Mars radius,and s denotes the downrange.

The objective of the longitudinal control law lies in the generation of the necessary amplitude of the bank angle that tracks the nominal trajectory.Thus the problem of tracking trajectory can be transformed into an optimization problem with the following the performance index to minimize

where s∗and h∗denote the downrange and altitude of the reference trajectory respectively. Based on the Pontryagin’s minimum principle, the Hamiltonian of the optimal control problem with the performance index in(15),is subject to the reduced dynamics in(9)–(13)as given by

where f is the state differential equation vector representing the right-hand sides of(9)–(13).pT= [pspVpγphpu]is the costate vector,and the dynamics of costates become

with the boundary conditions

The costate variables can also be called influence functions as Bryson[16]did.Thus the variation of the performance index can be obtained by

The goal of the longitudinal control law is to track the reference trajectory with little or even no deviation,so the following optimal condition can be obtained

Substituting(27)and(14)into(28),one can obtain

From(29),the commanded amplitude of the bank angle can be obtained when the actual states of the vehicle deviate from the referenced ones at each control cycle.The optimal condition in(29)can also be seen as a kind of feedback control and the feedback terms are variables along the reference trajectory.With the boundary conditions in(22)–(26)and costate dynamics in(17)–(21),all the influence functions can be solved along the reference trajectory by using the technique of backward integration.Note that the expressions of feedback terms are not unique and can be transformed into forms of interesting deviations,such as the drag acceleration and its rate which can be easily obtained by the accelerometer.In this paper, considering that the terms of velocity, drag acceleration and its rate are necessary to track the drag-vs-velocity profile[17],we also take the downrange into consideration to improve the tracking accuracy of flying range.Thus,deviations of downrange s,velocity V,drag acceleration D and its rateare selected as feedback terms.Thus the control equation can be obtained by

where K1,K2,K3and K4are the corresponding influence functions.It should be noted that(30)is a kind of error dynamics,which is asymptotically(or,more precisely,exponentially)stable as verified by[18].

From(7),the rate of the drag acceleration can be obtained by

where α denotes the derivative of β(h)with respect to altitude.

Because the flight path angle γ is usually small in the process of vehicles entering the Mars atmosphere,the drag acceleration is much greater than the term g sinγ and(31)can be approximated by

Thus the variation of˙D can be obtained by

The variation of D can be readily derived from(7):

By substituting(34)into(33),one can obtain

where

From(34)and(35),one can obtain

By substituting(39)into(29),one can finally obtain

3.2 Lateral control strategy

As can be seen from(6),the direction of the lift vector depends on the bank angle sign. Thus the lateral deviation can be modified by reversing the bank angle as the Apollo entry trajectory control did[19].The lateral control scheme is to design a corridor,which consists of a pair of longitudinal plane symmetrical dead band boundaries.The boundary is a linear function of velocity as defined by

where K5and K6are corresponding coefficients.

Then the crossrange error can be obtained by

with the crossrange variable χ as defined by

where Rtis the range to go,Ψ is the line-of-sight azimuth angle along the great circle from current site to the de-ployment point,where Ψ0denotes the angle from the entry point to the deployment point.

4.Simulation results and discussions

In this section,the entry trajectory control law using the OFB technique presented in the preceding section is applied to a specific Mars entry mission.The FBL and NPC control laws are introduced for comparison.The simulation setup parameters can refer to Benito[14].The partial vehicle parameters are referred to the Mars Science Laboratory(MSL)as Table 1 shows.As referred in[14],the trim angle of attack usually remains fixed to obtain the stable aerodynamic coefficients with respect to velocity,i.e.Mach as shown in Fig.1.Because the entry phase,which starts at the entry point and ends at the deployment altitude,manages the most energy of the entry vehicle,the entry guidance and control method would meet challenges brought by the high speed and high altitude, which are usually more than 3 Mach and 8 km,respectively.Therefore,for simulation convenience,the aerodynamic coefficients can be approximated by constants which can well represent the dynamic aerodynamic coefficients in supersonic and hypersonic phases under the trim angle of attack.The nominal entry and parachute deployment conditions are shown in Table 2 and Table 3 respectively.When the altitude reaches hf,the simulation is terminated.

Table 1 Vehicle parameters

Fig.1 Aerodynamic coefficients vs.Mach

Table 2 Nominal entry states

Table 3 Parachute deployment site

The optimization tool general pseudo spectral optimal control software (GPOPS) is used to obtain the optimal reference trajectory [20].With the parameters in Tables 1 – 3,the constrains of g-load, heating rate and dynamic pressureare considered in the generation of the reference trajectory.Because the tracked trajectories will be along the nominal one if the deviations among them are small, the path constrains would also be satisfied by the tracked trajectories.

The tracking accuracy of the proposed OFB-based control is evaluated by single simulation under nominal conditions.Simulation results are presented in Figs.2–11.The OFB gains in(40)are shown in Figs.2–5.The green hollow points in Figs.6–9 represent the actual trajectory and the blue lines represent the referenced one.As can be seen in the following figures,the proposed control law is able to track the reference trajectory with small deviations of altitude,velocity,drag acceleration and the rate of drag acceleration.Thus,as mentioned above,the path constrains imposed on the nominal trajectory can be satisfied well in the actual simulation scenario.

The flight path angle in Fig.10 is no more than 20°and illustrates the assumption used in(32).It should be noted that the bank angle command in Fig.11 varies between–90°and+90°.The constraints of the amplitude,rate and acceleration of the bank angle are also considered in the generation of the reference trajectory using the optimization method.

Fig.2 K1vs.time

Fig.3 K2vs.time

Fig.4 K3vs.time

Fig.5 K4vs.time

Fig.6 Referenced and actual altitude histories

Fig.7 Referenced and actual velocity histories

Fig.8 Referenced and actual drag acceleration histories

Fig.9 Referenced and actual rate of drag acceleration histories

Fig.10 Actual flight path angle history

Fig.11 Actual bank angle history

The rate and acceleration of the bank angle are limited within20°/s and10°/s2,respectively.Besides,to alleviate the burden of reaction control system,the number of bank angle reversal is limited.

To better show the robustness and accuracy of the OFB,the OFB is compared with the classic FBL and an NPC method in Monte Carlo simulations,in which random dispersions with Gauss distribution in initial entry states,aerodynamic coefficients and atmospheric density,are introduced.Table 4 shows the specific deviation values.The methodology of FBL-based control law can be found in[14],and will not be described in this paper.The NPC method is referred to[21]and is modified to satisfy the requirement of Mars entry.These control laws are tested under the same simulation setup.It should be noted that OFB and FBL use the same reference trajectory.The parameters in Tables 1–3 are used in the Monte Carlo simulations.

The simulation results using the OFB,FBL and NPC are shown in Figs.12–14 and Table 5.The figures show the deployment sites with blue vacant spots and the targeting point with red intersection in the longitude-vs-latitude plane.Three circles with the size of 2 km,5 km and 10 km away from the parachute deployment site are shown in these figures.

Table 4 Dispersions in Monte Carlo simulations

With the comparison of Fig.12 and Fig.13,it can be seen that the deployment sites using FBL spread wider than the ones using OFB.It also can be seen from Fig.12 and Fig.13 that the OFB has more narrow distribution than the FBL with respect to the longitudinal plane,and the deployment sites distributions using both methods are not symmetric with respect to the longitudinal plane.This phenomenon can be explained by Fig.11,in which at the tail of the control profile,the required command is near the limitation of the bank angle.Therefore,the vehicle has relatively obvious lateral motion,especially in the final phase of entry,in which the relatively dense atmosphere results in the great aerodynamic force.

Fig.12 Deployment sites using OFB

Fig.13 Deployment sites using FBL

Fig.14 Deployment sites using NPC

Due to the relationship between the bank angle and the drag profile,the tracking accuracy of the drag profile will largely affect the crossrange of the deployment sites.However,this situation does not happen in the case of NPC because there is no nominal trajectory to track.It can be seen from the statistic information in Table 5 that the OFB has higher accuracy and robustness with the existence of external disturbances and model uncertainties than the FBL.The OFB even performs a little better than the NPC.

Table 5 Deployment sites statistics

It should be noted that because the OFB uses the semi-analytic control law,the OFB has lower requirement for computation capacity than the NPC,which needs numerical integration of trajectory in each control cycle.

5.Conclusions

A trajectory controllaw based on OFB has been developed to track the reference trajectory for Mars entry.The amplitude and the reversals of the bank angle are controlled by the longitudinal and lateral control stragety.At each control cycle,the amplitude of the bank angle is obtained by the OFB-based controller to minimize tracking errors.The bank reversals are executed if the cross range exceeds a predetermined corridor which is designed by setting a boundary profile.The accuracy and robustness of the proposed closed-loop optimal feedback based control law in tracking the reference trajectory is verified via 500 deviation simulations,in which modeling errors and external disturbances are considered.Besides,with the comparison of the FBL based control law and a numerical predictor-corrector control law,the performance of the proposed control law is further certified.

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