Mars Entry Guidance Based on Predictor-Corrector Algorithm.pdf

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http://www.paper.edu.cn 基于预测校正算法的火星进入段制导周鎏宇1，夏元清1，陈荣芳1，崔平远2 1 北京理工大学，自动化学院，北京，100081 2 北京理工大学，宇航学院，北京，100081 摘要：这篇文章介绍了基于预测校正算法的火星进入段制导方法的发展并对其进行评估。未来火星任务的要求是能确保精确和高海拔的火星表面着陆。现在主要有两种进入段制导算法：基于标称轨迹的制导和基于预测校正的制导。跟踪标称轨迹的制导算法相对不依赖于模型，并且不要求很高的在线计算能力。但是它对初始条件具有较高的敏感性。而预测校正算法对于初始扰动的敏感性较低，但是却要求较高的在线计算能力。这篇文章对预测校正，横向控制和航向调整算法给出了详细介绍，同时提出了一种改进的预测校正算法来减少在线计算量。仿真结果表明这种进入段制导算法在有很高不确定性的情况下仍然能体现出可靠性和鲁棒性。关键词：进入制导，预测校正，在线计算，横向控制，航向调整。中图分类号： TP1 Mars Entry Guidance Based on Predictor-Corrector Algorithm ZHOU Liuyu1, XIA Yuanqing1，CHEN Rongfang1，CUI Pingyuan2 1 School of Automation, Beijing Institute of Technology, Beijing 100081, China 2 School of Aeronautics, Beijing Institute of Technology, Beijing 100081, China Abstract: This paper addresses the development and evaluation of the numerical predictor-corrector algorithm for the Mars entry guidance. The requirement of future Mars missions is to enable pinpoint and high elevation Mars landing. There are two main entry guidance algorithms: guidance using a reference trajectory and guidance using predictor-corrector. Guidance algorithm that tracks the reference trajectory during entry, has some degree of model-independence and does not require fast on-board computation, but it is sensitive to initial conditions. However, the predictor-corrector algorithm can be less sensitive 基金项目： The work of Yuanqing Xia was supported by the National Basic Research Program of China (973 Program) (2012CB720000), the National Natural Science Foundation of China (61225015,60974011), the Ph.D. Programs Foundation of Ministry of Education of China (20091101110023, 20111101110012), and Beijing Municipal Natural Science Foundation (4102053,4101001). 作者简介： Zhou Liuyu (1991-) male, bachelor candidate, major research direction: guidance and control, predictor-corrector. E-mail:liuyuzhou1@gmail.com. Correspondence author: Xia Yuanqing (1971-) male, professor, major research direction: spacecraft system control, networked control systems, robust control, active disturbance re- jection control. E-mail:xia yuanqing@bit.edu.cn. Chen Rongfang (1988-) male, master candidate, major research di- rection: guidance and control system design for the precise landing of the Mars lander. E-mail:xiaxiao432@gmail.com. Cui Pingyuan (1961-) male, professor, major research direction: deep space detector autonomous technology and track design, aircraft ight dynamics and control, intelligent information processing and autonomous navigation. E-mail:cuipy@bit.edu.cn. - 1 -

http://www.paper.edu.cn to initial dispersions, but needs fast on-board computation. Detailed descriptions of predictor-corrector algorithm, lateral control logic and heading alignment are given and an improved predictor-corrector algorithm is provided to shorten the on-board computational time. Simulation results show that this entry guidance algorithm demonstrates reliable and robust performance in situations with high uncertainties. Key words: entry guidance, predictor-corrector, on-board computation, lateral control, heading alignment. 0 Introduction The entry guidance purpose is to deliver the lander to a desired parachute deployment point in three-dimensional position space (longitude, latitude and altitude). Many considerable eﬀorts of entry guidance had been made. Harpold J. C. and Graves C. A. [1] carried out a shuttle entry guidance based on the drag-to-velocity proﬁle. Lu P. presented a quasi-equilibrium Glide guidance in lifting entry ﬂight [2]. A.Saraf et. al [3] presented an entry guidance algorithm called evolved acceleration guidance logic for entry (EAGLE) with potential to reduce the amount of pre-mission design eﬀort. The main challenge of Martian entry, however, is Mars’ comparatively thin atmosphere [4]. The Martian atmosphere is approximately 100 times thinner than that on Earth, which is insuﬃcient to provide adequate deceleration at high altitudes. Under these conditions the vehicle’s control authority will be low. Entry guidance algorithms to be used on Mars have to account for the reduced control authority to ensure satisfactory delivery accuracy. Carman et al. [5] and Mendeck and Carman [6] presented a reference-path entry guidance method using the guidance technique of Apollo second entry phase adapted for Mars landing conditions. Kluever [7] carried out the comparative performance analysis of pre-planned reference-path and predictive path-planning entry guidance methods for a lander modelled on the Mars Science Laboratory(MSL) mission. Powell [8] evaluated the performance of the numerical predictor- corrector technique to guide a Mars lander to both the aerocapture objective and precision landing one, namely, to deploy a parachute within 10km of a target. Xia Y., and Fu M. [9] developed methodology combined of the active disturbance rejection control (ADRC) and slide mode control(SMC) methods, which can be used as the feedback control laws in Mars entry guidance which have shown better performances than the traditional proportional-integral- derivative (PID) law. This paper describes guidance algorithms that use numerical predictor-corrector techniques to guide the spacecraft to the proper objectives, which is a modiﬁcation of the skip entry guidance for lunar-return mission carried out by Christopher W. Brunner and Ping Lu [10]. The main advantage of predictor-corrector guidance is that it can always ensure a feasible - 2 -

http://www.paper.edu.cn trajectory that satisﬁes the required conditions from the current position to the target deploy site and ﬁnd suitable command bank angles to ﬂy. However, this advantage is undermined by the increased computational time of integration and iteration. The total number of lines of code is much greater than that of other guidance algorithms. So, an improved predictor-corrector algorithm is provided in this paper to shorten the on-board computational time. This paper is organized as follows. The entry dynamic problem is formulated in Section 2. The predictor-corrector algorithm is presented in Section 3. Lateral control logic is analyzed in Section 4. Section 5 presents heading alignment algorithm. An improved predictor-corrector algorithm is provided in Section 6. Section 7 gives simulation results and some discussions. Finally, some concluding remarks are given in Section 8. 1 Entry Dynamic Since entry duration is short and the vehicle is unpowered during the entry phase, the lander is modeled as a point mass ﬂying in a planet-ﬁxed coordinate frame. Without taking into account of the Mars’ rotation, the standard simpliﬁed equations of motion for entry are [11] ˙θ = ˙ϕ = V r V r cos γ cos ψ cos ϕ cos γ sin ψ ˙r = V sin γ ˙V = −D − g sin γ [ [ L cos σ − (g − V 2 r 1 V ˙γ = ˙ψ = − 1 ] ) cos γ L sin σ + V cos γ V 2 r cos2 γ cos ψ tan ϕ (1) (2) (3) (4) (5) (6) ] where θ is the longitude, ϕ is the latitude, r is the radial distance from center of the Mars. V is the velocity and γ is the ﬂight path angle. ψ is the heading angle which is the angle between the projection of the velocity on the horizontal plane and the East direction, following the counter-clockwise (ccw) from East convention (hence Eastward direction is ψ = 0 , Northward direction is ψ = +90 deg and Southward direction is ψ = −90 deg ) [11]. The angles γ and ψ completely deﬁne the orientation of the velocity vector. σ is the bank angle deﬁned as the angle from the vertical direction to the lift vector, such that a bank to the right is positive and zero bank corresponds to the lift vector directed upward in the longitudinal plane. The gravitational acceleration is modeled as g = µ/r2, where µ is the planet’s gravitational constant. In Mars, µ is approximated to 4.284e + 013. The aerodynamic forces acting on the vehicle can be decomposed into lift and drag. The lift and drag accelerations can be obtained - 3 -

http://www.paper.edu.cn by D = L = 1 2 1 2 ρV 2 S m ρV 2 S m CD CL (7) (8) where ρ is the atmospheric density, S stands for the vehicle reference surface area and m is the mass of the lander. CD and CL represent the drag and lift coeﬃcients, respectively. The aerodynamic coeﬃcients are typically function of the Mach number and the angle of attack. In this paper, an exponential function of the atmospheric density proﬁle is obtained as follow ρ = ρ0 exp rr0 hs (9) where ρ0 is the density at the reference radius r0 and hs is the constant scale height. The entry capsule and aerodynamic forces are shown in Fig. 1, where CM represents the center of mass. The bank angle and lift force components are shown in Fig. 2. 图 1: Entry Capsule and Aerodynamic Forces. 2 Predictor-Corrector Algorithm The predictor-corrector algorithm discussed here is a downrange predictor-corrector one, which consists of predictor part and corrector part. Apollo also uses a predictor-corrector approach, but it is for lateral control. The predicted crossrange miss is compared to the estimated lateral control capability, and a reversal is performed when the predicted miss is equal to half of the lateral control capability. In this paper another lateral control logic will be discussed later. The function of predictor-corrector algorithm is to calculate the error between the current range-to-go and the predicted range which is also called miss distance in every - 4 -

http://www.paper.edu.cn 图 2: Bank Angle and Lift Force Components. guidance cycle via the predictor, then ﬁnd and output a command bank angle magnitude that satisﬁes the requirement that the current range-to-go and the predicted range are equal via corrector. 2.1 Predictor The main purpose in the predictor process is to calculate the range-to-go Stogo and the predicted range Sf . The range-to-go is decided by the longitude and latitude of the current place and the target deployment place. Let C be the current position and S be the target position. The range-to-go can be obtained by [10] Stogo = arccos [sin θC sin θS + cos θC cos θS cos (ϕC − ϕS)] (10) where θC, θS, ϕC, ϕS are the longitude and latitude of point C and S respectively. By integrating the equation ˙s = (11) V cos γ r and the entry dynamic equations(Eqs.1-6) from the current position to the point when the velocity is below the deployment requirement(400m/s in this paper), the predicted range Sf can be obtained. Then we can get the error between the range-to-go Stogo and the predicted range Sf . Note that the bank angle is needed in every guidance cycle of prediction in order to integrate the dynamic equations and get the predicted range, so a future bank angle planning is needed in predictor process, which can be called bank angle strategy. The simplest strategy is a constant bank angle one, which applies the current command bank angle to every guidance cycle of prediction in the future. C. W. Brunner and Ping Lu [10] proposed a more robust strategy, shown in Fig. 3. Ss and σs represent the range-to-go and bank angle in a certain guidance cycle of prediction. Sf and σf are constants which need to be designed. The algorithm for this strategy is as follows 1) If Stogo ≤ Sf , the constant bank angle strategy will be applied. - 5 -

http://www.paper.edu.cn 图 3: Bank Angle Strategy. 2) If Stogo > Sf , then in every guidance cycle of prediction, the range-to-go Ss has to be calculated. If Ss > Sf , σs = σf + Ss − Sf Stogo − Sf (σc − σf ) (12) If Ss ≤ Sf , the constant bank angle strategy will be applied, that is σs = σf . The integral is terminated when the predicted velocity is below the deployment velocity requirement. So we get △S = Stogo − Sp (13) This strategy has a better performance of robustness because in the last part of the entry phase, the vehicle ﬂies with a constant bank angle. Usually σf is set to 60-70 deg. If experiencing disturbance in the ﬁnal phase of entry, the speed of the vehicle will suddenly increase or decrease, so does the predicted range. The bank angle should be adjusted to meet the range error requirement. If σf is chosen to 60-70 deg, the longitudinal control part is cosσ ≈ 0.5, which can ensure the biggest control margin. The threshold Sf , should be set to a value depending on speciﬁc situations. Please note that the range-to-go Stogo has units of radians. So does Sf . 2.2 Corrector The purpose of the corrector process is that when the downrange error does not meet the requirement, the bank angle should be corrected to make the downrange error small enough. Simulation results show that the predicted range for vehicle is reciprocal to the current com- mand bank angle magnitude. So command bank angle magnitude should be decreased in order to increase the predicted range Sp. A secant iteration scheme is employed to ﬁnd the required - 6 -

http://www.paper.edu.cn bank-angle magnitude. The iterates are generated by the equation [10] cos σn+1 = cos σn − cos σn − cos σn1 △S(σn) − △S(σn1) △S (σn) (14) It is clear from the above equation that two predicted downrange errors are needed to get the new bank angle magnitude in this process. To start the secant iteration scheme appropriately for the ﬁrst time, an initial guess of the bank angle is required. Considering that large bank angles can meet the constraints on heat load, heat rate, and deceleration load, the algorithm begins by assuming an initial bank angle of 80 deg in the numerical simulation of the trajectory. After the ﬁrst guidance cycle, the previously calculated bank angle is used as an initial guess rather than beginning with a 80 deg bank angle. If this angle is still suﬃcient to bring the downrange error within accepted tolerance, the secant method is skipped and the previous bank angle is ﬂown. Otherwise, the secant method proceeds as already noted and determines a new bank-angle magnitude. The other parameter which needs to be concerned is the increase of bank angle in the second prediction of every guidance cycle. Since the corrector process needs two former predictions to have a new one, the second prediction of bank angle in every guidance cycle is obtained by increasing the ﬁrst predicted bank angle by a certain magnitude. From simulation results, it is found that the increase is better to set between 30-50 deg, which can signiﬁcantly reduce the iteration times. 图 4: Predictor-Corrector Algorithm Flow Chart. 3 Lateral Control Logic The reliability of the longitudinal predictor-corrector algorithm discussed above rests on an important assumption that the crossrange during entry is always enough small and can be - 7 -

http://www.paper.edu.cn neglected. So the predicted range can be equal to the predicted downrange. And the range-to- go is just the downrange-to-go deﬁned as the great-circle distance between the current position and the target deploy site. The ideal predictor-corrector algorithm is to make the predicted downrange and the range-to-go equal. But as predicted downrange is hard to compute, pre- dicted range is often used to substitute it in practice. So in order to actually ﬂy toward the target conditions, the lateral control logic should be used to conﬁne the crossrange deviation to an acceptable bound via bank angle reversals. There are mainly two methods of lateral control for low lift-to-drag (L/D) ratios vehicles: 1) Bank reversal based on crossrange error. 2) Bank reversal based on azimuth angle error. These two methods are alike in that when the error(crossrange error or azimuth angle error) exceeds the pre-specify threshold, the sign of bank angle is reversed. A crossrange and azimuth angle threshold is deﬁned as a quadratic or simply a linear function of the dimensionless velocity [10] χc = c1V + c0 The crossrange can be calculated as χ = arcsin [sin Stogo sin(ψ − Ψ)] (15) (16) where Stogo is the range-to-go, ψ is the current heading angle, and Ψ is the line-of-sight azimuth angle along a great circle to the landing site. All the variables have units of radians. Note that the deﬁnition of heading angle discussed in this paper is diﬀerent from that in some other works(e.g. Ref. [8]-[10]), the computation method of heading angle error will be given as follows [11] 图 5: Spherical Triangle. In Fig. 5 the point N corresponds to the North Pole of the planet, characterized by an indeterminate longitude and latitude of 90 deg. The distance d12 can be found by d12 = arccos [cos ϕ1 cos ϕ2 cos(θ1 − θ2) + sin ϕ1 sin ϕ2] (17) - 8 -

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