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Throttled Explicit Guidance to Realize Pinpoint Landing Under a Bounded Thrust Magnitude

2020; American Institute of Aeronautics and Astronautics; Volume: 44; Issue: 4 Linguagem: Inglês

10.2514/1.g005577

1533-3884

Takahiro Ito, Shin‐ichiro Sakai,

Astro and Planetary Science

Open AccessEngineering NotesThrottled Explicit Guidance to Realize Pinpoint Landing Under a Bounded Thrust MagnitudeTakahiro Ito and Shin-ichiro SakaiTakahiro ItoJapan Aerospace Exploration Agency, Sagamihara, 252-5210, Japan and Shin-ichiro SakaiJapan Aerospace Exploration Agency, Sagamihara, 252-5210, JapanPublished Online:29 Dec 2020https://doi.org/10.2514/1.G005577SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail AboutI. IntroductionCrewed and uncrewed campaigns for lunar and planetary soft landings have been actively explored for more than half a century. However, the landing accuracy in previous lunar and planetary soft landing missions was on the order of a few kilometers to more than tens of kilometers from the target [1,2]. Recent successful lunar and planetary missions [3–5] have subsequently provided information regarding associated terrains, and potential areas of investigation for subsequent campaigns have been identified, which has led to renewed interest in realizing lunar and planetary landings within 100 m from the target [6,7].Several early analytical studies on the fuel-optimal control problem for space flights [8–10] have indicated that, under specific assumptions, a fuel-optimal solution enables the realization of a maximum of only two bang-bang thrust magnitude switches and that the optimal thrust direction follows the bilinear tangent steering law. Powered flight guidance, which is an indirect method, often employs this concept to determine the optimal solution by solving the two-point boundary-value problem based on the Hamiltonian and first-order optimality condition [11]. A few powered descent guidance (PDG) strategies based on indirect methods have been proposed to achieve a simultaneous search of the optimal thrust direction and thrust switching. A recent study proposed the use of universal powered guidance (UPG) [12], whereby the first burn arc t1 is specified before flight through offline optimization, whereas the second burn initiation t2 is determined via online optimization. Owing to the offline optimization of t1 required for the UPG scheme, as a prerequisite, propellant expenditure should not be sensitive to variations around the precomputed t1; however, this prerequisite is not necessarily applicable to a wide variety of fuel-optimal pinpoint landing scenarios, particularly when a large trajectory correction is required.PDG-based on convex optimization [13–18], which is a direct method, is another potential onboard guidance scheme for pinpoint landing missions. This strategy converts the fuel-optimal control problem into a second-order cone programming problem as a parameter optimization problem. The advantage of this approach is that it can deal with multiple constraints on control and state variables. Nevertheless, a tradeoff relationship generally exists between the performance and computation time of selected algorithms [11]. In this context, compared with onboard guidance based on direct methods, guidance based on indirect methods is simpler and has fewer unknowns; it is more suitable for missions prioritizing faster computations rather than flexible constraint treatments.In this regard, the authors previously developed a guidance scheme based on an indirect method for the bounded thrust acceleration problem [19]. The developed scheme, termed "throttled explicit guidance" (TEG), can numerically determine the optimal thrust direction, thrust switching, and final time simultaneously through predictor–corrector iterations. However, the assumption on which the scheme is based—specifically, that thrust acceleration is constrained between the minimum and maximum bounds—can be problematic for most landers with throttling engines, where the thrust magnitude should be constrained to the corresponding bounds. For such landers, the TEG algorithm with bounded thrust acceleration cannot offer fuel-optimal solutions. Therefore, herein, this TEG scheme is extended to the fuel-optimal pinpoint-landing problem under a bounded thrust magnitude.The remainder of this Note is organized as follows: Section II provides a review of the fuel-optimal control problem and its solution under a bounded thrust magnitude. Section III describes the theoretical analysis of the thrust magnitude switching function performed to approximate the function in a simpler form by applying certain reasonable assumptions for a pinpoint landing problem. Section IV elaborates the TEG scheme for a bounded thrust magnitude problem, and Sec. V describes the testing of the TEG scheme via simulations for lunar pinpoint landing.II. Optimal Pinpoint Landing ProblemIn the considered fuel-optimal control problem, it is assumed that the gravity field is uniform, atmospheric effects are negligible, and the final time is free. The objective function J is to maximize the final mass of the vehicle M(tf), such that maximizeJ=M(tf)(1)The equations of motion for the vehicle in an inertial frame are R˙=V,V˙=TMu+gm,M˙=−m(2)where R=[x,y,z]T is the position; V is the velocity; u=[−cosψsinθ,sinθsinψ,cosθ]T is the unit thrust direction (where θ and ψ denote the zenith and azimuth angles, respectively); T=cm is the thrust magnitude (where c is the constant effective exhaust velocity); m is the rate of fuel expenditure; and gm is the acceleration of gravity. Note that the Euclidean norm of u is constrained to ‖u‖=1. The control variables for our problem are u and m.The thrust magnitude is constrained between its constant minimum and maximum values, i.e., Tmin and Tmax, respectively: Tmin≤T(t)≤Tmax(3)such that mmin≤m(t)≤mmax(4)where mmin and mmax denote the constant minimum and maximum bounds of the fuel consumption rate, respectively. Note that the thrust acceleration bounds (amin and amax) are functions of time: amin(t)≤a(t)≤amax(t)(5)The prescribed initial and final conditions are R(0)=R0,V(0)=V0,M(0)=M0(6)R(tf)=Rf,V(tf)=Vf(7)where R0, V0, and M0 denote the prescribed initial position, velocity, and mass, respectively, Rf and Vf denote the prescribed final position and velocity, respectively, and tf denotes the free final time. For a soft-landing problem, the target final velocity is zero.This fuel-optimal control problem has been studied extensively in early works pertaining to powered flights [8–10]. This problem can be solved analytically using the standard optimal control theory. By applying Pontryagin's maximum principle with the costate equations and transversality conditions, we obtain the optimal thrust direction: u(t)=λV(t)λV(t)(8)where λV is the costate for the velocity and λV is the Euclidean norm of the costate for the velocity (λV=‖λV‖). Note that the costate for the velocity is expressed as λV(t)=−νR(t−tf)+νV(9)where νR and νV are the constant Lagrange multipliers.Contrarily, the thrust magnitude switching function is defined as follows: κ(t)=cM(t)λV(t)−λM(t)(10)where λM is the costate for the mass, expressed as λM(t)=∂J∂M(tf)−∫ttfλ˙M dt=1−∫ttfcmM2λVTu dt(11)The optimal thrust magnitude can be categorized according to the value of the switching function: κ(t)≥0 when m(t)=mmax, κ(t)≤0 when m(t)=mmin, and κ(t)=0 when m(t)=m¯ (mmin<m¯ 0, we obtain κ^(t)>0 when κ(t)>0, κ^(t)<0 when κ(t)<0, and κ^(t)=0 when κ(t)=0. These expressions indicate that κ^(t) can be alternatively used to evaluate thrust switching. The derivative of K(t) is K˙(t)=−m(t)λM(t)+cm(t)M(t)λV(t)=m(t)κ(t)(15)Thus, K(t) can be expressed as K(t)=K(tf)−∫ttfK˙ dt=M(tf)[1−∫ttfmκ dtM(tf)](0≤t≤tf)(16)If the latter term in Eq. (16) is sufficiently small, K(t) can be approximated to a constant value Kc as K(t)≃Kc=M(tf)(17)which is equivalent to the final mass M(tf). By representing the approximated switching function as κ^a(t)=cλV(t)−Kc(18)thrust switching can be directly evaluated considering the quadratic switching equation for κ^a(t)=0: A(t−tf)2−B(t−tf)+C−(Kcc)2=0(19)where A=νRTνR,B=2νRTνV,C=νVTνV(20)Before evaluating whether the latter term in Eq. (16) is sufficiently small, it is assumed that the amount of fuel consumed between t(0≤t≤tf) and tf is small with respect to the final mass, such that (0≤)∫ttfm dtM(tf)=ϵ(t)≪1(21)where ϵ(t) is considerably smaller than 1 (e.g., ϵ∼0.1, or smaller). This assumption is realistic for a pinpoint landing guidance strategy, where a large diversion maneuver is executed during the later descent phase within a short time period.Subsequently, we determine the values of κ(t) at specific points. The minimum value of the switching function between 0≤t≤tf should be equivalent to and greater than κ(taxis), when 0≤taxis≤tf and taxis tf, respectively, where taxis is the axis of κ that meets κ˙(taxis)=0, expressed as taxis=tf+B2A(22)In this case, the switching function exhibits its minimum value κmin: κmin≥κ(taxis)=cMC−B24A−λM≥−λM(23)because C−B24A=(νRTνR)(νVTνV)−(νRTνV)2νRTνR≥0(24)Furthermore, λM is a monotonically increasing function and takes its maximum value at t=tf between 0≤t≤tf because λ˙M=(cm/M2)λV is greater than zero. Thus, from Eq. (11), the following expression can be obtained: κmin≥−1(25)Another key value is the one obtained at t=tf, which can be expressed as κ(tf)=cCM(tf)−1(26)If the landing guidance ends at a minimum thrust arc, by definition, the switching function at the final time must be equivalent to or less than zero, such that κ(tf)≤0when T(tf)=Tmin(27)In contrast, if the landing guidance ends at a maximum thrust arc, the switching function at the final time must be equivalent to or greater than zero, and C in Eq. (26) should be evaluated to determine the upper limit of κ(tf). To determine this limit, the Hamiltonian for our problem developed in Sec. II is defined as follows: H(t)=λRTV+λVT(cmMu+gm)−λMm(28)Thereafter, the constant Hamiltonian condition for the free final time problem is applied to Eq. (28) at the final time: H(tf)=νRTVf+[amax(tf)C−mmax]+νVTgm=0(29)where amax(tf) is the maximum thrust acceleration magnitude at the final time, defined as amax(tf)=TmaxM(tf)(30)Considering the soft-landing condition (Vf=0), H(tf)=[amax(tf)C−mmax]+νVTgm=0(31)As νVTgm≥−‖νV‖⋅‖gm‖=−Cgm, C[amax(tf)−gm]≤mmax(32)Furthermore, we assume that the maximum thrust acceleration at the final time is greater than gravitational acceleration. This assumption is reasonable because the maximum thrust acceleration of a lander is expected to possess sufficient vertical braking capability against gravity throughout the descent trajectory. Therefore, Eq. (32) can be rewritten as C≤mmaxamax(tf)−gm(33)Thus, the switching function at the final time must satisfy the following condition: κ(tf)≤cmmaxM(tf)1amax(tf)−gm−1=gmamax(tf)−gm(34)Assuming that amax(tf) is at least equivalent to or greater than 2gm, Eq. (34) becomes κ(tf)≤1whena(tf)=amax(tf)≥2gm(35)Subsequently, the latter term in Eq. (16) is assessed. The discussion indicates that the range of the switching function κ(t) can be constrained according to the specified time duration and thrust switching pattern. Table 1 presents the categorization of the κ(t) constraints according to the specified time duration and thrust switching pattern. Excluding the case where the solution has a maximum thrust structure and taxis≥tf, the switching function has the range −1≤κ(t)≤1(36)for the specified time duration. Therefore, the latter term in Eq. (16) has the range −ϵ(t)≤∫ttfmκ dtM(tf)≤ϵ(t)(37)for the specified time duration, except when the solution has a maximum thrust structure and taxis≥tf. As ϵ(t) is sufficiently smaller than one, the latter term in Eq. (16) can be approximated to zero; thus, the approximated switching function in Eq. (18) is applicable for the specified time duration. Furthermore, the specified time duration covers a sufficient range to evaluate accurately whether the approximated quadratic switching equation has any roots and the corresponding values (if any) between 0≤t≤tf. Thus, a suitable thrust switching configuration can be determined using the approximated switching equation.However, for the solution with the maximum thrust configuration and taxis≥tf, the inequality condition in Eq. (37) cannot be applied. Nevertheless, because m(t) and κ(t) are both greater than zero (and thus, K˙(t)>0), K(t) is equivalent to Kc when t=tf and smaller than Kc when 0≤t<tf. Consequently, κ^a(t)<κ^(t)(0≤t tf and κ^a(tf)>0, Eq. (38) leads to κ^(t)>0 for 0≤t≤tf. Thus, thrust switching can be accurately evaluated by checking whether the root values of the approximated switching equation in Eq. (19) are in the range 0≤t≤tf.These expressions indicate that the approximated quadratic switching equation specified in Eq. (19) can be used to evaluate accurately all thrust switching patterns under specific assumptions. The assumptions and statements established in this section can be summarized as follows:Assumption 1:The total fuel expenditure is sufficiently small compared with the final mass.Assumption 2:The target final velocity is zero.Assumption 3:The maximum thrust capability at the final time is sufficiently larger than gravitational force.Statement 1:The product of the mass costate and mass function can be approximated as a constant of the final mass.Statement 2:The equation of the switching function equaling zero can be approximated as a quadratic equation of time.Assumptions 1–3 are reasonable for a pinpoint landing guidance scheme. Statements 1 and 2 are derived analytically on the basis of these three assumptions. To the best of the authors' knowledge, these characteristics of the switching function have not been reported thus far.Finally, assumption 2 is discussed. From a practical perspective, statements 1 and 2 are expected to hold even when the target final velocity is nonzero, provided that the term νRTVf is relatively small compared with the other terms in Eq. (29). This efficacy has been demonstrated through several pinpoint landing simulations of the TEG scheme [21], in which a small lunar lander was guided from above a landing site with a target final altitude of 3000 m and downward velocity of 60 m/s. Nevertheless, it is possible that this efficacy varies for different cases; the effectiveness of the TEG scheme when the final target velocity is nonzero should also be verified for each case.IV. Throttled Explicit GuidanceThe original TEG scheme [19] was developed by the authors for the bounded thrust acceleration problem. However, when the thrust switching equation is approximated as a quadratic function of time, it can be extended to the bounded thrust magnitude problem, making it possible to apply practically this approach to landing missions.The TEG algorithm adopts a predictor–corrector approach, in which the final state errors on the final position, velocity, and Hamiltonian are predicted, and the unknowns are subsequently corrected to nullify the final state errors. The TEG scheme is unique because it considers thrust switching as a function of seven unknowns: x=[νRT,νVT,tf]T.The TEG algorithm includes two key functions: the switching operator and the final state predictor. The switching operator employs the quadratic switching equation to evaluate the number of times and when thrust switching must be performed. The final state predictor evaluates the final state errors on the basis of the seven unknowns and the number of times and when thrust switching should occur. In the numerical search process, the equations tend to be nonlinear particularly when the thrust switching structure changes (e.g., from min-max to max-min-max). To ensure robust convergence, the damped Newton's method [22] is applied to solve the simultaneous nonlinear equations. These aspects are described in the following subsections.A. Switching OperatorThe switching operator generalizes all possible thrust switching patterns into a max-min-max thrust switching configuration and generates t1 and t2 from the inputs of νR, νV, and tf. Thus, both t1 and t2 are regarded as a function of νR, νV, and tf.The discriminant of the quadratic switching Eq. (19), Dsw, is defined such that Dsw=B2−4A[C−(Kcc)2](39)If Dsw≥0, two real roots t1* and t2* (t1*≤t2*, thereby allowing the double root) exist, and they can be easily determined. If Dsw 0. In this case, t is assigned as the value on the extremum of κ(t); consequently, t1*=t2*=taxis. Subsequently, the switching operator outputs t1 and t2 between zero and the final time: t1=0(t1* tf), and t2=0(t2* tf).B. Final State PredictorThe final state predictor evaluates the final state errors: h(x)=(wR[R(tf)−Rf]wV[V(tf)−Vf]wHH(tf))(40)where wR=[wx000wy000wz],wV=[wu000wv000ww](41)Here, wx, wy, and wz; wu, wv, and ww; and wH denote the weighting constants for the final state errors of the position, velocity, and Hamiltonian, respectively. V(tf) and R(tf) can be calculated as V(tf)=V(t2)+(tf−t2)gm+cmmax∫t2tfu(t)M(t) dt(42)R(tf)=R(t2)+(tf−t2)V(t2)+12(tf−t2)2gm+cmmax∫t2tf(∫t2tu(τ)M(τ) dτ) dt(43)where V(t2)=V(t1)+(t2−t1)gm+cmmin∫t1t2u(t)M(t) dt(44)R(t2)=R(t1)+(t2−t1)V(t1)+12(t2−t1)2gm+cmmin∫t1t2(∫t1tu(τ)M(τ) dτ) dt(45)V(t1)=V0+t1gm+cmmax∫0t1u(t)M(t) dt(46)R(t1)=R0+t1V0+12t12gm+cmmax∫0t1(∫0tu(τ)M(τ) dτ) dt(47)The final state predictor first computes Eqs. (46) and (47), followed by Eqs. (44) and (45), and finally Eqs. (42) and (43). However, this computation is not necessary when the duration between two successive instants is zero; for example, if t1=t2, we can simply set R(t2)=R(t1) and V(t2)=V(t1) instead of computing Eqs. (44) and (45) to reduce the runtime of the TEG scheme. Furthermore, because the integrations of the thrust direction divided by the mass in Eqs. (42–47) can be obtained analytically and their analytical forms are known [23,24], rapid computation can be achieved.C. Damped Newton's MethodThe damped Newton's method is applied to solve the considered problem with h(x)=0. This approach comprises three stages: direction search, line search, and evaluation.In the direction search stage, the Jacobian matrix of h(x) is numerically computed, followed by the search for the optimal direction to reduce the final state errors. In the line search stage, the seven unknowns for the next iteration step are updated: xk+1=xk+αkdk(48)where αk is the scaling factor with 0<αk≤1, and dk is the optimal direction at the kth iteration step. αk is selected such that the norm of the final state errors decreases; specifically, αk is initially set as αk,0=1 and reduced such that αk,l+1=ραk,l(49)until the norm of the final state errors decrease. Note that ρ is a scalar with 0<ρ<1. The TEG algorithm imposes an additional terminal condition for the maximum number of updates, lmax, in case the identified search direction dk is not the optimal direction to reduce the final state errors. In the evaluation stage, the termination of the TEG algorithm is considered. Specifically, the algorithm is terminated when ‖h(xk+1)‖<ϵh(50)or when the predetermined maximum number of iterations is attained.V. Numerical TestsThe TEG scheme was evaluated by performing terminal descent simulations for a lunar pinpoint landing. Specifically, we examined its performance in a MATLAB programming environment (Release 2019b, released by The Mathworks), focusing on how the approximated switching function would affect the fuel optimal solution.Table 2 lists the parameter settings for the TEG and the lander specifications. The lander specifications in Table 2 assume an unmanned small lunar lander such as the Japanese smart lander for investigating the moon (SLIM) [25]. As mentioned previously, the constant Kc should be selected to be similar to the final mass of the lander. As the fuel-optimal final mass for the selected cases (presented in Table 4) was approximately 225–230 kg, Kc should be set around these values in principle. On the other hand, we can only make a guess on the final mass before the solution is found; therefore, the fuel optimality associated with the selection of Kc should be sufficiently robust against the uncertainty of the final mass. Hence, we investigated the sensitivity of the wide choice of Kc against the fuel optimality to range from 180 to 260 at intervals of 20. Note that this range covers the values of Kc that are far from the good estimates on the final mass; for example, Kc=260 is the condition greater than the initial mass; similarly, Kc=180 is the condition where the fuel is used more than twice as much as the fuel-optimal expenditure.Table 3 presents the initial and final conditions of the reference trajectory and their optimal solutions. The seven optimal unknowns in Table 3 were set as initial estimates to impose a cold start on the algorithm (which is generally the most difficult condition for convergence), and the algorithm was executed once per case. Four typical cases with different initial positions and velocities were considered. Table 4 lists the initial conditions for these four typical cases and the corresponding optimal fuel consumptions. Compared with those of cases 1 and 2, the initial dispersions of cases 3 and 4 are four times greater, which requires a larger trajectory correction. Figure 1 shows the fuel-optimal trajectories.Fig. 1 Fuel-optimal trajectories for four typical cases. Optimal trajectories were computed under a flat-moon, constant-gravity (constant value of gm listed in Table 2) model.The simulation settings in Tables 2–4 satisfy all the three assumptions presented in Sec. III; therefore, the approximated K(t) in Eq. (17) and the approximated switching function in Eq. (18) are applicable.After completing the simulations, it was noted that the TEG scheme successfully determined the optimal solution in an efficient manner. Figure 2 presents the thrust command profiles obtained using the TEG scheme for all cases. Figure 3 shows the excessive fuel usage against the optimal consumption for all cases. The TEG results exhibited a smooth convergence for all the cases, with a maximum of 12 repetitions (case 4 when Kc=180 and Kc=200) of the predictor–corrector procedure. As shown in Fig. 2, the thrust command profiles differed somewhat in terms of the value of Kc. Nonetheless, as shown in Fig. 3, the solutions determined using the TEG scheme exhibited fuel optimality, with maximum deviations of 0.020, 0.004, 0.093, and 0.042 kg from the optimal values for cases 1–4, indicating an increase of 0.1, 0.02, 0.5, and 0.2% from the optimal expenditure, respectively. These findings indicate that the precise tuning of Kc is not a prerequisite for executing TEG. This feature endows the strategy with robustness in realizing onboard guidance for an actual space mission, wherein the final mass of a lander is uncertain and cannot be predetermined accurately.Fig. 2 Thrust command profiles obtained using TEG for all cases.Fig. 3 Excessive fuel usage with respect to optimal consumption for all cases.Moreover, the performance in terms of fuel optimality was the highest, and the unknown values were closest to the optimal values for all the cases when Kc was at 220 (approximately equal to the final mass). This observation can be explained by Fig. 4, which presents the calculation results of the true switching function κ(t), mass costate λM(t), and K(t) for the optimal solutions for all the cases. Figure 4 was obtained independently from the TEG scheme. First, the optimization problems were numerically solved by minimizing fuel usage subject to the constraints of h(x)=0 and 0≤t1≤t2≤tf against the nine unknowns: x=[νRT,νVT,t1,t2,tf]T. Subsequently, the profiles in this figure were reproduced via numerical integration with the obtained nine unknowns. As expected, the profile of K(t) remained relatively constant around the final mass; therefore, the assumption of a constant Kc close to the final mass could be considered reasonable.Fig. 4 Computation results for time profiles of switching function κ(t), mass costate λM(t), and K(t)=λM(t)M(t) for optimal solutions of all cases.Finally, we evaluated the convergence procedure. Figures 5 and 6 show the iteration profile of the damped Newton's method and that of the thrust switching times and final time when Kc=220, respectively. Note that the thrust switching structure changed from min-max to max-min-max at the second iteration step for cases 1 and 3 and fifth iteration step for case 4, whereas it remained min-max for case 2 during their iteration procedure. Figure 5 indicates that the TEG algorithm tended to encounter the small values of the scaling factor αk in the line search of the damped Newton's method when the thrust switching structure was about to change (this tendency was also observed in the TEG version for the bounded thrust acceleration problem [19]). It implies that the equations to be solved were highly nonlinear in those iteration steps. Nevertheless, once the TEG algorithm met the correct thrust switching structure, it started to converge to the solution rapidly. The same trend was observed for the other cases with the different values of Kc, and all of them were able to find their solutions. These results support the assertion that the damped Newton's method in the TEG scheme is effective for the robust convergence even if the change of the thrust switching structures occurs during the numerical search.Fig. 5 Iteration profile of the damped Newton's method when Kc=220.Fig. 6 Iteration profile of the thrust switching times and final time when Kc=220.VI. ConclusionsTo realize pinpoint landings, it is necessary to use the control capability of the thrust direction and switching. Therefore, the TEG scheme for the bounded thrust acceleration problem was extended to the bounded thrust magnitude. To achieve prompt computations of thrust switching, the equation of the thrust magnitude switching function was approximated to zero as a quadratic equation of time, with the product of the mass function and its costate approximated as a constant value close to the final mass of the lander. 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Google ScholarTablesTable 1 Categorization of κ(t) constraints according to specified time duration and thrust switching patternThrust patternConstraint on κ(t)DurationMax (taxis≤0)0≤κ(t)≤10≤t≤tfMax (0<taxis<tf)0≤κ(t)≤1taxis≤t≤tfMax (taxis≥tf)κ(tf)≤κ(t)0≤t≤tfMin−1≤κ(t)≤00≤t≤tfMin-max−1≤κ(t)≤10≤t≤tfMax-min−1≤κ(t)≤0t1≤t≤tfMax-min-max−1≤κ(t)≤1t1≤t≤tfTable 2 Parameter settings for the TEG and the lander specificationsWeighting constant for final state errorsParameters for Newton's method wx,wy,wzwu,wv,wwwHρlmaxϵhConstant value, Kc10−211030.51010−6180–260 (in intervals of 20)Thrust rangeSpecific impulse Lander's initial mass Constant gravityTmin [N]Tmax [N]Isp [s] M0 [kg] gm [m/s2]300750320 250 [0,0,−1.61]TTable 4 Four typical cases with dispersions applied to initial position and velocity along with their optimal fuel consumptionsCase numberInitial position dispersions [m]Initial velocity dispersions [m/s]Fuel consumption [kg]Case 1[500,500,500]T[1,1,1]T20.516Case 2[−500,500,−500]T[−1,1,−1]T18.760Case 3[2000,2000,2000]T[4,4,4]T24.631Case 4[−2000,2000,−2000]T[−4,4,−4]T20.126Table 3 Initial and final conditions for reference trajectory and their optimal solutionsInitial positionInitial velocityTarget positionTarget velocityR0 [m]V0 [m/s]Rf [m]Vf [m/s][−5000,0,5000]T[120,0,−60]T[0,0,0]T[0,0,0]TLagrange multiplier constantFinal timeThrust switchingFuel consumptionνRνVtf [s]t1 [s]t2 [s]M0−M(tf) [kg][−0.00040,0.00000,−0.00135]T[−0.027,0.000,0.143]T93.300.0020.2719.404 Previous article Next article FiguresReferencesRelatedDetailsCited byPropellant-Optimal Powered Descent Guidance RevisitedPing Lu and Ryan Callan3 January 2023 | Journal of Guidance, Control, and Dynamics, Vol. 46, No. 2Optimal Powered Descent Guidance Under Thrust Pointing ConstraintTakahiro Ito and Shin-ichiro Sakai1 February 2023 | Journal of Guidance, Control, and Dynamics, Vol. 0, No. 0Propellant-Optimal Powered Descent Guidance RevisitedPing Lu and Ryan Callan19 January 2023Analytical Costate Estimation by a Reference Trajectory-Based Least-Squares MethodDi Wu, Lin Cheng , Fanghua Jiang and Junfeng Li31 March 2022 | Journal of Guidance, Control, and Dynamics, Vol. 45, No. 8 What's Popular Volume 44, Number 4April 2021 CrossmarkInformationCopyright © 2020 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-3884 to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsAerodynamicsAeronautical EngineeringAeronauticsAerospace SciencesBoundary Element MethodComputational Fluid DynamicsFluid DynamicsNumerical Analysis KeywordsThrustPontryagin's Maximum PrincipleNumerical IntegrationPowered Descent GuidanceSoft Landing ConditionsFuel ConsumptionExhaust VelocityPlanetary MissionsLunar LanderPropellantPDF Received31 July 2020Accepted23 November 2020Published online29 December 2020