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Desensitized Optimal Trajectory for Multi-phase Lunar Landing 1
Proceedings of the 20th World Congress Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of 20th The International Federation of Congress Automatic Control Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 The International Federation of Control The International of Automatic Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse,
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Desensitized Optimal Trajectory for Desensitized Optimal Trajectory for Desensitized Optimal Trajectory for Multi-phase Lunar Landing Multi-phase Lunar Landing Multi-phase Lunar Landing
S. Mathavaraj ∗∗ Radhakant Padhi ∗∗ ∗∗ S. ∗∗ S. Mathavaraj Mathavaraj ∗∗ Radhakant Radhakant Padhi Padhi ∗∗ S. Mathavaraj Radhakant Padhi ∗ ∗ Flight Dynamics Group, ISRO Satellite Center, Bangalore, India Dynamics (e-mail: Group, ISRO Satellite Center, Bangalore, India ∗ ∗ Flight Flight Group, Satellite [email protected]) Flight Dynamics Dynamics (e-mail: Group, ISRO ISRO Satellite Center, Center, Bangalore, Bangalore, India India [email protected]) ∗∗ (e-mail: [email protected]) Eng., Indian Institute of Science, India (e-mail: [email protected]) ∗∗ Department of Aerospace ∗∗ Department of Aerospace Eng., Indian Institute of Science, India ∗∗ Department of of(e-mail: Aerospace Eng., Indian Indian Institute Institute of of Science, Science, India India [email protected]) Department Aerospace Eng., (e-mail: (e-mail: [email protected]) [email protected]) (e-mail: [email protected]) Abstract: Lunar landing problem has been formulated as desensitized optimal control problem Abstract: Lunar landing problem has been formulated as desensitized control Abstract: Lunar landing problem formulated as optimal control problem and solved by Legendre method. The problem has beenoptimal split into threeproblem phases Abstract: Lunar landingPseudospectral problem has has been been formulated as desensitized desensitized optimal control problem and solved by Legendre Pseudospectral method. The problem has been split into three phases and solved by Legendre Pseudospectral method. The problem has been split into three phases to account for mission constraints, namely the braking with rough navigation stage (from 18km and solved by Legendre Pseudospectral method. The problem has been split into three phases to account for mission constraints, namely the braking with rough navigation stage (from 18km account for mission mission constraints, namely the braking braking with rough navigation stage (from (from 18km to account 7km), attitude holdconstraints, stage (holding the attitude for with 35sec) for anavigation typical mission scenario. In for namely the rough stage 18km to attitude hold (holding the for for mission In to 7km), 7km), of attitude hold stage stage (holding the attitude attitude for 35sec) 35sec) for aaa typical typical mission scenario. In presence uncertainties, following the open loop reference trajectory using the closedscenario. loop linear to 7km), attitude hold stage (holding the attitude for 35sec) for typical mission scenario. In presence of uncertainties, following the open loop reference trajectory using the closed loop linear presence of uncertainties, following the open loop reference trajectory using the closed loop linear quadratic regulator contribute greatly in trajectory dispersions. Theusing goal the is toclosed desensitize this presence of uncertainties, following the open loop reference trajectory loop linear quadratic regulator greatly in trajectory The goal is to desensitize this quadratic regulator contribute greatly in dispersions. The is to this multi-phase optimal contribute trajectory with reduced error in dispersions. presence of initial state thrust error, quadratic regulator contribute greatly in trajectory trajectory dispersions. The goal goal is error, to desensitize desensitize this multi-phase optimal trajectory with reduced error in presence of initial state error, thrust error, multi-phase optimal trajectory with reduced error in presence presence of initial initial state error,isthrust thrust error, Moon’s gravity uncertainty. Towith achieve this, error the fuel minimization coststate function augmented multi-phase optimal trajectory reduced in of error, error, Moon’s gravity uncertainty. this, fuel minimization function is Moon’s gravity uncertainty. To achieve this, the fuel loop minimization cost functionFollowing is augmented augmented with closed loop covarianceTo to achieve generate thethe open reference cost trajectory. this Moon’s gravity uncertainty. To achieve this, the fuel minimization cost function is augmented with closed loop covariance to generate the open loop reference trajectory. Following this with closed loop covariance to generate the open loop reference trajectory. Following this reference trajectory using closed loop linear quadratic regulator, shows significant reduction with closed loop covariance to generate the open loop reference trajectory. Following this reference trajectory using closed loop linear quadratic regulator, shows significant reduction reference trajectory using closed loop linear quadratic regulator, shows significant reduction in landing error. The amount of extra fuel consumed by desensitizing optimal trajectory is less reference trajectory using closed loop linear quadratic regulator, shows significant reduction in amount extra by optimal trajectory is in landing landing error. error. The amounttoof ofimprovement extra fuel fuel consumed consumed by desensitizing desensitizing optimal trajectory is less less significant when The compared in landing accuracy to meet mission constraints in landing error. The amount of extra fuel consumed by desensitizing optimal trajectory is less significant compared to in accuracy to meet constraints significant when compared to improvement improvement in landing landing accuracy to landing meet mission mission constraints assuring thewhen desensitized optimal control a viable technique for lunar trajectory design. significant when compared to improvement in landing accuracy to meet mission constraints assuring the desensitized optimal control a viable technique for lunar landing trajectory design. assuring assuring the the desensitized desensitized optimal optimal control control aa viable viable technique technique for for lunar lunar landing landing trajectory trajectory design. design. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Lunar Landing, Desensitized Optimal Control, Nonlinear Programming, Legendre Keywords: Lunar Lunar Landing, Landing, Desensitized Optimal Optimal Control, Nonlinear Nonlinear Programming, Legendre Legendre Keywords: Pseudospectral Keywords: Lunar Landing, Desensitized Desensitized Optimal Control, Control, Nonlinear Programming, Programming, Legendre Pseudospectral Pseudospectral Pseudospectral 1. INTRODUCTION jectory is of maximum minimum maximum structure. The 1. INTRODUCTION INTRODUCTION jectory is minimum structure. The 1. jectory is of of maximum maximum minimum maximum maximum structure. The main objective of the desensitized optimal control (DOC) 1. INTRODUCTION jectory is of maximum minimum maximum structure. The main objective of the desensitized optimal control (DOC) main objective of the desensitized optimal control (DOC) strategy is to reduce the sensitivity to state uncertainties Moon, the closest celestial body to our planet, is often main objective of the desensitized optimal control (DOC) is to the sensitivity to statethe uncertainties Moon, the the as closest celestial body to to our ourspace planet, is often often strategy strategy is reduce the sensitivity to uncertainties well as to reduce minimize To describe trajectory Moon, closest body planet, is considered a basecelestial for conducting technology strategy is to to reduce the fuel. sensitivity to state statethe uncertainties Moon, the as closest celestial body tonew ourspace planet, is often as as well as to minimize fuel. To describe trajectory considered a base for conducting new technology as well as to minimize fuel. To describe the trajectory dispersions in the formulation, the DOC strategy uses considered as a base for conducting new space technology demonstrations. Further to that, the desire is to soft land as well as to minimize fuel. To describe the trajectory considered as a base for conducting new space technology dispersions in the formulation, the DOC strategy uses demonstrations. Further to that, the desire is to soft land dispersions in the thetechnique. formulation, thethe DOC strategy uses linear covariance Then linear covariance demonstrations. to that, is to soft land ademonstrations. spacecraft withFurther / without man the has desire been the immediate dispersions in formulation, the DOC strategy uses Further to that, the desire is to soft land linear covariance technique. Then the linear covariance a spacecraft spacecraft with without manforhas has been the immediate immediate linear technique. the linear covariance the covariance trajectory is calculatedThen by integration and called agoal with // without man been the for several countries. Also such landing missions, of linear covariance technique. Then the linear covariance a spacecraft with / without man has been the immediate the trajectory calculated by integration and called goal for for several several countries. Also for such such landing missions, of of the is calculated by and of timesis the optimization process. Finally, usgoal countries. for landing missions, accurate landing on site Also is much essential demanding of the trajectory trajectory isin calculated by integration integration and called called goal for several countries. Also for such landing missions, thousands thousands of times in the optimization process. Finally, usaccurate landing on site is much essential demanding thousands of times in the optimization process. Finally, using the covariance and fuel consumption, the performance accurate landing on site is much essential demanding on-board autonomy. A critical difficulty of such an authousands of times in the optimization process. Finally, usaccurate landing on Asite is much essential demanding ing the covariance and fuel consumption, the performance on-board autonomy. critical difficulty of such an auing the theiscovariance covariance andThe fueloptimal consumption, the performance performance index constructed. trajectory obtained by on-board autonomy. A critical of such autonomous mission, however, is difficulty the fact that the an lander ing and fuel consumption, the on-board autonomy. A critical difficulty of such an auis constructed. The optimal obtained by tonomousthe mission, however, island the on factthe that the surface lander index index is The trajectory obtained by this performance indextrajectory is expected to be fuel tonomous mission, however, the fact that the lander carrying rover must soft is moon index is constructed. constructed. The optimal optimal trajectory obtained by tonomous mission, however, island the on factthe that the surface lander minimizing minimizing this performance index is expected to be fuel carrying the rover must soft moon minimizing this performance index is expected to be fuel saving and desensitized to the uncertainties as well. carrying the rover must soft land on the moon surface with the help of on-board sensors, processors and actuators minimizing this performance index is expected to be fuel carrying the rover must soft land on the moon surface with the the any helphuman of on-board on-board sensors, processors processors and actuators actuators saving saving and and desensitized desensitized to to the the uncertainties uncertainties as as well. well. with help of sensors, without intervention. desensitized to the uncertainties well. with the any helphuman of on-board sensors, processors and and actuators saving In real and scenario, the trajectory design shouldasaccount for without intervention. without any human intervention. In real scenario, the trajectory design should account for without any human intervention. In real real scenario, the trajectory trajectory design should accounttime for all mission constraints such as design camerashould look angle, Seywald (2003) proposed desensitized optimal trajectory In scenario, the account for all mission constraints such as camera look angle, time Seywald (2003) proposed desensitized optimal trajectory all mission constraints such as camera look angle, time for image processing which has not been considered in the Seywald (2003) proposed desensitized optimal trajectory design that enables to design trajectories insensitive to all mission constraints such as camera look angle, time Seywald (2003) proposed desensitized optimal trajectory image processing which has considered in the designperturbations that enables to design design trajectories insensitive to for for image processing which not been considered in above mentioned literatures. In not thisbeen paper, two stages are design that enables to trajectories to state encountered at any time insensitive along the trafor image processing which has has not been considered in the the design that enables to design trajectories insensitive to above mentioned literatures. In this paper, two stages are state perturbations perturbations encountered at time the above mentioned literatures. In this paper, two stages are to account for such mission constraints. The state encountered at any any covariance time along along shaping the tratra- considered jectory. Saunders (2012) implemented above mentioned literatures. In this paper, two stages are state perturbations encountered at any time along the traconsidered to account for such mission constraints. The jectory. Saunders (2012) implemented covariance shaping considered to account for such mission constraints. The problem has been addressed in orbital frame of reference jectory. Saunders (2012) implemented covariance shaping as a tool in reference trajectory design for atmospheric considered to account for such mission constraints. The jectory. Saunders (2012) implemented covariance shaping problem has in frame reference as aa tool tool in in reference reference trajectory design for for atmospheric atmospheric problem has been been addressed addressed in orbital orbital frame of of reference which is advantageous in defining the mission constraints as trajectory design reentry to reduce the sensitivity trajectories problem has been addressed in orbital frame of reference as a toolvehicles in reference trajectory design forof atmospheric which is advantageous in defining the mission constraints reentry vehicles to reduce the sensitivity of trajectories which is advantageous in defining the mission constraints directly. Then, this multi-phase lunar landing problem is reentry vehicles to the sensitivity of trajectories to uncertainties. To reduce maximize the probability of mission which is advantageous in defining the mission constraints reentry vehicles to reduce the sensitivity of trajectories directly. Then, this multi-phase lunar landing problem is to uncertainties. To maximize the probability of mission directly. Then, this multi-phase lunar landing problem is formulated as desensitized optimal control and solved by to uncertainties. maximize the of mission mission success, RobinsonTo (2013) introduced a new approach which directly. Then, this multi-phase lunar landing problem is to uncertainties. To maximize the probability probability of formulated as desensitized optimal control and solved by success, Robinson (2013) introduced a new approach which formulated as desensitized optimal control and solved by a viable technique Legendre pseudospectral method. success, Robinson (2013) introduced a new approach which minimize the variance of trajectory dispersions subject to formulated as desensitized optimal control and solved by success, Robinson (2013) introduced a new approach which the variance variance of trajectory trajectory dispersions subject to aaa viable viable technique technique Legendre Legendre pseudospectral pseudospectral method. method. minimize the of dispersions subject to aminimize fuel consumption constraint. For Mars atmospheric entry viable technique Legendre pseudospectral method. minimize the variance of trajectory dispersions subject to a fuel consumption constraint. For Mars atmospheric entry 2. DESENSITIZED TRAJECTORY OPTIMIZATION avehicle, fuel consumption For Mars entry Xu and Cuiconstraint. (2015) presented a atmospheric robust trajectory a fuel consumption constraint. For Mars atmospheric entry 2. vehicle,algorithm Xu and and Cui (2015)the presented robust trajectory 2. DESENSITIZED DESENSITIZED TRAJECTORY TRAJECTORY OPTIMIZATION OPTIMIZATION 2. DESENSITIZED TRAJECTORY OPTIMIZATION vehicle, Xu (2015) presented aa trajectory design to reduce sensitivity to uncertainties vehicle, Xu and Cui Cui (2015)the presented a robust robust trajectory design algorithm to reduce sensitivity to uncertainties Spacecraft is de-boosted from a circular orbit of 100 km design algorithm to reduce the sensitivity to uncertainties in states, aerodynamic density, aerodynamic coefficient. design algorithm to reduce the sensitivity to uncertainties Spacecraft is de-boosted from aa circular orbit ofHohmann 100 km in states, states, aerodynamic density, aerodynamic coefficient. Spacecraft is de-boosted from circular orbit 100 altitude. De-boosted spacecraft enters into Spacecraft is de-boosted from a circular orbitaa of ofHohmann 100 km km in aerodynamic density, aerodynamic coefficient. Hu et al. (2016) presented the landing on the small bodies in states, aerodynamic density, aerodynamic coefficient. altitude. De-boosted spacecraft enters into Hu et et al. al. in (2016) presented the landing on the the small small bodies altitude. De-boosted spacecraft trajectory from 100 kmenters to 18 into km. aaAtHohmann perilune altitude. De-boosted spacecraft enters into Hohmann Hu (2016) the landing on bodies problem whichpresented the thrust profile of optimal landing tra- transfer Hu et al. (2016) presented the landing on the small bodies from 100 18 At problem in in which the the thrust profile profile of optimal optimal landing tratra- transfer transferof trajectory trajectory fromenters 100 km km to 18 km. km. At perilune perilune 18 km , probe into to powered descent phase. transfer trajectory from 100 km to 18 km. At perilune problem problem in which which the thrust thrust profile of of optimal landing landing tra- height height of 18 km , probe enters into powered descent phase. Authors are thankful to DST-FIST for its financial support in heightthis of 18 18 km ,,the probe enters into into powered descenton phase. Near phase, propulsive engines are turned and height of km probe enters powered descent phase. Authors are thankful to DST-FIST for its financial support in Near this phase, the propulsive engines are turned on and AuthorsDIDO Near this phase, the are turned on and the probe velocity ispropulsive decreasedengines in order to enable the acquiring software to are thankful thankful to DST-FIST DST-FIST for for its its financial financial support support in in Near this phase, the propulsive engines are turned on and Authors are the probe velocity is decreased in order to enable the acquiring DIDO software the probe velocity is decreased in order to enable acquiring DIDO software the probe velocity is decreased in order to enable the the acquiring DIDO software Copyright 2017 IFAC 6294Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Copyright 2017 responsibility IFAC 6294Control. Peer review© of International Federation of Automatic Copyright © 2017 6294 Copyright ©under 2017 IFAC IFAC 6294 10.1016/j.ifacol.2017.08.1391
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lander to soft land on the surface of the Moon. There are four distinct phases that need to be performed in order to land the lander on the surface of moon. They are: (i) Deorbit maneuver phase (ii) Transfer orbit phase or coasting phase and (iii) Powered descent phase (iv) Vertical descent phase. The parking orbit of the spacecraft is planned to be 100 km circular for this study. Otherwise, the method developed is general in nature and can be applied to other types of orbit as well. Powered descent phase starts from the terminal condition of the transfer orbit phase. This phase is split into four stages due to mission constraint as shown in the Fig. 1. The absolute sensor information is available only from the start of the barking with precise navigation stage. So till the end of attitude hold phase, the open loop reference trajectory has to be followed using the inertial navigation system. The tracking of optimal reference trajectory generated (Mathavaraj et al. (2016)) using trajectory linearization quadratic method in presence of uncertainties contribute greatly in trajectory dispersions at the end of attitude hold phase. The aim is to generate the optimal reference trajectory insensitive to these uncertainties using the desensitized optimal control methodology which in turn tracked using linear quadratic regulator results in significant reduction in error ellipse.
Fig. 1. Powered descent phase mission scenario The system dynamics, as in (1) then can be represented as X˙ = f (X, U, t)
(3)
The spacecraft fuel consumption is to be minimized if it is to carry a enhanced payload mass for useful lunar explorations. So the cost function for the optimal control problem is selected as
J0 =
tf
T /m
dt
(4)
t0
2.1 System Dynamics 2.3 Closed Loop Linear Quadratic Regulator In order to describe the motion of a dynamic system it is necessary to define a suitable coordinate system and formulate equations for the motion in accordance with physical laws governing the system. Considering the vehicle motion’s as a point pass flying inside a planetary atmosphere, the equation of motion is defined as r˙ = w u θ˙ = r µ u2 T sin β − 2+ w˙ = m r r uw T cos β − u˙ = m r T m ˙ = − Isp g
δX = X − X ∗ δU = U − U ∗
(5)
So from Hamiltonian approach the deviation control δU using feedback gain matrix K(t) is obtained as (1)
δU = −K (t) δX
(6)
The updated control is obtained, as in (7)
The dynamics, as in (1) represent the two dimensional motion of the point mass model, where r represents the radial distance from the center of the moon, θ represents the longitude, u, w represents the tangential velocity and radial velocity components respectively and m represents the instantaneous mass of the lander. Also µ is the gravitational parameter of planet considered, g represents earth’s gravity and Isp is specific impulse of the engine considered for simulation. The control parameters considered are thrust T and its orientation β. For further details one can refer to Vinh et al. (1980) 2.2 Lunar Landing Optimal Trajectory Design The state vector X and control vector U considered as X = [r θ w u m] T U = [T β]
In the time interval between t0 to tf , let δX and δU be the deviation from the nominal trajectory (X ∗ , U ∗ ) respectively, as in (5).
T
(2)
U = U ∗ − η (t) K (t) δX
(7)
where, η(t) is multiplicative factor defined, as in (8) η (t) =
4 (T (t) − Tmin ) . (Tmax − T ∗ (t)) (Tmax − Tmin )
2
(8)
The multiplicative factor assures that if the updated thrust T (t) reaches the saturation limit then the updated and nominal control remains same. 2.4 Sensitivity Penalties In reality, the perturbations in initial condition and gravity uncertainties along the trajectory causes the spacecraft deviate from the nominal path. For safe pinpoint landing, excessive deviations from the nominal path may lead to mission failure. For successful mission, one has to account these uncertainties in optimal reference trajectory
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∂f ∂f G= − η (t) K (t) ∂X X ∗ ,U ∗ ∂U X ∗ ,U ∗
formulation. One way to address this, is to model the gravity and thrust uncertainties as stochastic process noise acting along the lander, as in (9).
where,
ˆ˙ = f X, ˆ U, t + W X ˆ = [ r θ w u m ]T X T W = [ 0 0 W r Wθ 0 ]
where, K(t) is feedback gain matrix determined from LQR technique. Also, Q is the power spectral density of the stochastic noise term W and it has following relation
(10)
W(t) ∼ (0, Q) E W (t) W T (t + τ ) = Qδ (τ )
The error in thrust magnitude and direction (δT, δβ) is assumed to obey normal distribution with mean of zeros and standard deviation of σT ,σβ respectively. According to property of stochastic variables, the mean of δTr ,δTθ are also zeros and the variance can be obtained as 2
δTr mN
tf 4 N 2 J1 = ci P (i, j) dt i=1 j=1
(19)
t0
ci ≥ 0
where, P (i, j) represents the covariance matrix elements, ˆ ci N is the dimension of the uncertain parameter vector X, is to weight the final dispersions of position and velocity. So the final performance index including the fuel consumption minimization is given by
(12)
(J0 + c0 J1 ) J = min ∗
(13)
So the desensitized optimal control formulation aims not only to minimize the lander’s fuel consumption but also reduces the sensitivity of optimal trajectory to uncertainties and perturbation.
where, Cg obey standard normal distribution, σg represents the gravity standard deviation parameter, and gN is the nominal Moon’s gravity. In radial direction, the thrust and gravity stochastic term can be added up and represented as Wr = C g σ g g N +
(18)
To include effect of uncertainties in optimal control formulation the following cost function J1 is to minimized since the linear covariance describes the degree of trajectory dispersions
2
The gravity uncertainties can be represented as g = (1 + Cg σg ) gN
(17)
(9)
The linear covariance of constructed stochastic state equation (9) can be propagated which represents the trajectory dispersions about the nominal trajectory. The thrust error δTr ,δTθ in two direction by variational approach is expressed as δTr TN cos βN = δβ δTθ βN −TN sin (11) sin βN +δT cos βN
V ar (δTr ) = (TN cos βN σβ ) + (sin βN σT ) 2 2 V ar (δTθ ) = (TN sin βN σβ ) + (cos βN σT )
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(14)
The variance of the stochastic acceleration terms in radial and tangential direction is obtained as
(20)
U
3. PROBLEM SPECIFIC FORMULATION As explained in section 2.2, the optimal control problem is to be solved which has dynamical constraints, equality constraints, inequality constraints and integral cost function. An efficient way to address this problem is by Legendre Pseudospectral method. For more details on Legendre Pseudospectral method refer Fahroo and Ross (2004). 3.1 Dynamical Constraints
V ar (δTr ) V ar (Wr ) = (σg gN ) + mN 2 V ar (δTθ ) V ar (Wθ ) = mN 2 2
(15)
2.5 Closed Loop Linear Covariance Shaping
3.2 Braking with Rough Navigation
From the stochastic state equations, the linear covariance state equation (P˙ ) is obtained as P˙ = GP + P GT + Q
The dynamical equation used for all phase is formulated into non-linear programming problem by Legendre pseudospectral method (Mathavaraj et al. (2016))
(16)
The objective of this stage 1 is to break the orbital velocity as much as possible and ensuring the lander is at proper altitude and attitude for camera imaging.
where, G is derived from the jacobian of the state equations (1) with feedback linear quadratic control is given by 6296
T
[ r1 (ti ) − r1si w1 (ti ) − w1si u1 (ti ) − u1si ] = 0 T r1 (tf ) − r1sf β1 (tf ) − β1sf = 0 (21)
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3.3 Attitude Hold The objective of this stage 2 is to image the lunar surface, process it and feed the navigation information to guidance algorithm for guiding the lander to hazard free landing site. During this stage, the thrust firing and attitude has to be with held to avoid jittering during image capturing.
T2 (t) − T1sf β2 (t) − β1sf
T
=0
(22)
4. RESULTS FROM SIMULATION EXPERIMENTS Initial and final condition for the point mass dynamics is tabulated in Table 1. In formulation, these condition become constraint for optimal control algorithm. As seen in table, each stage has its own mission constraint. Stage 1 terminal condition (T.C.) in altitude and attitude has been specified by the sensor team. Stage 2 constraint emphases on avoiding jittering during imaging. Detailed description of these constraints has been discussed in the section 3. Table 2 gives the parameter uncertainties considered for the Monte-carlo simulation. Table 1. Mission Constraints for Lunar Lander Stage
1
2
Variables Radial Vel. Tangent Vel. Altitude Thrust Orientation Radial Vel. Tangent Vel. Altitude Thrust Orientation
Initial 0 1.69 km/s 18.2 km Free Stage 1 T.C. Stage 1 T.C. Stage 1 T.C. 140 deg
Fig. 2. Lander Trajectory Implying Descent of Landercraft to Landing Site
Final Free Free 7 km 140 deg Free Free Free 140 deg
Table 2. Simulation Parameter Uncertainties Variables σT σβ σg Initial position Intial velocity
Distribution Normal Normal Normal Uniform Uniform
Value 0.5% Tmax 0.5 deg 10% gN ±500 m ±1 m/s
As explained in section 2.2, the optimal control problem is solved using the cost function, as in (4) addressing the fuel optimal. The obtained optimal reference trajectory satisfies all mission constraints specified, as in table (1). In presence of uncertainties and perturbations, to achieve the desired terminal position and velocity a closed loop feedback control is designed as explained in section 2.3. Using the feedback control law, Monte-Carlo analysis is carried out till the terminal flight time of attitude hold phase. Simulation result shows the error ellipse obtained is not within the mission constraints bound. In order to improve the terminal accuracy of the lander, desensitized optimal control trajectory design as explained in section 2.5 is carried out. As shown in the figures below, the trajectory as well as its flight duration is altered to account for perturbation. Using the DOC trajectory, Monte-carlo analysis is simulated with closed loop feedback control law to examine the effect on error ellipse at terminal condition of attitude hold phase. Fig. 2 shows the altitude of the lander decreases as the downrange reduces to desired position for no-DOC and DOC trajectory formulation satisfying mission constraint
Fig. 3. Thrust Orientation Profile Ensuring Camera Look Angle of Landercraft i.e. first stage terminal constraint of 7 km. Fig. 3 depicts the thrust vector direction with terminal constraints satisfied i.e. first stage terminal constraint of 140 deg making sure camera looking at landing site. Fig. 4 is the demanded thrust throttling percentage. Note that DOC formulation results in significant change in thrust demanded compared to no-DOC, to account for closed loop feedback control law in trajectory design. Note that the additional propellant consumption compared to no-DOC trajectory design is negligible. Also, note that in stage 2 the throttling percentage and thrust vector direction of former phase is maintained throughout as mission required, as in (22). Fig. 5-6 shows the significant reduction in position as well as velocity error ellipse respectively at terminal condition of attitude hold phase due to desensitized optimal control reference trajectory. 5. CONCLUSION Thus the lunar landing problem has been formulated as desensitized optimal control problem and solved by Legen-
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dre Pseudospectral method. Simulation analysis shows the reference trajectory satisfies all the mission constraints like thrust upper-lower bound, camera look angle constraint and time for image processing. Simulation analysis in presence of uncertainties and initial condition perturbation demonstrates the efficiency of the desensitized optimal control formulation over nominal trajectory optimization methodology for lunar landing problem. Monte Carlo simulations show that position errors of less than 100 m and velocity errors of less than 0.2 m/s are achieved. However the amount of extra fuel consumed is less significant when compared to improvement in landing accuracy assuring the desensitized optimal control is viable for lunar landing trajectory design. ACKNOWLEDGEMENTS
Fig. 4. Demanded Thrust Firing for Achieving Mission Goals
The authors are thankful to the Indian Space Research Organisation (ISRO) for giving an opportunity to work on this challenging problem. REFERENCES
Fig. 5. Monte-carlo Position Dispersions by DOC and NoDOC Trajectory
Fahroo, F. and Ross, M. (2004). Pseudospectral method knotting methods for solving optimal control problems. Journal of Guidance, Control, and Dynamics, 27, 397– 405. Hu, H., Zhu, S., and Cui, P. (2016). Desensitized optimal trajectory for landing on small bodies with reduced landing error. Journal of Aerospace Science and Technology, 48, 178–185. Mathavaraj, S., Pandiyan, R., and Padhi, R. (2016). Optimal trajectory planning for multiphase lunar landing. In Proceedings of the Advances in Control and Optimization of Dynamical Systems, volume 49, 124–129. Elsevier. Robinson, S.B. (2013). Spacecraft Guidance Techniques for Maximizing Mission Success. Ph.D. thesis, Utah State University. Saunders, B.R. (2012). Optimal trajectory design under uncertainty. Ph.D. thesis, Massachusetts Institute of Technology. Seywald, H. (2003). Desensitized optimal trajectories with control constraints. Advances in the Astronautical Sciences, 114, 737–743. Vinh, N., Busemann, A., and Culp, R. (1980). Hypersonic and planetary entry flight mechanics. Univ. of Michigan Press, Ann Arbor. Wang, H., Xie, M., Zhang, P., and Li, Y. (2014). Research of asteroid landing trajectory optimization based on gauss pseudo-spectral method. In Proceedings of the Intelligent Human-Machine Systems and Cybernetics, volume 2, 26–29. IEEE. Xu, H. and Cui, H. (2015). Robust trajectory design scheme under uncertainties and perturbations for mars entry vehicle. In Proceedings of the Computational Intelligence & Communication Technology, 762–766. IEEE.
Fig. 6. Monte-carlo Velocity Dispersions by DOC and NoDOC Trajectory 6298
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Desensitized Optimal Trajectory for Desensitized Optimal Trajectory for Desensitized Optimal Trajectory for Multi-phase Lunar Landing Multi-phase Lunar Landing Multi-phase Lunar Landing
S. Mathavaraj ∗∗ Radhakant Padhi ∗∗ ∗∗ S. ∗∗ S. Mathavaraj Mathavaraj ∗∗ Radhakant Radhakant Padhi Padhi ∗∗ S. Mathavaraj Radhakant Padhi ∗ ∗ Flight Dynamics Group, ISRO Satellite Center, Bangalore, India Dynamics (e-mail: Group, ISRO Satellite Center, Bangalore, India ∗ ∗ Flight Flight Group, Satellite [email protected]) Flight Dynamics Dynamics (e-mail: Group, ISRO ISRO Satellite Center, Center, Bangalore, Bangalore, India India [email protected]) ∗∗ (e-mail: [email protected]) Eng., Indian Institute of Science, India (e-mail: [email protected]) ∗∗ Department of Aerospace ∗∗ Department of Aerospace Eng., Indian Institute of Science, India ∗∗ Department of of(e-mail: Aerospace Eng., Indian Indian Institute Institute of of Science, Science, India India [email protected]) Department Aerospace Eng., (e-mail: (e-mail: [email protected]) [email protected]) (e-mail: [email protected]) Abstract: Lunar landing problem has been formulated as desensitized optimal control problem Abstract: Lunar landing problem has been formulated as desensitized control Abstract: Lunar landing problem formulated as optimal control problem and solved by Legendre method. The problem has beenoptimal split into threeproblem phases Abstract: Lunar landingPseudospectral problem has has been been formulated as desensitized desensitized optimal control problem and solved by Legendre Pseudospectral method. The problem has been split into three phases and solved by Legendre Pseudospectral method. The problem has been split into three phases to account for mission constraints, namely the braking with rough navigation stage (from 18km and solved by Legendre Pseudospectral method. The problem has been split into three phases to account for mission constraints, namely the braking with rough navigation stage (from 18km account for mission mission constraints, namely the braking braking with rough navigation stage (from (from 18km to account 7km), attitude holdconstraints, stage (holding the attitude for with 35sec) for anavigation typical mission scenario. In for namely the rough stage 18km to attitude hold (holding the for for mission In to 7km), 7km), of attitude hold stage stage (holding the attitude attitude for 35sec) 35sec) for aaa typical typical mission scenario. In presence uncertainties, following the open loop reference trajectory using the closedscenario. loop linear to 7km), attitude hold stage (holding the attitude for 35sec) for typical mission scenario. In presence of uncertainties, following the open loop reference trajectory using the closed loop linear presence of uncertainties, following the open loop reference trajectory using the closed loop linear quadratic regulator contribute greatly in trajectory dispersions. Theusing goal the is toclosed desensitize this presence of uncertainties, following the open loop reference trajectory loop linear quadratic regulator greatly in trajectory The goal is to desensitize this quadratic regulator contribute greatly in dispersions. The is to this multi-phase optimal contribute trajectory with reduced error in dispersions. presence of initial state thrust error, quadratic regulator contribute greatly in trajectory trajectory dispersions. The goal goal is error, to desensitize desensitize this multi-phase optimal trajectory with reduced error in presence of initial state error, thrust error, multi-phase optimal trajectory with reduced error in presence presence of initial initial state error,isthrust thrust error, Moon’s gravity uncertainty. Towith achieve this, error the fuel minimization coststate function augmented multi-phase optimal trajectory reduced in of error, error, Moon’s gravity uncertainty. this, fuel minimization function is Moon’s gravity uncertainty. To achieve this, the fuel loop minimization cost functionFollowing is augmented augmented with closed loop covarianceTo to achieve generate thethe open reference cost trajectory. this Moon’s gravity uncertainty. To achieve this, the fuel minimization cost function is augmented with closed loop covariance to generate the open loop reference trajectory. Following this with closed loop covariance to generate the open loop reference trajectory. Following this reference trajectory using closed loop linear quadratic regulator, shows significant reduction with closed loop covariance to generate the open loop reference trajectory. Following this reference trajectory using closed loop linear quadratic regulator, shows significant reduction reference trajectory using closed loop linear quadratic regulator, shows significant reduction in landing error. The amount of extra fuel consumed by desensitizing optimal trajectory is less reference trajectory using closed loop linear quadratic regulator, shows significant reduction in amount extra by optimal trajectory is in landing landing error. error. The amounttoof ofimprovement extra fuel fuel consumed consumed by desensitizing desensitizing optimal trajectory is less less significant when The compared in landing accuracy to meet mission constraints in landing error. The amount of extra fuel consumed by desensitizing optimal trajectory is less significant compared to in accuracy to meet constraints significant when compared to improvement improvement in landing landing accuracy to landing meet mission mission constraints assuring thewhen desensitized optimal control a viable technique for lunar trajectory design. significant when compared to improvement in landing accuracy to meet mission constraints assuring the desensitized optimal control a viable technique for lunar landing trajectory design. assuring assuring the the desensitized desensitized optimal optimal control control aa viable viable technique technique for for lunar lunar landing landing trajectory trajectory design. design. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Lunar Landing, Desensitized Optimal Control, Nonlinear Programming, Legendre Keywords: Lunar Lunar Landing, Landing, Desensitized Optimal Optimal Control, Nonlinear Nonlinear Programming, Legendre Legendre Keywords: Pseudospectral Keywords: Lunar Landing, Desensitized Desensitized Optimal Control, Control, Nonlinear Programming, Programming, Legendre Pseudospectral Pseudospectral Pseudospectral 1. INTRODUCTION jectory is of maximum minimum maximum structure. The 1. INTRODUCTION INTRODUCTION jectory is minimum structure. The 1. jectory is of of maximum maximum minimum maximum maximum structure. The main objective of the desensitized optimal control (DOC) 1. INTRODUCTION jectory is of maximum minimum maximum structure. The main objective of the desensitized optimal control (DOC) main objective of the desensitized optimal control (DOC) strategy is to reduce the sensitivity to state uncertainties Moon, the closest celestial body to our planet, is often main objective of the desensitized optimal control (DOC) is to the sensitivity to statethe uncertainties Moon, the the as closest celestial body to to our ourspace planet, is often often strategy strategy is reduce the sensitivity to uncertainties well as to reduce minimize To describe trajectory Moon, closest body planet, is considered a basecelestial for conducting technology strategy is to to reduce the fuel. sensitivity to state statethe uncertainties Moon, the as closest celestial body tonew ourspace planet, is often as as well as to minimize fuel. To describe trajectory considered a base for conducting new technology as well as to minimize fuel. To describe the trajectory dispersions in the formulation, the DOC strategy uses considered as a base for conducting new space technology demonstrations. Further to that, the desire is to soft land as well as to minimize fuel. To describe the trajectory considered as a base for conducting new space technology dispersions in the formulation, the DOC strategy uses demonstrations. Further to that, the desire is to soft land dispersions in the thetechnique. formulation, thethe DOC strategy uses linear covariance Then linear covariance demonstrations. to that, is to soft land ademonstrations. spacecraft withFurther / without man the has desire been the immediate dispersions in formulation, the DOC strategy uses Further to that, the desire is to soft land linear covariance technique. Then the linear covariance a spacecraft spacecraft with without manforhas has been the immediate immediate linear technique. the linear covariance the covariance trajectory is calculatedThen by integration and called agoal with // without man been the for several countries. Also such landing missions, of linear covariance technique. Then the linear covariance a spacecraft with / without man has been the immediate the trajectory calculated by integration and called goal for for several several countries. Also for such such landing missions, of of the is calculated by and of timesis the optimization process. Finally, usgoal countries. for landing missions, accurate landing on site Also is much essential demanding of the trajectory trajectory isin calculated by integration integration and called called goal for several countries. Also for such landing missions, thousands thousands of times in the optimization process. Finally, usaccurate landing on site is much essential demanding thousands of times in the optimization process. Finally, using the covariance and fuel consumption, the performance accurate landing on site is much essential demanding on-board autonomy. A critical difficulty of such an authousands of times in the optimization process. Finally, usaccurate landing on Asite is much essential demanding ing the covariance and fuel consumption, the performance on-board autonomy. critical difficulty of such an auing the theiscovariance covariance andThe fueloptimal consumption, the performance performance index constructed. trajectory obtained by on-board autonomy. A critical of such autonomous mission, however, is difficulty the fact that the an lander ing and fuel consumption, the on-board autonomy. A critical difficulty of such an auis constructed. The optimal obtained by tonomousthe mission, however, island the on factthe that the surface lander index index is The trajectory obtained by this performance indextrajectory is expected to be fuel tonomous mission, however, the fact that the lander carrying rover must soft is moon index is constructed. constructed. The optimal optimal trajectory obtained by tonomous mission, however, island the on factthe that the surface lander minimizing minimizing this performance index is expected to be fuel carrying the rover must soft moon minimizing this performance index is expected to be fuel saving and desensitized to the uncertainties as well. carrying the rover must soft land on the moon surface with the help of on-board sensors, processors and actuators minimizing this performance index is expected to be fuel carrying the rover must soft land on the moon surface with the the any helphuman of on-board on-board sensors, processors processors and actuators actuators saving saving and and desensitized desensitized to to the the uncertainties uncertainties as as well. well. with help of sensors, without intervention. desensitized to the uncertainties well. with the any helphuman of on-board sensors, processors and and actuators saving In real and scenario, the trajectory design shouldasaccount for without intervention. without any human intervention. In real scenario, the trajectory design should account for without any human intervention. In real real scenario, the trajectory trajectory design should accounttime for all mission constraints such as design camerashould look angle, Seywald (2003) proposed desensitized optimal trajectory In scenario, the account for all mission constraints such as camera look angle, time Seywald (2003) proposed desensitized optimal trajectory all mission constraints such as camera look angle, time for image processing which has not been considered in the Seywald (2003) proposed desensitized optimal trajectory design that enables to design trajectories insensitive to all mission constraints such as camera look angle, time Seywald (2003) proposed desensitized optimal trajectory image processing which has considered in the designperturbations that enables to design design trajectories insensitive to for for image processing which not been considered in above mentioned literatures. In not thisbeen paper, two stages are design that enables to trajectories to state encountered at any time insensitive along the trafor image processing which has has not been considered in the the design that enables to design trajectories insensitive to above mentioned literatures. In this paper, two stages are state perturbations perturbations encountered at time the above mentioned literatures. In this paper, two stages are to account for such mission constraints. The state encountered at any any covariance time along along shaping the tratra- considered jectory. Saunders (2012) implemented above mentioned literatures. In this paper, two stages are state perturbations encountered at any time along the traconsidered to account for such mission constraints. The jectory. Saunders (2012) implemented covariance shaping considered to account for such mission constraints. The problem has been addressed in orbital frame of reference jectory. Saunders (2012) implemented covariance shaping as a tool in reference trajectory design for atmospheric considered to account for such mission constraints. The jectory. Saunders (2012) implemented covariance shaping problem has in frame reference as aa tool tool in in reference reference trajectory design for for atmospheric atmospheric problem has been been addressed addressed in orbital orbital frame of of reference which is advantageous in defining the mission constraints as trajectory design reentry to reduce the sensitivity trajectories problem has been addressed in orbital frame of reference as a toolvehicles in reference trajectory design forof atmospheric which is advantageous in defining the mission constraints reentry vehicles to reduce the sensitivity of trajectories which is advantageous in defining the mission constraints directly. Then, this multi-phase lunar landing problem is reentry vehicles to the sensitivity of trajectories to uncertainties. To reduce maximize the probability of mission which is advantageous in defining the mission constraints reentry vehicles to reduce the sensitivity of trajectories directly. Then, this multi-phase lunar landing problem is to uncertainties. To maximize the probability of mission directly. Then, this multi-phase lunar landing problem is formulated as desensitized optimal control and solved by to uncertainties. maximize the of mission mission success, RobinsonTo (2013) introduced a new approach which directly. Then, this multi-phase lunar landing problem is to uncertainties. To maximize the probability probability of formulated as desensitized optimal control and solved by success, Robinson (2013) introduced a new approach which formulated as desensitized optimal control and solved by a viable technique Legendre pseudospectral method. success, Robinson (2013) introduced a new approach which minimize the variance of trajectory dispersions subject to formulated as desensitized optimal control and solved by success, Robinson (2013) introduced a new approach which the variance variance of trajectory trajectory dispersions subject to aaa viable viable technique technique Legendre Legendre pseudospectral pseudospectral method. method. minimize the of dispersions subject to aminimize fuel consumption constraint. For Mars atmospheric entry viable technique Legendre pseudospectral method. minimize the variance of trajectory dispersions subject to a fuel consumption constraint. For Mars atmospheric entry 2. DESENSITIZED TRAJECTORY OPTIMIZATION avehicle, fuel consumption For Mars entry Xu and Cuiconstraint. (2015) presented a atmospheric robust trajectory a fuel consumption constraint. For Mars atmospheric entry 2. vehicle,algorithm Xu and and Cui (2015)the presented robust trajectory 2. DESENSITIZED DESENSITIZED TRAJECTORY TRAJECTORY OPTIMIZATION OPTIMIZATION 2. DESENSITIZED TRAJECTORY OPTIMIZATION vehicle, Xu (2015) presented aa trajectory design to reduce sensitivity to uncertainties vehicle, Xu and Cui Cui (2015)the presented a robust robust trajectory design algorithm to reduce sensitivity to uncertainties Spacecraft is de-boosted from a circular orbit of 100 km design algorithm to reduce the sensitivity to uncertainties in states, aerodynamic density, aerodynamic coefficient. design algorithm to reduce the sensitivity to uncertainties Spacecraft is de-boosted from aa circular orbit ofHohmann 100 km in states, states, aerodynamic density, aerodynamic coefficient. Spacecraft is de-boosted from circular orbit 100 altitude. De-boosted spacecraft enters into Spacecraft is de-boosted from a circular orbitaa of ofHohmann 100 km km in aerodynamic density, aerodynamic coefficient. Hu et al. (2016) presented the landing on the small bodies in states, aerodynamic density, aerodynamic coefficient. altitude. De-boosted spacecraft enters into Hu et et al. al. in (2016) presented the landing on the the small small bodies altitude. De-boosted spacecraft trajectory from 100 kmenters to 18 into km. aaAtHohmann perilune altitude. De-boosted spacecraft enters into Hohmann Hu (2016) the landing on bodies problem whichpresented the thrust profile of optimal landing tra- transfer Hu et al. (2016) presented the landing on the small bodies from 100 18 At problem in in which the the thrust profile profile of optimal optimal landing tratra- transfer transferof trajectory trajectory fromenters 100 km km to 18 km. km. At perilune perilune 18 km , probe into to powered descent phase. transfer trajectory from 100 km to 18 km. At perilune problem problem in which which the thrust thrust profile of of optimal landing landing tra- height height of 18 km , probe enters into powered descent phase. Authors are thankful to DST-FIST for its financial support in heightthis of 18 18 km ,,the probe enters into into powered descenton phase. Near phase, propulsive engines are turned and height of km probe enters powered descent phase. Authors are thankful to DST-FIST for its financial support in Near this phase, the propulsive engines are turned on and AuthorsDIDO Near this phase, the are turned on and the probe velocity ispropulsive decreasedengines in order to enable the acquiring software to are thankful thankful to DST-FIST DST-FIST for for its its financial financial support support in in Near this phase, the propulsive engines are turned on and Authors are the probe velocity is decreased in order to enable the acquiring DIDO software the probe velocity is decreased in order to enable acquiring DIDO software the probe velocity is decreased in order to enable the the acquiring DIDO software Copyright 2017 IFAC 6294Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Copyright 2017 responsibility IFAC 6294Control. Peer review© of International Federation of Automatic Copyright © 2017 6294 Copyright ©under 2017 IFAC IFAC 6294 10.1016/j.ifacol.2017.08.1391
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lander to soft land on the surface of the Moon. There are four distinct phases that need to be performed in order to land the lander on the surface of moon. They are: (i) Deorbit maneuver phase (ii) Transfer orbit phase or coasting phase and (iii) Powered descent phase (iv) Vertical descent phase. The parking orbit of the spacecraft is planned to be 100 km circular for this study. Otherwise, the method developed is general in nature and can be applied to other types of orbit as well. Powered descent phase starts from the terminal condition of the transfer orbit phase. This phase is split into four stages due to mission constraint as shown in the Fig. 1. The absolute sensor information is available only from the start of the barking with precise navigation stage. So till the end of attitude hold phase, the open loop reference trajectory has to be followed using the inertial navigation system. The tracking of optimal reference trajectory generated (Mathavaraj et al. (2016)) using trajectory linearization quadratic method in presence of uncertainties contribute greatly in trajectory dispersions at the end of attitude hold phase. The aim is to generate the optimal reference trajectory insensitive to these uncertainties using the desensitized optimal control methodology which in turn tracked using linear quadratic regulator results in significant reduction in error ellipse.
Fig. 1. Powered descent phase mission scenario The system dynamics, as in (1) then can be represented as X˙ = f (X, U, t)
(3)
The spacecraft fuel consumption is to be minimized if it is to carry a enhanced payload mass for useful lunar explorations. So the cost function for the optimal control problem is selected as
J0 =
tf
T /m
dt
(4)
t0
2.1 System Dynamics 2.3 Closed Loop Linear Quadratic Regulator In order to describe the motion of a dynamic system it is necessary to define a suitable coordinate system and formulate equations for the motion in accordance with physical laws governing the system. Considering the vehicle motion’s as a point pass flying inside a planetary atmosphere, the equation of motion is defined as r˙ = w u θ˙ = r µ u2 T sin β − 2+ w˙ = m r r uw T cos β − u˙ = m r T m ˙ = − Isp g
δX = X − X ∗ δU = U − U ∗
(5)
So from Hamiltonian approach the deviation control δU using feedback gain matrix K(t) is obtained as (1)
δU = −K (t) δX
(6)
The updated control is obtained, as in (7)
The dynamics, as in (1) represent the two dimensional motion of the point mass model, where r represents the radial distance from the center of the moon, θ represents the longitude, u, w represents the tangential velocity and radial velocity components respectively and m represents the instantaneous mass of the lander. Also µ is the gravitational parameter of planet considered, g represents earth’s gravity and Isp is specific impulse of the engine considered for simulation. The control parameters considered are thrust T and its orientation β. For further details one can refer to Vinh et al. (1980) 2.2 Lunar Landing Optimal Trajectory Design The state vector X and control vector U considered as X = [r θ w u m] T U = [T β]
In the time interval between t0 to tf , let δX and δU be the deviation from the nominal trajectory (X ∗ , U ∗ ) respectively, as in (5).
T
(2)
U = U ∗ − η (t) K (t) δX
(7)
where, η(t) is multiplicative factor defined, as in (8) η (t) =
4 (T (t) − Tmin ) . (Tmax − T ∗ (t)) (Tmax − Tmin )
2
(8)
The multiplicative factor assures that if the updated thrust T (t) reaches the saturation limit then the updated and nominal control remains same. 2.4 Sensitivity Penalties In reality, the perturbations in initial condition and gravity uncertainties along the trajectory causes the spacecraft deviate from the nominal path. For safe pinpoint landing, excessive deviations from the nominal path may lead to mission failure. For successful mission, one has to account these uncertainties in optimal reference trajectory
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∂f ∂f G= − η (t) K (t) ∂X X ∗ ,U ∗ ∂U X ∗ ,U ∗
formulation. One way to address this, is to model the gravity and thrust uncertainties as stochastic process noise acting along the lander, as in (9).
where,
ˆ˙ = f X, ˆ U, t + W X ˆ = [ r θ w u m ]T X T W = [ 0 0 W r Wθ 0 ]
where, K(t) is feedback gain matrix determined from LQR technique. Also, Q is the power spectral density of the stochastic noise term W and it has following relation
(10)
W(t) ∼ (0, Q) E W (t) W T (t + τ ) = Qδ (τ )
The error in thrust magnitude and direction (δT, δβ) is assumed to obey normal distribution with mean of zeros and standard deviation of σT ,σβ respectively. According to property of stochastic variables, the mean of δTr ,δTθ are also zeros and the variance can be obtained as 2
δTr mN
tf 4 N 2 J1 = ci P (i, j) dt i=1 j=1
(19)
t0
ci ≥ 0
where, P (i, j) represents the covariance matrix elements, ˆ ci N is the dimension of the uncertain parameter vector X, is to weight the final dispersions of position and velocity. So the final performance index including the fuel consumption minimization is given by
(12)
(J0 + c0 J1 ) J = min ∗
(13)
So the desensitized optimal control formulation aims not only to minimize the lander’s fuel consumption but also reduces the sensitivity of optimal trajectory to uncertainties and perturbation.
where, Cg obey standard normal distribution, σg represents the gravity standard deviation parameter, and gN is the nominal Moon’s gravity. In radial direction, the thrust and gravity stochastic term can be added up and represented as Wr = C g σ g g N +
(18)
To include effect of uncertainties in optimal control formulation the following cost function J1 is to minimized since the linear covariance describes the degree of trajectory dispersions
2
The gravity uncertainties can be represented as g = (1 + Cg σg ) gN
(17)
(9)
The linear covariance of constructed stochastic state equation (9) can be propagated which represents the trajectory dispersions about the nominal trajectory. The thrust error δTr ,δTθ in two direction by variational approach is expressed as δTr TN cos βN = δβ δTθ βN −TN sin (11) sin βN +δT cos βN
V ar (δTr ) = (TN cos βN σβ ) + (sin βN σT ) 2 2 V ar (δTθ ) = (TN sin βN σβ ) + (cos βN σT )
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(14)
The variance of the stochastic acceleration terms in radial and tangential direction is obtained as
(20)
U
3. PROBLEM SPECIFIC FORMULATION As explained in section 2.2, the optimal control problem is to be solved which has dynamical constraints, equality constraints, inequality constraints and integral cost function. An efficient way to address this problem is by Legendre Pseudospectral method. For more details on Legendre Pseudospectral method refer Fahroo and Ross (2004). 3.1 Dynamical Constraints
V ar (δTr ) V ar (Wr ) = (σg gN ) + mN 2 V ar (δTθ ) V ar (Wθ ) = mN 2 2
(15)
2.5 Closed Loop Linear Covariance Shaping
3.2 Braking with Rough Navigation
From the stochastic state equations, the linear covariance state equation (P˙ ) is obtained as P˙ = GP + P GT + Q
The dynamical equation used for all phase is formulated into non-linear programming problem by Legendre pseudospectral method (Mathavaraj et al. (2016))
(16)
The objective of this stage 1 is to break the orbital velocity as much as possible and ensuring the lander is at proper altitude and attitude for camera imaging.
where, G is derived from the jacobian of the state equations (1) with feedback linear quadratic control is given by 6296
T
[ r1 (ti ) − r1si w1 (ti ) − w1si u1 (ti ) − u1si ] = 0 T r1 (tf ) − r1sf β1 (tf ) − β1sf = 0 (21)
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3.3 Attitude Hold The objective of this stage 2 is to image the lunar surface, process it and feed the navigation information to guidance algorithm for guiding the lander to hazard free landing site. During this stage, the thrust firing and attitude has to be with held to avoid jittering during image capturing.
T2 (t) − T1sf β2 (t) − β1sf
T
=0
(22)
4. RESULTS FROM SIMULATION EXPERIMENTS Initial and final condition for the point mass dynamics is tabulated in Table 1. In formulation, these condition become constraint for optimal control algorithm. As seen in table, each stage has its own mission constraint. Stage 1 terminal condition (T.C.) in altitude and attitude has been specified by the sensor team. Stage 2 constraint emphases on avoiding jittering during imaging. Detailed description of these constraints has been discussed in the section 3. Table 2 gives the parameter uncertainties considered for the Monte-carlo simulation. Table 1. Mission Constraints for Lunar Lander Stage
1
2
Variables Radial Vel. Tangent Vel. Altitude Thrust Orientation Radial Vel. Tangent Vel. Altitude Thrust Orientation
Initial 0 1.69 km/s 18.2 km Free Stage 1 T.C. Stage 1 T.C. Stage 1 T.C. 140 deg
Fig. 2. Lander Trajectory Implying Descent of Landercraft to Landing Site
Final Free Free 7 km 140 deg Free Free Free 140 deg
Table 2. Simulation Parameter Uncertainties Variables σT σβ σg Initial position Intial velocity
Distribution Normal Normal Normal Uniform Uniform
Value 0.5% Tmax 0.5 deg 10% gN ±500 m ±1 m/s
As explained in section 2.2, the optimal control problem is solved using the cost function, as in (4) addressing the fuel optimal. The obtained optimal reference trajectory satisfies all mission constraints specified, as in table (1). In presence of uncertainties and perturbations, to achieve the desired terminal position and velocity a closed loop feedback control is designed as explained in section 2.3. Using the feedback control law, Monte-Carlo analysis is carried out till the terminal flight time of attitude hold phase. Simulation result shows the error ellipse obtained is not within the mission constraints bound. In order to improve the terminal accuracy of the lander, desensitized optimal control trajectory design as explained in section 2.5 is carried out. As shown in the figures below, the trajectory as well as its flight duration is altered to account for perturbation. Using the DOC trajectory, Monte-carlo analysis is simulated with closed loop feedback control law to examine the effect on error ellipse at terminal condition of attitude hold phase. Fig. 2 shows the altitude of the lander decreases as the downrange reduces to desired position for no-DOC and DOC trajectory formulation satisfying mission constraint
Fig. 3. Thrust Orientation Profile Ensuring Camera Look Angle of Landercraft i.e. first stage terminal constraint of 7 km. Fig. 3 depicts the thrust vector direction with terminal constraints satisfied i.e. first stage terminal constraint of 140 deg making sure camera looking at landing site. Fig. 4 is the demanded thrust throttling percentage. Note that DOC formulation results in significant change in thrust demanded compared to no-DOC, to account for closed loop feedback control law in trajectory design. Note that the additional propellant consumption compared to no-DOC trajectory design is negligible. Also, note that in stage 2 the throttling percentage and thrust vector direction of former phase is maintained throughout as mission required, as in (22). Fig. 5-6 shows the significant reduction in position as well as velocity error ellipse respectively at terminal condition of attitude hold phase due to desensitized optimal control reference trajectory. 5. CONCLUSION Thus the lunar landing problem has been formulated as desensitized optimal control problem and solved by Legen-
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dre Pseudospectral method. Simulation analysis shows the reference trajectory satisfies all the mission constraints like thrust upper-lower bound, camera look angle constraint and time for image processing. Simulation analysis in presence of uncertainties and initial condition perturbation demonstrates the efficiency of the desensitized optimal control formulation over nominal trajectory optimization methodology for lunar landing problem. Monte Carlo simulations show that position errors of less than 100 m and velocity errors of less than 0.2 m/s are achieved. However the amount of extra fuel consumed is less significant when compared to improvement in landing accuracy assuring the desensitized optimal control is viable for lunar landing trajectory design. ACKNOWLEDGEMENTS
Fig. 4. Demanded Thrust Firing for Achieving Mission Goals
The authors are thankful to the Indian Space Research Organisation (ISRO) for giving an opportunity to work on this challenging problem. REFERENCES
Fig. 5. Monte-carlo Position Dispersions by DOC and NoDOC Trajectory
Fahroo, F. and Ross, M. (2004). Pseudospectral method knotting methods for solving optimal control problems. Journal of Guidance, Control, and Dynamics, 27, 397– 405. Hu, H., Zhu, S., and Cui, P. (2016). Desensitized optimal trajectory for landing on small bodies with reduced landing error. Journal of Aerospace Science and Technology, 48, 178–185. Mathavaraj, S., Pandiyan, R., and Padhi, R. (2016). Optimal trajectory planning for multiphase lunar landing. In Proceedings of the Advances in Control and Optimization of Dynamical Systems, volume 49, 124–129. Elsevier. Robinson, S.B. (2013). Spacecraft Guidance Techniques for Maximizing Mission Success. Ph.D. thesis, Utah State University. Saunders, B.R. (2012). Optimal trajectory design under uncertainty. Ph.D. thesis, Massachusetts Institute of Technology. Seywald, H. (2003). Desensitized optimal trajectories with control constraints. Advances in the Astronautical Sciences, 114, 737–743. Vinh, N., Busemann, A., and Culp, R. (1980). Hypersonic and planetary entry flight mechanics. Univ. of Michigan Press, Ann Arbor. Wang, H., Xie, M., Zhang, P., and Li, Y. (2014). Research of asteroid landing trajectory optimization based on gauss pseudo-spectral method. In Proceedings of the Intelligent Human-Machine Systems and Cybernetics, volume 2, 26–29. IEEE. Xu, H. and Cui, H. (2015). Robust trajectory design scheme under uncertainties and perturbations for mars entry vehicle. In Proceedings of the Computational Intelligence & Communication Technology, 762–766. IEEE.
Fig. 6. Monte-carlo Velocity Dispersions by DOC and NoDOC Trajectory 6298