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Optimal Orbital Transfers to Asteroids
5th International Conference on Advances in Control and 5th Conference on Optimization of Dynamical Systems 5th International International Conference on Advances Advances in in Control Control and and 5th International Conference on Advances in Control and Optimization of Systems February 18-22, 2018. Hyderabad, India Optimization of Dynamical Dynamical Systems Available online at www.sciencedirect.com Optimization of Dynamical Systems 5th International Conference on Advances in Control and February 18-22, Hyderabad, India February 18-22, 2018. 2018. Hyderabad, India February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
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IFAC PapersOnLine 51-1 (2018) 638–643
Optimal Optimal Orbital Orbital Transfers Transfers to to Asteroids Asteroids Optimal Orbital Transfers to Asteroids ∗ ∗∗ Mangal Kothari OptimalShribharath Orbital BTransfers to Asteroids ∗∗ Shribharath B ∗∗ Mangal Kothari ∗∗
Shribharath Shribharath B B ∗ Mangal Mangal Kothari Kothari ∗∗ ∗ Graduate Student, Dept. of Aerospace Engineering, Institute Shribharath B Mangal Kothari ∗∗Indian Graduate Student, Dept. of Aerospace Engineering, Indian Institute Graduate Student, Dept. of Aerospace Engineering, Indian of Technology Kanpur, India (e-mail: [email protected]) Graduate Student,Kanpur, Dept. of India Aerospace Engineering, Indian Institute Institute (e-mail: [email protected]) ∗∗ of Technology Technology Kanpur, India (e-mail: [email protected]) Assistant Professor, Dept. of Aerospace Engineering, Indian ∗ ∗∗ of Graduate Student, Dept. Dept. of India Aerospace Engineering, IndianIndian Institute Technology Kanpur, (e-mail: [email protected]) ∗∗ of Assistant Professor, of Aerospace Engineering, Assistant Professor, Dept. of Aerospace Engineering, Indian Institute of Technology Kanpur, India (e-mail: [email protected]) ∗∗ of Technology Kanpur, India (e-mail: [email protected]) Assistant Professor, Dept. of Aerospace Engineering, Indian Institute of Technology Kanpur, India (e-mail: [email protected]) Institute of (e-mail: [email protected]) ∗∗ Assistant Professor,Kanpur, Dept. ofIndia Aerospace Engineering, Indian Institute of Technology Technology Kanpur, India (e-mail: [email protected])
∗ ∗ ∗ ∗
Institute Technology Kanpur, India Mars (e-mail: Abstract: Many of theof smaller asteroids between [email protected]) Jupiter have poorly determined Abstract: Many of the the smaller smaller asteroids between Marsaand and Jupiter have poorly poorly determined Abstract: Many of asteroids between Mars Jupiter have orbits with high uncertainties. For successfully completing transfer missions consuming Abstract: Many of the smaller asteroids between Marsaand Jupiter have without poorly determined determined orbits with high uncertainties. For successfully completing transfer missions without consuming orbits with high uncertainties. For successfully completing a transfer missions without consuming unacceptable amounts of fuel, a robust optimal design is needed. In this paper, the spacecraft Abstract: Many of the smaller asteroids between Mars and Jupiter have poorly determined orbits with high uncertainties. successfully completing transferIn missions without unacceptable amounts of fuel, fuel,For a robust robust optimal design control is aneeded. needed. this The paper, the consuming spacecraft unacceptable amounts of a optimal design is In this paper, the spacecraft trajectory planning problem is posed as an optimal problem. solutions of the orbits with high uncertainties. For successfully completing a transfer missions without consuming unacceptable amounts of fuel, a robust optimal design is needed. In this paper, the spacecraft trajectory planning problem is posed as an optimal control problem. The solutions of the trajectory planning problem is posed as an optimal control problem. The solutions of the problem is obtained by employing a computationally efficient algorithm, Model Predictive Static unacceptable amounts of fuel, isa robust optimal design is needed. In this paper, the spacecraft trajectory planning problem posed as an optimal control problem. The solutions of the problem is obtained by employing a computationally efficient algorithm, Model Predictive Static problem is by employing a efficient algorithm, Model Predictive Static Programming (MPSP). The algorithm isasused generating nominal and adaptive trajectories. trajectory planning problem is posed an for optimal control problem. The solutions of the problem is obtained obtained by employing a computationally computationally efficient algorithm, Model Predictive Static Programming (MPSP). The algorithm is used used for generating nominal and adaptive trajectories. Programming (MPSP). The algorithm is for generating nominal and adaptive trajectories. A comparison study is carried out with classical solutions. The results show that MPSP is problem is obtained by employing a computationally efficient algorithm, Model Predictive Static Programming (MPSP). The algorithm is used for generating nominal and adaptive trajectories. A comparison study is carried out with classical solutions. The results show that MPSP is A comparison study is carried out with classical solutions. The results show that MPSP is highly reliable (MPSP). in generating computationally efficient low continuous thrust transfer trajectories Programming The algorithm is used for generating nominal and adaptive trajectories. A comparison study is carried out with classical solutions. The results show that MPSP is highly reliable in generating computationally efficient low continuous thrust transfer trajectories highly reliable generating computationally efficient low continuous thrust transfer particularly in in situations where there are stringent time and robustness is MPSP a need.is A comparison study is carried out with classical solutions. The results show thattrajectories highly reliable generating computationally efficient lowconstraints continuous thrust transfer trajectories particularly in in situations where there are stringent stringent time constraints and robustness is aa need. need. particularly in situations where there are time constraints and robustness is highly in generating computationally efficient lowHosting continuous thrust transfer trajectories particularly situations where there stringent time constraints and robustness is areserved. need. © 2018,reliable IFACin(International Federation of are Automatic Control) by Elsevier Ltd. All rights Keywords: Orbital transfers, Trajectory Asteroids, Optimal control particularly in situations where there aredesign, stringent time constraints and robustness is a need. Keywords: Orbital transfers, Trajectory design, Asteroids, Optimal control Keywords: Orbital transfers, Trajectory design, Asteroids, Optimal control Keywords: Orbital transfers, Trajectory design, Asteroids, Optimal control 1. INTRODUCTION algorithmOptimal to perform well under the presence of uncerKeywords: Orbital transfers, Trajectory design, Asteroids, control 1. INTRODUCTION algorithm to perform well under the presence of uncer1. INTRODUCTION algorithm to perform well inunder under the and presence of(2007)]. uncertainties was demonstrated [Kothari Padhiof 1. INTRODUCTION algorithm todemonstrated perform well the presence uncertainties was in [Kothari and Padhi (2007)]. tainties was demonstrated in [Kothari and Padhi (2007)]. The MPSP algorithm, unlike impulsive transfers, gives 1. INTRODUCTION perform well the and presence uncertainties wastodemonstrated inunder [Kothari Padhiof(2007)]. Interplanetary orbital transfers is an exhaustively analysed algorithm The MPSP algorithm, unlike impulsive transfers, gives The MPSP algorithm, unlike impulsive transfers, gives continuous solution which can be implemented through Interplanetary orbital transfers is an exhaustively analysed wassolution demonstrated incan [Kothari andtransfers, Padhi through (2007)]. The MPSP algorithm, unlike impulsive gives Interplanetary orbital transfers iseight an exhaustively exhaustively analysed domain in space missions. Theis planets in the solar tainties continuous which be implemented Interplanetary orbital transfers an analysed continuous solution which can be implemented through low thrust engines. This has become very popular in recent domain in space missions. The eight planets in the solar MPSPengines. algorithm, unlike impulsive transfers, gives continuous solution which can be implemented through domain have in space space missions. Theisset eight planets in the the solar The system a orbital well established of data such as mass, low thrust This has become very popular in recent Interplanetary transfers anof exhaustively analysed domain in missions. The eight planets in solar low thrust thrust engines. This has become very popular in recent times due to the reduced size of be theimplemented engine itself.in system have a well established set data such as mass, continuous solution which can through low engines. This has become very popular recent system have a well established set of data such as mass, size, and orbit. Given the The fact set that its position can be times due to the reduced size of the engine itself. domain inorbit. space missions. eight in the solar system have a well established of planets data such ascan mass, times due to to the reduced reduced size of the the very engine itself.in recent size, and Given the fact that its position be thrust engines. This has become popular times due the size of engine itself. size, and and orbit. Given the given fact set that its position can be low accurately known at any point ofposition time, optimal In this work, the MPSP algorithm is employed for gensystem have a well established of data such as mass, size, orbit. Given the fact that its can be accurately known at any given point of time, optimal In this work, the MPSP algorithm is employed for gentimes due torobust the reduced size of the engine itself. accurately known at any given point of time, optimal In this work, the MPSP algorithm is employed for gentrajectories can be generated beforehand and the mission eration of transfer trajectory to a hypothetical size, and orbit. Given the given fact that can be In accurately known at any pointitsofposition time, optimal this work, the MPSP algorithm is employed for gentrajectories can be generated beforehand and the mission eration of robust transfer trajectory to a hypothetical trajectories can be generated beforehand and the mission eration of robust transfer trajectory to a hypothetical control involves only a minor corrections [Battin (1999)]. asteroid near Mars with high orbital uncertainties. The accurately known at given point ofand time, optimal trajectories can be generated beforehand the (1999)]. mission In this work, the MPSP algorithm is uncertainties. employed for generation of robust transfer trajectory to a hypothetical control involves only aaany minor corrections [Battin asteroid near Mars with high orbital The control involves only minor corrections [Battin (1999)]. asteroid near Mars with high orbital uncertainties. Thea But the trajectory planning becomes highly challenging algorithm first generates nominal trajectory assuming trajectories can be generated beforehand and the mission control involves onlyplanning a minor becomes corrections [Battin (1999)]. eration of robust transfer trajectory to a hypothetical asteroid near Mars with high orbital uncertainties. The But the trajectory highly challenging algorithm first generates nominal trajectory assuming a But the trajectory planning becomes highly challenging algorithm first generates nominal trajectory assuming when the target orbit has high uncertainties such as astermean position of an asteroid and updates the trajectory control involves onlyplanning ahas minor corrections [Battin (1999)]. But the trajectory becomes highly challenging asteroid near with high orbital uncertainties. Theaa algorithm firstMars generates nominal trajectory assuming when the target orbit high uncertainties such as astermean position of an asteroid and updates the trajectory whenthe the trajectory target orbit has high high uncertainties such as asterastermean position position of an asteroid asteroid and updates theassuming trajectory oids. There is a orbit wellplanning known belt of asteroids between the algorithm when it receives the updated asteroid information. Thea But becomes highly challenging when the target has uncertainties such as first of generates nominal trajectory mean an and updates the trajectory oids. There is aa well known belt of asteroids between the when it receives the updated asteroid information. The oids. There is well known belt of asteroids between the when it receives the updated asteroid information. The planets Mars and Jupiter whose origins are still a subject algorithm is first compared with classical solutions such as when theMars target has high uncertainties such asteroids. There isand a orbit well known belt of asteroids between the mean position ofcompared an asteroid and updates the trajectory when it receives the updated asteroid information. The planets Jupiter whose origins are still aaassubject algorithm is first with classical solutions such as planets Mars and Jupiter whose origins areasteroids still subject algorithm is first compared with classical solutions such as of dispute in astrophysics. Several large have Hohmann transfer. The Hohmann transfer has been well oids. There is a well known belt of asteroids between the planets Mars and Jupiter whose origins are still a subject when it receives the updated asteroid information. The algorithm is first compared with classical solutions such as of dispute in astrophysics. Several large asteroids have Hohmann transfer. The Hohmann transfer has been well of dispute in astrophysics. Several large asteroids have Hohmann transfer. The Hohmann transfer has been well been identified and their orbits have been well determined. established as the analytical optimum for co-planar cirplanets Mars Jupiter whose origins are still a subject of dispute in and astrophysics. Several large asteroids have algorithm is first compared with classical solutions such as Hohmann transfer. The Hohmann transfer has been well been identified and their orbits have been well determined. established as the analytical optimum for co-planar cirbeen identified and their asteroids, orbits havewhich been well determined. established as the the analytical optimum for has co-planar cirHowever, several smaller are highly valued cular transfers [Battin (1999)]. Thetransfer solutions obtained by of dispute in astrophysics. Several large asteroids have been identified and their orbits have been well determined. Hohmann transfer. The Hohmann been well established as analytical optimum for co-planar cirHowever, several smaller asteroids, which are highly valued cular transfers [Battin (1999)]. The solutions obtained by However, several smaller asteroids, which are highly valued cular transfers transfers [Battin (1999)]. The solutions solutions obtained by for potential scientific inspection have very less number of established MPSP is then compared with optimum gradient method, a popular been identified and their orbits have been well determined. However, several smaller asteroids, which are highly valued as compared the analytical for co-planar circular [Battin (1999)]. The obtained by for potential scientific inspection have very less number of MPSP is then with gradient method, a popular for potential scientific inspection have very less number of MPSP is then compared with gradient method, a popular clear observations and hence a much poorly determined numerical method for obtaining control solutions [Bryson However, several smaller asteroids, which are highly valued for potential scientific inspection have very less number of cular transfers [Battin (1999)]. The solutions obtained by MPSP is then compared with gradient method, a popular clear observations and hence much poorly determined numerical method for obtaining control solutions [Bryson clear observations andinspection hence aaainmuch much poorly determined numerical method for obtaining control solutionsa popular [Bryson orbits. With high uncertainties the final target position, and Ho is(1975)]. It isfor demonstrated usingmethod, numerical simulafor potential scientific have very lessdetermined number of MPSP clear observations and hence poorly then compared with gradient numerical method obtaining control solutions [Bryson orbits. With high uncertainties in the final target position, and Ho (1975)]. It is demonstrated using numerical simulaorbits. With high uncertainties in the final target position, and Ho (1975)]. It is demonstrated using numerical simulaa one-go solution approach does not work here unlike the tions that the proposed approach is efficient in generating clear observations and hence ainmuch poorly determined orbits. With high uncertainties the work final target position, numerical method obtaining control solutions [Bryson and Ho (1975)]. It isfor demonstrated using numerical simulaacase one-go solution approach does not here unlike the tions that the proposed approach is efficient in generating a one-go solution approach does not work here unlike the tions that the proposed approach is efficient in generating of planets. An adaptive trajectory has to be designed solutions and converges quickly. orbits. With high uncertainties innot the work final target position, acase one-go solution approach does here unlike the and Ho (1975)]. It is demonstrated numerical simulations that the converges proposed approach isusing efficient in generating of planets. An adaptive trajectory has to be designed solutions and quickly. case of planets. Anapproach adaptive trajectory has to be be designed solutions and converges quickly. is efficient in generating with minimal online corrective efforts tohas reach the target. a one-go solution does not work here unlike the tions case of planets. An adaptive trajectory to designed thatand the proposed approach solutions converges quickly. The rest of this paper is organized as follows: Section 2 with minimal online corrective efforts to reach the target. with of minimal online corrective efforts to tohas reach the target. solutions The rest of this paper is organized as follows: Section 22 case planets. Antheory adaptive to be designed with minimal online corrective efforts reach the target. converges quickly. The rest rest and of this paper is organized as follows: Section Optimal control hastrajectory been extensively used for The discusses orbital transfers and background in brief. Then, of this transfers paper is organized as follows: Section 2 Optimal control theory has been used for orbital and background in brief. Then, with minimal online corrective effortsextensively toHowever, reach the target. Optimal control theory hasspacecraft. been extensively used fora discusses discusses orbital transfers and background background in brief. brief. Then,23 trajectory generations forhas such the optimal trajectory transfer problem is posed. Section Optimal control theory been extensively used for The rest orbital of trajectory this transfers papertransfer is organized as is follows: Section discusses and in Then, trajectory generations for spacecraft. However, such a the optimal problem posed. Section 3 trajectorycontrol generations for spacecraft. However, suchforaa gives the optimal optimal trajectory transfer problem is posed. posed. Section framework leads to twofor point boundary value problems a brief description ofand twobackground algorithms to brief. solve TPBOptimal theory hasspacecraft. been extensively used trajectory generations However, such discusses orbital transfers in Then, the trajectory transfer problem is Section 33 framework leads to two point boundary value problems gives a brief description of two algorithms to solve TPBframework generations leads to computationally twofor point boundary valueforproblems problems givesoptimal brieftrajectory description of two twoproblem algorithms todiscusses solve TPB(TPBVPs), that is intensive explicit VPs. Section 4 present transfer numerical resultsisand the3 trajectory spacecraft. However, such a the framework leads to two point boundary value posed. Section gives aa brief description of algorithms to solve TPB(TPBVPs), that is computationally intensive for explicit VPs. Section 44 present numerical results and discusses the (TPBVPs), that is computationally intensive for explicit VPs. Section present numerical results and discusses the solutions. A sub-optimal algorithm called Model Predictive results. Finally, the paper is concluded in Section 5. framework leads to computationally two algorithm point boundary valuefor problems (TPBVPs), that is intensive explicit gives a brief description ofistwo algorithms todiscusses solve5.TPBVPs. Section 4 present numerical resultsinand the solutions. A sub-optimal called Model Predictive results. Finally, the paper concluded Section solutions. A sub-optimal algorithm called Model Predictive results. Finally, the paper is concluded in Section 5. Static Programming (MPSP) was proposed to solved TPB(TPBVPs), that is computationally intensive for explicit solutions. A sub-optimal algorithm called Model Predictive VPs. Section 4 present numerical resultsinand discusses results. Finally, the paper is concluded Section 5. the Static Programming (MPSP) was proposed to solved TPBStaticefficiently Programming (MPSP) was proposed toApproximate solved TPB- results. Finally, 2. PROBLEM FORMULATION VPs by combining philosophies ofto solutions. A sub-optimal algorithm called Model Predictive Static Programming (MPSP) was proposed solved TPBthe paper is concluded in Section 5. 2. PROBLEM FORMULATION VPs efficiently by combining philosophies of Approximate 2. PROBLEM PROBLEM FORMULATION FORMULATION VPs efficiently efficiently by combining combining philosophies oftoApproximate Approximate Dynamic Programming andwas Model Predictive Control Static Programming (MPSP) proposed solved TPB2. VPs by philosophies of Dynamic Programming and Model Predictive Control Dynamic Programming andphilosophies Model Predictive Control by [Padhi and by Kothari (2009)]. The algorithm has been In this section, we first briefly introduce orbital transfer 2. PROBLEM FORMULATION VPs efficiently combining of Approximate Dynamic Programming and Model Predictive Control by [Padhi and Kothari (2009)]. The algorithm has been this section, we first briefly introduce orbital transfer by [Padhi and Kothari (2009)]. The algorithm has been In In this this section, westate first the briefly introduce orbital orbital transfer transfer applied to solve various guidance problems from aerospace and then formally problem. Dynamic Programming and Model Predictive Control by [Padhi and Kothari (2009)]. The algorithm has been In section, we first briefly introduce applied to solve various guidance problems from aerospace and then formally state the problem. applied to solve various guidance problems from aerospace and then formally state the problem. industry [Dwivedi et al. (2011); Padhi et al. (2014); Varma by [Padhi and Kothari (2009)].Padhi The algorithm hasVarma been In section, westate first the briefly introduce orbital transfer applied to[Dwivedi solve various guidance problems from aerospace andthis then formally problem. industry et al. (2011); et al. (2014); industry [Dwivedi et al. al. guidance (2011); Padhi et al. al.recently (2014); Varma and 2.1 Orbital transfers et al. (2016)]. The algorithm hasPadhi been also applied applied to solve various problems from aerospace then formally state the problem. industry [Dwivedi et (2011); et (2014); Varma 2.1 Orbital transfers et al. (2016)]. The algorithm has been also recently applied 2.1 Orbital Orbital transfers transfers et al. al. (2016)]. The algorithm algorithm hasPadhi been also recently applied to landing a spacecraft on moon [Banerjee et al. applied (2015)] industry [Dwivedi et al. (2011); et al.recently (2014); Varma 2.1 et (2016)]. The has been also to landing aa spacecraft on moon [Banerjee et al. (2015)] to landing spacecraft on moon [Banerjee et al. (2015)] Orbital transfers can be broadly classified into two types where the MPSP was combined with a neural network to 2.1 Orbital transfers et al. (2016)]. The was algorithm has been also recently to landing a spacecraft on moon [Banerjee et network al. applied (2015)] Orbital transfers can be broadly classified into two types where the MPSP combined with aa neural to Orbital transfers can becontinuous broadly classified classified into two two types where the MPSP was combined with neural network to namely impulsive andbe low thrust transfers. find global optimal solutions. The ability of the MPSP to landing aoptimal spacecraft on moon [Banerjee et network al. (2015)] transfers can broadly into types where the MPSP wassolutions. combined with a neural to Orbital namely impulsive and continuous low thrust transfers. find global The ability of the MPSP namely impulsive and continuous low thrust transfers. find global optimal solutions. The ability of the MPSP broadly classified into two types where the MPSP wassolutions. combinedThe withability a neural network to Orbital namely transfers impulsivecan andbecontinuous low thrust transfers. find global optimal of the MPSP
Copyright © 2018, 2018 IFAC 670Hosting namelybyimpulsive continuous find global optimal solutions. The abilityof of the MPSP 2405-8963 © IFAC (International Federation Automatic Control) Elsevier Ltd.and All rights reserved. Copyright © 2018 IFAC 670 Copyright ©under 2018 responsibility IFAC 670Control. Peer review of International Federation of Automatic Copyright © 2018 IFAC 670 10.1016/j.ifacol.2018.05.107 Copyright © 2018 IFAC 670
low thrust transfers.
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is the patched conic approximation. The motion of a spacecraft under the influence of a central body is a conic. In our solar system, the spacecrafts are often in the proximity of a single planet or solely under the influence of the Sun. Thus, a rendezvous mission starting from the Earth to a target body, in our case an asteroid, can be broadly split into three phases as follows: (i) Escape from the Earth: In this phase, the spacecraft begins from an earth bound orbit and escapes from the Earth’s gravity; (ii) Approach phase: This is the longest phase of the mission. In this phase, the spacecraft is mainly influenced by the Sun’s gravity; and (iii) Insertion Phase: This phase begins when the spacecraft has successfully entered the region of influence of the asteroid. Fig. 1. Hohmann Transfer
2.3 Problem Statement
In a continuous low thrust transfer where the speed of the spacecraft is slowly increased, the total ∆v required would be simply the change in the orbital speed of the two orbits. Impulsive transfers on the other hand use occasional large pulses of thrust. They make the use of the gravity of the sun itself to effect some part of the speed changes. Hence impulsive transfers are always more efficient than the continuous low thrust transfers [Battin (1999)]. However when the transfer time is not a constraint and only low thrust is required, we only need a smaller engine with less thrust capabilities which in turn lead to fuel savings. Further for missions to outer planets the impulsive transfers become impractical due to the requirement of large impulses. Hence continuous low thrust engines become the only and more efficient option for such missions. Hohmann transfer is a two tangential impulse transfer between two co-planar circular orbits. It is a popular option often used in space missions and is also known to be the have the optimal ∆v for radii ratio below the critical value. Beyond this ratio three impulsive bi-elliptic transfer becomes optimal [Battin (1999)]. However for close pair of orbits like Earth and Mars or Earth and Jupiter Hohmann transfer is the simplest option available in hand. The transfer is pictorially represented in Fig. 1. The magnitude of the two impulses required to achieve this transfer can be calculated using the readily available formula for the orbital speeds of the two circular and the elliptical transfer orbit as follows:
2r2 −1 r1 + r2 µ 2r1 1− ∆v2 = r2 r1 + r2
∆v1 =
µ r1
(1) (2)
In this paper, Hohmann transfer is used as benchmark to compare the fuel efficiency of different algorithms while having the same transfer time as that of Hohmann transfer. 2.2 Background The motion of the spacecraft inside solar system has to be modeled as an n-body problem including the forces from all the planets. But for the purpose of rendezvous missions an acceptable and established approximation 671
This work considers the problem of optimizing the second phase. The phase considered is pictorially represented in Fig. 2. The mission commences just outside the region of influence of the earth and terminates just outside the region of influence of the asteroid. The main purpose of this work is to design spacecraft trajectories under orbit(al) uncertainties of an asteroid using an optimal algorithm. The following assumptions are made in this work: (i) The spacecraft is assumed to have escaped the gravitational potential well of the earth; and (ii) During the approach phase the planetary perturbations are neglected. These assumptions simplify the mathematical model and make computations simpler. The final state of the asteroid is considered to be uncertain and is assumed to be a random variable with a known mean. The simplified equations of motion of a spacecraft are based on the Newton’s law of gravitation and are given as follows: x¨ = y¨ = z¨ =
−µx 3
+ ax
(3)
3
+ ay
(4)
(x2 + y 2 + z 2 ) 2 −µy
(x2 + y 2 + z 2 ) 2 −µz
(5) 3 + az (x2 + y 2 + z 2 ) 2 where µ is the gravitational constant of the central body and ax , ay , az are the control forces by means of continuous low thrust with thrust vectoring at all angles. The goal is to take the state vector, which contains quantities denoting the position x, y, z and the velocities which are the rates of the x, y, and z positions, from an initial value to a final value while minimizing the control effort which directly affects the fuel consumption. The problem can be posed as an optimal control problem with the minimization of the following cost function 1 tf J = Φ(Xf ) + L(X(t), U (t), t)dt (6) 2 to subject to Eqs. (3)-(5). Here, Φ(Xf ) = 21 (Xf −Xd )M (Xf − Xd ), where Xd is the desired final state and M is a matrix with large diagonal elements and L(X(t), U (t), t) = 1 T 2 U (t) RU (t), R > 0. By setting the first variation of J to zero one can obtain the state, co-state, and optimal control equations to be solved along with the boundary conditions to be satisfied [Bryson and Ho (1975)]. As
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3.2 Model Predictive Static Programming To outline MPSP, we consider general nonlinear systems in discrete form, the state and output dynamics of which are given by Xk+1 = Fk (Xk , Uk )
(11)
Yk = h(Xk )
(12)
where X ∈ ℜn , U ∈ ℜm , Y ∈ ℜp and k = 1, 2, . . . , N are the time steps. The primary objective is to obtain a suitable control history Uk , k = 1, 2, . . . , N −1, so that the output at the final time step YN goes to a desired value YN∗ , i.e. YN → YN∗ . In addition, we aim to achieve this task with minimum control effort. Fig. 2. Pictorial representation stated before, the solution leads to TPBVPs and therefore computationally intensive. In this work, we employ MPSP algorithm to solve the above problem. 3. OPTIMAL CONTROL ALGORITHMS In this section, we briefly describe two optimal control algorithms to solve TPBVPs posed in the previous section. 3.1 Gradient Method First we present a numerical technique, gradient method, to solve the TPBVPs. The approach requires a guess history that usually does not satisfy the necessary conditions of optimality. In this approach, the state, costate, and the boundary conditions are forced to be satisfied and the strategy is to satisfy the optimal control equation iteratively. The first variation of the augmented cost functional is tf ∂Φ T T ∂H ˙ ¯ δ J = δXf +λ + δX − λf + ∂Xf ∂X t0 T ∂H T ∂H ˙ ... δU + δλ − X dt (7) ∂U ∂λ Since the state, costate and boundary conditions are satisfied, we get tf T ∂H ¯ δJ = dt (8) δU ∂U t0 Selecting δU = −τ ∂H ∂U and substituting into the previous equation, we get T tf ∂H ∂H dt (9) −τ δ J¯ = ∂U ∂U t0 The updated control is given by ∂H dt (10) U (p + 1) = U (p) − τ ∂U where p is the iteration variable. It can be observed from the last equation that with this updated, the first variation decreases with each iteration and eventually approaches zero. Hence, all the necessary conditions get satisfied. The value of τ for convergence however was be found by trial and error. 672
Starting with a guess value of control history which need not meet the objective. MPSP serves to compute an error history of the control variable, which needs to be subtracted from the previous history to get an improved control history. Expanding YN about YN∗ using Taylor series expansion and using small error approximation, we get ∂YN dXN △YN ∼ (13) = dYN = ∂XN However from (11), we can write the error in state at time step (k + 1) as dXk+1
∂Fk = ∂Xk
∂Fk dXk + ∂Uk
(14)
dUk
where dXk and dUk are the error of state and control at time step k respectively. Expanding dXN as in (14) (for k = N − 1) and substituting it in (13), we get dYN = A dX1 + B1 dU1 + B2 dU2 + . . .+ BN −1 dUN −1 (15) where ∂YN ∂FN −1 ∂F1 A ... ∂XN ∂XN −1 ∂X1 ∂FN −1 ∂Fk+1 ∂Fk ∂YN ... Bk ∂XN ∂XN −1 ∂Xk+1 ∂Uk for k = 1 . . . N − 1. Since the initial condition is specified, there is no error in the first term; which means dX1 = 0. With this (15) reduces to dYN = B1 dU1 + B2 dU2 + . . . + BN −1 dUN −1 =
N −1
Bk dUk
k=1
(16) Now Bk can be computed recursively. For doing this, first 0 we define BN −1 as follows 0 BN −1 =
∂YN ∂XN
(17)
Next we compute Bk0 , k = (N − 2), (N − 3), . . . , 1 as Bk0
=
0 Bk+1
∂Fk+1 ∂Xk+1
(18)
Finally, Bk , k = (N − 2), (N − 3), . . . , 1 can be computed as
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Table 1. Orbital Parameters used in Simulation Parameter a (AU) e (no unit) i (deg)
Earth 1 0.016 1.5787
Target 1.5236 0.0934 1.67
Fig. 3. Rendezvous without observations Bk = Bk0
∂Fk ∂Uk
(19)
The set of equations get closed by setting an additional the objective function which is given by
J=
N −1 1 0 (Uk − dUk )T Rk (Uk0 − dUk ) 2
Table 2. Performance Comparison of Gradient Method and MPSP (20)
k=1
where Uk0 , k = 1, . . . , (N − 1) is the previous control history solution and dUk is the corresponding error in the control history. The cost function in (20) needs to be minimized subjected to the constraint in (16), where Rk > 0 (a positive definite matrix) is the weighting matrix, which needs to be chosen judiciously by the control designer. The selection of such a cost function is motivated by the fact that we are interested in finding a l2 -norm minimizing control history, since (Uk0 −dUk ) is the updated control value at k (see (23)). Equations (16) and (20) formulate an appropriate constrained static optimization problem. The augmented cost function is given by N −1 1 0 J¯ = (Uk − dUk )T Rk (Uk0 − dUk ) + 2 k=1
λT (dYN −
N −1
Bk dUk )
(21)
k=1
Using optimization theory, we get dUk = Rk−1 BkT λ + Uk0
Fig. 4. Rendezvous with observations
(22)
Hence, the updated control at time step k = 1, 2, . . . , (N − 1) is given by
Method Hohmann Transfer Gradient Method MPSP
Fuel Consumption (x10−3 AU/day) 3.2 4.6 4.6
4. RESULTS AND DISCUSSION 4.1 Orbital transfer trajectory to an asteroid In JPL’s small body database the asteroids are rated on a scale from 0 to 9 for measuring the orbital uncertainty. Such uncertainties occur because the motion of the asteroids are complicated by shape and spin dependent phenomena like Yarkovsky effect [Nesvorn` y and Bottke (2004)]. In this work, the asteroid is chosen to be a fictitious asteroid near Mars. The following trajectories using the standard orbital parameters of Earth and Mars given in Table 1 were obtained using MPSP without feedback. As it is seen from Fig. 3, the spacecraft approaches the asteroid but still a finite amount of error is present at the final position. In practical situations, this can occur due to the modeling approximations and the errors in the measurement of the position of the asteroid and the spacecraft. This error could be of the order of a few thousand kilometers. This is normally is of the same order of the size of the region of influence of the asteroid. In order to assure convergence MPSP can be re-initiated on board at definite intervals and corrections to the control inputs based on the measurements of the current spacecraft state vector. 4.2 Comparison of MPSP with Gradient Method
Uk =
Uk0
− dUk =
Rk−1 BkT A−1 λ
(dYN − bλ )
(23)
It is clear that the updated control history solution is a closed form solution, and hence, control solution can be updated with very minimal computational requirement. 673
The efficiency of MPSP and Gradient Method are compared with Hohmann Transfer, the theoretical optimal solution. The total ∆v required for a Hohmann transfer orbit from Earth to Mars can be directly calculated as shown in Section 2. The transfer time is simply taken as
5th International Conference on Advances in Control and 642 Shribharath B et al. / IFAC PapersOnLine 51-1 (2018) 638–643 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
Table 3. Speed comparison of Gradient Method and MPSP Method Hohmann Transfer Gradient Method MPSP
No.of Iterations 180668 7
Fig. 7. Rendezvous in presence of uncertainties
Fig. 5. Control Solution with equal weights
Fig. 8. Control Solution for adaptive trajectory R matrix and hence convergence is negatively affected. Also the total fuel consumption of the MPSP solution increasing from 4.6 to 4.9 in the same units as in the table. Hence, it is convenient to get different types of solution using the MPSP algorithm. Fig. 6. Control Solution with high initial weights
4.3 Orbital transfer in presence of uncertainties
the Hohmann transfer time and the obtained results are tabulated in Tables 2-3. The following points are inferred from the results: (i) The continuous thrust transfer expectedly consumes more fuel in comparison with the Hohmann transfer with a same transfer time; (ii) The gradient method gives a similar solution as MPSP, but its speed of convergence is very less. (iii) MPSP on the other hand has a very high speed of convergence; and (iv) MPSP has its final conditions as hard constraints unlike Gradient method which has it as soft constraints. This feature attracts MPSP for situations where stringent requirements regarding the final boundary value are present. The control solution provided by the MPSP often has large magnitudes of inputs in the beginning period. Although this aids to the quick rate of convergence, this need not be desirable at all circumstances. Sometimes a gradual build up to higher values of thrust is desirable. This can be achieved using the R matrix in the cost function chosen. By smartly changing the elements of the matrix for different time steps as required one may obtain smoother solutions using MPSP. To demonstrate this, for the same problem for which the above comparisons were made, high weights were placed for initial time steps and the weights were decreased exponentially. This changed the qualitative nature of the solutions as displayed in the following figures. A caution for this approach is that it may require more iterations to converge when heavy weights are used in the 674
Initially, the spacecraft is allowed to follow the nominal control history based in the mean of the estimated state vector of the target at the final time. As the spacecraft approaches closer to the target multiple observations are made at definite intervals and new control commands are generated based on the new mean of the estimated target position. In order to simulate the unpredicted motion of the asteroid, a random change in the expected final position of the asteroid is assumed. The results show that the MPSP algorithm works and the uncertainty in the final position of the asteroid can be easily handled by MPSP. The convergence of the algorithm depends on the range of the radar device and the number of observations made. The results are shown in Figs 7 and 8. The final phase shows the adaptive changes made in the control to achieve the rendezvous. But the fuel consumed to make these corrections is proportional to the deviation in the position of the asteroid from the expected position. 4.4 Nominal Trajectory Design When the orbit of a target in space is known accurately, the trajectory design is straight forward and the rendezvous can be achieved easily. But the presence of uncertainties has led us into splitting the mission into the nominal trajectory following and the corrective maneuvers at the end. The mean of the known information about the target orbit can generally be used to generate the nominal
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the nature of the solution thus handling the problem satisfactorily, providing fast-converging and robust solutions. REFERENCES
Fig. 9. Response curves for different values of Qi s Table 4. Comparison of fuel consumption for different values of Qi Qi 0 1 10
Total ∆V (km/s) 5.76 8.35 12.41
trajectory. In generating a nominal trajectory, once again the fuel consumed can be minimized using the gradient method. One drawback of this method is that the fuel consumed for the corrective phase is inversely proportional to the time available for correction. In situations where the uncertainties are very high, the final corrections can require large control inputs and hence, the over all mission becomes less fuel efficient. To tackle this problem, the integral separation of the spacecraft and the target is included into the objective function of gradient method. The advantage of this representation is that, now the Hamiltonian of the problem formulation can include xT Qx in addition to the fuel costs. This aims to bring the state close to the target as fast as required. In this work, the matrix Q is chosen as follows: Q = [Q1 00; 0Q20; 00Q3; 000; 000; 000] where Q1 ,Q2 and Q3 are positive. This aids to add costs to the time integral of the distance of separation between the spacecraft and the target leaving free the relative velocity. Gradient method was used to obtain optimal trajectories for different Q matrices of the modified Hamiltonian and the results are summarized in Fig. 9. As seen from the graph, as the value of the elements Q1 , Q2 and Q3 are increased the spacecraft reaches closer to the target sooner. But this comes with an additional cost in the total fuel consumed. The fuel consumption for the different cases are given in Table 4. 5. CONCLUSIONS In this work, MPSP was applied to find computationally efficient suboptimal and robust solutions for orbital transfers with high target position uncertainties. It was shown that MPSP is efficient in generating adaptive orbital transfer trajectories. Comparisons were made with the gradient method for speed and optimality using Hohmann transfer as the benchmark. The MPSP solution was found to be having an optimality ratio equivalent to gradient based algorithms. The MPSP also allowed additional control over 675
Banerjee, A., Padhi, R., and Vatsal, V. (2015). Optimal guidance for accurate lunar soft landing with minimum fuel consumption using model predictive static programming. In American Control Conference (ACC), 2015, 1861–1866. IEEE. Battin, R.H. (1999). An introduction to the mathematics and methods of astrodynamics. Aiaa. Bryson, A. and Ho, Y. (1975). Applied Optimal Control: Optimization, Estimation and Control. Hemisphere Publishing Corporation. Dwivedi, P.N., Bhattacharya, A., Padhi, R., et al. (2011). Suboptimal midcourse guidance of interceptors for highspeed targets with alignment angle constraint. Journal of Guidance Control and Dynamics, 34(3), 860. Kothari, M. and Padhi, R. (2007). A hybrid energyinsensitive explicit guidance scheme for long range flight vehicles with solid motors. IFAC Proceedings Volumes, 40(7), 651–656. Nesvorn` y, D. and Bottke, W.F. (2004). Detection of the yarkovsky effect for main-belt asteroids. Icarus, 170(2), 324–342. Padhi, R., Chawla, C., and Das, P.G. (2014). Partial integrated guidance and control of interceptors for highspeed ballistic targets. Journal of Guidance, Control, and Dynamics. Padhi, R. and Kothari, M. (2009). Model predictive static programming: a computationally efficient technique for suboptimal control design. International journal of innovative computing, information and control, 5(2), 399–411. Varma, S.A., Parwana, H., and Kothari, M. (2016). A pitch controlled suboptimal guidance scheme with impact-angle-constraints. In AIAA Guidance Navigation and Control Conference, SciTech.
ScienceDirect
IFAC PapersOnLine 51-1 (2018) 638–643
Optimal Optimal Orbital Orbital Transfers Transfers to to Asteroids Asteroids Optimal Orbital Transfers to Asteroids ∗ ∗∗ Mangal Kothari OptimalShribharath Orbital BTransfers to Asteroids ∗∗ Shribharath B ∗∗ Mangal Kothari ∗∗
Shribharath Shribharath B B ∗ Mangal Mangal Kothari Kothari ∗∗ ∗ Graduate Student, Dept. of Aerospace Engineering, Institute Shribharath B Mangal Kothari ∗∗Indian Graduate Student, Dept. of Aerospace Engineering, Indian Institute Graduate Student, Dept. of Aerospace Engineering, Indian of Technology Kanpur, India (e-mail: [email protected]) Graduate Student,Kanpur, Dept. of India Aerospace Engineering, Indian Institute Institute (e-mail: [email protected]) ∗∗ of Technology Technology Kanpur, India (e-mail: [email protected]) Assistant Professor, Dept. of Aerospace Engineering, Indian ∗ ∗∗ of Graduate Student, Dept. Dept. of India Aerospace Engineering, IndianIndian Institute Technology Kanpur, (e-mail: [email protected]) ∗∗ of Assistant Professor, of Aerospace Engineering, Assistant Professor, Dept. of Aerospace Engineering, Indian Institute of Technology Kanpur, India (e-mail: [email protected]) ∗∗ of Technology Kanpur, India (e-mail: [email protected]) Assistant Professor, Dept. of Aerospace Engineering, Indian Institute of Technology Kanpur, India (e-mail: [email protected]) Institute of (e-mail: [email protected]) ∗∗ Assistant Professor,Kanpur, Dept. ofIndia Aerospace Engineering, Indian Institute of Technology Technology Kanpur, India (e-mail: [email protected])
∗ ∗ ∗ ∗
Institute Technology Kanpur, India Mars (e-mail: Abstract: Many of theof smaller asteroids between [email protected]) Jupiter have poorly determined Abstract: Many of the the smaller smaller asteroids between Marsaand and Jupiter have poorly poorly determined Abstract: Many of asteroids between Mars Jupiter have orbits with high uncertainties. For successfully completing transfer missions consuming Abstract: Many of the smaller asteroids between Marsaand Jupiter have without poorly determined determined orbits with high uncertainties. For successfully completing transfer missions without consuming orbits with high uncertainties. For successfully completing a transfer missions without consuming unacceptable amounts of fuel, a robust optimal design is needed. In this paper, the spacecraft Abstract: Many of the smaller asteroids between Mars and Jupiter have poorly determined orbits with high uncertainties. successfully completing transferIn missions without unacceptable amounts of fuel, fuel,For a robust robust optimal design control is aneeded. needed. this The paper, the consuming spacecraft unacceptable amounts of a optimal design is In this paper, the spacecraft trajectory planning problem is posed as an optimal problem. solutions of the orbits with high uncertainties. For successfully completing a transfer missions without consuming unacceptable amounts of fuel, a robust optimal design is needed. In this paper, the spacecraft trajectory planning problem is posed as an optimal control problem. The solutions of the trajectory planning problem is posed as an optimal control problem. The solutions of the problem is obtained by employing a computationally efficient algorithm, Model Predictive Static unacceptable amounts of fuel, isa robust optimal design is needed. In this paper, the spacecraft trajectory planning problem posed as an optimal control problem. The solutions of the problem is obtained by employing a computationally efficient algorithm, Model Predictive Static problem is by employing a efficient algorithm, Model Predictive Static Programming (MPSP). The algorithm isasused generating nominal and adaptive trajectories. trajectory planning problem is posed an for optimal control problem. The solutions of the problem is obtained obtained by employing a computationally computationally efficient algorithm, Model Predictive Static Programming (MPSP). The algorithm is used used for generating nominal and adaptive trajectories. Programming (MPSP). The algorithm is for generating nominal and adaptive trajectories. A comparison study is carried out with classical solutions. The results show that MPSP is problem is obtained by employing a computationally efficient algorithm, Model Predictive Static Programming (MPSP). The algorithm is used for generating nominal and adaptive trajectories. A comparison study is carried out with classical solutions. The results show that MPSP is A comparison study is carried out with classical solutions. The results show that MPSP is highly reliable (MPSP). in generating computationally efficient low continuous thrust transfer trajectories Programming The algorithm is used for generating nominal and adaptive trajectories. A comparison study is carried out with classical solutions. The results show that MPSP is highly reliable in generating computationally efficient low continuous thrust transfer trajectories highly reliable generating computationally efficient low continuous thrust transfer particularly in in situations where there are stringent time and robustness is MPSP a need.is A comparison study is carried out with classical solutions. The results show thattrajectories highly reliable generating computationally efficient lowconstraints continuous thrust transfer trajectories particularly in in situations where there are stringent stringent time constraints and robustness is aa need. need. particularly in situations where there are time constraints and robustness is highly in generating computationally efficient lowHosting continuous thrust transfer trajectories particularly situations where there stringent time constraints and robustness is areserved. need. © 2018,reliable IFACin(International Federation of are Automatic Control) by Elsevier Ltd. All rights Keywords: Orbital transfers, Trajectory Asteroids, Optimal control particularly in situations where there aredesign, stringent time constraints and robustness is a need. Keywords: Orbital transfers, Trajectory design, Asteroids, Optimal control Keywords: Orbital transfers, Trajectory design, Asteroids, Optimal control Keywords: Orbital transfers, Trajectory design, Asteroids, Optimal control 1. INTRODUCTION algorithmOptimal to perform well under the presence of uncerKeywords: Orbital transfers, Trajectory design, Asteroids, control 1. INTRODUCTION algorithm to perform well under the presence of uncer1. INTRODUCTION algorithm to perform well inunder under the and presence of(2007)]. uncertainties was demonstrated [Kothari Padhiof 1. INTRODUCTION algorithm todemonstrated perform well the presence uncertainties was in [Kothari and Padhi (2007)]. tainties was demonstrated in [Kothari and Padhi (2007)]. The MPSP algorithm, unlike impulsive transfers, gives 1. INTRODUCTION perform well the and presence uncertainties wastodemonstrated inunder [Kothari Padhiof(2007)]. Interplanetary orbital transfers is an exhaustively analysed algorithm The MPSP algorithm, unlike impulsive transfers, gives The MPSP algorithm, unlike impulsive transfers, gives continuous solution which can be implemented through Interplanetary orbital transfers is an exhaustively analysed wassolution demonstrated incan [Kothari andtransfers, Padhi through (2007)]. The MPSP algorithm, unlike impulsive gives Interplanetary orbital transfers iseight an exhaustively exhaustively analysed domain in space missions. Theis planets in the solar tainties continuous which be implemented Interplanetary orbital transfers an analysed continuous solution which can be implemented through low thrust engines. This has become very popular in recent domain in space missions. The eight planets in the solar MPSPengines. algorithm, unlike impulsive transfers, gives continuous solution which can be implemented through domain have in space space missions. Theisset eight planets in the the solar The system a orbital well established of data such as mass, low thrust This has become very popular in recent Interplanetary transfers anof exhaustively analysed domain in missions. The eight planets in solar low thrust thrust engines. This has become very popular in recent times due to the reduced size of be theimplemented engine itself.in system have a well established set data such as mass, continuous solution which can through low engines. This has become very popular recent system have a well established set of data such as mass, size, and orbit. Given the The fact set that its position can be times due to the reduced size of the engine itself. domain inorbit. space missions. eight in the solar system have a well established of planets data such ascan mass, times due to to the reduced reduced size of the the very engine itself.in recent size, and Given the fact that its position be thrust engines. This has become popular times due the size of engine itself. size, and and orbit. Given the given fact set that its position can be low accurately known at any point ofposition time, optimal In this work, the MPSP algorithm is employed for gensystem have a well established of data such as mass, size, orbit. Given the fact that its can be accurately known at any given point of time, optimal In this work, the MPSP algorithm is employed for gentimes due torobust the reduced size of the engine itself. accurately known at any given point of time, optimal In this work, the MPSP algorithm is employed for gentrajectories can be generated beforehand and the mission eration of transfer trajectory to a hypothetical size, and orbit. Given the given fact that can be In accurately known at any pointitsofposition time, optimal this work, the MPSP algorithm is employed for gentrajectories can be generated beforehand and the mission eration of robust transfer trajectory to a hypothetical trajectories can be generated beforehand and the mission eration of robust transfer trajectory to a hypothetical control involves only a minor corrections [Battin (1999)]. asteroid near Mars with high orbital uncertainties. The accurately known at given point ofand time, optimal trajectories can be generated beforehand the (1999)]. mission In this work, the MPSP algorithm is uncertainties. employed for generation of robust transfer trajectory to a hypothetical control involves only aaany minor corrections [Battin asteroid near Mars with high orbital The control involves only minor corrections [Battin (1999)]. asteroid near Mars with high orbital uncertainties. Thea But the trajectory planning becomes highly challenging algorithm first generates nominal trajectory assuming trajectories can be generated beforehand and the mission control involves onlyplanning a minor becomes corrections [Battin (1999)]. eration of robust transfer trajectory to a hypothetical asteroid near Mars with high orbital uncertainties. The But the trajectory highly challenging algorithm first generates nominal trajectory assuming a But the trajectory planning becomes highly challenging algorithm first generates nominal trajectory assuming when the target orbit has high uncertainties such as astermean position of an asteroid and updates the trajectory control involves onlyplanning ahas minor corrections [Battin (1999)]. But the trajectory becomes highly challenging asteroid near with high orbital uncertainties. Theaa algorithm firstMars generates nominal trajectory assuming when the target orbit high uncertainties such as astermean position of an asteroid and updates the trajectory whenthe the trajectory target orbit has high high uncertainties such as asterastermean position position of an asteroid asteroid and updates theassuming trajectory oids. There is a orbit wellplanning known belt of asteroids between the algorithm when it receives the updated asteroid information. Thea But becomes highly challenging when the target has uncertainties such as first of generates nominal trajectory mean an and updates the trajectory oids. There is aa well known belt of asteroids between the when it receives the updated asteroid information. The oids. There is well known belt of asteroids between the when it receives the updated asteroid information. The planets Mars and Jupiter whose origins are still a subject algorithm is first compared with classical solutions such as when theMars target has high uncertainties such asteroids. There isand a orbit well known belt of asteroids between the mean position ofcompared an asteroid and updates the trajectory when it receives the updated asteroid information. The planets Jupiter whose origins are still aaassubject algorithm is first with classical solutions such as planets Mars and Jupiter whose origins areasteroids still subject algorithm is first compared with classical solutions such as of dispute in astrophysics. Several large have Hohmann transfer. The Hohmann transfer has been well oids. There is a well known belt of asteroids between the planets Mars and Jupiter whose origins are still a subject when it receives the updated asteroid information. The algorithm is first compared with classical solutions such as of dispute in astrophysics. Several large asteroids have Hohmann transfer. The Hohmann transfer has been well of dispute in astrophysics. Several large asteroids have Hohmann transfer. The Hohmann transfer has been well been identified and their orbits have been well determined. established as the analytical optimum for co-planar cirplanets Mars Jupiter whose origins are still a subject of dispute in and astrophysics. Several large asteroids have algorithm is first compared with classical solutions such as Hohmann transfer. The Hohmann transfer has been well been identified and their orbits have been well determined. established as the analytical optimum for co-planar cirbeen identified and their asteroids, orbits havewhich been well determined. established as the the analytical optimum for has co-planar cirHowever, several smaller are highly valued cular transfers [Battin (1999)]. Thetransfer solutions obtained by of dispute in astrophysics. Several large asteroids have been identified and their orbits have been well determined. Hohmann transfer. The Hohmann been well established as analytical optimum for co-planar cirHowever, several smaller asteroids, which are highly valued cular transfers [Battin (1999)]. The solutions obtained by However, several smaller asteroids, which are highly valued cular transfers transfers [Battin (1999)]. The solutions solutions obtained by for potential scientific inspection have very less number of established MPSP is then compared with optimum gradient method, a popular been identified and their orbits have been well determined. However, several smaller asteroids, which are highly valued as compared the analytical for co-planar circular [Battin (1999)]. The obtained by for potential scientific inspection have very less number of MPSP is then with gradient method, a popular for potential scientific inspection have very less number of MPSP is then compared with gradient method, a popular clear observations and hence a much poorly determined numerical method for obtaining control solutions [Bryson However, several smaller asteroids, which are highly valued for potential scientific inspection have very less number of cular transfers [Battin (1999)]. The solutions obtained by MPSP is then compared with gradient method, a popular clear observations and hence much poorly determined numerical method for obtaining control solutions [Bryson clear observations andinspection hence aaainmuch much poorly determined numerical method for obtaining control solutionsa popular [Bryson orbits. With high uncertainties the final target position, and Ho is(1975)]. It isfor demonstrated usingmethod, numerical simulafor potential scientific have very lessdetermined number of MPSP clear observations and hence poorly then compared with gradient numerical method obtaining control solutions [Bryson orbits. With high uncertainties in the final target position, and Ho (1975)]. It is demonstrated using numerical simulaorbits. With high uncertainties in the final target position, and Ho (1975)]. It is demonstrated using numerical simulaa one-go solution approach does not work here unlike the tions that the proposed approach is efficient in generating clear observations and hence ainmuch poorly determined orbits. With high uncertainties the work final target position, numerical method obtaining control solutions [Bryson and Ho (1975)]. It isfor demonstrated using numerical simulaacase one-go solution approach does not here unlike the tions that the proposed approach is efficient in generating a one-go solution approach does not work here unlike the tions that the proposed approach is efficient in generating of planets. An adaptive trajectory has to be designed solutions and converges quickly. orbits. With high uncertainties innot the work final target position, acase one-go solution approach does here unlike the and Ho (1975)]. It is demonstrated numerical simulations that the converges proposed approach isusing efficient in generating of planets. An adaptive trajectory has to be designed solutions and quickly. case of planets. Anapproach adaptive trajectory has to be be designed solutions and converges quickly. is efficient in generating with minimal online corrective efforts tohas reach the target. a one-go solution does not work here unlike the tions case of planets. An adaptive trajectory to designed thatand the proposed approach solutions converges quickly. The rest of this paper is organized as follows: Section 2 with minimal online corrective efforts to reach the target. with of minimal online corrective efforts to tohas reach the target. solutions The rest of this paper is organized as follows: Section 22 case planets. Antheory adaptive to be designed with minimal online corrective efforts reach the target. converges quickly. The rest rest and of this paper is organized as follows: Section Optimal control hastrajectory been extensively used for The discusses orbital transfers and background in brief. Then, of this transfers paper is organized as follows: Section 2 Optimal control theory has been used for orbital and background in brief. Then, with minimal online corrective effortsextensively toHowever, reach the target. Optimal control theory hasspacecraft. been extensively used fora discusses discusses orbital transfers and background background in brief. brief. Then,23 trajectory generations forhas such the optimal trajectory transfer problem is posed. Section Optimal control theory been extensively used for The rest orbital of trajectory this transfers papertransfer is organized as is follows: Section discusses and in Then, trajectory generations for spacecraft. However, such a the optimal problem posed. Section 3 trajectorycontrol generations for spacecraft. However, suchforaa gives the optimal optimal trajectory transfer problem is posed. posed. Section framework leads to twofor point boundary value problems a brief description ofand twobackground algorithms to brief. solve TPBOptimal theory hasspacecraft. been extensively used trajectory generations However, such discusses orbital transfers in Then, the trajectory transfer problem is Section 33 framework leads to two point boundary value problems gives a brief description of two algorithms to solve TPBframework generations leads to computationally twofor point boundary valueforproblems problems givesoptimal brieftrajectory description of two twoproblem algorithms todiscusses solve TPB(TPBVPs), that is intensive explicit VPs. Section 4 present transfer numerical resultsisand the3 trajectory spacecraft. However, such a the framework leads to two point boundary value posed. Section gives aa brief description of algorithms to solve TPB(TPBVPs), that is computationally intensive for explicit VPs. Section 44 present numerical results and discusses the (TPBVPs), that is computationally intensive for explicit VPs. Section present numerical results and discusses the solutions. A sub-optimal algorithm called Model Predictive results. Finally, the paper is concluded in Section 5. framework leads to computationally two algorithm point boundary valuefor problems (TPBVPs), that is intensive explicit gives a brief description ofistwo algorithms todiscusses solve5.TPBVPs. Section 4 present numerical resultsinand the solutions. A sub-optimal called Model Predictive results. Finally, the paper concluded Section solutions. A sub-optimal algorithm called Model Predictive results. Finally, the paper is concluded in Section 5. Static Programming (MPSP) was proposed to solved TPB(TPBVPs), that is computationally intensive for explicit solutions. A sub-optimal algorithm called Model Predictive VPs. Section 4 present numerical resultsinand discusses results. Finally, the paper is concluded Section 5. the Static Programming (MPSP) was proposed to solved TPBStaticefficiently Programming (MPSP) was proposed toApproximate solved TPB- results. Finally, 2. PROBLEM FORMULATION VPs by combining philosophies ofto solutions. A sub-optimal algorithm called Model Predictive Static Programming (MPSP) was proposed solved TPBthe paper is concluded in Section 5. 2. PROBLEM FORMULATION VPs efficiently by combining philosophies of Approximate 2. PROBLEM PROBLEM FORMULATION FORMULATION VPs efficiently efficiently by combining combining philosophies oftoApproximate Approximate Dynamic Programming andwas Model Predictive Control Static Programming (MPSP) proposed solved TPB2. VPs by philosophies of Dynamic Programming and Model Predictive Control Dynamic Programming andphilosophies Model Predictive Control by [Padhi and by Kothari (2009)]. The algorithm has been In this section, we first briefly introduce orbital transfer 2. PROBLEM FORMULATION VPs efficiently combining of Approximate Dynamic Programming and Model Predictive Control by [Padhi and Kothari (2009)]. The algorithm has been this section, we first briefly introduce orbital transfer by [Padhi and Kothari (2009)]. The algorithm has been In In this this section, westate first the briefly introduce orbital orbital transfer transfer applied to solve various guidance problems from aerospace and then formally problem. Dynamic Programming and Model Predictive Control by [Padhi and Kothari (2009)]. The algorithm has been In section, we first briefly introduce applied to solve various guidance problems from aerospace and then formally state the problem. applied to solve various guidance problems from aerospace and then formally state the problem. industry [Dwivedi et al. (2011); Padhi et al. (2014); Varma by [Padhi and Kothari (2009)].Padhi The algorithm hasVarma been In section, westate first the briefly introduce orbital transfer applied to[Dwivedi solve various guidance problems from aerospace andthis then formally problem. industry et al. (2011); et al. (2014); industry [Dwivedi et al. al. guidance (2011); Padhi et al. al.recently (2014); Varma and 2.1 Orbital transfers et al. (2016)]. The algorithm hasPadhi been also applied applied to solve various problems from aerospace then formally state the problem. industry [Dwivedi et (2011); et (2014); Varma 2.1 Orbital transfers et al. (2016)]. The algorithm has been also recently applied 2.1 Orbital Orbital transfers transfers et al. al. (2016)]. The algorithm algorithm hasPadhi been also recently applied to landing a spacecraft on moon [Banerjee et al. applied (2015)] industry [Dwivedi et al. (2011); et al.recently (2014); Varma 2.1 et (2016)]. The has been also to landing aa spacecraft on moon [Banerjee et al. (2015)] to landing spacecraft on moon [Banerjee et al. (2015)] Orbital transfers can be broadly classified into two types where the MPSP was combined with a neural network to 2.1 Orbital transfers et al. (2016)]. The was algorithm has been also recently to landing a spacecraft on moon [Banerjee et network al. applied (2015)] Orbital transfers can be broadly classified into two types where the MPSP combined with aa neural to Orbital transfers can becontinuous broadly classified classified into two two types where the MPSP was combined with neural network to namely impulsive andbe low thrust transfers. find global optimal solutions. The ability of the MPSP to landing aoptimal spacecraft on moon [Banerjee et network al. (2015)] transfers can broadly into types where the MPSP wassolutions. combined with a neural to Orbital namely impulsive and continuous low thrust transfers. find global The ability of the MPSP namely impulsive and continuous low thrust transfers. find global optimal solutions. The ability of the MPSP broadly classified into two types where the MPSP wassolutions. combinedThe withability a neural network to Orbital namely transfers impulsivecan andbecontinuous low thrust transfers. find global optimal of the MPSP
Copyright © 2018, 2018 IFAC 670Hosting namelybyimpulsive continuous find global optimal solutions. The abilityof of the MPSP 2405-8963 © IFAC (International Federation Automatic Control) Elsevier Ltd.and All rights reserved. Copyright © 2018 IFAC 670 Copyright ©under 2018 responsibility IFAC 670Control. Peer review of International Federation of Automatic Copyright © 2018 IFAC 670 10.1016/j.ifacol.2018.05.107 Copyright © 2018 IFAC 670
low thrust transfers.
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is the patched conic approximation. The motion of a spacecraft under the influence of a central body is a conic. In our solar system, the spacecrafts are often in the proximity of a single planet or solely under the influence of the Sun. Thus, a rendezvous mission starting from the Earth to a target body, in our case an asteroid, can be broadly split into three phases as follows: (i) Escape from the Earth: In this phase, the spacecraft begins from an earth bound orbit and escapes from the Earth’s gravity; (ii) Approach phase: This is the longest phase of the mission. In this phase, the spacecraft is mainly influenced by the Sun’s gravity; and (iii) Insertion Phase: This phase begins when the spacecraft has successfully entered the region of influence of the asteroid. Fig. 1. Hohmann Transfer
2.3 Problem Statement
In a continuous low thrust transfer where the speed of the spacecraft is slowly increased, the total ∆v required would be simply the change in the orbital speed of the two orbits. Impulsive transfers on the other hand use occasional large pulses of thrust. They make the use of the gravity of the sun itself to effect some part of the speed changes. Hence impulsive transfers are always more efficient than the continuous low thrust transfers [Battin (1999)]. However when the transfer time is not a constraint and only low thrust is required, we only need a smaller engine with less thrust capabilities which in turn lead to fuel savings. Further for missions to outer planets the impulsive transfers become impractical due to the requirement of large impulses. Hence continuous low thrust engines become the only and more efficient option for such missions. Hohmann transfer is a two tangential impulse transfer between two co-planar circular orbits. It is a popular option often used in space missions and is also known to be the have the optimal ∆v for radii ratio below the critical value. Beyond this ratio three impulsive bi-elliptic transfer becomes optimal [Battin (1999)]. However for close pair of orbits like Earth and Mars or Earth and Jupiter Hohmann transfer is the simplest option available in hand. The transfer is pictorially represented in Fig. 1. The magnitude of the two impulses required to achieve this transfer can be calculated using the readily available formula for the orbital speeds of the two circular and the elliptical transfer orbit as follows:
2r2 −1 r1 + r2 µ 2r1 1− ∆v2 = r2 r1 + r2
∆v1 =
µ r1
(1) (2)
In this paper, Hohmann transfer is used as benchmark to compare the fuel efficiency of different algorithms while having the same transfer time as that of Hohmann transfer. 2.2 Background The motion of the spacecraft inside solar system has to be modeled as an n-body problem including the forces from all the planets. But for the purpose of rendezvous missions an acceptable and established approximation 671
This work considers the problem of optimizing the second phase. The phase considered is pictorially represented in Fig. 2. The mission commences just outside the region of influence of the earth and terminates just outside the region of influence of the asteroid. The main purpose of this work is to design spacecraft trajectories under orbit(al) uncertainties of an asteroid using an optimal algorithm. The following assumptions are made in this work: (i) The spacecraft is assumed to have escaped the gravitational potential well of the earth; and (ii) During the approach phase the planetary perturbations are neglected. These assumptions simplify the mathematical model and make computations simpler. The final state of the asteroid is considered to be uncertain and is assumed to be a random variable with a known mean. The simplified equations of motion of a spacecraft are based on the Newton’s law of gravitation and are given as follows: x¨ = y¨ = z¨ =
−µx 3
+ ax
(3)
3
+ ay
(4)
(x2 + y 2 + z 2 ) 2 −µy
(x2 + y 2 + z 2 ) 2 −µz
(5) 3 + az (x2 + y 2 + z 2 ) 2 where µ is the gravitational constant of the central body and ax , ay , az are the control forces by means of continuous low thrust with thrust vectoring at all angles. The goal is to take the state vector, which contains quantities denoting the position x, y, z and the velocities which are the rates of the x, y, and z positions, from an initial value to a final value while minimizing the control effort which directly affects the fuel consumption. The problem can be posed as an optimal control problem with the minimization of the following cost function 1 tf J = Φ(Xf ) + L(X(t), U (t), t)dt (6) 2 to subject to Eqs. (3)-(5). Here, Φ(Xf ) = 21 (Xf −Xd )M (Xf − Xd ), where Xd is the desired final state and M is a matrix with large diagonal elements and L(X(t), U (t), t) = 1 T 2 U (t) RU (t), R > 0. By setting the first variation of J to zero one can obtain the state, co-state, and optimal control equations to be solved along with the boundary conditions to be satisfied [Bryson and Ho (1975)]. As
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3.2 Model Predictive Static Programming To outline MPSP, we consider general nonlinear systems in discrete form, the state and output dynamics of which are given by Xk+1 = Fk (Xk , Uk )
(11)
Yk = h(Xk )
(12)
where X ∈ ℜn , U ∈ ℜm , Y ∈ ℜp and k = 1, 2, . . . , N are the time steps. The primary objective is to obtain a suitable control history Uk , k = 1, 2, . . . , N −1, so that the output at the final time step YN goes to a desired value YN∗ , i.e. YN → YN∗ . In addition, we aim to achieve this task with minimum control effort. Fig. 2. Pictorial representation stated before, the solution leads to TPBVPs and therefore computationally intensive. In this work, we employ MPSP algorithm to solve the above problem. 3. OPTIMAL CONTROL ALGORITHMS In this section, we briefly describe two optimal control algorithms to solve TPBVPs posed in the previous section. 3.1 Gradient Method First we present a numerical technique, gradient method, to solve the TPBVPs. The approach requires a guess history that usually does not satisfy the necessary conditions of optimality. In this approach, the state, costate, and the boundary conditions are forced to be satisfied and the strategy is to satisfy the optimal control equation iteratively. The first variation of the augmented cost functional is tf ∂Φ T T ∂H ˙ ¯ δ J = δXf +λ + δX − λf + ∂Xf ∂X t0 T ∂H T ∂H ˙ ... δU + δλ − X dt (7) ∂U ∂λ Since the state, costate and boundary conditions are satisfied, we get tf T ∂H ¯ δJ = dt (8) δU ∂U t0 Selecting δU = −τ ∂H ∂U and substituting into the previous equation, we get T tf ∂H ∂H dt (9) −τ δ J¯ = ∂U ∂U t0 The updated control is given by ∂H dt (10) U (p + 1) = U (p) − τ ∂U where p is the iteration variable. It can be observed from the last equation that with this updated, the first variation decreases with each iteration and eventually approaches zero. Hence, all the necessary conditions get satisfied. The value of τ for convergence however was be found by trial and error. 672
Starting with a guess value of control history which need not meet the objective. MPSP serves to compute an error history of the control variable, which needs to be subtracted from the previous history to get an improved control history. Expanding YN about YN∗ using Taylor series expansion and using small error approximation, we get ∂YN dXN △YN ∼ (13) = dYN = ∂XN However from (11), we can write the error in state at time step (k + 1) as dXk+1
∂Fk = ∂Xk
∂Fk dXk + ∂Uk
(14)
dUk
where dXk and dUk are the error of state and control at time step k respectively. Expanding dXN as in (14) (for k = N − 1) and substituting it in (13), we get dYN = A dX1 + B1 dU1 + B2 dU2 + . . .+ BN −1 dUN −1 (15) where ∂YN ∂FN −1 ∂F1 A ... ∂XN ∂XN −1 ∂X1 ∂FN −1 ∂Fk+1 ∂Fk ∂YN ... Bk ∂XN ∂XN −1 ∂Xk+1 ∂Uk for k = 1 . . . N − 1. Since the initial condition is specified, there is no error in the first term; which means dX1 = 0. With this (15) reduces to dYN = B1 dU1 + B2 dU2 + . . . + BN −1 dUN −1 =
N −1
Bk dUk
k=1
(16) Now Bk can be computed recursively. For doing this, first 0 we define BN −1 as follows 0 BN −1 =
∂YN ∂XN
(17)
Next we compute Bk0 , k = (N − 2), (N − 3), . . . , 1 as Bk0
=
0 Bk+1
∂Fk+1 ∂Xk+1
(18)
Finally, Bk , k = (N − 2), (N − 3), . . . , 1 can be computed as
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Table 1. Orbital Parameters used in Simulation Parameter a (AU) e (no unit) i (deg)
Earth 1 0.016 1.5787
Target 1.5236 0.0934 1.67
Fig. 3. Rendezvous without observations Bk = Bk0
∂Fk ∂Uk
(19)
The set of equations get closed by setting an additional the objective function which is given by
J=
N −1 1 0 (Uk − dUk )T Rk (Uk0 − dUk ) 2
Table 2. Performance Comparison of Gradient Method and MPSP (20)
k=1
where Uk0 , k = 1, . . . , (N − 1) is the previous control history solution and dUk is the corresponding error in the control history. The cost function in (20) needs to be minimized subjected to the constraint in (16), where Rk > 0 (a positive definite matrix) is the weighting matrix, which needs to be chosen judiciously by the control designer. The selection of such a cost function is motivated by the fact that we are interested in finding a l2 -norm minimizing control history, since (Uk0 −dUk ) is the updated control value at k (see (23)). Equations (16) and (20) formulate an appropriate constrained static optimization problem. The augmented cost function is given by N −1 1 0 J¯ = (Uk − dUk )T Rk (Uk0 − dUk ) + 2 k=1
λT (dYN −
N −1
Bk dUk )
(21)
k=1
Using optimization theory, we get dUk = Rk−1 BkT λ + Uk0
Fig. 4. Rendezvous with observations
(22)
Hence, the updated control at time step k = 1, 2, . . . , (N − 1) is given by
Method Hohmann Transfer Gradient Method MPSP
Fuel Consumption (x10−3 AU/day) 3.2 4.6 4.6
4. RESULTS AND DISCUSSION 4.1 Orbital transfer trajectory to an asteroid In JPL’s small body database the asteroids are rated on a scale from 0 to 9 for measuring the orbital uncertainty. Such uncertainties occur because the motion of the asteroids are complicated by shape and spin dependent phenomena like Yarkovsky effect [Nesvorn` y and Bottke (2004)]. In this work, the asteroid is chosen to be a fictitious asteroid near Mars. The following trajectories using the standard orbital parameters of Earth and Mars given in Table 1 were obtained using MPSP without feedback. As it is seen from Fig. 3, the spacecraft approaches the asteroid but still a finite amount of error is present at the final position. In practical situations, this can occur due to the modeling approximations and the errors in the measurement of the position of the asteroid and the spacecraft. This error could be of the order of a few thousand kilometers. This is normally is of the same order of the size of the region of influence of the asteroid. In order to assure convergence MPSP can be re-initiated on board at definite intervals and corrections to the control inputs based on the measurements of the current spacecraft state vector. 4.2 Comparison of MPSP with Gradient Method
Uk =
Uk0
− dUk =
Rk−1 BkT A−1 λ
(dYN − bλ )
(23)
It is clear that the updated control history solution is a closed form solution, and hence, control solution can be updated with very minimal computational requirement. 673
The efficiency of MPSP and Gradient Method are compared with Hohmann Transfer, the theoretical optimal solution. The total ∆v required for a Hohmann transfer orbit from Earth to Mars can be directly calculated as shown in Section 2. The transfer time is simply taken as
5th International Conference on Advances in Control and 642 Shribharath B et al. / IFAC PapersOnLine 51-1 (2018) 638–643 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
Table 3. Speed comparison of Gradient Method and MPSP Method Hohmann Transfer Gradient Method MPSP
No.of Iterations 180668 7
Fig. 7. Rendezvous in presence of uncertainties
Fig. 5. Control Solution with equal weights
Fig. 8. Control Solution for adaptive trajectory R matrix and hence convergence is negatively affected. Also the total fuel consumption of the MPSP solution increasing from 4.6 to 4.9 in the same units as in the table. Hence, it is convenient to get different types of solution using the MPSP algorithm. Fig. 6. Control Solution with high initial weights
4.3 Orbital transfer in presence of uncertainties
the Hohmann transfer time and the obtained results are tabulated in Tables 2-3. The following points are inferred from the results: (i) The continuous thrust transfer expectedly consumes more fuel in comparison with the Hohmann transfer with a same transfer time; (ii) The gradient method gives a similar solution as MPSP, but its speed of convergence is very less. (iii) MPSP on the other hand has a very high speed of convergence; and (iv) MPSP has its final conditions as hard constraints unlike Gradient method which has it as soft constraints. This feature attracts MPSP for situations where stringent requirements regarding the final boundary value are present. The control solution provided by the MPSP often has large magnitudes of inputs in the beginning period. Although this aids to the quick rate of convergence, this need not be desirable at all circumstances. Sometimes a gradual build up to higher values of thrust is desirable. This can be achieved using the R matrix in the cost function chosen. By smartly changing the elements of the matrix for different time steps as required one may obtain smoother solutions using MPSP. To demonstrate this, for the same problem for which the above comparisons were made, high weights were placed for initial time steps and the weights were decreased exponentially. This changed the qualitative nature of the solutions as displayed in the following figures. A caution for this approach is that it may require more iterations to converge when heavy weights are used in the 674
Initially, the spacecraft is allowed to follow the nominal control history based in the mean of the estimated state vector of the target at the final time. As the spacecraft approaches closer to the target multiple observations are made at definite intervals and new control commands are generated based on the new mean of the estimated target position. In order to simulate the unpredicted motion of the asteroid, a random change in the expected final position of the asteroid is assumed. The results show that the MPSP algorithm works and the uncertainty in the final position of the asteroid can be easily handled by MPSP. The convergence of the algorithm depends on the range of the radar device and the number of observations made. The results are shown in Figs 7 and 8. The final phase shows the adaptive changes made in the control to achieve the rendezvous. But the fuel consumed to make these corrections is proportional to the deviation in the position of the asteroid from the expected position. 4.4 Nominal Trajectory Design When the orbit of a target in space is known accurately, the trajectory design is straight forward and the rendezvous can be achieved easily. But the presence of uncertainties has led us into splitting the mission into the nominal trajectory following and the corrective maneuvers at the end. The mean of the known information about the target orbit can generally be used to generate the nominal
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the nature of the solution thus handling the problem satisfactorily, providing fast-converging and robust solutions. REFERENCES
Fig. 9. Response curves for different values of Qi s Table 4. Comparison of fuel consumption for different values of Qi Qi 0 1 10
Total ∆V (km/s) 5.76 8.35 12.41
trajectory. In generating a nominal trajectory, once again the fuel consumed can be minimized using the gradient method. One drawback of this method is that the fuel consumed for the corrective phase is inversely proportional to the time available for correction. In situations where the uncertainties are very high, the final corrections can require large control inputs and hence, the over all mission becomes less fuel efficient. To tackle this problem, the integral separation of the spacecraft and the target is included into the objective function of gradient method. The advantage of this representation is that, now the Hamiltonian of the problem formulation can include xT Qx in addition to the fuel costs. This aims to bring the state close to the target as fast as required. In this work, the matrix Q is chosen as follows: Q = [Q1 00; 0Q20; 00Q3; 000; 000; 000] where Q1 ,Q2 and Q3 are positive. This aids to add costs to the time integral of the distance of separation between the spacecraft and the target leaving free the relative velocity. Gradient method was used to obtain optimal trajectories for different Q matrices of the modified Hamiltonian and the results are summarized in Fig. 9. As seen from the graph, as the value of the elements Q1 , Q2 and Q3 are increased the spacecraft reaches closer to the target sooner. But this comes with an additional cost in the total fuel consumed. The fuel consumption for the different cases are given in Table 4. 5. CONCLUSIONS In this work, MPSP was applied to find computationally efficient suboptimal and robust solutions for orbital transfers with high target position uncertainties. It was shown that MPSP is efficient in generating adaptive orbital transfer trajectories. Comparisons were made with the gradient method for speed and optimality using Hohmann transfer as the benchmark. The MPSP solution was found to be having an optimality ratio equivalent to gradient based algorithms. The MPSP also allowed additional control over 675
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