Skip to main content
SpringerLink
Log in

Optimal low-thrust trajectories to asteroids through an algorithm based on differential dynamic programming

  • Original Article
  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript

Abstract

In this paper an optimisation algorithm based on Differential dynamic programming is applied to the design of rendezvous and fly-by trajectories to near Earth objects. Differential dynamic programming is a successive approximation technique that computes a feedback control law in correspondence of a fixed number of decision times. In this way the high dimensional problem characteristic of low-thrust optimisation is reduced into a series of small dimensional problems. The proposed method exploits the stage-wise approach to incorporate an adaptive refinement of the discretisation mesh within the optimisation process. A particular interpolation technique was used to preserve the feedback nature of the control law, thus improving robustness against some approximation errors introduced during the adaptation process. The algorithm implements global variations of the control law, which ensure a further increase in robustness. The results presented show how the proposed approach is capable of fully exploiting the multi-body dynamics of the problem; in fact, in one of the study cases, a fly-by of the Earth is scheduled, which was not included in the first guess solution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

a :

Coefficient matrix of the Runge–Kutta–Fehlberg integration scheme, or acceleration vector

A k :

Matrix of the DDP algorithm at stage k

b :

Coefficient matrix of the Runge–Kutta–Fehlberg integration scheme

B k :

Matrix of the DDP algorithm at stage k

c :

Constant between 0 and 1

c :

Coefficient matrix of the Runge–Kutta–Fehlberg integration scheme

C k :

Matrix of the DDP algorithm at stage k

D k :

Matrix of the DDP algorithm at stage k

E k :

Matrix of the DDP algorithm at stage k

f :

Discrete-time state transition function

\({{\tilde {\bf f}}}\) :

Function containing the continuous dynamics equations

g :

Scalar stage-wise loss function

h k :

Discretisation step

H k :

Matrix of the DDP algorithm at stage k

I(m):

Identity matrix of size m

I sp :

Specific impulse of the spacecraft engine

j :

Integer number

J :

Cost function of the minimisation problem

k :

Integer number indicating the generic stage of DDP and the decision time of the trajectory at which the control law is allowed to change

k lim :

State from which the new control law is adopted for the integration of the dynamics

K k :

Matrix of the DDP algorithm at stage k

l :

Number of components of the Lagrange multiplier vector

m :

Number of components of the control vector, or mass of the spacecraft

n :

Number of components of the state vector

N :

Total number of decision times or control stages

P k :

Matrix of the DDP algorithm at stage k

Q k :

Matrix of the DDP algorithm at stage k

r :

Position vector

R Earth :

Radius of the Earth

reltol :

Relative tolerance

reltol mesh :

Relative tolerance on the mesh selection

R k :

Matrix of the DDP algorithm at stage k

s :

State vector

S k :

Matrix of the DDP algorithm at stage k

t :

Time

T :

Thrust vector

tol r :

Absolute tolerance on the position error

tol v :

Absolute tolerance on the velocity error

u :

Control vector

v :

Velocity vector

V :

Optimal return function

w :

Weight parameter

\({\tilde {w}}\) :

Weight parameter

x :

Cartesian coordinate along the x axis

y :

Cartesian coordinate along the y axis

z :

Cartesian coordinate along the z axis

Z k :

Matrix of the DDP algorithm at stage k

α :

In-plane angle of the velocity vector with respect to the Earth inertial reference frame

\({{\boldsymbol\beta_{k}}}\) :

Coefficient vector of the feedback control law component proportional to the variation of the state vector at stage k

\({{\boldsymbol\gamma_{k}}}\) :

Coefficient vector of the feedback control law component proportional to the variation of the Lagrange multiplier vector at stage k

δ :

Out-of plane angle of the velocity vector with respect to the Earth inertial reference frame

\({\varepsilon }\) :

Constant between 0 and 1

\({{\boldsymbol\lambda }}\) :

Vector of Lagrange multipliers

\({\mu _{{\rm Earth}}}\) :

Gravitational constant of the Earth

μ Sun :

Gravitational constant of the Sun

\({\varphi }\) :

Scalar function representing the constrains on the final stage

\({\Theta _k }\) :

Difference between the optimal return function at state k applying the new control, and the optimal return function at state k applying the nominal control

1:

Initial condition of a variable

k :

Stage of the DDP procedure

out:

Threshold value to exit a computational loop

target:

Variable related to the target body

x :

Vector component along the Cartesian x axis

y :

Vector component along the Cartesian y axis

z :

Vector component along the Cartesian z axis

*:

New nominal control for the algorithm with global variation in control

k :

Stage of the DDP procedure

\({\square}\) :

Variable

\({\bar{{\square}}}\) :

Nominal value of \({\square}\)

\({\left\{ {\square_k }\right\}}\) :

Sequence of variable \({\square}\) in time

\({\square^{T}}\) :

Transposed

\({\delta \square}\) :

Differential variation of \({\square}\)

\({\delta \square}\) :

Finite difference variation of \({\square}\)

\({{\it QP}\left[ \square \right]}\) :

Linear quadratic part of the Taylor expansion of the function \({\square}\)

\({\square_s }\) :

Gradient of the scalar function \({\square}\), or Jacobian of the vector function \({\square}\) with respect to the state s

\({\square_{ss}}\) :

Block components of the Hessian matrix of the scalar or the vector function \({\square}\) with respect to the state s

\({\square_u }\) :

Gradient of the scalar function \({\square}\), or Jacobian of the vector function \({\square}\) with respect to the state u

\({\square_{uu}}\) :

Block components of the Hessian matrix of the scalar or the vector function \({\square}\) with respect to the state u

\({\frac{d\square}{dt}}\) :

Derivative of \({\square}\) over time

\({\leftarrow }\) :

Assignment (in an algorithm)

\({\left\| \square \right\|_\infty }\) :

Norm infinity of the vector \({\square}\)

References

  • Bellman R.: Dynamic Programming. Princeton University Press, Princeton (1957)

    Google Scholar 

  • Bertsekas, D.P.: Dynamic Programming and Optimal Control, 3rd edn. Athena Scientific, Belmont (2005–2007)

  • Betts J.T.: Survey of numerical methods for trajectory optimization. J. Guid. Control Dyn. 21(2), 193–207 (1998). doi:10.2514/2.4231

    Article  MATH  Google Scholar 

  • Betts, J.T.: Practical methods for optimal control using nonlinear programming. SIAM, Philadelphia (2001)

  • Betts J.T., Erb S.O.: Optimal low thrust trajectories to the moon. SIAM J. Appl. Dyn. Syst. 2(2), 144–170 (2003)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Casalino L., Colasurdo G., Pastrone D.: Optimal low-thrust escape trajectories using gravity assist. J. Guid. Control Dyn. 22(5), 637–642 (1999)

    Article  Google Scholar 

  • Colasurdo G., Casalino L.: Trajectories towards near-earth-objects using solar electric propulsion. Adv. Astronaut. Sci. 103, 593–608 (1999)

    Google Scholar 

  • Coleman T.F. , Li Y.: On the convergence of reflective newton methods for large-scale nonlinear minimization subject to bounds. Math. Program. 67(2), 189–224 (1994)

    Article  MathSciNet  Google Scholar 

  • Coleman T.F., Li Y.: An interior, trust region approach for nonlinear minimization subject to bounds. SIAM J. Optimi. 6, 418–445 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Colombo C., Vasile M., Radice G.: Semi-analytical solution for the optimal low-thrust deflection of near earth objects. J. Guid. Control Dyn. 32(3), 796–809 (2009). doi:10.2514/1.40363

    Article  Google Scholar 

  • Conway B., Chilan C., Wall B.: Evolutionary principles applied to mission planning problems. Celest. Mech. Dyn. Astron. 97(2), 73–86 (2007). doi:10.1007/s10569-006-9052-7

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • De Boor C.: A Practical Guide to Splines. Springer, New York (1978)

    MATH  Google Scholar 

  • Dormand J.R., Prince P.J.: A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  • Enright P.J., Conway B.A.: Optimal finite-thrust spacecraft trajectories using collocation and nonlinear programming. J. Guid. Control Dyn. 14(5), 981–985 (1991). doi:10.2514/3.20739

    Article  Google Scholar 

  • Gershwin S.B., Jacobson D.H.: A discrete-time differential dynamic programming with application to optimal orbit transfer. AIAA J. 8(9), 1616–1626 (1970)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Gill P.E., Murray W., Wright M.H.: Practical Optimization. Academic Press, New York (1981)

    MATH  Google Scholar 

  • Guelman M.: Earth-to-moon transfer with a limited power engine. J. Guid. Control Dyn. 18(5), 1113–1138 (1995)

    Article  Google Scholar 

  • Herman A.L., Spencer D.B.: Optimal, low-thrust earth-orbit transfers using higher-order collocation methods. J. Guid. Control Dyn. 25(1), 40–47 (2002). doi:10.2514/2.4873

    Article  Google Scholar 

  • Jacobson D.H., Mayne D.Q.: Differential Dynamic Programming. American Elsevier, New York (1969)

    Google Scholar 

  • Kluever C.A.: Optimal low-thrust interplanetary trajectories by direct method techniques. J. Astronaut. Sci. 45(3), 247–262 (1997)

    MathSciNet  Google Scholar 

  • Kluever C.A., Pierson B.L.: Optimal low-thrust three-dimensional earth-moon trajectories. J. Guid. Control Dyn. 18(4), 830–837 (1995). doi:10.2514/3.21466

    Article  Google Scholar 

  • Lantoine, G., Russel, R.P.: A hybrid differential dynamic programming algorithm for low-thrust optimization. AIAA/AAS Astrodynamics Specialist Conference and Exhibit, 18–21 August 2008, Honoululu, Hawaii (2008)

  • Liao, L., Shoemaker, C.A.: Advantages of differential dynamic programming over newton’s method for discrete-time optimal control problems. Technical Report Cornell University. Available at http://ecommons.library.cornell.edu/handle/1813/5474 (1992)

  • Mayne D.A.: A second order gradient method for determining optimal trajectories for nonlinear discrete-time systems. Int. J. Control 3, 85–95 (1966)

    Article  Google Scholar 

  • Murray, D.M.: Differential dynamic programming for the efficient solution of optimal control problems. Ph.D. Dissertation, Department of Mathematics, University of Arizona, University Microfilm Inc, Tucson (1978)

  • Murray D.M., Yakowitz S.J.: Differential dynamic programming and Newton’s method for discrete optimal control problem. J. Optim. Theory Appl. 43, 395–414 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  • Nah R.S., Vadali S.R., Braden E.: Fuel-optimal, low-thrust, three-dimensional Earth–Mars trajectories. J. Guid. Control Dyn. 24(6), 1100–1107 (2001). doi:10.2514/2.4844

    Article  Google Scholar 

  • Olympio, J.T.: Algorithm for low-thrust optimal interplanetary transfers with escape and capture phases. AIAA/AAS Astrodynamics Specialist Conference and Exhibit, 18–21 August 2008, Honoululu, Hawaii (2008)

  • Perozzi E., Casalino L., Colasurdo G., Rossi A., Valsecchi G.B.: Resonant fly-by missions to near earth asteroids. Celest. Mech. Dyn. Astron. 83(1), 49–62 (2002). doi:10.1023/A:1020122511548

    Article  MATH  ADS  Google Scholar 

  • Pierson B.L., Kluever C.A.: Three-stage approach to optimal low-thrust earth-moon trajectories. J. Guid. Control Dyn. 17(6), 1275–1282 (1994). doi:10.2514/3.21344

    Article  ADS  Google Scholar 

  • Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishenko E.F.: The Mathematical Theory of Optimal Processes. Wiley Interscience, New York (1962)

    MATH  Google Scholar 

  • Price K., Storn R., Lampinen J.: Differential Evolution—a Practical Approach to Global Optimization. Springer, Berlin (2005)

    MATH  Google Scholar 

  • Racca G.D.: New challenges to trajectory design by the use of electric propulsion and other new means of wandering in the solar system. Celest. Mech. Dyn. Astron. 85(1), 1–24 (2003). doi:10.1023/A:1021787311087

    Article  MATH  MathSciNet  ADS  Google Scholar 

  • Ranieri C.L., Ocampo C.A.: Optimization of roundtrip, time-constrained, finite burn trajectories via an indirect method. J. Guid. Control Dyn. 28(2), 306–314 (2005). doi:10.2514/1.5540

    Article  Google Scholar 

  • Ranieri C.L., Ocampo C.A.: Indirect optimization of spiral trajectories. J. Guid. Control Dyn. 29(6), 1360–1366 (2006). doi:10.2514/1.19539

    Article  Google Scholar 

  • Scheel W.A., Conway B.A.: Optimization of very-low-thrust, many-revolution spacecraft trajectories. J. Guid. Control Dyn. 17(6), 1185–1192 (1994). doi:10.2514/3.21331

    Article  ADS  Google Scholar 

  • Tang S., Conway B.A.: Optimization of low-thrust interplanetary trajectories using collocation and nonlinear programming. J. Guid. Control Dyn. 18(3), 599–604 (1995). doi:10.2514/3.21429

    Article  Google Scholar 

  • Vadali S.R., Nah R.S., Braden E., Johnson I.L. Jr: Fuel-optimal planar earth–mars trajectories using low-thrust exhaust-modulated propulsion. J. Guid. Control Dyn. 23(3), 476–482 (2000). doi:10.2514/2.4553

    Article  Google Scholar 

  • Vasile, M., Bernelli-Zazzera, F.: Targeting a heliocentric orbit combining low-thrust propulsion and gravity assist manoeuvres’. Oper. Res. Space Air. 79. Book Series in Applied Optimization, Kluwer. ISBN 1-4020-1218-7, (2003)

  • Vasile M., Bernelli-Zazzera F.: Optimizing low-thrust and gravity assist maneuvres to design interplanetary trajectories. J. Astronaut. Sci. 51(1), 13–35 (2003)

    MathSciNet  Google Scholar 

  • Whiffen, G.J.: Static/dynamic control for optimizing a useful objective. No. Patent 6,496,741, Dec (2002)

  • Whiffen, G.J., Sims, J.A.: Application of a novel optimal control algorithm to low-thrust trajectory optimization. No. AAS 01-209, Feb. (2001)

  • Whiffen, G.J., Sims, J.A.: Application of the SDC optimal control algorithm to low-thrust escape and capture including fourth-body effects. In: 2nd International Symposium on Low-Thrust Trajectories, Toulouse, 18–20 June (2002)

  • Yakowitz S.: Theoretical and computational advances in differential dynamic programming. Control Cybern. 17(2-3), 173–189 (1988)

    MATH  MathSciNet  Google Scholar 

  • Yakowitz S., Rutherford B.: Computational aspects of discrete-time optimal control. Appl. Math. Comput. 15, 29–45 (1984)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Aerospace Engineering, University of Glasgow, James Watt South Building, G12 8QQ, Glasgow, UK

    Camilla Colombo, Massimiliano Vasile & Gianmarco Radice

Authors
  1. Camilla Colombo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Massimiliano Vasile
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gianmarco Radice
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Camilla Colombo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Colombo, C., Vasile, M. & Radice, G. Optimal low-thrust trajectories to asteroids through an algorithm based on differential dynamic programming. Celest Mech Dyn Astr 105, 75–112 (2009). https://doi.org/10.1007/s10569-009-9224-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10569-009-9224-3

Keywords

Search

Navigation