Optimal Guidance Strategy for Low Earth Orbit Based on the Laser Solutions

Transcription

1 Optimal Guidance Strategy for Low Earth Orbit Based on the Laser Solutions Ting Jin, Jian Cai Institute of Microelectronics of Chinese Academy of Science, Beijing, China 28 April 2015

2 Laboratory Research Area The interaction between laser and material research Laser propulsion technology research Spacecraft trajectory design and optimization technology research Navigation and control technology research

3 Introduction A rapid increase in the amount of space debris has a growing threat to human space flight activities. The existing monitor and remove technology have no idea about the debris of 1 to 10cm diameter which is the most harmful to the spacecraft. Therefore, it is urgently needed to develop new monitor and remove technology such as laser solution.

4 R Guidance strategy Guided trajectory R Risk of collision how to optimize the thrust direction of laser u u,, x uy uz T Unguided trajectory Norminal trajectory same mass consumption same time period Debris orbit Spacecraft orbit max R

5 Guidance Process Set parameters of the mission Model construction Opitimal model treatment Opitimal model solution Numerical simulation and analysis Dynamic equation Performance index Task constraints Maximum principle Distribution method Global optimization : differential evolution algorithm + Local optimization : SQP method Obtain optimal thrust direction

6 Model construction heliocentric ecliptic inertial coordinate system X Z r x z f T Y Because of the simple and intuitive format of the aircraft motion equation under heliocentric ecliptic inertial coordinate system. It has been widely used in the orbit design and optimization of all kinds of missions. In the two body system which is composed of gravitational body and aircraft, if the gravitational body assumed to be original point of heliocentric ecliptic inertial coordinate system, the equation of aircraft motion in this coordinate system can be described as follows:

7 Model construction Optimal model of trajectory X z O Z y u t u n a T x u debris Y reference coordinate system orbital coordinate system With the equation of mass consumption is considered, the dynamic model of debris in heliocentric ecliptic inertial coordinate system is written as: r& v n P r r - ri ri v& S 3 i 3 3 r i1 r - ri r T m& gi 0 sp T h [ u is the unit vector of thrust r, ut, un ] acceleration in orbital coordinate system, which can be described with pitch angle and yaw angle as: h T T [ ur, ut, un] [cos cos,cos sin,sin ] T m h

8 Model construction Performance index maximum deflection distance (same mass consumption,same time period) Task constraints J R R max Φ L debris r v spacecraft t L --impact time of the laser thrust is when the debris reach the dangerous area of collision tl rl tl t v t L L L 0

9 Guidance Process Set parameters of the mission Model construction Opitimal model treatment Opitimal model solution Numerical simulation and analysis Dynamic equation Performance index Task constraints Maximum principle Collocation method Global optimization : differential evolution algorithm + Local optimization : SQP method Obtain optimal thrust direction

10 Opitimal model treatment Opitimal model treatment method Indirect method Due to differential equations two-point boundary value problems, to obtain the optimal control law Due to the strong nonlinear characteristic, this method is extremely sensitive to conjugate status initial value Hybrid method Transversality condition is abandoned, dynamics equation constraints is only considered, it has been changed into a parameter optimization problem Initial value of associate state variable should be guessed Direct method Control variables are discretization processed, it has been changed into a nonlinear programming problem Principle of the direct method is simple and the convergence of this method is good Direct distribution optimization algorithm based on quadratic polynomial approximation

11 Opitimal model treatment In this research, the entire deflection orbit is divided into sections by discrete strategy. Within each discrete arc, quadratic polynomial is used to approximate thrust direction angles: X t a b t t c t t i i i i0 i i0 t d e t t f t t i i i i0 i i0 L tl, v, ta, a1, b1, c1, d1, e1, f1,..., a, b, c, d, e, f N N N N N N In addition, to ensure the continuity of the thrust direction angles and its rate of change at the discrete arc stitching point, the following constraints are needed to be satisfied: ai +1 ai bi t cit di+1 di ei t fit Φi = bi +1 bi 2cit ei +1 ei 2 fit T

12 Guidance Process Set parameters of the mission Model construction Opitimal model treatment Opitimal model solution Numerical simulation and analysis Dynamic equation Performance index Task constraints Maximum principle Collocation method Global optimization : differential evolution algorithm + Local optimization : SQP method Obtain optimal thrust direction

13 Opitimal model solution Global optimization : differential evolution algorithm The compute process of the differential evolution algorithm is: 1) Initialization 2) Mutation operation 3) Cross operation X R ( X X ) X U 0 S max min min V X F( X X ) i 1 i i i c c c c i1 i1 c, Sc R c V i c, V R H or ck other i 1 1 1, ( i ) ( i i U c J U c J Xc) X c 4) Select operation X i c, other every new results should be estimated in the calculation process. The estimate principle is that the current value and the best interaction results which is computed before should be compared. The termination conditions in this research is: J J now before Compared with the traditional random algorithm, differential evolution has better adaptive ability. It can speed up the convergence speed and improve compute precision to solve the optimization problem.

14 Opitimal model solution Local optimization : sequential quadratic programming algorithm(sqp) Interaction format of SQP algorithm can be described as follows: 1) Set initial value of parameter 0,select Positive definite matrices H, T T and k 0 J 2) Solve the quadratic programming problem(qp) If dk 0, interaction stop, where J is the performance index of the optimization problem, g is constraints, d is the direction of interaction research; 3) Set,where s is decided by linear research; X X s d k1 k k k k 4) Correct, make sure is positive definite ; H H k k 1 5) Set k k1,return to step 2 X 0 min d ( X ) d H d / 2, k T s. t. g ( X ) d g ( X ) 0, g j k j k T ( X ) d g ( X ) 0 j k j k k

15 Guidance Process Set parameters of the mission Model construction Opitimal model treatment Opitimal model solution Numerical simulation and analysis Dynamic equation Performance index Task constraints Maximum principle Collocation method Global optimization : differential evolution algorithm + Local optimization : SQP method Obtain optimal thrust direction

16 mt sp minpc S D Initial conditions Numerical simulation and analysis Mission parameters Initial mass of spacecraft Mass of space debris Value m S /kg 1000 m D /kg 1 Work efficiency of laser propulsion 0.7 Impact time for deflection T /s Specific impulse for deflection I sp /s 300 Number of discrete arcs N 50 Average power of the laser P /W 10 Coupling coefficient C/(N/W) 10-4 Propulsion time for each discrete arcs /s 90

17 Pitch angle/rad Yaw angle/rad Numerical simulation and analysis Simulation result of the thrust direction angle 0 Pitch angle law 3 Yaw angle law time/s x time/s x 10 4 Time histories of pitch angle Time histories of yaw angle

18 Thrust direction angle/rad Numerical simulation and analysis Simulation result of the thrust direction angle Thrust direction angle law 3 Pitch 2.5 Yaw time/s x 10 4 It can be seen that time histories of thrust direction angle has kept continuous except one or two times saltus in the middle. It can improve the task reliability and reduce the operation difficulty of laser solution. Time histories of thrust direction angle

19 Relative Range /km Numerical simulation and analysis Simulation result of the relative range dangerous area debris orbit with thrust dangerous range Relative Range( Debris-Spacecraft) debris orbit without thrust time/s x 10 4 Time histories of relative range safety area It can be seen that when the orbital space debris reaches the nearest point to the dangerous area of collision, the laser solution works on the debris. The maximum deflection distance of debris is almost 70km through a long working time for laser solution.

20 Conclusion Conclusion

21 Thank you for your attention! Best regards Ting Jin