Method and apparatus for spacecraft attitude control using polynomial interpolation

1. A vehicle guidance system for implementing attitude control of a vehicle, comprising:

a command reduction module for generating a reduced data set corresponding to a time-varying attitude command trajectory for a vehicle, comprising

circuitry configured to receive at least one time-varying attitude command

trajectory defining an attitude of the vehicle, and

circuitry configured to reduce the at least one time-varying attitude

command trajectory into a vector of polynomial coefficients; and

a command reconstruction module for generating data corresponding to the time-varying attitude command trajectory, comprising

circuitry configured to receive a vector of polynomial coefficients, and

circuitry configured to reconstruct at least one time-varying attitude command trajectory by utilizing the vector of polynomial coefficients and an affine transformation to perform a polynomial interpolation operation.

2. The system according to claim 1 , further comprising a first communication interface located remote from the vehicle and a second communication interface located on the vehicle, the first communication interface configured to transmit the vector of polynomial coefficients to the second communication interface.

3. The system according to claim 1 , further comprising:

an attitude control system including an input for receiving a command trajectory; and

a switching device including an input operatively coupled to the command reconstruction module, and an output operatively coupled to the attitude control system input, the switching device configured to selectively provide the reconstructed time-varying attitude command trajectory to the attitude control system input.

4. A method for implementing attitude control of a vehicle, comprising:

receiving at a location remote from the vehicle data corresponding to at least one time-varying attitude command trajectory defining an attitude of the vehicle; and

using circuitry of a command reduction module located remote from the vehicle to reduce the at least one time-varying attitude command trajectory into a vector of polynomial coefficients.

5. The method according to claim 4 further comprising:

receiving at the vehicle the vector of polynomial coefficients; and

using circuitry of a command reconstruction module to reconstruct the at least one time-varying attitude command trajectory by utilizing the vector of polynomial coefficients and an affine transformation to perform a polynomial interpolation operation.

6. The vehicle guidance system according to claim 1 , wherein the circuitry configured to reduce the at least one time-varying attitude command trajectory into the vector polynomial coefficients is configured to perform the reduction based on a user input.

7. The vehicle guidance system according to claim 6 , wherein the user input comprises at least one of

an initial maneuver time or an end maneuver time for the vehicle, time histories of vehicle attitude commands,

a predetermined number N of polynomial coefficients to compute, N being an integer,

a tolerance for an acceptable polynomial approximation error,

a set of mathematical basis functions b(t), or

a set of distinct time points or nodes.

8. The vehicle guidance system according to claim 1 , wherein the circuitry is configured to generate a maneuver to minimize or maximize an objective function.

9. The vehicle guidance system according to claim 1 , wherein the circuitry configured to reduce the at least one time-varying attitude command trajectory is configured to

expand N+1 mathematical basis functions b(t) over a selected time grid, compute the values of a set of polynomial coefficients c for N+1 basis functions, and

evaluate the infinity norm, wherein N is a non-negative integer.

10. The vehicle guidance system according to claim 9 , wherein the circuitry configured to reduce the at least one time-varying attitude command trajectory is configured to

compare the evaluated infinity norm to a predetermined tolerance,

when the evaluated infinity norm is greater than the predetermined tolerance, increment N and repeat the expansion, and evaluation steps, and

when the evaluated infinity norm is less than the predetermined tolerance, conclude that the reduction process is complete.

11. The vehicle guidance system according to claim 9 , wherein the at least one time-varying attitude command trajectory is represented in the form, y N (t)= j=0 N c j b j (t), where y N (t) is an approximation of the at least one time-varying attitude trajectory having arbitrary accuracy, N is an integer representing a number N+1 of basis functions, b j (t) denotes the j th basis function, and c j is a polynomial coefficient for the j th basis function.

12. The vehicle guidance system according to claim 11 , wherein values of the coefficients c j are selected to minimize the infinity norm, ∥y(t)−y N (t)∥=∥y(t)− j=0 N c j b j (t)∥≦ε, where ε is a maximum tolerance on polynomial approximation error, y(t) is the original time-history of the at least one time-varying attitude command trajectory.

13. The vehicle guidance system according to claim 9 , wherein the circuitry is configured to utilize Lagrange interpolating polynomials, b j (t)= i=0,ij N t−t i /t j −t i , as the basis functions.

14. The vehicle guidance system according to claim 1 , wherein the circuitry configured to reconstruct the at least one time-varying attitude command trajectory is configured to perform the reconstruction based on at least one of an initial maneuver time for the vehicle, an end maneuver time for the vehicle, an attitude control system sample time, a clock time of the vehicle control system, a set of mathematical basis functions, or a set of distinct time points or nodes.

15. The vehicle guidance system according to claim 14 , wherein the circuitry configured to reconstruct the at least one time-varying attitude command trajectory is configured to use Lagrange polynomials as the basis functions b j (t).

16. The vehicle guidance system according to claim 1 , wherein the circuitry configured to reconstruct the at least one time-varying attitude command trajectory is configured to map a current sample time to interval [−1, +1] via an affine transform.

17. The vehicle guidance system according to claim 1 , wherein the circuitry configured to reconstruct the at least one time-varying attitude command trajectory is configured to perform polynomial interpolation using the vector of polynomial coefficients and the equation y N (t)= j=0 N c j b j (t), where N is an integer representing a number N+1 of basis functions, b j (t) is the j th basis function, and c j is polynomial coefficient for the j th basis function.

18. The vehicle guidance system according to claim 1 , wherein the circuitry configured to reconstruct the at least one time-varying attitude command trajectory is configured to perform Barycentric polynomial interpolation using the vector of polynomial coefficients and the equation y N (t)=l(t) j=0 N w j /t−t j c j , where t is time, t j is a j th time point, w j is a Barycentric weight for the j th time point, c j is the j th polynomial coefficient, N is an integer representing a number N+1 of basis functions, and l(t) is the quantity l(t)=(t−t 0 )(t−t 1 ) . . . (t−t N ).