• Journal of applied mathematics & informatics
• /
• v.30 no.3_4
• /
• pp.455-462
• /
• 2012

The cyclic group $Cn={\langle}(12{\cdots}n){\rangle}$ acts on the set ($^{[n]}_k$) of all $k$-subsets of [$n$]. In this action of $C_n$ the number of orbits of size $d$, for $d|n$, is $$O^{n,k}_d=\frac{1}{d}\sum_{\frac{n}{d}|s|n}{\mu}(\frac{ds}{n})(^{n/s}_{k/s})$$. Stanton and White[7] generalized the above identity to construct the orbit polynomials $$O^{n,k}_d(q)=\frac{1}{[d]_{q^{n/d}}}\sum_{\frac{n}{d}|s|n}{\mu}(\frac{ds}{n})[^{n/s}_{k/s}]{_q}^s$$ and conjectured that $O^{n,k}_d(q)$ have non-negative coefficients. In this paper we give a combinatorial proof for the positivity of coefficients of the orbit polynomial $O^{n,3}_d(q)$.

• Korean Journal of Mathematics
• /
• v.25 no.3
• /
• pp.349-358
• /
• 2017

The cyclic group $C_n={\langle}(12{\cdots}n){\rangle}$ acts on the set $(^{[n]}_k)$ of all k-subsets of [n]. In this action of $C_n$ the number of orbits of size d, for d | n, is $$O^{n,k}_d={\frac{1}{d}}{\sum\limits_{{\frac{n}{d}}{\mid}s{\mid}n}}{\mu}({\frac{ds}{n}})(^{n/s}_{k/s})$$. Stanton and White [6] generalized the above identity to construct the orbit polynomials $$O^{n,k}_d(q)={\frac{1}{[d]_{q^{n/d}}}}{\sum\limits_{{\frac{n}{d}}{\mid}s{\mid}n}}{\mu}({\frac{ds}{n}})[^{n/s}_{k/s}]_{q^s}$$ and conjectured that $O^{n,k}_d(q)$ have non-negative coefficients. In this paper we give a constructive proof for the positivity of coefficients of the orbit polynomial $O^{n,2}_d(q)$.

• International Journal of Aeronautical and Space Sciences
• /
• v.17 no.2
• /
• pp.240-252
• /
• 2016

In this paper, an on-board orbit propagator and compressing trajectory method based on B-spline for a lunar explorer are proposed. An explorer should recognize its own orbit for a successful mission operation. Generally, orbit determination is periodically performed at the ground station, and the computed orbit information is subsequently uploaded to the explorer, which would generate a heavy workload for the ground station and the explorer. A high-performance computer at the ground station is employed to determine the orbit required for the explorer in the parking orbit of Earth. The method not only reduces the workload of the ground station and the explorer, but also increases the orbital prediction accuracy. Then, the data was compressed into coefficients within a given tolerance using B-spline. The compressed data is then transmitted to the explorer efficiently. The data compression is maximized using the proposed methods. The methods are compared with a fifth order polynomial regression method. The results show that the proposed method has the potential for expansion to various deep space probes.

• Korean Journal of Remote Sensing
• /
• v.14 no.2
• /
• pp.137-148
• /
• 1998

An algorithm developed for the precision correction of high resolution satellite images is introduced in this paper. In general, the polynomial warping algorithm which derives polynomial equations between GCPs extracted from an image and a base map requires many GCPs well-distributed over the image. The precision correction algorithm described in this paper is based on a sensor-orbit-Earth geometry, and therefore, it is capable of correcting a raw image using only 2-3 GCPs. This algorithm estimates the errors on the orbit determination and the attitude of the satellite by using a Kalman filter. This algorithm was implemented, tested and integrated into the KITSAT-3 image preprocessing software.

• International Journal of Aeronautical and Space Sciences
• /
• v.8 no.1
• /
• pp.122-128
• /
• 2007

In this paper, a method of station keeping strategy using relative orbital motion and numerical optimization technique is presented for geostationary satellite. Relative position vector with respect to an ideal geostationary orbit is generated using high precision orbit propagation, and compressed in terms of polynomial and trigonometric function. Then, this relative orbit model is combined with optimization scheme to propose a very efficient and flexible method of station keeping planning. Proper selection of objective and constraint functions for optimization can yield a variety of station keeping methods improved over the classical ones. Nonlinear simulation results have been shown to support such concept.

• Journal of the Chungcheong Mathematical Society
• /
• v.16 no.1
• /
• pp.7-14
• /
• 2003

Let $P_c(z)=z^n+c$ be a complex polynomial with an integer $n{\geq}2$. We derive a criterion that the critical orbit of $P_c$ escapes to infinity and investigate its applications to the degree-n bifurcation set. The intersection of the degree-n bifurcation set with the real line as well as with a typical symmetric axis is explicitly written as a function of n. A well-defined escape-time algorithm is also included for the improved construction of the degree-n bifurcation set.

• Aerospace Engineering and Technology
• /
• v.8 no.2
• /
• pp.61-67
• /
• 2009

This paper proposes an on-board orbit data generation algorithm for geostationary satellites. The concept of the proposed algorithm is as follows. From the ground, the position and velocity deviations with respect to the assumed reference orbit are computed for 48 hours of time duration in 30 minutes interval, and the generated data are up-loaded to the satellite to be stored. From the table, three nearest data sets are selected to compute position and velocity deviation for asked epoch time by applying $2^{nd}$ order polynomial interpolation. The computed position and velocity deviation data are added to reference orbit to recover absolute orbit information. Here, the reference orbit is selected to be ideal geostationary orbit with a zero inclination and zero eccentricity. Thanks to very low computational burden, this algorithm allows us to generate orbit data at 1Hz or even higher. In order to support 48 hours autonomy, maximum 3K byte memory is required as orbit data storage. It is estimated that this additional memory requirement is acceptable for geostationary satellite application.

• Journal of the Korean Society for Precision Engineering
• /
• v.21 no.7
• /
• pp.76-82
• /
• 2004

The solutions of neighboring optimal control are typically obtained using the sweep method or transition matrices. Due to the numerical integration, however, the gain matrix can become infinite as time go to final one in the transition matrices, and the Riccati solution can become infinite when the final time free. To overcome these disadvantages, this paper proposes the pseudospectral Legendre method which is to first discreteize the linear boundary value problem using the global orthogonal polynomial, then transforms into an algebraic equations. Because this method is not necessary to take any integration of transition matrix or Riccati equation, it can be usefully used in real-time operation. Finally, its performance is verified by the numerical example for the space vehicle's orbit transfer.

• Journal of the Korea Institute of Information and Communication Engineering
• /
• v.22 no.11
• /
• pp.1448-1454
• /
• 2018

In an optical observing system, three pairs of observations at equal time intervals are required for the orbit determination method to obtain orbital information of an unknown space objects. In this paper, we propose a method of estimating a streak for acquiring three pairs of observations using one streak image information. Satellite trajectory simulation data were generated for nine cases using the STK program in order to verify the characteristics of the orbit of space object and estimation performance. Simulation was performed by applying three approaches that can estimate the next streak position after a few seconds from one streak image information, and the estimation performance was evaluated. Linear vector method and Kalman Filter method based on the linear assumption tend to increase the estimation error in the region where the nonlinearity is large. However estimation method using the polynomial curve fitting based on the least square method showed smaller and uniform error result than the previous methods.

• Proceedings of the KSRS Conference
• /
• 2003.11a
• /
• pp.750-752
• /
• 2003

The objective of this investigation is to compare the geometric precision of Rigorous Sensor Model and Rational Function Model for QuickBird images. In rigorous sensor model, we use the on-board data and ground control points to fit an orbit; then, a least squares filtering technique is applied to collocate the orbit. In rational function model, we first use the rational polynomial coefficients provided by the satellite company. Then the systematic bias of the coefficients is compensated by an affine transformation using ground control points. Experimental results indicate that, the RFM provides a good approximation in the position accuracy.