1. Introduction
The ability of pinpoint soft landing on a lunar surface, where we want to examine, broadens the range of scientific and exploration missions. In order to perform the pinpoint landing, measure-ment of relative velocity against the surfaces is considered to be essential. The ISAS/JAXA has been developing a landing radar for future lunar missions (e.g., SELENE-2). The C-band pulse radar provides not only altitude but also velocity information. The integrated configuration of the altimeter and the velocity meter is mainly be-cause of restricted weight budget in expected missions. Required accuracy of the velocity measurement is less than 5% or 10 cm/s.
Until now, we have proposed a novel method based on Doppler measurement at multiple points within a received pulse [1]. The method was im-plemented on our developing radar BBM (Bread Board Model), and tested in the helicopter-based field experiments. On the start of the next devel-opment phases (i.e., EM and PFM) soon, it is important to understand how the parameters in the radar design affect the accuracy of the veloc-ity measurement. We have developed a radar si-mulator, with which we can examine the relations between the measurement accuracy and several design parameters: Beam width (range/azimuth), beam tilt angle, pulse width, A/D sampling fre-quency, point selection, and so on. In the follow-ing, after a brief review of the proposed method for velocity measurement, the results of the si-mulation study are shown in detail.
2. Velocity Measurement With the
Multi-Points Method
The landing radar can be classified as a pulse-Doppler radar in a broad sense. It is em-ploying C-band (4.3 GHz) due to the integrated configuration with the altimeter. So the beam is as broad as 15 degrees; the value seems a kind of limit when considering an antenna size on the expected lunar lander. Furthermore, it has to be considered that attitude of the lander might fluc-tuate between }15 degrees against the surface.
Every Doppler radar intrinsically measures the Doppler frequency in the line-of-sight direction. The properties of the landing radar give difficul-ties that the line-of-sight direction is not constant, because the received pulse is extended in the time domain corresponding to the beam width, or the line-of-sight is biased by the attitude fluctuation. Assuming that the surface is flat, a look angle, ?, corresponding to a time sample, t, is easily cal-culated with altitude information, h, such that
(1)
where c is the speed of light. The Doppler fre-quency, fd, at the time has a following relation to horizontal velocity against the surface, v,
(2)
where ? denotes the carrier wavelength. The re-ceived wave at a time instant, however, is coher-ent summation of reflection from a part of the doughnut-shaped surface. So the Doppler fre-quency in (2) is distributed over the range
(3)
where ? is the pulse width. Therefore, a straight-forward estimation of v suffers from the ambigu-ity related to (3).
In order to overcome the problem, a new method of the velocity measurement has been proposed [1]. It is based on an idea that the most probable curve is able to be determined from (t, fd) plots in the plane of the range time and the Doppler frequency. It is referred as the multi-points method with some dozen of time samples. In this method, the Doppler frequencies at the multiple time samples, where the S/N is comparatively good without a notch of the level, are estimated through FFT or DFT. Though each Doppler frequency also has the ambiguity caused by the range resolution, an optimum velocity is derived by the least-squares regression (i.e., curve fitting)
(4)
Equation (4) contains the altitude h. Hence, the velocity would have to be influenced by the alti-tude error, if calculated straightforwardly. To avoid this, the altitude h is swept in a searching manner without using the immediate value in the multi-points method. The objective function is summation of the residual in the curve fitting as
(5)
The velocity corresponding to the altitude that minimizes G(h) is outputted.
3. Simulator Design
The accuracy of the multi-points method de-pends on several design parameters such as beam width, beam tilt angle, pulse width, A/D sampling frequency, and point selection. We have devel-oped a radar simulator to evaluate sensitivity of the measurement accuracy to each parameter. Fig. 1 (a) and (b) are results of helicopter-based ex-periments in March 2006 and 2007, respectively. A difference of random errors in each case is not negligible: 1.90 % in Fig. 1 (a) and 3.80 % in (b). The flight configuration was almost same; the altitude was about 700 ft, and the target velocity was 30 m/s. Only the beam tilt angle was changed from 30?@to 25? between the tests. Though it can be expected that the narrow width of reflected pulses with decrease in the tilt angle gives influence on the random error, verification is required for the development of EM and PFM in the future. The simulation study in this paper is motivated by the verification requirements.
The simulation starts on the plane of the range time and the Doppler frequency. Input values into the simulator are:
- Altitude, h,
- Velocity, vx and vy,
- Beam tilt angle,
- Beam width (range/azimuth),
- Beam direction relative to x-y coordinate, ?,
- Sampling interval of slant range (i.e., A/D sampling frequency),
- Pulse width, ?,
- Maximum number of points for curve fitting.
First, measurable slant range is calculated from the altitude, the beam tilt angle, and the beam width in the range direction. Next, candidate samples are decided considering the sampling interval of the slant range. If the number of the candidates is larger than the maximum number of points for fitting, the remainder is randomly eliminated. Then the Doppler frequency at each sample is given by
(6)
where R is the slant range, and
(7)
(8)
Since tcontr, contributing time to the reflection, is generated as a random number between 0 and ?, and ? is generated as a random number within the azimuth beam width, the computed fd has a cer-tain ambiguity. Finally, the velocity vector, ?vx? and ?vy?, is estimated by the fitting scheme of (t, fd) in four beams.
Fig. 2 is an example of output of the simulator. The configuration is almost same as Fig. 1 (b). It is noted that the random error provided by the simulator, 1.98 % in Fig. 2, is a little smaller than the results of the field experiments, probably be-cause the simulator does not take account of some error factors originating from the helicop-ter-based experiments.
4. Results and Discussion
In this section, several results of the simulation are shown, and the sensitivity of the measurement accuracy to each parameter is discussed.
Beam tilt angle (Fig. 3)
The random error in the longitudinal direction, where the target velocity of 30 m/s is given, is considerably sensitive to the tilt angle. The dif-ference between Fig. 1 (a) and (b) is explainable using the result in Fig. 2. On the other hand, the error in the lateral direction, where the target ve-locity is 0 m/s, seems constant independent of the beam tilt angle. This is because the error is due to crosstalk from the longitudinal direction.
Pulse width (Fig. 4)
As the pulse width becomes narrow, the random error in the longitudinal direction is decreased. The result is reasonable, considering that a main factor of the Doppler ambiguity is the pulse width. The error in the lateral direction does not depend upon the pulse width, too.
Azimuth beam width (Fig. 5)
As might be expected, the error in the lateral direction can be reduced by narrowing the beam width azimuthally. The beam width is the only parameter to decrease the crosstalk between the lateral and longitudinal directions.
Target velocity (Fig. 6)
Of course the random errors of the measured velocity exhibit a proportional relation with the target velocity. So the requirement of 10 cm/s accuracy at the hovering phase prior to landing is not so severe.
Altitude/Mode (Fig. 7)
A dedicated mode for measurements at short range, named eshort modef, is prepared in our landing radar. The differences from the nominal emiddlef mode are the pulse width (50 ? 15 ns) and 16 times finer sampling of the slant range. Fig. 7 says that both the pulse width and the coarse range sampling degrade the accuracy of the middle mode at short range. Also the figure is suggestive concerning the switching altitude from middle to short mode (? 150 m or so).
References
[1] S. Fukuda, T. Sakai, and T. Mizuno, gA novel method of velocity measurement for the lu-nar/planetary landing radar,h Proc. 16th Workshop on JAXA Astrodynamics and Flight Mechanics, pp.101-106, 2006.