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# Multibeam Altimeter Navigation Update Using Faceted Shape Model

- Monday, 24 November 2008

### The model is applicable to a body having almost any complex shape.

A method of incorporating information, acquired by a multibeam laser or radar altimeter system, pertaining to the distance and direction between the system and a nearby target body, into an estimate of the state of a vehicle upon which the system is mounted, involves the use of a faceted model to represent the shape of the target body. In the original intended application, the vehicle would be a spacecraft and the target body would be an asteroid, comet, or similar body that the spacecraft was required to approach. The method could also be used in navigating aircraft at low altitudes over terrain that is rough and/or occupied by objects of significant structure.

Fundamentally, what one seeks to measure is the distance from the vehicle to the target body. The present method is the product of a generalization of a prior method of altimetry, in which the target body has a simple shape represented by a spherical or ellipsoidal model. In principle, the estimate of distance or altitude obtained by use of a multibeam altimeter can be more robust than that obtained by use of a single-beam altimeter, but if the surface of the target body has a complex and/or irregular shape, then it becomes more difficult to define the distance and compute the distance from readings of a multibeam altimeter.

The faceted shape model of the present method facilitates the definition and computation of distance to a target object having almost any shape, no matter how irregular and complex. The use of faceted shape models to represent complex three-dimensional objects is common in the computer-graphics literature and in the movie and video-game industries. In this method, the distance to be measured is defined as the length of the vector (ρ) from the center of mass of the multifaceted shape model to the center of mass of the vehicle, as depicted in the upper part of the figure.

The state-update information derived
from the most recent set of multibeam-altimeter
measurements is listed systematically
in a range-measurement table
(RMT), depicted in the lower part of the
figure, in which the planar facets of the
shape model are represented in Hesse’s
normal form. Each row of the table contains
the data from one of the altimeter
beams. The first column contains the row index (*i*), which is the cardinal number
of the affected beam. The second
column contains a number, between 0
and 1, representing the degree of confidence
in the measurements. At the present
state of development of the method,
the confidence is taken to be either 0
(signifying complete rejection) or 1
(representing complete acceptance) of
the data in the row. The third column
contains the scalar range measurement
|**r**| of the *i*th beam; the
fourth column contains the standard
deviation (σ) of the range
measurement.

The fifth column contains the
Cartesian components [*N*_{x}, *N*_{y},
*N*_{z}] of the transpose of the unit
vector (N^{T}) normal to the model
facet containing the intersection
of the *i*th laser beam with the surface
of the target object.
Typically, this intersection point
is not known exactly and must be
estimated, on the basis of the current
state estimate, by a previously
developed method that lies
beyond the scope of this article.
The sixth column contains the
facet constant, κ (the perpendicular
distance from the center of
mass of the target body to the
affected facet). The seventh column
contains the Cartesian components
[*d*_{x}, *d*_{y}, *d*_{z}] of the unit
vector along the *i*th laser beam.
The seventh column contains the
Cartesian components [*c*_{x}, *c*_{y}, *c*_{z}]
of the position vector from the
center of mass of the vehicle to
the origin of the *i*th laser beam.

The entries in the RMT are mapped into a measurement equation for use by a Kalman filter that incorporates altimetry information into the final estimate of the state of a spacecraft or other vehicle maneuvering in the vicinity of a target body. The relative position vector, ρ, is part of the state vector that is updated by use of the Kalman filter.

*This work was done by David S. Bayard,
Paul Brugarolas, and Steve Broschart
of Caltech for NASA’s Jet Propulsion
Laboratory. For more information, contact
This email address is being protected from spambots. You need JavaScript enabled to view it.
. NPO-44428*

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