Accelerating the pace of engineering and science

# Documentation

### Fundamental Coordinate System Concepts

Coordinate systems allow you to keep track of an aircraft or spacecraft's position and orientation in space. The Aerospace Blockset™ coordinate systems are based on these underlying concepts from geodesy, astronomy, and physics.

#### Definitions

The blockset uses right-handed (RH) Cartesian coordinate systems. The right-hand rule establishes the x-y-z sequence of coordinate axes.

An inertial frame is a nonaccelerating motion reference frame. In an inertial frame, Newton's second law holds: force = mass × acceleration. Loosely speaking, acceleration is defined with respect to the distant cosmos, and an inertial frame is often said to be nonaccelerated with respect to the "fixed stars." Because the Earth and stars move so slowly with respect to one another, this assumption is a very accurate approximation.

Strictly defined, an inertial frame is a member of the set of all frames not accelerating relative to one another. A noninertial frame is any frame accelerating relative to an inertial frame. Its acceleration, in general, includes both translational and rotational components, resulting in pseudoforces (pseudogravity, as well as Coriolis and centrifugal forces).

The blockset models the Earth's shape (the geoid) as an oblate spheroid, a special type of ellipsoid with two longer axes equal (defining the equatorial plane) and a third, slightly shorter (geopolar) axis of symmetry. The equator is the intersection of the equatorial plane and the Earth's surface. The geographic poles are the intersection of the Earth's surface and the geopolar axis. In general, the Earth's geopolar and rotation axes are not identical.

Latitudes parallel the equator. Longitudes parallel the geopolar axis. The zero longitude or prime meridian passes through Greenwich, England.

#### Approximations

The blockset makes three standard approximations in defining coordinate systems relative to the Earth.

• The Earth's surface or geoid is an oblate spheroid, defined by its longer equatorial and shorter geopolar axes. In reality, the Earth is slightly deformed with respect to the standard geoid.

• The Earth's rotation axis and equatorial plane are perpendicular, so that the rotation and geopolar axes are identical. In reality, these axes are slightly misaligned, and the equatorial plane wobbles as the Earth rotates. This effect is negligible in most applications.

• The only noninertial effect in Earth-fixed coordinates is due to the Earth's rotation about its axis. This is a rotating, geocentric system. The blockset ignores the Earth's acceleration around the Sun, the Sun's acceleration in the Galaxy, and the Galaxy's acceleration through the cosmos. In most applications, only the Earth's rotation matters.

This approximation must be changed for spacecraft sent into deep space, i.e., outside the Earth-Moon system, and a heliocentric system is preferred.

#### Motion with Respect to Other Planets

The blockset uses the standard WGS-84 geoid to model the Earth. You can change the equatorial axis length, the flattening, and the rotation rate.

You can represent the motion of spacecraft with respect to any celestial body that is well approximated by an oblate spheroid by changing the spheroid size, flattening, and rotation rate. If the celestial body is rotating westward (retrogradely), make the rotation rate negative.

### Coordinate Systems for Modeling

Modeling aircraft and spacecraft is simplest if you use a coordinate system fixed in the body itself. In the case of aircraft, the forward direction is modified by the presence of wind, and the craft's motion through the air is not the same as its motion relative to the ground.

See Equations of Motion for further details on how the Aerospace Blockset product implements body and wind coordinates.

#### Body Coordinates

The noninertial body coordinate system is fixed in both origin and orientation to the moving craft. The craft is assumed to be rigid.

The orientation of the body coordinate axes is fixed in the shape of body.

• The x-axis points through the nose of the craft.

• The y-axis points to the right of the x-axis (facing in the pilot's direction of view), perpendicular to the x-axis.

• The z-axis points down through the bottom the craft, perpendicular to the xy plane and satisfying the RH rule.

Translational Degrees of Freedom.  Translations are defined by moving along these axes by distances x, y, and z from the origin.

Rotational Degrees of Freedom.  Rotations are defined by the Euler angles P, Q, R or Φ, Θ, Ψ. They are:

 P or Φ Roll about the x-axis Q or Θ Pitch about the y-axis R or Ψ Yaw about the z-axis

Unless otherwise specified, by default the software uses ZYX rotation order for Euler angles.

#### Wind Coordinates

The noninertial wind coordinate system has its origin fixed in the rigid aircraft. The coordinate system orientation is defined relative to the craft's velocity V.

The orientation of the wind coordinate axes is fixed by the velocity V.

• The x-axis points in the direction of V.

• The y-axis points to the right of the x-axis (facing in the direction of V), perpendicular to the x-axis.

• The z-axis points perpendicular to the xy plane in whatever way needed to satisfy the RH rule with respect to the x- and yaxes.

Translational Degrees of Freedom.  Translations are defined by moving along these axes by distances x, y, and z from the origin.

Rotational Degrees of Freedom.  Rotations are defined by the Euler angles Φ, γ, χ. They are:

Unless otherwise specified, by default the software uses ZYX rotation order for Euler angles.

Modeling aerospace trajectories requires positioning and orienting the aircraft or spacecraft with respect to the rotating Earth. Navigation coordinates are defined with respect to the center and surface of the Earth.

#### Geocentric and Geodetic Latitudes

The geocentric latitude λ on the Earth's surface is defined by the angle subtended by the radius vector from the Earth's center to the surface point with the equatorial plane.

The geodetic latitude µ on the Earth's surface is defined by the angle subtended by the surface normal vector n and the equatorial plane.

#### NED Coordinates

The north-east-down (NED) system is a noninertial system with its origin fixed at the aircraft or spacecraft's center of gravity. Its axes are oriented along the geodetic directions defined by the Earth's surface.

• The x-axis points north parallel to the geoid surface, in the polar direction.

• The y-axis points east parallel to the geoid surface, along a latitude curve.

• The z-axis points downward, toward the Earth's surface, antiparallel to the surface's outward normal n.

Flying at a constant altitude means flying at a constant z above the Earth's surface.

#### ECI Coordinates

The Earth-centered inertial (ECI) system is a mixed inertial system. It is oriented with respect to the Sun. Its origin is fixed at the center of the Earth. (See figure following.)

• The z-axis points northward along the Earth's rotation axis.

• The x-axis points outward in the Earth's equatorial plane exactly at the Sun. (This rule ignores the Sun's oblique angle to the equator, which varies with season. The actual Sun always remains in the xz plane.)

• The y-axis points into the eastward quadrant, perpendicular to the xz plane so as to satisfy the RH rule.

Earth-Centered Coordinates

#### ECEF Coordinates

The Earth-center, Earth-fixed (ECEF) system is a noninertial system that rotates with the Earth. Its origin is fixed at the center of the Earth. (See figure preceding.)

• The z′-axis points northward along the Earth's rotation axis.

• The x′-axis points outward along the intersection of the Earth's equatorial plane and prime meridian.

• The y′-axis points into the eastward quadrant, perpendicular to the x-z plane so as to satisfy the RH rule.

### Coordinate Systems for Display

Several display tools are available for use with the Aerospace Blockset product. Each has a specific coordinate system for rendering motion.

#### MATLAB Graphics Coordinates

MATLAB Graphics uses this default coordinate axis orientation:

• The x-axis points out of the screen.

• The y-axis points to the right.

• The z-axis points up.

#### FlightGear Coordinates

FlightGear is an open-source, third-party flight simulator with an interface supported by the blockset.

The FlightGear coordinates form a special body-fixed system, rotated from the standard body coordinate system about the y-axis by -180 degrees:

• The x-axis is positive toward the back of the vehicle.

• The y-axis is positive toward the right of the vehicle.

• The z-axis is positive upward, e.g., wheels typically have the lowest z values.

#### AC3D Coordinates

AC3D is a low-cost, widely used, geometry editor available from www.ac3d.org. Its body-fixed coordinates are formed by inverting the three standard body coordinate axes:

• The x-axis is positive toward the back of the vehicle.

• The y-axis is positive upward, e.g., wheels typically have the lowest y values.

• The z-axis is positive to the left of the vehicle.

### References

Recommended Practice for Atmospheric and Space Flight Vehicle Coordinate Systems, R-004-1992, ANSI/AIAA, February 1992.

Mapping Toolbox documentation, The MathWorks, Inc., Natick, Massachusetts. Mapping Toolbox.

Rogers, R. M., Applied Mathematics in Integrated Navigation Systems, AIAA, Reston, Virginia, 2000.

Sobel, D., Longitude, Walker & Company, New York, 1995.

Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, 2nd ed., Aircraft Control and Simulation, Wiley-Interscience, New York, 2003.

Thomson, W. T., Introduction to Space Dynamics, John Wiley & Sons, New York, 1961/Dover Publications, Mineola, New York, 1986.

World Geodetic System 1984 (WGS 84), http://earth-info.nga.mil/GandG/wgs84/.