matlab-convert-aerospace-coordinates

Aerospace Fundamentals

Core Aerospace Toolbox functions for unit conversions, time conversions, coordinate transformations, and rotation representations.

When to Use

• Converting between aerospace unit systems (SI, imperial, nautical)
• Computing Julian dates, modified Julian dates, decimal years, or TDB time
• Transforming positions between ECEF, ECI, LLA, or flat Earth frames
• Converting between geodetic and geocentric latitude
• Building DCMs for frame transformations (ECI↔ECEF, ECEF↔NED, body↔wind, body↔stability)
• Converting between Euler angles, DCMs, quaternions, and Rodrigues vectors
• Using the quaternion object for rotation math, interpolation, or composition
• Identifying which coordinate frame data is expressed in

When NOT to Use

• Atmosphere models (atmosisa, atmoscoesa) — use matlab-compute-aerospace-environment
• Airspeed corrections (correctairspeed) — not covered here
• Orbit propagation or satellite maneuvers — use Aerospace Toolbox, Aerospace Blockset or Satellite Communications Toolbox
• Navigation-specific transforms (lla2enu, lla2ned) — use Navigation Toolbox
• Simulink blocks — use /model-based-design-core:building-simulink-models with Aerospace Blockset

Workflow

1. Identify the coordinate frame — Determine what frame your data is in and what frame you need. See references/coordinate-systems.md for frame definitions and decision guide.
2. Convert units first — Ensure inputs match the function's expected units before calling transforms.
3. Apply the transformation — Use the appropriate function with correct argument ordering.
4. Verify — Round-trip the result back to the original frame; error should be < 1e-10.

Key Functions

Unit Conversions

Function	Converts	Units
convlength	Length	'ft', 'm', 'km', 'in', 'mi', 'naut mi'
convvel	Velocity	'ft/s', 'm/s', 'km/s', 'in/s', 'km/h', 'mph', 'kts', 'ft/min'
convang	Angle	'deg', 'rad', 'rev'
convacc	Acceleration	'ft/s^2', 'm/s^2', 'km/s^2', 'in/s^2', 'km/h-s', 'mph/s', 'G''s'
convangacc	Angular acceleration	'deg/s^2', 'rad/s^2', 'rpm/s'
convangvel	Angular velocity	'deg/s', 'rad/s', 'rpm'
convforce	Force	'N', 'lbf'
convmass	Mass	'kg', 'lbm', 'slug'
convpres	Pressure	'Pa', 'psi', 'psf', 'atm'
convtemp	Temperature	'K', 'R', 'F', 'C'
convdensity	Density	'kg/m^3', 'slug/ft^3', 'lbm/ft^3', 'lbm/in^3'

All conversion functions use the same signature: output = convXXX(value, fromUnit, toUnit)

Time Conversions

Function	Purpose	Since
juliandate	Calendar → Julian Date	R2006b
mjuliandate	Calendar → Modified Julian Date (JD − 2400000.5)	R2006b
decyear	Calendar → decimal year	R2006b
leapyear	Test if year is leap year	R2006b
tdbjuliandate	Terrestrial Time → TDB Julian Date	R2015a

Coordinate Transformations

Function	From	To	Since
lla2ecef	LLA (geodetic)	ECEF	R2006b
ecef2lla	ECEF	LLA (geodetic)	R2006b
lla2eci	LLA	ECI	R2014a
eci2lla	ECI	LLA	R2014a
ecef2eci	ECEF (pos/vel/acc)	ECI	R2019a
eci2ecef	ECI (pos/vel/acc)	ECEF	R2019a
eci2aer	ECI	AER (azimuth, elevation, range)	R2015a
lla2flat	LLA	Flat Earth	R2011a
flat2lla	Flat Earth	LLA	R2011a
geod2geoc	Geodetic latitude	Geocentric latitude	R2006b
geoc2geod	Geocentric latitude	Geodetic latitude	R2006b
ijk2keplerian	ECI position/velocity	Keplerian elements	R2019a
keplerian2ijk	Keplerian elements	ECI position/velocity	R2019a
siderealTime	UTC Julian date	GMST (deg) / GAST (s)	R2021a
dcmeci2ecef	—	ECI-to-ECEF DCM	R2013b
dcmecef2ned	—	ECEF-to-NED DCM	R2006b
dcm2latlon	ECEF-to-NED DCM	Lat/Lon	R2006b
dcmbody2wind	Alpha, Beta	Body-to-Wind DCM	R2006b
dcm2alphabeta	Body-to-Wind DCM	Alpha, Beta	R2006b
dcmbody2stability	Alpha	Body-to-Stability DCM	R2022a

Rotation Representations

Function	From	To	Since
angle2dcm	Euler angles	DCM	R2006b
dcm2angle	DCM	Euler angles	R2006b
angle2quat	Euler angles	Quaternion (1×4)	R2006b
quat2angle	Quaternion (1×4)	Euler angles	R2007b
dcm2quat	DCM	Quaternion (1×4)	R2006b
quat2dcm	Quaternion (1×4)	DCM	R2006b
angle2rod	Euler angles	Rodrigues vector	R2017a
rod2angle	Rodrigues vector	Euler angles	R2017a
dcm2rod	DCM	Rodrigues vector	R2017a
rod2dcm	Rodrigues vector	DCM	R2017a
quat2rod	Quaternion (1×4)	Rodrigues vector	R2017a
rod2quat	Rodrigues vector	Quaternion (1×4)	R2017a

Quaternion Object

Method	Purpose
quaternion(E,'eulerd',RS,PF)	Create from Euler angles (degrees)
quaternion(E,'euler',RS,PF)	Create from Euler angles (radians)
quaternion(RM,'rotmat',PF)	Create from rotation matrix
quaternion(RV,'rotvec')	Create from rotation vector (radians)
compact(q)	Extract [w x y z] array
eulerd(q,RS,PF)	Convert to Euler angles (degrees)
euler(q,RS,PF)	Convert to Euler angles (radians)
rotmat(q,PF)	Convert to rotation matrix
rotvec(q) / rotvecd(q)	Convert to rotation vector (rad/deg)
rotatepoint(q,pts)	Rotate points (active rotation)
rotateframe(q,pts)	Rotate frame (passive rotation)
normalize(q)	Normalize to unit quaternion
slerp(q1,q2,t)	Spherical linear interpolation
meanrot(q)	Mean rotation of array
dist(q1,q2)	Angular distance (radians)
angvel(q,dt,PF)	Angular velocity from quaternion array
randrot(n)	Uniform random rotations

Quaternion Math (Array-Based)

Function	Purpose	Since
quatmultiply(q,r)	Quaternion product (compose rotations)	R2006b
quatconj(q)	Conjugate (negate vector part)	R2006b
quatinv(q)	Inverse (conjugate / norm²)	R2006b
quatnormalize(q)	Normalize to unit quaternion	R2006b
quatnorm(q)	Squared norm (q·q)	R2006b
quatmod(q)	Modulus (sqrt of norm)	R2006b
quatrotate(q,v)	Rotate vector by quaternion	R2006b
quatdivide(q,r)	Divide quaternion by quaternion	R2006b
quatinterp(p,q,f,method)	Interpolate ('slerp', 'lerp', 'nlerp')	R2016a
quatexp(q)	Exponential of quaternion	R2016a
quatlog(q)	Natural logarithm of quaternion	R2016a
quatpower(q,pow)	Quaternion raised to a power	R2016a

Patterns

Unit Conversion

% Always: convXXX(value, 'from', 'to')
alt_m = convlength(35000, 'ft', 'm');        % 10668.0 m
speed_ms = convvel(250, 'kts', 'm/s');       % 128.61 m/s
angle_rad = convang(45, 'deg', 'rad');       % 0.7854 rad
accel_g = convacc(9.81, 'm/s^2', 'G''s');   % 1.0 G
omega_rpm = convangvel(360, 'deg/s', 'rpm'); % 60 rpm

Time Conversion

% Julian Date from components (year, month, day, hour, min, sec)
jd = juliandate(2024, 6, 15, 12, 0, 0);     % 2460477.0

% Modified Julian Date
mjd = mjuliandate(2024, 6, 15, 12, 0, 0);   % 60476.5

% From datetime objects
dt = datetime(2024, 6, 15, 12, 0, 0);
jd = juliandate(dt);

% Decimal year
dy = decyear(2024, 6, 15);                   % 2024.4536

% TDB Julian Date from Terrestrial Time [yr mo day hr min sec]
tt = [2024 6 15 12 0 0];
jdTDB = tdbjuliandate(tt);

LLA ↔ ECEF

% LLA is [latitude_deg, longitude_deg, altitude_m]
lla = [40, -74, 0];
ecef = lla2ecef(lla);           % [1348613.0, -4703172.4, 4077985.6] m

% Round-trip verification
lla_check = ecef2lla(ecef);     % [40.0, -74.0, 0.0]

LLA ↔ ECI (time-dependent)

% ECI transforms require UTC time
lla = [40, -74, 1000];
utc = [2024 6 15 12 0 0];
posECI = lla2eci(lla, utc);

% Back to LLA
lla_check = eci2lla(posECI, utc);

Keplerian Elements ↔ ECI (IJK)

% Convert ECI position/velocity to Keplerian orbital elements
rijk = [-2981784; 5207055; 3161595];   % ECI position (m), 3x1
vijk = [-3384; -4887; 4843];           % ECI velocity (m/s), 3x1
[a, ecc, incl, RAAN, argp, nu] = ijk2keplerian(rijk, vijk);
% a=m, angles in degrees

% Convert Keplerian elements back to ECI position/velocity
[rijk_back, vijk_back] = keplerian2ijk(a, ecc, incl, RAAN, argp, nu);

% For a different central body (e.g., Mars)
[a_mars, ecc_mars, incl_mars, RAAN_mars, argp_mars, nu_mars] = ...
    ijk2keplerian(rijk, vijk, CentralBody="Mars");

Sidereal Time

% Compute Greenwich Mean Sidereal Time (GMST) and Apparent (GAST)
utcJD = juliandate(2019, 1, 4, 12, 0, 0);  % UTC as Julian date
dUT1 = 0;    % UT1-UTC offset (seconds)
dAT  = 37;   % TAI-UTC offset (leap seconds)
[thGMST, thGAST] = siderealTime(utcJD, dUT1, dAT);
% thGMST in degrees, thGAST in seconds

Flat Earth Approximation

% Good for short-range simulations (< ~100 km from reference)
llo = [40, -74];    % reference lat/lon (deg)
psio = 0;           % angular direction of flat Earth x-axis (rad, 0=North)
href = 0;           % reference height (m)

lla_point = [40.01, -73.99, 100];
flatPos = lla2flat(lla_point, llo, psio, href);  % [x, y, z] in meters
lla_back = flat2lla(flatPos, llo, psio, href);

Euler Angles ↔ DCM

% CRITICAL: angle2dcm expects RADIANS, not degrees
yaw = deg2rad(30); pitch = deg2rad(10); roll = deg2rad(5);
dcm = angle2dcm(yaw, pitch, roll, 'ZYX');

% Extract angles back (returns radians)
[y, p, r] = dcm2angle(dcm, 'ZYX');
fprintf('Yaw=%.1f, Pitch=%.1f, Roll=%.1f deg\n', rad2deg(y), rad2deg(p), rad2deg(r));

DCM ↔ Quaternion (Aerospace Toolbox format)

% Aerospace Toolbox quaternion format: [q0 q1 q2 q3] = [scalar, vector]
q = dcm2quat(dcm);       % 1x4, scalar-first
dcm_back = quat2dcm(q);  % 3x3

% Verify orthogonality
err = max(abs(dcm' * dcm - eye(3)), [], 'all');
assert(err < 1e-14, 'DCM is not orthogonal');

Quaternion Object (Modern Approach)

% Create from Euler angles — specify sequence AND frame type
q = quaternion([30 10 5], 'eulerd', 'ZYX', 'frame');

% Rotate a point (active rotation)
pt = [1 0 0];
pt_rotated = rotatepoint(q, pt);   % [0.8529, 0.4924, -0.1736]

% Compose rotations by multiplication
q_total = q2 * q1;  % applies q1 first, then q2

% Always normalize after arithmetic accumulation
q = normalize(q);

Quaternion Interpolation (SLERP)

% Smooth interpolation between two orientations
q1 = quaternion([0 0 0], 'eulerd', 'ZYX', 'frame');
q2 = quaternion([90 0 0], 'eulerd', 'ZYX', 'frame');
t = linspace(0, 1, 5)';
q_interp = slerp(q1, q2, t);

% Verify: yaw progresses linearly for pure yaw rotation
e = eulerd(q_interp, 'ZYX', 'frame');
% e(:,1) = [0, 22.5, 45, 67.5, 90]

Quaternion Math (Array-Based Functions)

% Compose two rotations: q2 applied after q1
q1 = angle2quat(deg2rad(30), 0, 0, 'ZYX');
q2 = angle2quat(0, deg2rad(10), 0, 'ZYX');
q_total = quatmultiply(q2, q1);  % q2 * q1 (apply q1 first)

% Inverse rotation
q_inv = quatinv(q1);
q_identity = quatmultiply(q1, q_inv);  % [1 0 0 0]

% Rotate a vector
v = [1 0 0];
v_rot = quatrotate(q1, v);  % rotate v by q1

% Normalize after accumulation
q_accumulated = quatnormalize(q_total);

% Interpolation (slerp, lerp, nlerp)
q_mid = quatinterp(q1, q2, 0.5, 'slerp');

% Relative rotation: what rotation takes q1 to q2?
q_rel = quatdivide(q2, q1);  % q_rel such that q2 = q_rel * q1

% Exponential/logarithm (useful for angular velocity integration)
q_log = quatlog(q1);         % maps to tangent space
q_back = quatexp(q_log);     % back to quaternion

% Fractional rotation (half the rotation of q1)
q_half = quatpower(q1, 0.5);

Rodrigues Vector

% Compact 3-element representation (singular at 180 deg)
rod = dcm2rod(dcm);           % 1x3 vector
dcm_back = rod2dcm(rod);

% Convert between all representations
rod = quat2rod(q_array);      % q is M×4 (scalar-first)
q_back = rod2quat(rod);

Body-to-Wind and Stability Frames

% Body-to-wind DCM from angle of attack and sideslip
alpha = deg2rad(5);
beta = deg2rad(2);
dcm_bw = dcmbody2wind(alpha, beta);

% Extract alpha/beta from a DCM
[alpha_out, beta_out] = dcm2alphabeta(dcm_bw);

% Body-to-stability (alpha only, no sideslip)
dcm_bs = dcmbody2stability(alpha);

ECEF-to-NED Frame DCM

% Get the DCM to rotate vectors from ECEF to local NED
lat = deg2rad(40);
lon = deg2rad(-74);
dcm_ecef2ned = dcmecef2ned(lat, lon);

% Transform an ECEF velocity to NED
v_ecef = [10; 20; 30];
v_ned = dcm_ecef2ned * v_ecef;

ECI to AER (Azimuth, Elevation, Slant Range)

% Compute look angles from a ground station to a satellite in ECI
posECI = [-2981784, 5207055, 3161595];  % satellite ECI position (m)
utc = [2019 1 4 12 0 0];               % observation time
lla0 = [28.5, -80.5, 0];               % ground station [lat, lon, alt] (deg, deg, m)

aer = eci2aer(posECI, utc, lla0);
% aer = [azimuth_deg, elevation_deg, slant_range_m]

Conventions

• angle2dcm and angle2quat expect radians — always convert with deg2rad() or convang first
• Rotation sequence default is 'ZYX' (yaw-pitch-roll) — always specify explicitly for clarity
• Aerospace Toolbox quaternion format is [scalar, i, j, k] — the 1×4 array functions use scalar-first
• quaternion object also uses scalar-first (compact returns [w x y z])
• rotatepoint vs rotateframe — rotatepoint rotates the point (active); rotateframe rotates the frame (passive). They are inverses.
• LLA ordering is [lat, lon, alt] in degrees and meters
• geod2geoc and geoc2geod expect radians for latitude, meters for height/radius
• ECI functions require UTC time — results change with Earth's rotation
• Normalize quaternions after arithmetic operations to prevent drift
• Rodrigues vector is singular at 180° — use quaternions for arbitrary rotations

Common Mistakes

Mistake	Why It's Wrong	Correct Approach
Pass degrees to angle2dcm	Function expects radians	Use deg2rad() or convang(val,'deg','rad')
Omit rotation sequence	Default ZYX may not match your convention	Always pass 'ZYX', 'XYZ', etc. explicitly
Assume quaternion is [x y z w]	Aerospace Toolbox uses [w x y z] (scalar-first)	Check format; use compact(q) to verify
Use rotatepoint when meaning rotateframe	They are inverses — wrong one flips the rotation	Active rotation = rotatepoint; passive = rotateframe
Skip normalization after quaternion math	Quaternion drift causes non-unit norm, distorted rotations	Call normalize(q) after accumulating rotations
Pass degrees to geod2geoc	Expects radians for latitude	Convert: geod2geoc(deg2rad(lat), alt)
Forget UTC for ECI transforms	ECI position depends on Earth rotation at that instant	Always provide [yr mo day hr min sec]
Use Rodrigues near 180° rotation	Rodrigues vector has a singularity at π	Use quaternion representation instead
Ignore gimbal lock near ±90° pitch	Euler angle extraction loses a degree of freedom	Use quaternion or DCM directly for computations

Gimbal Lock

Euler angle representations lose one degree of freedom when the second rotation reaches ±90° (for ZYX: pitch = ±90°). Symptoms:

• dcm2angle returns unexpected yaw/roll values near pitch = ±90°
• Interpolating Euler angles produces erratic paths near singularity

Solution: Use quaternions for computation and interpolation. Only convert to Euler angles for display or human interpretation.

% Detect gimbal lock risk
[~, pitch, ~] = dcm2angle(dcm, 'ZYX');
if abs(pitch) > deg2rad(85)
    warning('Near gimbal lock — use quaternion representation');
end

References

• See references/coordinate-systems.md for detailed frame definitions (ECEF, ECI, NED, body, wind, stability), axis conventions, and a decision guide for choosing the right frame.
• See references/rotation-conventions.md for detailed rotation math: DCM properties, quaternion algebra, gimbal lock theory, and conversion paths between all representations.
• See references/unit-conversion-guide.md for complete unit string tables and chaining conversions.

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