Orbit mechanics functions and types, including Keplerian, spherical and cartesian elements. More...

Classes
struct	orb_data_s
	Structure for storing orbit data. More...

struct	gps_data_s
	Data structure for the GPS model. More...

struct	gps_orb_out_s
	Structure for storing the orbit data output of the GPS Model. More...

struct	orbit_closest_point_out
	Structure for storing the data output of the closest orbit. More...

Typedefs
typedef struct orb_data_s	orb_data_t
	Structure for storing orbit data.

typedef struct gps_data_s	gps_data_t
	Data structure for the GPS model.

typedef struct gps_orb_out_s	gps_orb_out_t
	Structure for storing the orbit data output of the GPS Model.

typedef struct orbit_closest_point_out	orbit_closest_point_out
	Structure for storing the data output of the closest orbit.

Functions
void	el_to_rv (const orb_data_t &data, ml_matrix &pos, ml_matrix &vel)
	Converts orbital elements to r and v for an elliptic orbit. More...

void	rv_to_nearest (const ml_matrix &pos, const ml_matrix &vel, const ml_matrix &pos_target, double jD, ml_matrix &distance, ml_matrix &time)
	Compute nearest distance and time to a set of points on the earth.

orb_data_t	rv_to_el (const ml_matrix &pos, const ml_matrix &vel, double mu=MU_EARTH)
	Converts position and velocity vectors to Keplerian orbital elements. More...

double	r_to_true (double a, double e, double r)
	Converts distance, sma and e to true anomaly. More...

double	rv_to_sma (double r_mag, double v_mag, double mu=MU_EARTH)
	Computes the semimajor axis given position and velocity. More...

double	mean_to_true (double ecc, double mean)
	Computes the true anomaly from the mean anomaly. More...

double	mean_to_ecc (double ecc, double mean)
	Generate the eccentric anomaly from the mean anomaly and the eccentricity. More...

double	ecc_to_true (double ecc, double e)
	Computes the true anomaly from the eccentric or hyperbolic anomaly. More...

double	ecc_to_mean (double ecc, double e)
	Converts eccentric anomaly to mean anomaly. More...

double	true_to_mean (double ecc, double nu)
	Converts true anomaly to mean anomaly. More...

double	mean_to_true_abs (double ecc, double mean_anom)
	Converts mean anomaly to true without wrapping. More...

double	true_to_ecc (double ecc, double nu)
	Converts true anomaly to eccentric or hyperbolic anomaly. More...

double	rarp_to_ecc (double ap, double per)
	Compute the eccentricity from apogee and perigee radii. More...

double	rarp_to_sma (double ap, double per)
	Computes the semi major axis from apogee and perigee radii. More...

double	slr (double ecc, double sma)
	Compute the semi-latus rectum. More...

ml_matrix	cp_to_i (double inc, double raan, double w)
	Converts the transformation matrix from the perifocal frame to the inertial frame. More...

double	orb_rate (double r, double a, double mu=MU_EARTH)
	Compute the orbital rate for non-circular orbits, or distance units other than [km]. More...

double	orb_rate (double r)
	Compute the orbital rate for circular orbits and distance units in [km]. More...

double	period (double a, double mu=MU_EARTH)
	Compute the orbital period. More...

void	lin_orb (double n, ml_matrix &a, ml_matrix &b, double dt)
	Computes the linearized orbit equations. More...

void	rv_orb_gen (const ml_matrix &el, const ml_matrix &t, ml_matrix &r, ml_matrix &v, double mu=MU_EARTH)
	Generate an orbit by propagating Keplerian elements. More...

void	rv_orb_gen (const orb_data_t &el, const ml_matrix &t, ml_matrix &r, ml_matrix &v, double mu=MU_EARTH)
	Generate an orbit by propagating Keplerian elements. More...

ml_matrix	cart_orb_rhs (const ml_matrix &x)
	Computes the right-hand-side of the unforced orbit equations about a mass point in cartesian coordinates. More...

ml_matrix	cart_orb_rhs (const ml_matrix &x, const ml_matrix &a)
	Right hand side of the cartesion equations of motion. More...

ml_matrix	satellite_visibility (const ml_matrix &from_pos, const ml_matrix &targets, double horizon_angle, double radius)
	Compute satellite visibility. More...

ml_matrix	planetary_accelerations (const ml_matrix &pos, double jd, int center)
	Compute perturbations due to other planets aside from the center. More...

ml_matrix	acc_planet (const ml_matrix &pos, double mu, const ml_matrix &rho)
	Acceleration due to a planet on a spacecraft within the sphere of influence of another body. More...

ml_matrix	orbit_acquisition_dv (ml_matrix &t, const orb_data_t &elI, const orb_data_t &elF, double smaTol=0.5, double incTol=1e-3, double raTol=1e-3)
	Compute delta-v sequence to acquire orbit. More...

double	phase_change_dv (double a0, double deltaA, double kTarget, double mu)
	Compute delta-v for a phase change. More...

ml_matrix	circular_maneuver (double p, double r, double m)
	Compute radial force and tangential impulse for a circular maneuver. More...

ml_matrix	gps_satellite (double jD)
	GPS satellite locations. More...

orbit_closest_point_out	orbit_closest_point (const ml_matrix &r_ef_target, const ml_int_array &t_target, const orb_data_s &el, int n_points=1000, int n_orbits=1)
	Closest points in an orbit. More...

ml_matrix	r_ef_from_el (const orb_data_t &el, int n_points=1000, int n_orbits=1)
	Closest r from elements.

ml_matrix	gps_eci_state_from_data (double longitude, double latitude, double heading, double velocity, double altitude, double jd)
	ECI state from GPS data.

void	print_orbit_data (orb_data_s orbData)
	Display the contents of the passed-in orb_data_s struct.

Detailed Description

Orbit mechanics functions and types, including Keplerian, spherical and cartesian elements.

Function Documentation

◆ el_to_rv()

void el_to_rv	(	const orb_data_t &	data,
		ml_matrix &	pos,
		ml_matrix &	vel
	)

Converts orbital elements to r and v for an elliptic orbit.

Parameters

data	Orbital elements for the satellite.
pos	position "r" matrix to be returned (km).
vel	velocity "v" matrix to be returned (km/s).

References cp_to_i(), orb_data_s::ecc, orb_data_s::inc, orb_data_s::mean_anom, mean_to_true(), MU_EARTH, orb_data_s::perigee, orb_data_s::raan, and orb_data_s::sma.

Referenced by FFEccDeltaElem2Hills(), FFEccHills2DeltaElem(), ImpulsiveManeuver(), orbit_acquisition_dv(), r_ef_from_el(), and galilean_moons::update().

◆ rv_to_el()

orb_data_t rv_to_el	(	const ml_matrix &	pos,
		const ml_matrix &	vel,
		double	mu
	)

Converts position and velocity vectors to Keplerian orbital elements.

Parameters

pos	position "r" matrix for the satellite (km).
vel	velocity "v" matrix for the satellite (km/s).
mu	gravitational constant for the central body of the orbit pair (km^3/sec^2)

Returns: The Keplerian orbital elements for the satellite.

References orb_data_s::ecc, orb_data_s::inc, orb_data_s::mean_anom, orb_data_s::orb_rate, orb_rate(), orb_data_s::perigee, PI, orb_data_s::raan, orb_data_s::sma, orb_data_s::true_anom, true_to_mean(), and TWO_PI.

Referenced by ECI2MeanElements(), FFEccHills2DeltaElem(), and GetHillsMats().

◆ r_to_true()

double r_to_true	(	double	a,
		double	e,
		double	r
	)

Converts distance, sma and e to true anomaly.

Parameters

a	semimajor axis
e	eccentricity
r	distance

Returns: true anomaly (rad)

◆ rv_to_sma()

double rv_to_sma	(	double	r_mag,
		double	v_mag,
		double	mu
	)

Computes the semimajor axis given position and velocity.

Parameters

r_mag	Position magnitude (length).
v_mag	Velocity magnitude (length/s).
mu	Gravitational parameter (length^3/s^2).

Returns: Semimajor axis (length).

◆ mean_to_true()

double mean_to_true	(	double	ecc,
		double	mean_anom
	)

Computes the true anomaly from the mean anomaly.

Parameters

ecc	Eccentricity.
mean_anom	Mean anomaly (rad).

Returns: True anomaly (rad).

Referenced by AutoFormGeometry(), DeltaEl2AlfriendMat(), El2Alfriend(), el_to_rv(), FFEccDiscreteHills(), FFEccEstimateCost(), GenerateTimeVector(), GVEErrorDynamics(), ImpulsiveLPManeuver(), IterativeImpulsiveManeuver(), LPEccentricGVE(), LPEccentricTimeWeight(), mean_to_true_abs(), MeanAnom2TrueLat(), MonitoringAlg(), rv_orb_gen(), and TargetTrueAnom().

◆ mean_to_ecc()

double mean_to_ecc	(	double	ecc,
		double	mean_anom
	)

Generate the eccentric anomaly from the mean anomaly and the eccentricity.

Parameters

ecc	Eccentricity.
mean_anom	Mean anomaly (rad).

Returns: Eccentric anomaly (rad).

◆ ecc_to_true()

double ecc_to_true	(	double	ecc,
		double	e
	)

Computes the true anomaly from the eccentric or hyperbolic anomaly.

Parameters

ecc	Eccentricity.
e	Hyperbolic or Eccentric anomaly (rad).

Returns: True anomaly (rad).

◆ ecc_to_mean()

double ecc_to_mean	(	double	ecc,
		double	e
	)

Converts eccentric anomaly to mean anomaly.

Parameters

ecc	Eccentricity.
hyp	Hyperbolic anomaly (rad).

Returns: Mean anomaly (rad).

Referenced by true_to_mean().

◆ true_to_mean()

double true_to_mean	(	double	ecc,
		double	nu
	)

Converts true anomaly to mean anomaly.

Parameters

ecc	Eccentricity.
nu	True anomaly (rad).

Returns: Mean anomaly (rad).

References ecc_to_mean(), and true_to_ecc().

Referenced by Alfriend2El(), FFEccDiscreteHills(), GenerateTimeVector(), LPEccentric(), LPEccentricGVE(), rv_to_el(), TargetTrueAnom(), and TimeUntilTheta().

◆ mean_to_true_abs()

double mean_to_true_abs	(	double	ecc,
		double	mean_anom
	)

Converts mean anomaly to true without wrapping.

Parameters

ecc	Eccentricity in radians.
mean_anom	Mean anomaly in radians.

Returns: The true anomaly in radians.

References mean_to_true(), PI, and TWO_PI.

◆ true_to_ecc()

double true_to_ecc	(	double	ecc,
		double	nu
	)

Converts true anomaly to eccentric or hyperbolic anomaly.

Parameters

ecc	Eccentricity in radians.
nu	True anomaly in radians.

Returns: The eccentric or hyperbolic anomaly in radians.

Referenced by true_to_mean().

◆ rarp_to_ecc()

double rarp_to_ecc	(	double	ap,
		double	per
	)

Compute the eccentricity from apogee and perigee radii.

Parameters

ap	Apogee radius.
per	Perigee radius.

Returns: Eccentricity.

◆ rarp_to_sma()

double rarp_to_sma	(	double	ap,
		double	per
	)

Computes the semi major axis from apogee and perigee radii.

Parameters

ap	Apogee radius.
per	Perigee radius.

Returns: The semi-major axis.

◆ slr()

double slr	(	double	ecc,
		double	sma
	)

Compute the semi-latus rectum.

Parameters

ecc	Eccentricity.
sma	Semi major axis (or perigee radius for parabola).

Returns: Semi-latus rectum.

Referenced by ImpulsiveManeuver(), and rv_orb_gen().

◆ cp_to_i()

ml_matrix cp_to_i	(	double	inc,
		double	raan,
		double	w
	)

Converts the transformation matrix from the perifocal frame to the inertial frame.

Parameters

inc	Inclination (rad).
raan	Right ascension of the ascending node (rad).
w	Argument of perigee (rad).

Returns: The inertial frame (3,3) transformation matrix.

Referenced by el_to_rv(), gps_satellite(), and rv_orb_gen().

◆ orb_rate() [1/2]

double orb_rate	(	double	r,
		double	sma,
		double	mu
	)

Compute the orbital rate for non-circular orbits, or distance units other than [km].

Parameters

r	Radius
sma	Semi major axis (inf for parabola)
mu	Gravitational parameter

Returns: w Angular velocity

Referenced by FFEccDeltaElem2Goals(), FFEccEstimateCost(), FFEccGenerateTeamGoals(), GenerateTimeVector(), Goals2Hills(), ImpulsiveLPManeuver(), ImpulsiveManeuver(), IterativeImpulsiveManeuver(), LPCircularTimeWeight(), LPEccentricGVE(), LPEccentricTimeWeight(), MonitoringAlg(), rv_to_el(), and TimeUntilTheta().

◆ orb_rate() [2/2]

double orb_rate ( double r )

Compute the orbital rate for circular orbits and distance units in [km].

Parameters

r Radius

Returns: w Angular velocity (rad/s)

References MU_EARTH.

◆ period()

double period	(	double	a,
		double	mu
	)

Compute the orbital period.

The units for a and mu must be consistent, for example [km] and [km3/s2]. FLT_MAX is substituted for infinity.

Parameters

a	Semi-major axis
mu	Gravitation parameter, default is Earth

Returns: p Orbital period (s)

References TWO_PI.

Referenced by GenerateTeamGoals(), orbit_closest_point(), and r_ef_from_el().

◆ lin_orb()

void lin_orb	(	double	n,
		ml_matrix &	a,
		ml_matrix &	b,
		double	dt
	)

Computes the linearized orbit equations.

If value of dT is not equal to -1 it will generate the discrete time equations.

Parameters

n	(1) orbit rate (rad/s).
a	(6,6) State transition matrix.
b	(6,3) Input matrix.
dt	(1) Time step (s).

Referenced by DiscreteHills(), and LPCircular().

◆ rv_orb_gen() [1/2]

void rv_orb_gen	(	const ml_matrix &	el,
		const ml_matrix &	t,
		ml_matrix &	pos,
		ml_matrix &	vel,
		double	mu
	)

Generate an orbit by propagating Keplerian elements.

Parameters

el	(6,1) Elements vector [a,i,W,w,e,M]
t	(1,n) Times from 0 to infinity (sec)
pos	(3,n) Position vectors for times t
vel	(3,n) Velocity vectors for times t
mu	(1,1) Gravitational potential

References orb_data_s::ecc, orb_data_s::inc, orb_data_s::mean_anom, orb_data_s::perigee, orb_data_s::raan, rv_orb_gen(), and orb_data_s::sma.

Referenced by orbit_acquisition_dv(), orbit_closest_point(), rv_orb_gen(), and Target().

◆ rv_orb_gen() [2/2]

void rv_orb_gen	(	const orb_data_t &	el,
		const ml_matrix &	t,
		ml_matrix &	pos,
		ml_matrix &	vel,
		double	mu
	)

Generate an orbit by propagating Keplerian elements.

Overloaded for calling with elements of type "orb_data_s"

Parameters

el	(1,1) Elements data structure
t	(1,n) Times from 0 to infinity (sec)
pos	(3,n) Position vectors for times t
vel	(3,n) Velocity vectors for times t
mu	(1,1) Gravitational potential

References cp_to_i(), orb_data_s::ecc, orb_data_s::inc, orb_data_s::mean_anom, mean_to_true(), orb_data_s::perigee, orb_data_s::raan, slr(), and orb_data_s::sma.

◆ cart_orb_rhs() [1/2]

ml_matrix cart_orb_rhs ( const ml_matrix & x )

Computes the right-hand-side of the unforced orbit equations about a mass point in cartesian coordinates.

Parameters

x	(6,1) State [x;y;z;dx/dt;dy/dt;dz/dt], pos: (km), vel: (km/s)

Returns: xDot (6,1) State derivative

References MU_EARTH.

◆ cart_orb_rhs() [2/2]

ml_matrix cart_orb_rhs	(	const ml_matrix &	x,
		const ml_matrix &	a
	)

Right hand side of the cartesion equations of motion.

Parameters

x	(6,1) Current state vector, pos: (km), vel: (km/s).
a	(3,1) Acceleration vector (km/s/s).

Returns: xDot (6,1) State derivative.

References MU_EARTH.

◆ satellite_visibility()

ml_matrix satellite_visibility	(	const ml_matrix &	from_pos,
		const ml_matrix &	targets,
		double	horizon_angle,
		double	radius
	)

Compute satellite visibility.

Parameters

from_pos	Position of the viewing satellite.
targets	Positions of possibly visible satellites.
horizon_angle	Angle above horizon for visibility
radius	Planet radius.

Returns: A ml_int_array with "1" for each visible satellite and "0" for each that is not visible.

◆ planetary_accelerations()

ml_matrix planetary_accelerations	(	const ml_matrix &	pos,
		double	jd,
		int	center
	)

Compute perturbations due to other planets aside from the center.

Currently only handles the earth, moon and sun.

Parameters

pos	(3,1) Position vector wrt center
jd	(1,1) Julian Date
center	1 for Earth and 2 for Moon

Returns: (3,1) Acceleration

References acc_planet(), moon_vector(), MU_EARTH, MU_MOON, MU_SUN, and sun_vector().

◆ acc_planet()

ml_matrix acc_planet	(	const ml_matrix &	pos,
		double	mu,
		const ml_matrix &	rho
	)

Acceleration due to a planet on a spacecraft within the sphere of influence of another body.

Parameters

pos	(3,1) Vector of the spacecraft from the central body
mu	(1) Planet's gravitational parameter
rho	(3,:) Vector of the planet from the central body

Returns: (3,1) Acceleration

Referenced by planetary_accelerations().

◆ orbit_acquisition_dv()

ml_matrix orbit_acquisition_dv	(	ml_matrix &	t,
		const orb_data_t &	elI,
		const orb_data_t &	elF,
		double	smaTol,
		double	incTol,
		double	raTol
	)

Compute delta-v sequence to acquire orbit.

Inclination and right ascension correction are a single burn. Semi-major axis correction is a two-burn Hohmann transfer set. The conditions for performing a SMA correction are:

The SMA error exceeds the tolerance, AND 2. a) The final SMA is smaller than the initial, OR b) Both the inc. and r.a. error are within tolerance. The times will include the wait time for the necessary Hohmann phasing.

Parameters

t	delta-V time (output)
elI	Initial orbital elements
elF	Final orbital elements
smaTol	Tolerance on semi-major axis
incTol	Tolerance on inclination
raTol	Tolerance on right ascension

Returns: Matrix of delta-V's (3xn)

References au_to_q(), el_to_rv(), orb_data_s::inc, orb_data_s::mean_anom, orb_data_s::perigee, PI, orb_data_s::raan, rv_orb_gen(), orb_data_s::sma, and WrapPhase().

◆ phase_change_dv()

double phase_change_dv	(	double	a0,
		double	deltaA,
		double	kTarget,
		double	mu
	)

Compute delta-v for a phase change.

Parameters

a0	(1,1) Initial semi-major axis
deltaA	(1,1) Phase change
kTarget	(1,1) Number of revolutions
mu	(1,1) Planet's gravitational parameter

Returns: (1,1) Delta-v

References PI.

◆ circular_maneuver()

ml_matrix circular_maneuver	(	double	p,
		double	r,
		double	m
	)

Compute radial force and tangential impulse for a circular maneuver.

Parameters

p	Period (min)
r	Radius of orbit (m)
m	Mass (kg)

Returns: [Radial force (N); Tangential impulse (Ns)]

References PI.

◆ gps_satellite()

ml_matrix gps_satellite ( double jD )

GPS satellite locations.

Models the GPS constellation very approximately using a static set of elements. Each column is the stacked [r;v] of one GPS satellite.

Parameters

jD	Julian date

Returns: Position and velocity of 24 GPS satellites, (6x24)

References cp_to_i().

◆ orbit_closest_point()

orbit_closest_point_out orbit_closest_point	(	const ml_matrix &	r_ef_target,
		const ml_int_array &	t_target,
		const orb_data_t &	el,
		int	n_points,
		int	n_orbits
	)

Closest points in an orbit.

Parameters

r_ef_target	Target positions in the earth fixed frame
el	Orbit of observing spacecraft
n_points	Number of points in the orbit to check
n_orbits	Number of orbits to check

Returns: Data structure with closest points, minimum distance and time of minimun

References orb_data_s::epoch, orbit_closest_point_out::id, jd_to_jcent(), orbit_closest_point_out::julian_date, period(), orbit_closest_point_out::r, rv_orb_gen(), orb_data_s::sma, and true_earth().

Classes

Typedefs

Functions

Detailed Description

Function Documentation

◆ el_to_rv()

◆ rv_to_el()

◆ r_to_true()

◆ rv_to_sma()

◆ mean_to_true()

◆ mean_to_ecc()

◆ ecc_to_true()

◆ ecc_to_mean()

◆ true_to_mean()

◆ mean_to_true_abs()

◆ true_to_ecc()

◆ rarp_to_ecc()

◆ rarp_to_sma()

◆ slr()

◆ cp_to_i()

◆ orb_rate() [1/2]

◆ orb_rate() [2/2]

◆ period()

◆ lin_orb()

◆ rv_orb_gen() [1/2]

◆ rv_orb_gen() [2/2]

◆ cart_orb_rhs() [1/2]

◆ cart_orb_rhs() [2/2]

◆ satellite_visibility()

◆ planetary_accelerations()

◆ acc_planet()

◆ orbit_acquisition_dv()

◆ phase_change_dv()

◆ circular_maneuver()

◆ gps_satellite()

◆ orbit_closest_point()