Classes
struct	RigidBodyState
struct	RigidBodyDerivatives
	Time derivatives of rigid body state. More...
struct	MassProperties

Functions
template<typename Scalar>
Vec3< Scalar >	position_dot (const Vec3< Scalar > &velocity)
template<typename Scalar>
Scalar	attitude_response_accel (const Scalar &angle, const Scalar &angle_dot, const Scalar &angle_cmd, const Scalar &omega_n, const Scalar &zeta)
template<typename Scalar>
Vec3< Scalar >	thrust_direction_btt (const Scalar &gamma, const Scalar &chi)
template<typename Scalar>
Vec3< Scalar >	lift_direction_btt (const Scalar &gamma, const Scalar &chi, const Scalar &phi)
template<typename Scalar>
Vec3< Scalar >	thrust_direction_stt (const Scalar &theta, const Scalar &psi)
template<typename Scalar>
Vec3< Scalar >	side_force_direction_stt (const Scalar &theta, const Scalar &psi)
template<typename Scalar>
Vec3< Scalar >	normal_force_direction_stt (const Scalar &theta, const Scalar &psi)
template<typename Scalar>
Vec3< Scalar >	velocity_dot_btt (const Scalar &thrust, const Scalar &drag, const Scalar &lift, const Scalar &mass, const Scalar &gravity, const Scalar &gamma, const Scalar &chi, const Scalar &phi)
template<typename Scalar>
Vec3< Scalar >	velocity_dot_stt (const Scalar &thrust, const Scalar &drag, const Scalar &normal_force, const Scalar &side_force, const Scalar &mass, const Scalar &gravity, const Scalar &theta, const Scalar &psi)
template<typename Scalar>
Scalar	gamma_dot (const Scalar &lift, const Scalar &weight, const Scalar &mass, const Scalar &velocity, const Scalar &gamma, const Scalar &phi)
template<typename Scalar>
Scalar	chi_dot_btt (const Scalar &lift, const Scalar &mass, const Scalar &velocity, const Scalar &gamma, const Scalar &phi)
template<typename Scalar>
Scalar	load_factor_from_lift (const Scalar &lift, const Scalar &weight)
template<typename Scalar>
Scalar	bank_for_turn_rate (const Scalar &velocity, const Scalar &chi_dot, const Scalar &gravity)
template<typename Scalar>
Vec2< Scalar >	planar_acceleration (const Scalar &force_x, const Scalar &force_y, const Scalar &mass)
template<typename Scalar>
Vec2< Scalar >	planar_acceleration (const Vec2< Scalar > &force, const Scalar &mass)
template<typename Scalar>
Vec2< Scalar >	planar_acceleration_gravity (const Scalar &force_x, const Scalar &force_y, const Scalar &mass, const Scalar &gravity)
template<typename Scalar>
Vec3< Scalar >	point_mass_acceleration (const Vec3< Scalar > &force, const Scalar &mass)
template<typename Scalar>
Scalar	speed (const Vec3< Scalar > &velocity)
template<typename Scalar>
Vec3< Scalar >	velocity_direction (const Vec3< Scalar > &velocity)
template<typename Scalar>
Vec3< Scalar >	point_mass_acceleration_ecef (const Vec3< Scalar > &position, const Vec3< Scalar > &velocity, const Vec3< Scalar > &force, const Scalar &mass, const Vec3< Scalar > &omega_earth)
template<typename Scalar>
Vec3< Scalar >	specific_force (const Vec3< Scalar > &acceleration)
template<typename Scalar>
Vec3< Scalar >	g_load (const Vec3< Scalar > &acceleration, const Scalar &g0=Scalar(9.80665))
template<typename Scalar>
Scalar	g_load_magnitude (const Vec3< Scalar > &acceleration, const Scalar &g0=Scalar(9.80665))
template<typename Scalar>
Scalar	flight_path_angle (const Vec3< Scalar > &velocity)
template<typename Scalar>
Scalar	heading_angle (const Vec3< Scalar > &velocity)
template<typename Scalar>
Vec3< Scalar >	velocity_from_angles (const Scalar &speed, const Scalar &gamma, const Scalar &chi)
template<typename Scalar>
Scalar	specific_kinetic_energy (const Vec3< Scalar > &velocity)
template<typename Scalar>
Scalar	specific_potential_energy (const Vec3< Scalar > &position, const Scalar &mu)
template<typename Scalar>
Scalar	specific_energy (const Vec3< Scalar > &position, const Vec3< Scalar > &velocity, const Scalar &mu)
template<typename Scalar>
Vec3< Scalar >	rail_direction_ned (const Scalar &azimuth, const Scalar &elevation)
template<typename Scalar>
MassProperties< Scalar >	aggregate_mass_properties (const std::vector< MassProperties< Scalar > > &components)
template<typename Scalar>
MassProperties< Scalar >	transform_mass_properties (const MassProperties< Scalar > &props, const Mat3< Scalar > &rotation, const Vec3< Scalar > &translation)

Function Documentation

◆ aggregate_mass_properties()

template<typename Scalar>

MassProperties< Scalar > vulcan::mass::aggregate_mass_properties ( const std::vector< MassProperties< Scalar > > & components )

Aggregate a collection of mass properties

Computes combined CG and inertia tensor about combined CG. Equivalent to sum() in Python.

Parameters

components Vector of component mass properties

Returns: Combined mass properties

◆ attitude_response_accel()

template<typename Scalar>

Scalar vulcan::dynamics::attitude_response_accel	(	const Scalar &	angle,
		const Scalar &	angle_dot,
		const Scalar &	angle_cmd,
		const Scalar &	omega_n,
		const Scalar &	zeta )

Compute attitude angle acceleration via 2nd-order transfer function

Implements: α̈ = ω²(α_cmd - α) - 2ζωα̇

This models the vehicle's attitude response to commands (e.g., from autopilot). The ω_n and ζ parameters characterize the closed-loop response.

Template Parameters

Scalar Scalar type (double or casadi::MX)

Parameters

angle	Current angle [rad]
angle_dot	Current angle rate [rad/s]
angle_cmd	Commanded angle [rad]
omega_n	Natural frequency of response [rad/s]
zeta	Damping ratio of response

Returns: Angular acceleration [rad/s²]

◆ bank_for_turn_rate()

template<typename Scalar>

Scalar vulcan::dynamics::bank_for_turn_rate	(	const Scalar &	velocity,
		const Scalar &	chi_dot,
		const Scalar &	gravity )

Compute required bank angle for coordinated turn

For a given turn rate and velocity, the required bank angle is: tan(φ) = V·χ̇ / g

Parameters

velocity	Speed [m/s]
chi_dot	Desired heading rate [rad/s]
gravity	Gravitational acceleration [m/s²]

Returns: Required bank angle [rad]

◆ chi_dot_btt()

template<typename Scalar>

Scalar vulcan::dynamics::chi_dot_btt	(	const Scalar &	lift,
		const Scalar &	mass,
		const Scalar &	velocity,
		const Scalar &	gamma,
		const Scalar &	phi )

Compute heading rate from lateral acceleration (bank-to-turn)

χ̇ = (L·sin(φ)) / (m·V·cos(γ))

Parameters

lift	Lift magnitude [N]
mass	Vehicle mass [kg]
velocity	Speed [m/s]
gamma	Flight path angle [rad]
phi	Bank angle [rad]

Returns: Heading rate [rad/s]

◆ flight_path_angle()

template<typename Scalar>

Scalar vulcan::dynamics::flight_path_angle ( const Vec3< Scalar > & velocity )

Compute flight path angle (gamma) from velocity

Flight path angle is the angle between the velocity vector and the local horizontal plane, positive upward.

$$\gamma = \arcsin(-v_z / |v|)$$ (for Z-down convention)

Parameters

velocity Velocity vector [m/s] (Z-down)

Returns: Flight path angle [rad]

◆ g_load()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::g_load	(	const Vec3< Scalar > &	acceleration,
		const Scalar &	g0 = Scalar(9.80665) )

Compute g-load from acceleration

Parameters

acceleration	Acceleration vector [m/s²]
g0	Standard gravity [m/s²] (default 9.80665)

Returns: Load factor [g's]

◆ g_load_magnitude()

template<typename Scalar>

Scalar vulcan::dynamics::g_load_magnitude	(	const Vec3< Scalar > &	acceleration,
		const Scalar &	g0 = Scalar(9.80665) )

Compute total g-load magnitude

Parameters

acceleration	Acceleration vector [m/s²]
g0	Standard gravity [m/s²]

Returns: Total load factor magnitude [g's]

◆ gamma_dot()

template<typename Scalar>

Scalar vulcan::dynamics::gamma_dot	(	const Scalar &	lift,
		const Scalar &	weight,
		const Scalar &	mass,
		const Scalar &	velocity,
		const Scalar &	gamma,
		const Scalar &	phi )

Compute flight path angle rate from vertical acceleration

γ̇ = (L·cos(φ) - W·cos(γ)) / (m·V)

For pseudo-5DOF, given the normal load factor, compute gamma_dot.

Parameters

lift	Lift magnitude [N]
weight	Weight (m*g) [N]
mass	Vehicle mass [kg]
velocity	Speed [m/s]
gamma	Current flight path angle [rad]
phi	Bank angle [rad]

Returns: Flight path angle rate [rad/s]

◆ heading_angle()

template<typename Scalar>

Scalar vulcan::dynamics::heading_angle ( const Vec3< Scalar > & velocity )

Compute heading angle (chi) from velocity

Heading angle is the angle from North to the horizontal projection of velocity, measured clockwise (for NED convention).

$$\chi = \arctan2(v_y, v_x)$$

Parameters

velocity Velocity vector [m/s] (in NED frame)

Returns: Heading angle [rad]

◆ lift_direction_btt()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::lift_direction_btt	(	const Scalar &	gamma,
		const Scalar &	chi,
		const Scalar &	phi )

Compute lift direction for bank-to-turn vehicle

Lift acts perpendicular to velocity, in the plane defined by bank angle. For zero bank, lift is "up" (opposite gravity). For non-zero bank, lift tilts toward the turn.

Parameters

gamma	Flight path angle [rad]
chi	Heading angle [rad]
phi	Bank angle [rad] (positive = right wing down)

Returns: Lift unit direction in NED frame

◆ load_factor_from_lift()

template<typename Scalar>

Scalar vulcan::dynamics::load_factor_from_lift	(	const Scalar &	lift,
		const Scalar &	weight )

Compute load factor from lift

n = L / W

Parameters

lift	Lift magnitude [N]
weight	Weight [N]

Returns: Load factor [g's]

◆ normal_force_direction_stt()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::normal_force_direction_stt	(	const Scalar &	theta,
		const Scalar &	psi )

Compute normal force direction for skid-to-turn vehicle

Normal force acts perpendicular to body X in the vertical body plane. This is the lift-like force generated by angle of attack (alpha).

Parameters

theta	Pitch angle [rad]
psi	Yaw angle [rad]

Returns: Normal force unit direction in NED frame (body -Z direction)

◆ planar_acceleration() [1/2]

template<typename Scalar>

Vec2< Scalar > vulcan::dynamics::planar_acceleration	(	const Scalar &	force_x,
		const Scalar &	force_y,
		const Scalar &	mass )

Compute 2-DOF planar point mass acceleration

Simple Newton's second law in 2D: $$\ddot{x} = F_x/m, \quad \ddot{y} = F_y/m$$

Template Parameters

Scalar Scalar type (double or casadi::MX)

Parameters

force_x	X-direction force [N]
force_y	Y-direction force [N]
mass	Point mass [kg]

Returns: Acceleration vector [ax, ay] [m/s²]

◆ planar_acceleration() [2/2]

template<typename Scalar>

Vec2< Scalar > vulcan::dynamics::planar_acceleration	(	const Vec2< Scalar > &	force,
		const Scalar &	mass )

Compute 2-DOF planar acceleration from force vector

Template Parameters

Scalar Scalar type (double or casadi::MX)

Parameters

force	Force vector [Fx, Fy] [N]
mass	Point mass [kg]

Returns: Acceleration vector [ax, ay] [m/s²]

◆ planar_acceleration_gravity()

template<typename Scalar>

Vec2< Scalar > vulcan::dynamics::planar_acceleration_gravity	(	const Scalar &	force_x,
		const Scalar &	force_y,
		const Scalar &	mass,
		const Scalar &	gravity )

Compute 2-DOF planar acceleration with gravity (Y-down convention)

Parameters

force_x	X-direction force [N] (not including gravity)
force_y	Y-direction force [N] (not including gravity)
mass	Point mass [kg]
gravity	Gravitational acceleration [m/s²] (positive down)

Returns: Acceleration vector [ax, ay] [m/s²]

◆ point_mass_acceleration()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::point_mass_acceleration	(	const Vec3< Scalar > &	force,
		const Scalar &	mass )

Compute 3-DOF point mass acceleration in inertial frame

Simple Newton's second law in 3D: $$\ddot{\mathbf{r}} = \mathbf{F}/m$$

Template Parameters

Scalar Scalar type (double or casadi::MX)

Parameters

force	Force vector in inertial frame [N]
mass	Point mass [kg]

Returns: Acceleration vector [m/s²]

◆ point_mass_acceleration_ecef()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::point_mass_acceleration_ecef	(	const Vec3< Scalar > &	position,
		const Vec3< Scalar > &	velocity,
		const Vec3< Scalar > &	force,
		const Scalar &	mass,
		const Vec3< Scalar > &	omega_earth )

Compute 3-DOF point mass acceleration in ECEF frame

Includes Coriolis and centrifugal terms for rotating reference frame: $$\ddot{\mathbf{r}} = \mathbf{F}/m - 2\boldsymbol{\omega}_e \times \mathbf{v} - \boldsymbol{\omega}_e \times (\boldsymbol{\omega}_e \times \mathbf{r})$$

Template Parameters

Scalar Scalar type (double or casadi::MX)

Parameters

position	Position in ECEF [m]
velocity	Velocity in ECEF [m/s]
force	Force vector in ECEF [N]
mass	Point mass [kg]
omega_earth	Earth rotation vector [rad/s] (typically {0, 0, 7.2921159e-5})

Returns: Acceleration vector in ECEF [m/s²]

◆ position_dot()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::position_dot ( const Vec3< Scalar > & velocity )

Compute position derivative (trivial kinematic relationship)

For pseudo-5DOF in NED frame: ṙ = v

Parameters

velocity Velocity vector in NED [m/s]

Returns: Position derivative in NED [m/s]

◆ rail_direction_ned()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::rail_direction_ned	(	const Scalar &	azimuth,
		const Scalar &	elevation )

Compute rail direction unit vector from elevation and azimuth

Parameters

azimuth	Rail azimuth from North [rad]
elevation	Rail elevation from horizontal [rad]

Returns: Unit direction vector in NED frame

◆ side_force_direction_stt()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::side_force_direction_stt	(	const Scalar &	theta,
		const Scalar &	psi )

Compute side force direction for skid-to-turn vehicle

Side force acts perpendicular to body X-axis in the horizontal body plane. This is the force generated by sideslip (beta).

Parameters

theta	Pitch angle [rad]
psi	Yaw angle [rad]

Returns: Side force unit direction in NED frame (body Y direction)

◆ specific_energy()

template<typename Scalar>

Scalar vulcan::dynamics::specific_energy	(	const Vec3< Scalar > &	position,
		const Vec3< Scalar > &	velocity,
		const Scalar &	mu )

Compute total specific mechanical energy

Parameters

position	Position vector [m]
velocity	Velocity vector [m/s]
mu	Gravitational parameter [m³/s²]

Returns: Specific mechanical energy [J/kg]

◆ specific_force()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::specific_force ( const Vec3< Scalar > & acceleration )

Compute specific force from acceleration (a = F/m)

Parameters

acceleration Acceleration vector [m/s²]

Returns: Specific force [m/s²] = [N/kg]

◆ specific_kinetic_energy()

template<typename Scalar>

Scalar vulcan::dynamics::specific_kinetic_energy ( const Vec3< Scalar > & velocity )

Compute specific kinetic energy

Parameters

velocity Velocity vector [m/s]

Returns: Specific kinetic energy [J/kg] = [m²/s²]

◆ specific_potential_energy()

template<typename Scalar>

Scalar vulcan::dynamics::specific_potential_energy	(	const Vec3< Scalar > &	position,
		const Scalar &	mu )

Compute specific potential energy (gravitation)

Parameters

position	Position vector from center of attracting body [m]
mu	Gravitational parameter [m³/s²]

Returns: Specific potential energy [J/kg] (negative for bound orbits)

◆ speed()

template<typename Scalar>

Scalar vulcan::dynamics::speed ( const Vec3< Scalar > & velocity )

Compute velocity magnitude

Parameters

velocity Velocity vector [m/s]

Returns: Speed [m/s]

◆ thrust_direction_btt()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::thrust_direction_btt	(	const Scalar &	gamma,
		const Scalar &	chi )

Compute thrust direction for bank-to-turn vehicle

Bank-to-turn: Vehicle banks (rolls) to turn. Thrust is along body X-axis. Lift acts perpendicular to velocity, rotated by bank angle.

Parameters

gamma	Flight path angle [rad] (positive = climbing)
chi	Heading angle [rad] (from North)

Returns: Thrust unit direction in NED frame

◆ thrust_direction_stt()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::thrust_direction_stt	(	const Scalar &	theta,
		const Scalar &	psi )

Compute thrust direction for skid-to-turn vehicle

Skid-to-turn: Vehicle yaws to turn (like a missile with cruciform fins). Thrust is along body X-axis, defined by pitch (theta) and yaw (psi).

Parameters

theta	Pitch angle [rad] (positive = nose up)
psi	Yaw angle [rad] (positive = nose right)

Returns: Thrust unit direction in NED frame

◆ transform_mass_properties()

template<typename Scalar>

MassProperties< Scalar > vulcan::mass::transform_mass_properties	(	const MassProperties< Scalar > &	props,
		const Mat3< Scalar > &	rotation,
		const Vec3< Scalar > &	translation )

Transform mass properties to a new coordinate frame

Parameters

props	Original mass properties
rotation	Rotation matrix from old frame to new frame
translation	Translation from old origin to new origin (in old frame)

Returns: Mass properties expressed in new frame

◆ velocity_direction()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::velocity_direction ( const Vec3< Scalar > & velocity )

Compute unit velocity direction

Parameters

velocity Velocity vector [m/s]

Returns: Unit velocity direction (normalized)

◆ velocity_dot_btt()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::velocity_dot_btt	(	const Scalar &	thrust,
		const Scalar &	drag,
		const Scalar &	lift,
		const Scalar &	mass,
		const Scalar &	gravity,
		const Scalar &	gamma,
		const Scalar &	chi,
		const Scalar &	phi )

Compute velocity derivative for bank-to-turn vehicle

Parameters

thrust	Thrust magnitude [N]
drag	Drag magnitude [N]
lift	Lift magnitude [N]
mass	Vehicle mass [kg]
gravity	Gravitational acceleration [m/s²]
gamma	Flight path angle [rad]
chi	Heading angle [rad]
phi	Bank angle [rad]

Returns: Velocity derivative in NED [m/s²]

◆ velocity_dot_stt()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::velocity_dot_stt	(	const Scalar &	thrust,
		const Scalar &	drag,
		const Scalar &	normal_force,
		const Scalar &	side_force,
		const Scalar &	mass,
		const Scalar &	gravity,
		const Scalar &	theta,
		const Scalar &	psi )

Compute velocity derivative for skid-to-turn vehicle

Parameters

thrust	Thrust magnitude [N]
drag	Drag magnitude [N]
normal_force	Normal force magnitude [N] (from alpha)
side_force	Side force magnitude [N] (from beta)
mass	Vehicle mass [kg]
gravity	Gravitational acceleration [m/s²]
theta	Pitch angle [rad]
psi	Yaw angle [rad]

Returns: Velocity derivative in NED [m/s²]

◆ velocity_from_angles()

template<typename Scalar>

Vec3< Scalar > vulcan::dynamics::velocity_from_angles	(	const Scalar &	speed,
		const Scalar &	gamma,
		const Scalar &	chi )

Compute velocity from speed and flight path angles

Parameters

speed	Speed magnitude [m/s]
gamma	Flight path angle [rad]
chi	Heading angle from North [rad]

Returns: Velocity vector in NED [m/s]

Classes

Functions

Function Documentation

◆ aggregate_mass_properties()

◆ attitude_response_accel()

◆ bank_for_turn_rate()

◆ chi_dot_btt()

◆ flight_path_angle()

◆ g_load()

◆ g_load_magnitude()

◆ gamma_dot()

◆ heading_angle()

◆ lift_direction_btt()

◆ load_factor_from_lift()

◆ normal_force_direction_stt()

◆ planar_acceleration() [1/2]

◆ planar_acceleration() [2/2]

◆ planar_acceleration_gravity()

◆ point_mass_acceleration()

◆ point_mass_acceleration_ecef()

◆ position_dot()

◆ rail_direction_ned()

◆ side_force_direction_stt()

◆ specific_energy()

◆ specific_force()

◆ specific_kinetic_energy()

◆ specific_potential_energy()

◆ speed()

◆ thrust_direction_btt()

◆ thrust_direction_stt()

◆ transform_mass_properties()

◆ velocity_direction()

◆ velocity_dot_btt()

◆ velocity_dot_stt()

◆ velocity_from_angles()