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Cowell Orbit Kinematic Model

Class name: D2S2.Model.Satellite.CowellOrbitKinematicModel Library: D2S2 Library

Inheritance CowellOrbitKinematicModel : ModelBase Implements ISatelliteKinematicModel,IInitialisable

Details

Provides orbit position and velocity, and attitude angles and angular rates to the satellite. This model numerically integrates the attitude (if the IntegrateAttitude property is set to true), as well as position and velocity by taking into account all forces and torque on the satellite.

Orbit kinematic state is represented as cartesian vectors for position and velocity, in ECI coordinate frame. The orbit state is propagated by numerically integrating the motion equation: \(\bm{\ddot{r}}=-\frac{\mu \bm{r}}{\bm{|r|}^{3}} + \bm{q} + \nabla V\)

Where \(\bm{r}\) is the position vector from the centre of mass of the Earth to the satellite, \(\bm{\ddot{r}}=\frac{d^{2}\bm{r}}{dt^{2}}\) is the acceleration vector, \(\mu = GM\) is the gravitational parameter for the Earth.

The vector \(\bm{q}\) is the acceleration due to non-conservative perturbing forces such as aerodynamic drag, solar radiation pressure (SRP) and electromagnetic forces. The acceleration vector \(\bm{q}\) is simply obtained by dividing the sum of the non-conservative forces by the satellite mass, \(m\). \(\bm{q} = \bm{F}_{aerodynamic} + \bm{F}_{solar} + \bm{F}_{components}\)

The effect of conservative perturbations are captured by the potential function, \(V\). The gradient of this potential function provides the acceleration due to third-body Sun and Moon, and central body (Earth) gravity. \(V(\bm{r}) = V_{Earth}(\bm{r}) + V_{Sun}(\bm{r}) + V_{moon}(\bm{r})\)

If the IntegrateAttitude option is enabled, the attitude of the satellite is propagated by numeric integration of the Euler dynamic equation \(\bm{I\dot\omega_{B}^{I}} = \bm{N_{GG}} + \bm{N_{aero}} + \bm{N_{solar}} + \bm{N_{components}} - \bm{\omega_{B}^{I}} \times (\bm{I\omega_{B}^{I}} + \bm{h_{internal}})\) where \(\bm{I}\) is the moment of inertia tensor of the satellite, \(\bm{\omega_{B}^{I}} = \bm{\omega_{B}^{O}} + A_{I->O}[0,-\omega_{O}, 0]^{T}\) is the is the angular rate of the satellite relative to the ECI coordinate frame \(\bm{N_{GG}}\) is the gravity gradient disturbance torque, \(\bm{N_{aero}}\) is the aerodynamic disturbance torque from the surface model, \(\bm{N_{solar}}\) is the SRP disturbance torque from the surface model, \(\bm{N_{components}}\) is the sum of torques produced by the satellite components, \(\bm{h_{internal}}\) is the sum of internal angular moment of all satellite components (reaction wheels, for instance).

The gravity gradient torque is due to a result of varying gravitational force on parts of the spacecraft depending on their distance to the Earth. It is calculated from the equation \(\bm{N_{GG}} = \frac{3GM}{R^{3}}(\bm{z_{O}^{B}} \times \bm{Iz_{O}^{B}})\) where \(G\) is the Earth gravitational parameter \(M\) is the mass of the Earth, \(R\) is the satellite distance from the Earth centre, and \(\bm{z_{O}^{B}}=A_{O->B}[0,0,1]^{T}\) is the nadir vector in the spacecraft body coordinate frame.

The initial attitude state is specified through the InitialRoll, InitialPitch, InitialYaw and InitialRates properties.

The initial orbit state is specified through the InitialOrbitState property.

The numeric intergration scheme can be changed through the NumericalIntegratorSelection property.

Initial Value Properties

Property Description
InitialRoll (Roll Angle) Initial roll angle [deg]
InitialPitch (Pitch Angle) Initial pitch angle [deg]
InitialYaw (Yaw Angle) Initial yaw angle [deg]
InitialRates (Angular Rates) Initial inertially referenced angular rate vector [deg/s]
InitialOrbitState (Orbit State) Inition orbit state
NumericalIntegratorSelection (Numerical Integrator) Numeric integration selection
IntegrateAttitude (Integrate Attitude) Option to enable numeric integration of attitude state