High Performance Simulation of Attitude and Translation Dynamics

Ivanka Pelivan, Stefanie Grotjan, Michel S. Guilherme, Silvia Scheithauer, Stephan Theil

ZARM / University of Bremen

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 1 2006 Outline

• Motivation • Objectives • Simulator Target Missions • Simulator – Architecture – Core Features – Dynamics – Disturbance Models – Verification • Summary and Outlook

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 2 2006 Motivation (1/2) Future scientific space missions (LISA/LISA Pathfinder, Gaia, MICROSCOPE, STEP, etc.) are scientifically and technically a new generation.

Reasons: – expected improvement of measurement accuracy by several orders of magnitude – utilisation of new key technologies – very close link between spacecraft dynamics and scientific measurements

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 3 2006 Motivation (2/2) • S/C developers and engineers need a tool for: – Analysis and engineering of AOCS – Performance validation of S/C systems • Scientific users need a tool for: – Error analysis and budgets – Development of the scientific data reduction – Development of scientific in-flight monitoring tools Development of a high-fidelity S/C dynamics simulator: ! Very accurate dynamics modelling ! Enhanced models of environment and disturbances

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 4 2006 Simulator Objectives • Provide comprehensive simulation of the real system including science signal and error sources • Provide simulation environment for control system performance validation • Generate data needed to test data reduction methods • Provide capability for identification of the satellite and instrument

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 5 2006 The Force and Torque Free Satellite Drag-free concept: Shield satellite payload (proof mass) from all external non-gravitational disturbances ! payload follows a purely gravitational orbit Disturbance Force Satellite Body Satellite is forced to follow the proof mass ! Distance between satellite and proof TM mass is controlled Disturbance Force Control Force ! Introduction of coupling forces and torques Control Force • First suggested and analyzed by B. Lange (1964) • First generation: TRIAD I (1971), TIP II (1974) • Second generation: Gravity Probe-B (2004), GOCE, LISA, MICROSCOPE, STEP, LISA Pathfinder

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 6 2006 Target Missions • Intended application to: – Drag-Free Missions: • Gravity Probe B – as a reference and for validation • MICROSCOPE – satellite dynamics model for independent scientific data reduction • LISA – simulator elements for the scientific data reduction • STEP – analysis and engineering tool – Astronomy missions: • Hipparcos – as a reference and for validation • Gaia – simulator elements for the scientific data reduction – Other missions: • Pioneer 10/11 – utilization of disturbance models for the re- analysis of the Doppler data

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 7 2006 General Simulator Structure

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 8 2006 Simulator Architecture • Modular Design in Matlab/Simulink and C/Fortran – Matlab/Simulink is wrapper for development and analysis. – Major blocks shall be coded in C/Fortran. – Modules are available as library. – Simulator for each mission is assembled from modules. – Initialisation and set-up through data files

• Possibility to integrate into data reduction process – A transition to pure C/Fortran code necessary – Interfaces for estimation algorithms

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 9 2006 Simulator Architecture • Modules: Dynamics Core, Environment and Disturbance, Sensor, Controller, Actuator, Transformations • User Interface: Matlab/Simulink • Core and most Environment/Disturbance modules: s-function blocks

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 10 2006 Satellite and Test Mass Dynamics

" Objects are treated as rigid bodies " 6 satellite degrees of freedom " 3 for translation FDist " 3 for rotation " GTM1 GTM2 6 degrees of freedom per test mass FControl " Driving force: GSat " Gravitation

• Satellite dynamics described in inertial and satellite body-fixed frame • Test mass dynamics described in sensor and test mass frame

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 11 2006 Coordinate Frames

inertial frame (ECI, i), body/satellite frame (b), mechanical/structural frame (m), accelerometer frame (a), sensitive axis frames for test masses 1 and 2 (sens1, sens2), test mass body frames for test masses 1 and 2

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 12 2006 Satellite Equations of Motion

• Translation of Satellite:

– Motion of a rigid body in a gravity field – Additional force = coupling force

• Rotation of Satellite:

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 13 2006 Test Mass Equations of Motion • Definition in sensor coordinate frame: satellite-fixed, non inertial Considers accelerations and rotations of system • Translation:

• Rotation: - Approach: Conservation of angular momentum

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 14 2006 Simulator Core Features • Simulation of full satellite and test mass dynamics in six degrees of freedom by numerical integration of the equations of motion (13 states: attitude rate, quaternion, position, velocity) • Up to four differential accelerometers utilizing two test masses each (8 (TMs) + 1 (Satellite) = 9x13 states = 117 states) • Consideration of linear and nonlinear coupling forces and torques between satellite and test masses as well as between test masses • Modelling of cross-coupling interaction • Earth gravity model up to 360th degree and order, influence of Sun, Moon and planets can be included • Gravity-gradient forces and torques • 5th order Runge-Kutta numerical integration • 128 bit numerical precision (`quad precision') on an ALPHA processor Several error sources are considered in the model: + misalignment and attitude errors + coupling biases + displacement errors

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 15 2006 Force & Torque Modeling (1/2) • Modeling of forces and torques acting on the satellite and test masses because of: – Gravitation and Gravity Gradients – Control (forces and torques applied by the control system) – Interaction with the upper layers of the Earth atmosphere – Electromagnetic radiation • heat, radio communication emission • Absorption and reflection of radiation incident (sun, Albedo, etc.) – Interaction with the magnetic field – Interaction (coupling) between satellite and test masses • Test mass sensing and actuation systems • Gravitational coupling

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 16 2006 Force & Torque Modeling (2/2) • Modeling approaches: 1. Utilization AND extension of standard models 2. Derivation of parametric models from detailed FEM analysis of specific effects

• Standard models used: – International Geomagnetic Reference Field (IGRF, IAGA) – Earth Gravity Model (EGM, NASA) – Mass Spectrometer Incoherent Scatter Model (MSIS, NRL) • Short-term variations of Earth atmospheric density (analysis of CHAMP mission data)

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 17 2006 Attitude Dependency of Gravitational Acceleration

• Origin: Gradient of gravity field • Center of Mass and Center of Gravity do not coincide. • Effects: – Gravity gradient torque – Attitude dependent gravity force

CoM CoM

CoG CoG Earth g g CoM CoG

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 18 2006 Gravity-Gradient Forces and Torques

• Force from Earth potential field dF = dm!dF

F = Fmono + FGG

• Gravity-gradient Torque: T = r dF 1) T = r Gr 2)

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 19 2006 Modeling of Density Variations (1/3) • Processing of data from CHAMP mission: – Orbit data: position, velocity – Sensor data: star tracker, accelerometers – Further data: geometry, area, mass • Computation of density:

• Analysis of the frequency spectrum • Design of a form filter in order to create the difference spectrum from white noise

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 20 2006 Modeling of Density Variations (2/3)

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 21 2006 Modeling of Density Variations (3/3)

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 22 2006 Modeling of Forces from Electromagnetic Radiation (1/2) • Effects: – EM radiation incident: • Sun light • Albedo light • Infrared (thermal) radiation of Earth – EM radiation emission: • Thermal radiation emission • Radio frequency emission

• Basic equations:

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 23 2006 Modeling of Forces from Electromagnetic Radiation (2/2) • Implementation: – Definition of elements representing the satellite surface – Determination of visibility • Back face determination • Shadowing – Computation of force for each element – Summation of total force and torque – Creation of look-up tables for total force and torque

(Method also applicable for atmospheric drag)

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 24 2006 Modeling of Torques from the Interaction with the External Magnetic Field (1/3) - Analytic simplified derivation of the torque assuming the shape of an ellipsoid - Numerical computation using Finite Elements (FEM) • Computation of magnetic dipole from element data:

; ;

• Derivation of a parametric model – !-Metal-Shield (STEP)

– Core of magnetic coils

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 25 2006 Modeling of Torques from the Interaction with the External Magnetic Field (2/3)

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 26 2006 Modeling of Torques from the Interaction with the External Magnetic Field (3/3)

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 27 2006 General Modeling of Coupling Forces

• Specific coupling forces:

• Coupling torques:

• Definition of general force:

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 28 2006 Verification of Simulator

• Comparison to analytical solution of simplified system – Simplified model renders ODE in Mathieu-Form – Verification by comparison of stability boundaries

• Comparison to Hill‘s Equation – Analytical description of uncoupled relative movement

• Comparison to other orbit propagators

• Verification of uncoupled attitude motion

• Test of dynamic coupling between satellite and test masses

• Verification with flight data

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 29 2006 Summary and Outlook • Multi-body mechanical system simulator (satellite and test masses). • Satellite and test mass dynamics is represented by a set of coupled ordinary differential equations (13 per body). • Modeling of disturbances include also smaller effects such as: – Density variation – Torques from magnetization of components – Radiation incident and EM radiation emission • Outlook: – Validation of simulator with flight data – Post-mission analysis of GP-B and Hipparcos – Model improvement based on post-mission analysis and flight data – Adaptation to STEP, MICROSCOPE, Gaia • Future additions: – Adaptation of dynamics core to L2-orbits – Implementation of Earth Albedo – Structural effects (thermo-elastic)

3rd Internationall Workshop on Astrodynamiics tools and Techniiques, Noordwiijk, 2 - 5 October 30 2006