J1b_6_-_PTD_Thrust_Measurement_Approach.pdf

Pathfinder Technology Demonstrator 02/10/2016

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1 SUMMARY

This document provides the measurement requirement needed to estimate the thrust produced by a continuous thrusting propulsion system on the Pathfinder Technology Demonstrator (PTD) nanosatellite in Low Earth Orbit (LEO). The total accuracy of the thrust measurement (environment and spacecraft uncertainty in addition to measurement estimation) is also discussed.

2 THRUST ESTIMATION ACCURACY

There are two primary methods conops variations to estimate the thrust through GPS measurements.

1. Measure GPS at constant frequency, periodically turn thruster on and off (~1 orbit period)

2. Measure GPS at constant frequency, leave thruster on

The latter conops method is substantially less accurate due to the uncertainty in the environment disturbance forces which is dominated by the atmospheric drag term. The former increases accuracy by estimating the disturbance force during the period in which the thruster is turned off which is then approximated to follow the same profile over the next thrusting period. The former is baselined for this document and reflected in requirement R2 below. Table below provides a budget for the thrust estimation accuracy for the Pathfinder Technology Demonstrator (PTD) nanosatellite.

Error Source Reference Section / Notes Error Value

Estimator Algorithm (Thrust)

4 Measurement Estimation Error 1.8%

Estimator Algorithm (Disturbance Environment)

4 Measurement Estimation Error 5 Uncertainty Analysis

1.8%

Margin Safety factor (simplified model) 100.0%

TOTAL 7.2%

3 MEASUREMENT REQUIREMENTS

R1) The Spacecraft shall take GPS measurements of position and velocity with an accuracy of 26m RMS and 0.67m/s RMS respectively at a frequency of once per minute during both thrusting and non-thrusting states.

R2) The Spacecraft shall thrust continuously for a minimum of one orbit, followed by a period of no thrust for minimum of one obit (with this cycled repeated for length of experiment).

R3) The Spacecraft shall store the GPS telemetry along with thrust/no-thrust time tagged state flags of accuracy less than 1 second.

R4) The Spacecraft shall downlink the gathered GPS and state flag telemetry and provide to Ames operations team via MINX at a maximum interval of once per week.

R5) The Spacecraft GPS axes shall be known with respect to the Spacecraft body axes to within an error of 0.1deg and be provided to Ames as a part of the ICD.

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4 MEASUREMENT ESTIMATION ERROR

4.1 Estimation Error for Requirement 1

Below is the output for 100 Monte Carlo iterations of the least squares orbit estimation algorithm used to estimate thrust magnitude for the PTD cubesat model.

4.2 Model Description

System model: Spherical Earth gravitational model, Harris Priester atmospheric model with Earth oblateness, constant anti-velocity vector thrust, 500km circular 98.7deg inclination orbit, state vector of position and velocity.

Runge-Kutta 4 integrator with 0.5s step size

GPS measurement model: white Gaussian noise at select interval applied to integrated ‘truth’ orbit

Numerical Hamiltonian on thrust and initial state (position, velocity) (RK4) with constant step size

Iteration step up with weighted factor for GPS uncertainty 3sigma values

Least Squares estimator algorithm

Mean error: 0.42% Standard Deviation: 0.45% 3-Sigma error: 1.8%

GPS error: 26m RMS, 0.67m/s RMS CubeSat model: PTD 6U w deployables Orbit: 500km circular, 97.8deg Harris Priester magnitude: max Thrust: 150uN

#Iterations: 100 Varied parameters: GPS measurement

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5 UNCERTAINTY ANALYSIS

The environmental disturbances and spacecraft parameters modeled have uncertainties associated with them that reduce the accuracy of the thrust estimation (if known perfectly, the thrust estimation would be as accurate as the measurement).

5.1 Spacecraft Uncertainties

The spacecraft uncertainties contribute to the uncertainty of the affects of environmental disturbances, affecting the overall thrust estimation accuracy. A 6U CubeSat with 12U deployable surface area is used for reference under worst case configurations (maximum deployable surface towards disturbance). The environmental disturbances of concern are: atmospheric drag, and solar radiation pressure (SRP). Table below provides a rough budget for spacecraft sources of uncertainty pertaining to environment disturbance estimates.

Variable Environment Source Uncertainty Source Uncertainty Units %*

Surface area (�⃑⃑� ) Atmospheric drag, SRP

Geometry measurement ±6.5e-04 m2 0.5%

Deployment geometry wrt S/C body axes

±9.9e-08 m2 ~0%

Body axes wrt orbit axes ±9.9e-08 m2 ~0%

Drag coefficient (𝑪𝑫) Atmospheric drag Flat plate approximation Not analyzed - -

Emissivity (𝒒) SRP Emissivity approximation Not analyzed - -

*Percentage relative to worst case configuration described above

5.2 Environment Uncertainties

The environmental disturbance uncertainties are tabulated below in Table. Atmospheric drag dominates the uncertainties due to the low altitude orbit and geometry of the spacecraft. Table below simplifies the atmospheric density uncertainty to the upper and lower limits of the Harris-Priester model for rough approximation. Percentage variations of upper and lower limits against mean are shown in Figure below.

Environment Source Uncertainty Source Uncertainty Units %*

Atmospheric Drag Density model value (Harris-Priester)** ±8.3e-13 kg/m3 68%

Solar Radiation Pressure Solar constant (TIM SORCE – 2008) ±0.5 W/m2 ~0%

*Percentage relative to mean **For 500km altitude – see Harris-Priester table for full range

Figure 1 - Atmospheric Density Variation against Mean due to F10.7 parameter

20%

40%

60%

80%

100%

300 350 400 450 500 550 600 650 700P e rc e n ta ge v ar ia ti o n fr o m M e an

Altitude (km)