Transcript Lecture 7
ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 7: Linearization and the State Transition Matrix University of Colorado Boulder Homework 2 – Due September 11 Lecture Quiz 2 – Due Friday @ 5pm Future Lectures ◦ ◦ ◦ ◦ ◦ Lecture Lecture Lecture Lecture Lecture 8 – Friday 9/11 @ 9am 9 – Monday 9/14 @ 9am 10 – Monday 9/14 @ 4pm 11 – Monday 9/21 @ 9am 12 – Monday 9/21 @ 4pm University of Colorado Boulder 2 Lecture Quiz 1 Results Linearization – How we do it? (wrap-up) State Transition Matrix (STM) ◦ Derivation ◦ Solution Methods University of Colorado Boulder 3 Lecture Quiz 1 University of Colorado Boulder 4 Correct – 98% 2% 98% 0% 0% University of Colorado Boulder 5 Low accuracy (~1 km) ◦ Low resolution imaging, space surveillance Medium accuracy (~100 m) ◦ Medium resolution imaging, orbit prediction, laser tracking High accuracy (<10 cm) ◦ Relative motion/formation flying ◦ Scientific studies of the Earth University of Colorado Boulder 6 Correct – 80% We are modeling the forces acting on a spacecraft using only the twobody force and gravity perturbations (e.g., spherical harmonics model). Our ECF/ECI coordinate system transformation is the full model, i.e., it includes the Earth’s rotation, nutation/precession, bias, etc. The spherical harmonic model we use has degree and order 70. We propagate a test orbit, and see that the energy is not constant. This means that our orbit propagator has an error. ◦ True ◦ False University of Colorado Boulder 7 University of Colorado Boulder 8 University of Colorado Boulder Derived from a time-varying potential In the inertial frame, the potential varies due to the Earth’s rotation If we assume pure rotation and only zonal harmonics (Jn terms), there is no change in time 9 Correct – 76% Which of these time systems uses the definition of the second based on atomic time? ◦ ◦ ◦ ◦ Atomic Time (TAI) Terrestrial Time (TT) GPS Time UT1 University of Colorado Boulder 96% 36% 89% 16% 10 University of Colorado Boulder 11 Correct – 27% Which of the following are assumed when using the Newton-Raphson method of solving a nonlinear system ◦ The observations are nonlinear, but the dynamics are linear ◦ The equations describing the dynamics and the 36% observations are nonlinear 60% ◦ The observations (or known data) have no errors 64% ◦ The number of observations equals the number of unknowns 27% University of Colorado Boulder 12 Correct – 98% Atmospheric drag is a conservative force ◦ True ◦ False University of Colorado Boulder 13 Linearization – Why do we need it? (review) University of Colorado Boulder 14 How do we estimate X ? How do we estimate the errors εi? How do we account for force and observation model errors? University of Colorado Boulder 15 For now, let’s consider a linear problem: University of Colorado Boulder 16 This is the “normal equation” for the least squares estimator ◦ “hat” notation indicates vector that minimizes J(x) ◦ We treat this as the estimated state We assumed that the state-observation relationship was linear, but the orbit determination problems is nonlinear ◦ We will linearize the formulation of the problem University of Colorado Boulder 17 University of Colorado Boulder 18 Linearization – How do we do it? (continued) University of Colorado Boulder 19 University of Colorado Boulder 20 University of Colorado Boulder 21 Which terms are non-zero? University of Colorado Boulder 22 Which terms equal 1? What are the partials w.r.t. μ? University of Colorado Boulder 23 What is the final answer? University of Colorado Boulder 24 Computed, not measured values! University of Colorado Boulder 25 Linearization – State Transition Matrix University of Colorado Boulder 26 When x is linear (note lower case!) then there exists a solution to the linear, first order system of differential equations: The solution is of the form: Φ(t,ti) is the state transition matrix (STM) that maps x(ti) to the state x(t) at time t. University of Colorado Boulder 27 Constant! Why? What is the differential equation for the STM? University of Colorado Boulder 28 University of Colorado Boulder 29 There are four methods to generate the STM: ◦ Solve from the direct Taylor expansion ◦ If A is constant, use the Laplace Transform or eigenvector/value analysis ◦ Analytically integrate the differential equation directly ◦ Numerically integrate the equations (ode45) University of Colorado Boulder 30 State Transition Matrix – Alternative Derivation University of Colorado Boulder 31 Expand X(t) in a Taylor series about X*(t): University of Colorado Boulder 32 University of Colorado Boulder 33 University of Colorado Boulder 34 State Transition Matrix – Laplace Transform University of Colorado Boulder 35 Laplace Transforms are useful for analysis of linear time-invariant systems: ◦ ◦ ◦ ◦ ◦ electrical circuits, harmonic oscillators, optical devices, mechanical systems, even some orbit problems. Transformation from the time domain into the Laplace domain. Inverse Laplace Transform converts the system back. University of Colorado Boulder 36 University of Colorado Boulder 37 Solve the ODE We can solve this using “traditional” calculus: University of Colorado Boulder 38 Solve the ODE Or, we can solve this using Laplace Transforms: University of Colorado Boulder 39 Solve the ODE: University of Colorado Boulder 40 University of Colorado Boulder 41 University of Colorado Boulder 42