The B-Plane Interplanetary Mission Design

Transcription

1 The B-Plane Interplanetary Mission Design Collin Bezrouk 2/11/2015 2/11/2015 1

2 Contents 1. Motivation for B-Plane Targeting 2. Deriving the B-Plane 3. Deriving Targetable B-Plane Elements 4. How to Target the B-Plane 5. Patched Conics (Lambert) Example 6. Mars Odyssey (JPL) Example 2/11/2015 2

3 Motivation for the B-Plane 2/11/2015 3

4 Motivation for the B-Plane We have an interplanetary transfer from Earth to Mars Mars TCM We are constructing a Trajectory Correction Maneuver (TCM) to now target our desired Mars Insertion Sun Earth 2/11/2015 4

5 Motivation for the B-Plane Target: h p = 300 km i = Mars Mars Desired orbit insertion: h p = 300 km, inc = 93.5 TCM Our current path (w/o TCM): h p = 450,000 km, inc = 50 How do we build our TCM? Sun Earth ΔV = ΔV x, ΔV y, ΔV z T =? 2/11/2015 5

6 Target: h p = 300 km i = Mars V x The target parameters (h p and i) are highly non-linear with respect to the components of the ΔV. It is hard to predict how a change in ΔV will change the target values. This is remedied by the B-Plane! V y ( 800, 130) (1200, 135) (300, 93.5) (2500, 62) (300, 5) 2/11/ (not to scale)

7 Deriving the B-Plane 2/11/2015 7

8 Definition of the B-Plane The B-Plane is a planar coordinate system that allows easy targeting for hyperbolic, two-body trajectories. We will target the B-vector, which lies n the the intersection of the B-plane and the trajectory plane. It shows where the incoming hyperbolic asymptote intersects the B-Plane. 2/11/2015 8

9 B-Plane Coordinate System The B-Plane is the T- R plane, normal to S // V in. T lies in the ecliptic plane and is normal to S R completes the orthonormal coordinates. B = b B where b is the semi-minor axis (also called the impact parameter). 2/11/2015 9

10 Deriving the B-Vector 2/11/

11 Deriving the B-Vector We will target B instead of orbital elements or states of the flyby. There are two useful method for converting to B: Method 1: Used once you re beyond preliminary mission design. Navigation solutions are given using this method. Useful if you are propagating your trajectory in a multi-body dynamics model. Requires you have five target state variables (ex. r, v) or orbital elements (ex. a, e, i, Ω, ω) that can be converted into a, e, and h (angular momentum direction). Method 2: Used for patched conics models in preliminary mission design. For example, linking two Lambert solutions to make a flyby. Requires V in and V out 2/11/

12 Example of Method 2: New Horizons V in Asymptote B-Plane V out 2/11/

13 Deriving the B-Plane: Method 1 Method 1: (given r, v) h = r v r v Note: r, v must be relative to the flyby body. e = 1 μ v2 μ r r r v v cos β = 1 e k = Ecliptic Orbit Normal: 0,0,1 T S = cos β e e + sin β h e h e T = S k S k R = S T 2/11/

14 Deriving the B-Plane: Method 2 Method 2: (given V in, V out ) S = V in V in k = Ecliptic Orbit Normal: 0,0,1 T T = S k S k R = S T 2/11/

15 Computing the B-Vector The B-vector lies in the intersection of the B-Plane and the trajectory plane. B = b B where B = S h Find b using the Hyperbola Equation: b 2 = c 2 a 2 2/11/

16 Computing the Impact Parameter Method 1: (given r, v) b 2 = c 2 a 2 a = 2 r v2 μ c = ae 1 b 2 = ae 2 a 2 b 2 = a 2 e 2 1 b = a e 2 1 2/11/

17 Computing the Impact Parameter Method 2: (given V in, V out ) b 2 = c 2 a 2 b 2 = a 2 c2 a 2 1 a μ V 2 b 2 = μ2 V 4 c a 2 1 cos β = a c = 1 + V 2 r p μ 1 b 2 = μ2 V V 2 r p μ 2 1 (continued on next slide) 2/11/

18 Computing the Impact Parameter Method 2: (continued) b 2 = μ2 V V 2 r p μ 2 1 b = μ V V 2 r p μ 2 1 Where r p = μ V 2 cos 1 π ψ 2 1 and cos ψ = V in V out V in V out 2/11/

19 B-Plane Target Parameters We can target the B-vector using its Cartesian or polar components. Cartesian: B T = B T B R = B R Polar: b = B θ = tan 1 B R B T (Use atan2(br,bt) in programming to avoid quadrant ambiguity) 2/11/

20 How to Target the B-Plane Parameters 2/11/

21 The Targeting Setup We want our spacecraft to hit the B-plane at B T and B R. Begin with the state R, V where your TCM will be performed. Compute the reference B-Plane parameters: B T, B R = f(r, V) How do we calculate V = V + ΔV such that B T, B R = f(r, V )? Sun Mars Earth TCM 2/11/

22 Deriving the Targeting Equations Use a Taylor Series Expansion about the reference parameters: B T = B T + B B R B R ΔV V V + H. O. T Move the B-Plane terms to the RHS. ΔB = B B and ΔV = V V 0 ΔB T = B ΔB R ΔV ΔV x ΔV y ΔV z Jacobian Matrix Solve this matrix equation to estimate ΔV 2/11/

23 Solving the Linear Approximation ΔB = B ΔV ΔV 2 1 = 2 3 (3 1) The Jacobian matrix is not invertible, so how do we solve? Option 1 (Worst): Fix a component of ΔV and remove the corresponding column from the Jacobian matrix. The Jacobian is (2x2) and invertible. Option 2 (Better): The problem is underdetermined, and has infinite solutions. Find the solution where ΔV is minimized. Min. Norm solution to matrix problem A x = b is given by: x = A T AA T 1 b 2/11/

24 Solving the Linear Approximation Option 3 (Best): Add an additional element to ΔB to make the Jacobian 3x3. We use the linearized time of flight as an additional target. This is useful if you wish to land at a specific planetary location. ΔV = B ΔV 1 ΔB Now, we need to compute the Jacobian matrix. 2/11/

25 Numerically Computing the Jacobian The Jacobian shows how each element in B is affected by each element in ΔV. These partial derivatives are defined for the values in B and will be different as B changes. B T B T B T B ΔV = ΔV x B R ΔV x TOF ΔV y B R ΔV y TOF ΔV z B R ΔV z TOF ΔV x ΔV y ΔV z 2/11/

26 Numerically Computing the Jacobian We can calculate the partials with a forward Euler (first-order) finite difference method: u u x + Δx u(x) x = lim Δx 0 Δx Procedure: 1. [B T, B R, TOF] = f R, V 2. Perturb velocity in one direction: V + = V + ΔV x, 0,0 T 3. [B T +, B R +, TOF + ] = f R, V + 4. Compute partials: B T = B + T BT ΔV x ΔV x = B + R BR ΔV x ΔV x, B R This will provide the first column of the Jacobian. 5. Repeat with perturbations in ΔV y and ΔV z, etc. 2/11/

27 Numerically Computing the Jacobian Numerical Partial Value 2.5 x B-Plane Partials B T / V x B T / V y B R / V x B R / V y Perturbation Magnitude (km/s) Partials have a constant value for a large range of perturbing velocity magnitudes! This shows numerically that the B-Plane elements change linearly with the ΔV components. We can expect a very accurate solution from this setup. Why are the values not constant for ΔV < or ΔV > 10 0? 2/11/

28 Numerically Computing the Jacobian Numerical Partial Value 2.5 x B-Plane Partials B T / V x B T / V y B R / V x B R / V y Perturbation Magnitude (km/s) Machine Precision Round-off Errors Perturbation is outside of the linear range 2/11/

29 Completing the Targeting Recall that we ve linearized the problem by ignoring terms second order and higher in our Taylor series expansion. To remedy this, we ll need to iterate our solution until we ve reached our desired B-Plane values. Compute the ΔV, then update your reference velocity. Compute your new current B-Plane values, and repeat the procedure. V i+1 = V i + ΔV i Iterate until your B-Plane elements are within tolerance of your desired B-Plane elements. Computer your total TCM: ΔV TCM = V N V 0 = ΔV i 2/11/

30 Mars Odyssey Example Slides by Moriba Jah Formerly of CCAR 2/11/

31 Interplanetary Trajectory Earth at Arriv al 1.02 AU Vernal Equinox Launch: 07-APR AU 45.9û TCM-5 E-7 h Mars Arriv al 24-Oct-2001 TCM-4 E-12 d Mars at Launch TCM-1 L + 46 d TCM-2 L + 86 d TCM-3 E - 37 d 2/11/

32 Mission Constraints To achieve the mission, the spacecraft must: Be injected into an orbit with a period of less than 22 hours, while having a 300 km periapse altitude (+/- 25 km) and an inclination of 93.5º (+/- 0.2º), including MOI burn execution errors. This is equivalent to hitting a golf ball from NY to Paris and making it in the hole in only 4 swings. (achieved: 18:36 period km and 93.51º) Employ aerobraking over a 3-month period (walk-in, main phase, end-game/walk-out) in order to maximize payload mass and minimize propellant expense. By the end of aerobraking, stabilize in a 400 km circular, frozen, sun-synchronous orbit with a 2PM LMST AEQUAX. 2/11/

33 Mars Odyssey Navigation Navigation Major Events Injection TCM-1 TCM-2 TCM-3 TCM-4 TCM-5 (Contingency) MOI Period Reduction Maneuver 2/11/

34 Launch Target 2/11/

35 TCM-1 Design TCM-1 Execution Date: 23-May-01 2/11/

36 TCM-2 Design TCM-2 Execution Date: 02-July-01 2/11/

37 TCM-3 Design TCM-3 Execution Date: 17-Sept-01 2/11/

38 TCM-4 Design TCM-4 Execution Date: 12-Oct-01 Target Alt: 300 km Inc: Current Estimate (OD034) Alt: 324.1±11 km Inc: ±0.2 Current Miss (Est-Target) Alt: +24 km Inc: +0.6 TCM-4 to Correct Miss V: 0.08 m/s 2/11/

39 Final Navigation Solution After TCM-4 OD Knowledge at the time of TCM-4 Design (3s) 2/11/

40 Mars Orbit Insertion Burn Burn Start +00:00:00 Enter Earth Occult +00:09:37 Enter Solar Eclipse +00:09:59 Exit Solar Eclipse +00:11:46 Periapsis km +00:12:51 Burn End +00:19:44 Exit Earth Occult +00:29:24 2/11/

41 MOI Viewed From Earth Burn Start +00:00:00 Enter Earth Occult +00:09:37 Exit Earth Occult +00:29:24 2/11/