Vai's Uni Notes

Hill-Clohessy-Wiltshire Equations

7 min read

Linearized relative motion of a secondary spacecraft about a primary in a circular orbit (Curtis chapter 7).

Linearization

  • Primary 𝐴 (reference orbit) and secondary 𝐵 (possibly thrusting):
    • ̈𝐫𝐴=−𝜇𝑟3𝐴𝐫𝐴̈𝐫𝐵=−𝜇𝑟3𝐵𝐫𝐵+𝐅thrust
  • Relative position and its inertial acceleration:
    • 𝛿𝐫=𝐫𝐵−𝐫𝐴𝛿̈𝐫=−𝜇𝑟3𝐵𝐫𝐵+𝐅th+𝜇𝑟3𝐴𝐫𝐴
  • Assumption: 𝛿𝑟≪𝑟𝐵,𝑟𝐴, with 𝐫𝐵=𝐫𝐴+𝛿𝐫
  • Expand the 𝐫𝐵 term:
    • 𝐫𝐵𝑟3𝐵=𝐫𝐴+𝛿𝐫(𝑟2𝐴+2𝐫𝐴⋅𝛿𝐫+𝛿𝑟2)32=𝐫𝐴+𝛿𝐫𝑟3𝐴(1+2𝐫𝐴⋅𝛿𝐫𝑟2𝐴+…)−32
    • binomial expansion (1+𝑥)𝑚=1+𝑚𝑥+…:
      • 𝐫𝐵𝑟3𝐵≈𝐫𝐴+𝛿𝐫𝑟3𝐴(1−3𝐫𝐴⋅𝛿𝐫𝑟2𝐴)
      • (the notes write the remaining terms of 𝛿̈𝐫, +𝐅thrust+𝜇𝑟3𝐴𝐫𝐴, on the end of this line; they are combined in the next step)
  • Keep first-order terms in 𝛿𝐫:
    • 𝛿̈𝐫=−𝜇𝑟3𝐴(−̂𝐫𝐴3𝐫𝐴⋅𝛿𝐫𝑟𝐴+𝛿𝐫)+𝐅thrust
  • Components of 𝛿𝐫 in the RSW frame of 𝐴: 𝛿𝐫=[𝑥𝑦𝑧], with 𝑥 radial (so ̂𝐫𝐴⋅𝛿𝐫=𝑥):
    • 𝛿̈𝐫inertial=−𝜇𝑟3𝐴(−2𝑥̂𝐢+𝑦̂𝐣+𝑧̂𝐤)+𝐅thrust

Rotating Frame

  • The RSW frame rotates, so (transport theorem):
    • 𝛿̈𝐫=𝛿̈𝐫inertial−̇𝛀×𝛿𝐫⏟tangential acc.−2𝛀×𝛿̇𝐫⏟Coriolis−𝛀×(𝛀×𝛿𝐫)⏟centrifugal
  • 𝛀 = orbital angular velocity of the primary:
    • 𝛀=𝑛̂𝐤,𝑛=√𝜇𝑟3𝐴
    • For circular motion this is a constant, equal to the mean motion times the ̂𝐤 unit vector
    • The tangential acceleration is then 0
  • For the other two terms, expand in components:
    • 𝛀×𝛿̇𝐫=𝑛(−̇𝑦̂𝐢+̇𝑥̂𝐣)𝛀×(𝛀×𝛿𝐫)=−𝑛2(𝑥̂𝐢+𝑦̂𝐣)
    • and the gravity term, with 𝜇𝑟3𝐴=𝑛2:
      • −𝜇𝑟3𝐴(−3𝑥̂𝐢+𝛿𝐫)=−𝑛2(−2𝑥̂𝐢+𝑦̂𝐣+𝑧̂𝐤)
  • Altogether:
    • 𝛿̈𝐫=−𝑛2(−2𝑥̂𝐢+𝑦̂𝐣+𝑧̂𝐤)−2𝑛(−̇𝑦̂𝐢+̇𝑥̂𝐣)+𝑛2(𝑥̂𝐢+𝑦̂𝐣)+𝐅th

HCW Equations

Hill-Clohessy-Wiltshire (HCW) Equations

  • ̈𝑥−2𝑛̇𝑦−3𝑛2𝑥=𝑓𝑥̈𝑦+2𝑛̇𝑥=𝑓𝑦̈𝑧+𝑛2𝑧=𝑓𝑧
  • 𝑥 radial, 𝑦 along-track, 𝑧 cross-track; 𝑓𝑥,𝑓𝑦,𝑓𝑧: components of 𝐅thrust
  • Linear, constant coefficients (circular reference orbit)

Solving the HCW Equations

  • Assume no external forces: 𝑓𝑥=𝑓𝑦=𝑓𝑧=0
  • The one-dimensional 𝑧 equation is decoupled from the rest (simple harmonic motion):
    • 𝑧(𝑡)=𝑧0cos𝑛𝑡+̇𝑧0𝑛sin𝑛𝑡
  • Two coupled 𝑥-𝑦 equations in the orbital plane; differentiate the 𝑥 equation and substitute ̈𝑦=−2𝑛̇𝑥:
    • 𝑥(3)=2𝑛̈𝑦+3𝑛2̇𝑥=−4𝑛2̇𝑥+3𝑛2̇𝑥⇒𝑥(3)+𝑛2̇𝑥=0
  • With 𝑣=̇𝑥:
    • ̈𝑣+𝑛2𝑣=0⇒𝑣(𝑡)=𝑣0𝑥cos𝑛𝑡+𝑣0𝑦sin𝑛𝑡
    • 𝑥(𝑡)=∫𝑣(𝑡)d𝑡=𝑥0+𝑣0𝑥𝑛sin𝑛𝑡−𝑣0𝑦𝑛(cos𝑛𝑡−1)
  • Match initial conditions:
    • 𝑥(0)=𝑥0̇𝑥(𝑡)=𝑣0𝑥cos𝑛𝑡+𝑣0𝑦sin𝑛𝑡,̇𝑥(0)=̇𝑥0=𝑣0𝑥̈𝑥(𝑡)=−𝑛𝑣0𝑥sin𝑛𝑡+𝑛𝑣0𝑦cos𝑛𝑡,̈𝑥(0)=̈𝑥0=𝑛𝑣0𝑦
    • 𝑣0𝑥=̇𝑥0,𝑣0𝑦=̈𝑥0𝑛=2̇𝑦0+3𝑛𝑥0

Corrected from the notes

  • The handwriting writes 𝑣0𝑦=̈𝑥0𝑛=2̈𝑦0+3𝑛𝑥0 (double dot on 𝑦)
  • From the 𝑥 equation at 𝑡=0, ̈𝑥0=2𝑛̇𝑦0+3𝑛2𝑥0, so it is 2̇𝑦0; the following 𝑥(𝑡) and the typed solution on the next page both use ̇𝑦0
  • So:
    • 𝑥(𝑡)=4𝑥0+2̇𝑦0𝑛+̇𝑥0𝑛sin𝑛𝑡−(2̇𝑦0𝑛+3𝑥0)cos𝑛𝑡
  • Then 𝑦 from ̈𝑦=−2𝑛̇𝑥:
    • ̈𝑦=−2𝑛̇𝑥=−2𝑛(̇𝑥0cos𝑛𝑡+̈𝑥0𝑛sin𝑛𝑡)=−2𝑛̇𝑥0cos𝑛𝑡−(4𝑛̇𝑦0+6𝑛2𝑥0)sin𝑛𝑡
    • 𝑦(𝑡)=2̇𝑥0𝑛cos𝑛𝑡+(6𝑥0+4̇𝑦0𝑛)sin𝑛𝑡−(6𝑛𝑥0+3̇𝑦0)𝑡⏟secular+…

Secular drift

  • The secular term −(6𝑛𝑥0+3̇𝑦0)𝑡 grows without bound
  • This secular term arises only because of the linearization approximation, which is violated given enough time
  • It vanishes when
    • ̇𝑦0=−2𝑛𝑥0

Full Solution

  • 𝑥(𝑡)=4𝑥0+2̇𝑦0𝑛+̇𝑥0𝑛sin𝑛𝑡−(3𝑥0+2̇𝑦0𝑛)cos𝑛𝑡𝑦(𝑡)=2̇𝑥0𝑛cos𝑛𝑡+(6𝑥0+4̇𝑦0𝑛)sin𝑛𝑡−(6𝑛𝑥0+3̇𝑦0)𝑡−2̇𝑥0𝑛+𝑦0𝑧(𝑡)=𝑧0cos𝑛𝑡+̇𝑧0𝑛sin𝑛𝑡
  • ̇𝑥(𝑡)=̇𝑥0cos𝑛𝑡+(2̇𝑦0+3𝑛𝑥0)sin𝑛𝑡̇𝑦(𝑡)=−2̇𝑥0sin𝑛𝑡+(6𝑛𝑥0+4̇𝑦0)cos𝑛𝑡−(6𝑛𝑥0+3̇𝑦0)̇𝑧(𝑡)=−𝑧0𝑛sin𝑛𝑡+̇𝑧0cos𝑛𝑡

State Transition Matrix

  • These equations can be written as 3×3 sub-matrices of the state transition matrix:
    • [𝛿𝐫(𝑡)𝛿̇𝐫(𝑡)]=[Φ𝑟𝑟(𝑡)Φ𝑣𝑟(𝑡)Φ𝑟𝑣(𝑡)Φ𝑣𝑣(𝑡)][𝛿𝐫0𝛿̇𝐫0]
  • Φ𝑟𝑟(𝑡)=[4−3cos𝑛𝑡6(sin𝑛𝑡−𝑛𝑡)001000cos𝑛𝑡](7.53a)Φ𝑟𝑣(𝑡)=[1𝑛sin𝑛𝑡2𝑛(cos𝑛𝑡−1)02𝑛(1−cos𝑛𝑡)1𝑛(4sin𝑛𝑡−3𝑛𝑡)0001𝑛sin𝑛𝑡](7.53b)Φ𝑣𝑟(𝑡)=[3𝑛sin𝑛𝑡6𝑛(cos𝑛𝑡−1)000000−𝑛sin𝑛𝑡](7.53c)Φ𝑣𝑣(𝑡)=[cos𝑛𝑡−2sin𝑛𝑡02sin𝑛𝑡4cos𝑛𝑡−3000cos𝑛𝑡](7.53d)

Example Trajectories

  • Curtis Fig. 7.6: trajectory of 𝐵 relative to 𝐴 in the comoving frame during five of the target’s orbits, target eccentricity 0.1: loops that drift steadily along 𝑦 (about 40 km over five orbits)
  • Curtis Fig. 7.7: closed 2-by-1 relative ellipse when a_B = a_A (from Dr. Healy's notes)
    • Curtis Fig. 7.7: same, target eccentricity 0: a closed ellipse, 2 (along-track) by 1 (radial), annotated 𝑒𝐵=0, 𝑎𝐵=𝑎𝐴
    • Same semimajor axis ⇒ same period ⇒ no drift
  • Curtis Fig. 7.8: relative motion of a particle vs its Clohessy-Wiltshire approximation (from Dr. Healy's notes)
    • Curtis Fig. 7.8: the actual relative motion slowly departs from the Clohessy-Wiltshire prediction over hundreds of km: the linearization breaks down

Graph View

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