MIT OpenCourseWare http://ocw.mit.edu 16.346 Astrodynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Lecture 26 The Clohessy-Wiltshire Equations of Relative Motion Clohessy-Wiltshire Equations† We begin with the equations for the restricted three-body problem d2r dr  Gmm1 Gmm2 m dt2 +2ω × dt + ω × (ω × r) = − ρ13 ρ1 − ρ23 ρ2 where m1r1 + m2r2ρ1 = r − r1 ρ2 = r − r2 r = r3 − m1 + m2 ω = ω iζ with ω2 = G(m1 + 3 m2 + m) G(m13 + m2) r12 ≈ r12 With m1 and m2 on ξ -axis, then r1 = r1 iξ and r2 = r2 iξ To adapt these equations to the problem of a chase spacecraft m in pursuit of a target spacecraft m1 both moving about a central body of mass m2 , let both m and m1 become infinitesimal. As a result r2 will be zero so that r and ρ2 are the same vector. The vector ρ1 ≡ ρ is the position of the chase spacecraft relative to the target spacecraft. Further, the angular velocity is ω2 = Gm32 or ω2 r13 = Gm2 r1 so that the equations of motion of the chase spacecraft can be written as d2 ρ dρ ω2r13 +2ω × )] = − r+ ω × [ω × (ρ + r13dt2 dt rwhere ρ and r1 = r1 iξ are the position vectors of the chase and target spacecrafts, respectively. Note: r = ρ + r1 . This differential equation is non-linear because of the factor 1/r3 . However, with the use of the Taylor Series expansion, we write r2 =(ρ + r1) · (ρ + r1)= ρ2 +2ρ · r1 + r12 =1+2x iξ · i r1 + x 2 2 2 2r r r1 1 1 r11 = (1+2 iξ · i x + x 2)− 2 =1 − iξ · i x + r r1 r1 ··· ρ where x = . Therefore, r1 r13 =1 − 3 iξ · iρ ρ + O ρ22  r3 r1 r1 † W.H. Clohessy and R.S. Wiltshire, Journal of Aerospace Sciences, Vol. 27, No. 9, 1960, pp. 653–658. 16.346 Astrodynamics Lecture 26  and the equation will be linear if we ignore the higher order terms. Then d2 ρ dρ 1 dt2 +2ω × dt + ω × [ω × (ρ + r1)] = −ω2 1 − 3(iξ · ρ) r1 (ρ + r1) reduces to d2 ρ dρ+2ω × + ω × (ω × ρ)= −ω2 ρ +3ω2(iξ · ρ)iξ + O(ρ2)dt2 dt since the term with the factor (iξ · ρ)ρ is O(ρ2). Finally, ρ = ξiξ + η iη + ζiζ so that ω × (ω × ρ)= −ω2(ξiξ + η iη) and iξ · ρ = ξ Therefore, the differential equation for the motion of the chase spacecraft relative to the target spacecraft is d2 ρ dρ+2ω × = −ω2ζ iζ +3ω2ξ iξ + O(ρ2)dt2 dt or in scalar form d2ξ dη dt2 − 2ω dt − 3ω2ξ =0 d2η dξ+2ω =0 dt2 dt d2ζ + ω2ζ =0 dt2 It is sometimes convenient to express the position vector ρ ≡ r = x iθ + y i r − z i z i r1 = i r ω = −ω i z with x in the direction of motion iθ , y in the radial direction i r and i z = iθ × i r normal to the orbital plane. Then the equations of motion are‡ are d2x dy+2ω =0 dt2 dt d2y dx dt2 − 2ω dt − 3ω2 y =0 d2z + ω2 z =0 dt2 The Clohessy-Wiltshire equations are three simultaneous second-order, linear, constant-coefficient, coupled differential equations which are capable of exact solution. ‡ S.W. Shepperd, Journal of Guidance, Control, and Dynamics, Vol. 14, No. 6, 1991, pp. 1318–1322. 16.346 Astrodynamics Lecture 26  General Solution of the C-W Equations Introduce the dimensionless time variable τ = ωt so that the Clohessy-Wiltshire equations take the form d2x dy+2=0 dτ 2 dτ d2y dx dτ 2 − 2dτ − 3y =0 d2z + z =0 dτ2 The general solution of these equations, with initial conditions x0 , y0 , z0 ,˙x0 ,˙y0 and ˙z0 and using the notation dx =˙x, dy =˙y and dz =˙z ,is dτ dτ dτ ⎤ ⎥⎦ ⎡ ⎢⎣ ⎤ ⎥⎦ x 1 6 sin τ − 6τ 4 sin τ − 3τ 2 cos τ − 2 x0 y 0 4 − 3 cos τ 2 − 2 cos τ sin τy0= x˙ 0 6 cos τ − 6 4 cos τ − 3 −2 sin τx˙0 y˙ 0 3 sin τ 2 sin τ cos τy˙0 z = cos τ sin τ z0 z˙ − sin τ cos τz˙0 ⎡⎢⎣  ⎤⎥⎦ ⎡⎢⎣  16.346 Astrodynamics Lecture