M11-T2 — Relative Motion and Rendezvous

Learning objectives

After this lesson, you should be able to:

• State the Clohessy-Wiltshire equations and their closed-form state transition matrix.
• Explain the drift of a chaser from a radial offset and the condition for bounded relative motion.
• Plan a two-impulse transfer between two points of the relative frame and find its delta-v against the transfer time.
• Say how far the linear equations remain accurate.

The intuition

UAV M5-T4: Quaternion-based orientation treats the attitude of a vehicle; M11-T1: Orbits for Guidance: Elements, Energy and Delta-v gave the orbit of one spacecraft. Rendezvous asks about two: where is the chaser relative to the target, and what burn takes it to a chosen point? Hopper's landing is a rendezvous with a pad in a uniform gravity field; the orbital version is stranger, because the frame rotates.

The frame and the equations

Fix a frame to the target on a circular orbit: x radially outward, y along the direction of motion, z out of the plane. The target's mean motion is $n$ (1.131×10⁻³ rad/s at 400 km, a period of 92.6 min). For a chaser close to the target (distance much smaller than the orbit radius), subtracting the two gravity fields and including the Coriolis and centrifugal terms of the rotating frame gives a linear system with constant coefficients:

$\ddot{x} = 3n^2x + 2n\dot{y} + a_x$, $\ddot{y} = -2n\dot{x} + a_y$, $\ddot{z} = -n^2z + a_z$. These are the Clohessy-Wiltshire equations. The out-of-plane motion is a separate oscillator at the orbital rate.

The counter-intuitive drift

A chaser 100 m above the target with no relative velocity does not stay there or fall toward the target. After one orbit it is back at the same height but 3770 m behind: $\Delta y = -12\pi x_0$ per orbit. The reason is energy: at the higher radius the chaser has the target's speed but not the circular speed for that radius, so it falls into a different orbit with a different period. Bounded relative motion needs an along-track velocity of $\dot{y}_0 = -2nx_0$ (−0.226 m/s for 100 m): then the chaser moves on a 2:1 ellipse and returns to its start.

A transfer

Given the initial relative state and a time $t$, the equations are solved for the impulse that takes the chaser to a chosen position. Starting 1000 m behind and arriving at a hold point 20 m behind in half an orbit (46 min) needs a first burn of 0.277 m/s radially (down) and a second burn of the same size to stop: 0.554 m/s in total. Longer transfers are cheaper: a quarter orbit costs 1.51 m/s, three quarters 0.22 and a full orbit 0.12 m/s. The path swings 245 m below the target's orbit.

How far the linear equations hold

Checked against the nonlinear two-body equations after half an orbit, the difference is 0.20 m for a 100 m separation, 1.5 m for 1 km and 145 m for 10 km: the error grows as the square of the separation. The approximation is excellent for approach manoeuvres inside a few kilometres.

Near a circular orbit the relative motion is linear. A radial offset drifts you backward by 12π times the offset each orbit, bounded motion needs ẏ = −2nx, and a transfer between two relative positions has a closed-form delta-v that falls as the time grows.

Definitions

Target frame (LVLH)

Local vertical, local horizontal: x radially outward, y along the target's velocity, z along the orbit normal (completing the set).

Mean motion $n$

$n = \sqrt{\mu/R^3}$ of the target's circular orbit.

Clohessy-Wiltshire (Hill) equations

The linearized relative equations of motion in the target frame for a chaser near a circular orbit.

V-bar, R-bar

The lines along the target's velocity ($y$ axis) and along its radius ($x$ axis) used for approach.

Hold point

A point near the target at which the chaser stops relative to it before the final approach.

Derivation: the equations, the drift and the transfer

The relative acceleration in the rotating frame.

1. $$\ddot{\mathbf{r}}_c - \ddot{\mathbf{r}}_t = -\mu\left(\dfrac{\mathbf{r}_c}{r_c^3} - \dfrac{\mathbf{r}_t}{r_t^3}\right)$$

   The difference of the gravitational accelerations of chaser and target in the inertial frame (no thrust).

2. $$\dfrac{\mathbf{r}_c}{r_c^3} \approx \dfrac{\mathbf{r}_t + \boldsymbol\rho}{r_t^3}\left(1 - \dfrac{3\,\rho_x}{r_t}\right)$$

   Expand about the target for a separation $\boldsymbol\rho$ much smaller than the orbit radius, to first order.

3. $$\ddot{\boldsymbol\rho} + 2\boldsymbol\omega\times\dot{\boldsymbol\rho} + \boldsymbol\omega\times(\boldsymbol\omega\times\boldsymbol\rho) = -\dfrac{\mu}{r_t^3}\left(\boldsymbol\rho - 3\rho_x\hat{\mathbf{x}}\right)$$

   Transform to the frame rotating at $\boldsymbol\omega = n\hat{\mathbf{z}}$: Coriolis and centrifugal terms appear on the left.

4. $$\ddot{x} = 3n^2x + 2n\dot{y},\ \ddot{y} = -2n\dot{x},\ \ddot{z} = -n^2z$$

   Use $\mu/r_t^3 = n^2$ and write out the components: the CW equations.

The drift

1. $$y(t) = 6(\sin nt - nt)\,x_0\quad(\dot{x}_0 = \dot{y}_0 = 0, y_0 = 0)$$

   The along-track solution for a pure radial offset.

2. $$y(2\pi/n) = -12\pi\,x_0$$

   After one orbit $\sin nt = 0$ and $nt = 2\pi$: the chaser has drifted $12\pi x_0$ backward for an offset above the target (for $x_0 = 100$ m: 3770 m).

3. $$\dot{y}_0 = -2n\,x_0 \;\Rightarrow\; y(t) = \text{bounded}$$

   The secular term in y is $(-6nx_0 - 3n\dot{y}_0)\,t$, which vanishes when $\dot{y}_0 = -2nx_0$: the 2:1 ellipse of bounded relative motion.

The two-impulse transfer

1. $$\mathbf{r}(t) = \boldsymbol\Phi_{rr}\mathbf{r}_0 + \boldsymbol\Phi_{rv}\mathbf{v}_0^+$$

   The position at time t from the position and the velocity just after the first burn: $\boldsymbol\Phi_{rr}$ and $\boldsymbol\Phi_{rv}$ are the 2 × 2 blocks of the solution above.

2. $$\mathbf{v}_0^+ = \boldsymbol\Phi_{rv}^{-1}\left(\mathbf{r}_f - \boldsymbol\Phi_{rr}\mathbf{r}_0\right)$$

   Solve for the velocity that arrives at the target position $\mathbf{r}_f$. The first impulse is this minus the initial velocity; the second is minus the arrival velocity (to stop).

3. $$\Delta v_1 = \|\mathbf{v}_0^+ - \mathbf{v}_0\|,\ \ \Delta v_2 = \|\mathbf{v}_f^- - \mathbf{v}_f\|$$

   The two costs; they are equal here by the symmetry of the transfer.

The CW equations are the linearization of two-body gravity in a rotating frame. Their solution makes a transfer a 2 × 2 linear solve, and shows that along-track position is the quantity that drifts, at a rate set by the energy difference.

A worked example

A target on a circular 400 km orbit ($n = 1.1314\times10^{-3}$ rad/s, period 92.56 min).

1. Drift. A chaser 100 m above the target with no relative velocity is at +100.00 m radially and −3769.9 m along-track after one orbit (the formula $-12\pi x_0$ gives −3769.9 m).
2. Bounded motion. With the along-track speed −0.2263 m/s (= −2nx₀) the drift disappears: after one orbit the chaser is at (+100.00, +0.00) m.
3. Transfer. From 1000 m behind (at rest relative to the target) to 20 m behind: transfer time 0.25, 0.5, 0.75, 1.0, 1.5 orbits give total delta-v 1.508, 0.554, 0.224, 0.118, 0.554 m/s (23, 46, 69, 93, 139 min). The half-orbit transfer: first burn (−0.277, 0.000) m/s in (radial, along-track), a radial swing of 245 m.
4. Nonlinear check. After half an orbit with a 0.5 m/s radial kick, the CW prediction differs from the full two-body motion by 0.20 m (100 m separation), 1.53 m (1 km) and 145 m (10 km).
5. Straight-line approach. Holding a constant closing speed along the velocity bar needs a radial acceleration $a_x = 2nv$ of 2.26×10⁻⁴ m/s² (23 µg) at 0.1 m/s, 1.13×10⁻³ (115 µg) at 0.5 m/s and 2.26×10⁻³ (231 µg) at 1.0 m/s. Closing 980 m takes 163, 33 and 16 min, and the braking delta-v is about the closing speed.

import math
import gnckit as g

MU, RE = 3.986004418e14, 6378137.0
R0 = RE + 400e3
n = math.sqrt(MU / R0 ** 3)
Torb = 2 * math.pi / n
print(f'target on a circular 400 km orbit: mean motion n = {n:.4e} rad/s, period {Torb / 60:.2f} min')

def cw_phi(t):
    """Clohessy-Wiltshire state transition for (x radial up, y along track, xdot, ydot); the out-of-plane motion is a separate oscillator."""
    s, c = math.sin(n * t), math.cos(n * t)
    Prr = [[4 - 3 * c, 0.0], [6 * (s - n * t), 1.0]]
    Prv = [[s / n, 2 * (1 - c) / n], [2 * (c - 1) / n, (4 * s - 3 * n * t) / n]]
    Pvr = [[3 * n * s, 0.0], [6 * n * (c - 1), 0.0]]
    Pvv = [[c, 2 * s], [-2 * s, 4 * c - 3]]
    return Prr, Prv, Pvr, Pvv

def propagate(r0, v0, t):
    Prr, Prv, Pvr, Pvv = cw_phi(t)
    r = [sum(Prr[i][j] * r0[j] for j in range(2)) + sum(Prv[i][j] * v0[j] for j in range(2)) for i in range(2)]
    v = [sum(Pvr[i][j] * r0[j] for j in range(2)) + sum(Pvv[i][j] * v0[j] for j in range(2)) for i in range(2)]
    return r, v

print('Drift from a radial offset: a chaser 100 m above the target with no relative velocity')
r, v = propagate([100.0, 0.0], [0.0, 0.0], Torb)
print(f'  after one orbit the radial offset is {r[0]:+.2f} m and the along-track position {r[1]:+.1f} m (formula -12 pi x0 = {-12 * math.pi * 100:.1f} m per orbit)')
v0 = [0.0, -2 * n * 100.0]
r, v = propagate([100.0, 0.0], v0, Torb)
print(f'  with the along-track speed {v0[1]:+.4f} m/s (= -2 n x0) the drift disappears: after one orbit ({r[0]:+.2f}, {r[1]:+.2f}) m, and the motion is a 2:1 ellipse')

print('Two-impulse transfer from 1000 m behind the target (y = -1000 m, x = 0, at rest relative to it) to a hold point 20 m behind:')
rinit, rfin = [0.0, -1000.0], [0.0, -20.0]
def two_impulse(t):
    Prr, Prv, Pvr, Pvv = cw_phi(t)
    rhs = [rfin[i] - sum(Prr[i][j] * rinit[j] for j in range(2)) for i in range(2)]
    v1 = g.mat_solve(Prv, rhs)                       # the velocity just after the first burn
    rr, vv = propagate(rinit, v1, t)
    dv1 = math.hypot(*v1)
    dv2 = math.hypot(-vv[0], -vv[1])                 # stop at the hold point
    return dv1, dv2, v1, vv
print('  transfer time (fraction of an orbit)   dv1 (m/s)   dv2 (m/s)   total (m/s)   time (min)')
for frac in (0.25, 0.5, 0.75, 1.0, 1.5):
    t = frac * Torb
    dv1, dv2, v1, vv = two_impulse(t)
    print(f'  {frac:5.2f}                                {dv1:9.3f}   {dv2:9.3f}   {dv1 + dv2:10.3f}   {t / 60:9.1f}')
dv1, dv2, v1, vv = two_impulse(0.5 * Torb)
print(f'  half-orbit transfer: first burn ({v1[0]:+.3f}, {v1[1]:+.3f}) m/s in (radial, along-track); the path swings {max(abs(propagate(rinit, v1, k * Torb / 100)[0][0]) for k in range(51)):.0f} m radially')

print('Check against the nonlinear two-body motion (target on a circular orbit, chaser started from the same relative state):')
def two_body(state, dt):
    def f(t, s):
        r_ = math.hypot(s[0], s[1])
        return [s[2], s[3], -MU * s[0] / r_ ** 3, -MU * s[1] / r_ ** 3]
    return g.rk4_step(f, 0.0, state, dt)
def to_lvlh(tgt, cha, ang):
    """Relative position in the frame x = radial of the target, y = along its track (forward)."""
    dx, dy = cha[0] - tgt[0], cha[1] - tgt[1]
    ux, uy = tgt[0] / math.hypot(tgt[0], tgt[1]), tgt[1] / math.hypot(tgt[0], tgt[1])
    return dx * ux + dy * uy, -dx * uy + dy * ux
for sep in (100.0, 1000.0, 10000.0):
    # target at angle 0, speed along +y; the chaser is sep m behind along the track (y negative in the local frame)
    tgt = [R0, 0.0, 0.0, math.sqrt(MU / R0)]
    ang = -sep / R0
    cha = [R0 * math.cos(ang), R0 * math.sin(ang), -math.sin(ang) * math.sqrt(MU / R0) , math.cos(ang) * math.sqrt(MU / R0)]
    # give the chaser a 0.5 m/s radial kick so that the motion is not trivial
    cha[2] += 0.5 * math.cos(ang); cha[3] += 0.5 * math.sin(ang)
    t = 0.0; dt = 2.0
    # CW initial relative state
    rel0 = to_lvlh(tgt, cha, 0.0)
    cw_r0 = [rel0[0], rel0[1]]
    cw_v0 = [0.5, 0.0]
    while t < 0.5 * Torb:
        tgt = two_body(tgt, dt); cha = two_body(cha, dt); t += dt
    nl = to_lvlh(tgt, cha, 0.0)
    cw, _ = propagate(cw_r0, cw_v0, t)
    print(f'  separation {sep:7.0f} m, half an orbit: nonlinear ({nl[0]:+9.2f}, {nl[1]:+10.2f}) m, CW ({cw[0]:+9.2f}, {cw[1]:+10.2f}) m, difference {math.hypot(nl[0] - cw[0], nl[1] - cw[1]):8.2f} m')

print('Straight-line approach along the velocity bar (V-bar) at constant closing speed:')
for vc in (0.1, 0.5, 1.0):
    ax = 2 * n * vc
    print(f'  closing speed {vc:3.1f} m/s: radial acceleration needed to hold the line a_x = 2 n v = {ax:.2e} m/s2 ({ax / 9.80665 * 1e6:.0f} micro-g); time to close 980 m: {980 / vc / 60:6.1f} min; braking delta-v about {vc:.1f} m/s')

target on a circular 400 km orbit: mean motion n = 1.1314e-03 rad/s, period 92.56 min
Drift from a radial offset: a chaser 100 m above the target with no relative velocity
  after one orbit the radial offset is +100.00 m and the along-track position -3769.9 m (formula -12 pi x0 = -3769.9 m per orbit)
  with the along-track speed -0.2263 m/s (= -2 n x0) the drift disappears: after one orbit (+100.00, +0.00) m, and the motion is a 2:1 ellipse
Two-impulse transfer from 1000 m behind the target (y = -1000 m, x = 0, at rest relative to it) to a hold point 20 m behind:
  transfer time (fraction of an orbit)   dv1 (m/s)   dv2 (m/s)   total (m/s)   time (min)
   0.25                                    0.754       0.754        1.508        23.1
   0.50                                    0.277       0.277        0.554        46.3
   0.75                                    0.112       0.112        0.224        69.4
   1.00                                    0.059       0.059        0.118        92.6
   1.50                                    0.277       0.277        0.554       138.8
  half-orbit transfer: first burn (-0.277, +0.000) m/s in (radial, along-track); the path swings 245 m radially
Check against the nonlinear two-body motion (target on a circular orbit, chaser started from the same relative state):
  separation     100 m, half an orbit: nonlinear (    -0.74,   -1867.91) m, CW (    -0.60,   -1867.76) m, difference     0.20 m
  separation    1000 m, half an orbit: nonlinear (    -1.04,   -2767.91) m, CW (    -1.11,   -2766.38) m, difference     1.53 m
  separation   10000 m, half an orbit: nonlinear (   -10.69,  -11767.90) m, CW (   -52.23,  -11628.60) m, difference   145.36 m
Straight-line approach along the velocity bar (V-bar) at constant closing speed:
  closing speed 0.1 m/s: radial acceleration needed to hold the line a_x = 2 n v = 2.26e-04 m/s2 (23 micro-g); time to close 980 m:  163.3 min; braking delta-v about 0.1 m/s
  closing speed 0.5 m/s: radial acceleration needed to hold the line a_x = 2 n v = 1.13e-03 m/s2 (115 micro-g); time to close 980 m:   32.7 min; braking delta-v about 0.5 m/s
  closing speed 1.0 m/s: radial acceleration needed to hold the line a_x = 2 n v = 2.26e-03 m/s2 (231 micro-g); time to close 980 m:   16.3 min; braking delta-v about 1.0 m/s

In robotics

Every rendezvous and proximity operation uses these equations or a close relative.

• Approach corridors. Stations and servicing missions approach along the velocity bar (V-bar) or radial bar (R-bar), with a hold at about 20 m and a final approach at tens of centimetres per second. The equations give the delta-v and the drift the corridor must absorb.
• Passive safety. A trajectory on which a failed chaser drifts away without hitting the target is designed with the bounded-motion condition and the drift direction in mind: the chaser below the target drifts ahead, above drifts behind.
• Formation flying. A constellation maintained as a 2:1 relative ellipse needs only occasional corrections to cancel the along-track drift from small energy differences.
• Cost against time. The delta-v of a transfer falls as the time grows (0.55 m/s in 46 min, 0.12 m/s in 93 min), so mission planners trade fuel for time: the least-fuel choices are slow and use drift.
• Validity. The linear model is accurate to 1.5 m at 1 km after half an orbit but 145 m at 10 km: far-range phasing uses the orbital elements of the previous lesson, and near range the linear equations, with the correction applied by an onboard filter.

The lab M11-L2: Approach Along the Velocity Bar brings a chaser from 1 km to a hold point inside an approach corridor. The absolute orbits used for far-range phasing are in M11-T1: Orbits for Guidance: Elements, Energy and Delta-v; the navigation of the relative state uses the range and angle sensors of M7-T5: The Aiding Sensors of Aircraft and Spacecraft.

Common misconceptions

Key takeaways

1. Near a circular orbit, relative motion obeys $\ddot{x} = 3n^2x + 2n\dot{y}$, $\ddot{y} = -2n\dot{x}$, $\ddot{z} = -n^2z$, with a closed-form solution.
2. A radial offset x₀ produces an along-track drift of −12πx₀ per orbit: 3770 m for 100 m.
3. Bounded relative motion needs ẏ₀ = −2nx₀: −0.226 m/s for a 100 m offset.
4. A two-impulse transfer is a 2 × 2 solve: 1000 m behind to 20 m behind costs 0.554 m/s in half an orbit and 0.118 m/s in a full orbit.
5. The equations are accurate to 0.2 m at 100 m, 1.5 m at 1 km and 145 m at 10 km after half an orbit.
6. A constant-speed approach along V-bar needs a radial acceleration 2nv.

References and further reading

Each source is marked free or paid. Free to read does not mean free to reuse: the course resources page lists every source with its licence.

• W. H. Clohessy and R. S. Wiltshire, "Terminal guidance system for satellite rendezvous", Journal of the Aerospace Sciences 27(9), 1960, the original paper. Paid.
• H. D. Curtis, Orbital Mechanics for Engineering Students, 4th edition, Butterworth-Heinemann, 2020, the chapter on relative motion and rendezvous. Paid.
• W. Fehse, Automated Rendezvous and Docking of Spacecraft, Cambridge University Press, 2003, the practical reference on approach strategies. Paid.
• MIT 16.851 Satellite Engineering (OpenCourseWare) , which covers orbits and relative motion as part of satellite design. Free; licence CC BY-NC-SA 4.0.

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What’s next

Related content

Nothing below is required to finish this lesson.

• Before
  M11-T1: Orbits for Guidance: Elements, Energy and Delta-v

  Describe an orbit by its elements, use the vis-viva equation, and compute the delta-v of a Hohmann transfer and of a plane change.

• Lab
  M11-L2: Approach Along the Velocity Bar

  Bring a chaser from 1 km behind a station to a hold point at 20 m, inside an approach corridor, with the least delta-v, using the Clohessy-Wiltshire equations.

Other courses on this site treat neighbouring topics from their own angle:

• UAV M5-T4: Quaternion-based orientation
• UAV M5-T5: Attitude control
• Linear Algebra M7-T4: SO(3)
• Linear Algebra M10-T4: The Exponential Map
• SLAM M2-T3: Rotation Matrices, Quaternions, and Pose Representation

This course leaves the following to the pages that already teach them:

• Quaternion algebra and the rotation group
• Mission design, launch windows and interplanetary transfers (outside the scope of this site)
• Structural, thermal and propulsion design (outside the scope of this site)