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Two-impulse Rendezvous Planning about Thrusting Spacecraft on $\mathrm{SE}_2(3)$

Micah Condie, Abigaile Woodbury, James Goppert

arXiv:2609.28413v1eess.SYcs.SY

TL;DR

HCW assumes an unforced Keplerian reference, an assumption violated by missions requiring continuous thrust. The paper develops an SE2(3) rendezvous-planning framework with linearized gravity and control mismatch; in the circular-coasting special case, its residuals vanish identically, leaving only the gravity remainder as propagation error.

  • Problem

    HCW assumes an unforced Keplerian reference, an assumption violated by missions requiring continuous thrust.

    Source: §I · pp. 3–4

  • Method

    The framework develops two-impulse planning from SE2(3) log-error dynamics with linearized gravity and control mismatch.

    Source: §III · p. 11, §VIII · p. 41

  • Results

    In the circular-coasting special case, residuals ε vanish identically, leaving only the gravity remainder as propagation error; at typical proximity scales, gravity components are small fractions of the first-order tidal term.

    Source: E. Special Cases · p. 28, §VIII · p. 41

  • Takeaways & Limitations

    The stated approximation domain uses attitude-error magnitude θ in (0, π), with quantities extending continuously to θ = 0.

    Source: §III · p. 11

Abstract

from arXiv · show

The classical Hill--Clohessy--Wiltshire equations assume an unforced Keplerian reference trajectory, an assumption that is violated by missions requiring continuous thrust. We address this limitation with a relative motion framework on the $\mathrm{SE}_2(3)$ Lie group that encodes position, velocity, and attitude in a unified geometric state. For computational tractability we linearize both the gravity mismatch and the body-frame control mismatch between the two vehicles, deriving tight analytic upper bounds on the neglected higher-order terms in each case. Under circular coasting Keplerian assumptions the framework recovers the Hill--Clohessy--Wiltshire equations exactly, establishing classical rendezvous theory as a special case rather than an independent linearization. For thrusting reference trajectories the state transition matrix acquires off-diagonal attitude--translation coupling blocks absent from classical formulations, and absorbing the control mismatch re-centers the linearization at the mean of the two vehicles' inputs. A two-impulse rendezvous planner derived directly from the state transition matrix accounts for both effects. Numerical simulations confirm recovery of the classical equations to machine precision, demonstrate successful rendezvous about a thrusting reference where classical planners fail, and validate the gravity and control mismatch linearization bounds throughout the transfer.

Nomenclature

The nomenclature defines variables for the spacecraft states, inputs, error bounds, and planning system.

  • a and ā denote actual and reference body-frame accelerations, while a_m is their mean.Source: Nomenclature · p. 1
  • A, Ã and Φ, Φ̃ denote matched- and mismatched-control system matrices and free and forced state transition matrices.Source: Nomenclature · pp. 1, 3
  • b(T, t₀) and k_b denote the forced log-error system’s particular integral and its contribution to arrival velocity error.Source: Nomenclature · p. 2
  • K_p and K_R are arrival-velocity gains on initial position and attitude error, respectively.Source: Nomenclature · p. 2
  • The remaining symbols define position, velocity, attitude, gravity, Jacobian, and control-mismatch quantities used in the framework.These include Q and M blocks, T_c, α(θ), β(θ), γ₁, γ₂, c(θ), c_C, θ, φ, ν̃, and m̃.Source: Nomenclature · p. 2
  • Δa and Δω are actual-minus-reference acceleration and angular-velocity mismatches; ω_m is the mean body-frame angular velocity.Source: Nomenclature · p. 3

I. Introduction

The introduction motivates relative-motion planning for thrusting spacecraft and presents an SE2(3) framework that recovers HCW under coasting assumptions. It outlines the state, inputs, and relative-error representation used to develop that framework.

I. Introduction

  • HCW planning relies on an unforced Keplerian reference, an assumption violated by continuous station-keeping and low-thrust transfers.Source: §I · pp. 3–4
  • The paper proposes analytic bounds for gravity and control-mismatch linearizations and a two-impulse planner derived from the state transition matrix.Source: §I · pp. 4–5
  • Under circular coasting Keplerian assumptions, the linear time-varying system recovers HCW exactly, while thrusting references require attitude–translation coupling absent from classical formulations.Source: §I · p. 5

A. Lie-Group Preliminaries

  • SE2(3) groups attitude, velocity, and position into one spacecraft state, with configuration error mapped by Log to a vector in R9 for analysis.Source: A. Lie-Group Preliminaries · p. 6

C. Inputs and Equations of Motion

D. Configuration-Tracking Error

E. Log-Error Dynamics

The log-error dynamics separate kinematic and input-mismatch effects from gravity mismatch. The gravity contribution acts on velocity coordinates and is linearized with bounded attitude-induced and separation-dependent remainders.

E. Log-Error Dynamics

F. Action of the Gravity Term on Velocity Coordinates

III. Gravity Linearization and Error Bounds

A. Gravity Taylor Expansion

  • A second-order Taylor expansion of the two-body field about the reference position yields a linear gravity-gradient term and a separation-dependent remainder.Source: A. Gravity Taylor Expansion · pp. 11, 13

B. Linear Time-Varying Error System

The matched-control linear time-varying system uses the reference input and drops the bounded gravity remainders. Its state transition matrix is computed numerically for general thrusting references.

IV. Recovery of HCW as a Special Case

Under circular coasting Keplerian assumptions, the relative-motion system recovers HCW exactly, with the same gravity approximation as the classical derivation. The control-mismatch treatment extends the system to arbitrary known mismatch with an analytic residual bound.

IV. Recovery of HCW as a Special Case

  • Under the standard HCW assumptions, the LTV system recovers the classical Hill–Clohessy–Wiltshire equations exactly.The relative position is expressed in LVLH coordinates; the reference is circular and coasting, with zero initial attitude error.Source: §IV · p. 15
  • The SE2(3) recovery makes the same approximation as the classical derivation: it drops only the second-order gravity Taylor remainder.The zero-attitude-error set is invariant, so the recovery adds no attitude-induced approximation.Source: §IV · pp. 16–17

V. Control Mismatch Linearization

  • The control-mismatch term is linearized to first order in the state, with the residual bounded analytically.This yields a forced LTV system for arbitrary known mismatch; matched control is the special case where the mismatch is zero.Source: §V · p. 17

A. Block Structure of the Inverse Left Jacobian

The inverse left Jacobian’s block structure separates acceleration-mismatch effects from angular-velocity coupling. The control-mismatch linearization uses that structure to collect first-order terms and bound the residual.

A. Block Structure of the Inverse Left Jacobian

B. Linearization and Residual Bounds

  • The induced-norm bound for M is attained when its translation argument is parallel to the attitude error.The maximization over direction yields the bound stated in Lemma 4.Source: B. Linearization and Residual Bounds · pp. 19, 21
  • The first-order control-mismatch terms are collected into a linear state term with the input as coefficient.The expansion identifies this term with the Jacobian correction evaluated at half the mismatch input.Source: B. Linearization and Residual Bounds · p. 22

C. Forced Linear Time-Varying Error System

The control-mismatch linearization produces a forced LTV error system, with attitude error remaining autonomous. Its bounds support the system’s stated validity regime and feed into a rendezvous planner that includes attitude–translation coupling.

C. Forced Linear Time-Varying Error System

Dropping the three bounded remainders gives the forced linear time-varying system

  • Absorbing the first-order Jacobian correction re-centers the linearization at the mean of the two vehicles’ inputs.Matched control recovers the reference input as that mean.Source: Dropping the three bounded remainders… · p. 23
  • The forced LTV linearization requires small attitude error and relative separation compared with orbital radius.The logarithmic coordinate restriction is θ < π; sustained angular-velocity mismatch can drive the relative rotation to that limit.Source: Dropping the three bounded remainders… · pp. 23–24
  • The position residual scales with separation and is dominant at proximity-operations scales; the numerical study validates all three residual bounds along the transfer.The bounds are direction-maximized because the relative orientations needed to retain direction dependence are not known in advance; the stated constants are sharp.Source: Dropping the three bounded remainders… · p. 24
  • The rendezvous planner adds an attitude–translation coupling term that vanishes when both vehicles coast and dominates terminal error when either thrusts.The exact rendezvous condition is in log coordinates, so propagation is the planner’s only approximation.Source: Dropping the three bounded remainders… · p. 24

A. Problem Statement

The problem statement specifies impulsive velocity corrections over a transfer horizon from a given log-error state. The terminal log-coordinate conditions correspond exactly to physical rendezvous.

A. Problem Statement

  • Given the initial log-error state and transfer horizon, the problem seeks impulsive velocity corrections.The supplied statement identifies the initial state, horizon, and corrections as the planning inputs and variables.Source: A. Problem Statement · p. 25
  • The terminal position and velocity conditions in log coordinates are equivalent to physical rendezvous without approximation.The position and velocity mappings are linear bijections under the stated injectivity restriction.Source: A. Problem Statement · p. 25

B. State Transition Matrix and Forced Response

The state transition matrix captures attitude-to-translation coupling for thrusting references, while the impulse map converts between physical velocity increments and log coordinates. The resulting two-impulse solution includes an attitude correction and requires an invertible position–velocity transition block.

B. State Transition Matrix and Forced Response

  • For thrusting references, the mean-acceleration term propagates attitude error into translation, creating STM coupling blocks that vanish when the mean acceleration is zero.The attitude-to-position and attitude-to-velocity blocks are generally nonzero under thrusting; when both vehicles coast, translation and rotation decouple.Source: B. State Transition Matrix and Forced… · p. 25

C. Impulse Structure in Log Coordinates

  • The exact log-coordinate impulse map avoids an O(θ) velocity-mapping error; at θ = 5°, the naive mapping's transverse error is 137 times its magnitude error.The naive map misdirects the impulse by θ/2, while shortening its magnitude by the second-order factor 1 − θ^2/24 + O(θ^4).Source: C. Impulse Structure in Log Coordinates · p. 27

D. Departure and Arrival Impulses

  • The two-impulse solution is unique when the position–velocity STM block is invertible, and its departure impulse includes a correction for initial attitude error absent from classical formulations.The departure correction separates into cancellation of current velocity error, correction for initial separation, and correction for attitude error.Source: D. Departure and Arrival Impulses · pp. 27–28
  • For circular coasting transfers, degenerate horizons occur at nT = kπ within the first orbit; under thrusting or eccentric references, the singular roots must be found numerically.The condition number of the position–velocity block measures conditioning beyond the binary feasibility test.Source: D. Departure and Arrival Impulses · p. 28

E. Special Cases

The planner reduces to classical forms under matched-control and coasting assumptions, including the textbook HCW solution for circular coasting transfers. Under thrusting references, attitude–translation coupling can dominate translational error, with applicability depending on the coupling-to-tidal ratio and the transfer setting.

E. Special Cases

  • With matched control, control-mismatch residuals vanish and only the gravity remainder contributes propagation error; with a coasting reference, attitude error decouples from translation.For a thrusting chief and coasting deputy, the forcing depends on the reference trajectory and the accumulated position response distinguishes the planner from its matched-control form.Source: E. Special Cases · pp. 28–29
  • Under circular coasting Keplerian assumptions, the planner reduces to the classical HCW two-impulse solution and its textbook formula in LVLH coordinates.The planner identity and textbook formula agree to O(10^-16) relative in the numerical validation.Source: E. Special Cases · p. 29

F. Remarks on Scope and Extension

  • When κ ≳ 1, attitude–translation coupling can dominate translational error dynamics; when κ ≪ 1, an HCW-class planner suffices as a correction to the tidal term.The criterion does not itself predict the proposed planner's benefit; where both terminal errors are linear in θ, their ratio is independent of θ.Source: F. Remarks on Scope and Extension · p. 30
  • The planner's impulsive approximation requires burns short relative to the transfer horizon and, for thrusting references, effectively constant reference acceleration and angular velocity during each burn.Finite-burn corrections may be applied by centering the burn arc on the impulse time.Source: F. Remarks on Scope and Extension · p. 30

A. Setup

  • The numerical scenarios span nearly three orders of magnitude in initial κ and bracket the κ = 1 crossover; the setup uses comparators that isolate attitude coupling and circular-orbit assumptions.The proposed planner integrates the full 9 × 9 system, while the translational-only comparator zeros the attitude-coupling blocks; elliptical cases also include the Yamanaka–Ankersen comparator.Source: A. Setup · p. 32

B. Exact Recovery of the Classical Equations

Under circular coasting Keplerian assumptions, the framework recovers HCW exactly; across the tested scenarios, attitude–translation coupling explains when thrusting-reference planners need corrections beyond classical formulations. The experiments also assess conditioning, computational cost, mismatched control, and the bounds on discarded terms.

B. Exact Recovery of the Classical Equations

  • Under circular coasting Keplerian assumptions, the transition matrix agrees with textbook HCW to relative error 1.5 × 10^-16–1.3 × 10^-15 over nT ∈ (0, 2π].The textbook two-impulse formula agrees to O(10^-16) relative.Source: B. Exact Recovery of the Classical… · p. 32

C. Mechanism Isolation

  • The proposed planner corrects 98.2% of Row C terminal position error; in Row F, coupling-blind Yamanaka–Ankersen reaches 4523 m.Thrust alone does not degrade planning accuracy at zero attitude error; with nonzero attitude error, coupling is the channel by which thrust reaches translation.Source: C. Mechanism Isolation · p. 33, Table 2 · p. 33
  • In Row C, terminal velocity error is 1.18 × 10^-2 m/s for the proposed planner versus 0.833 m/s for the classical planner, a factor of 71.The planner enforces both terminal position and velocity conditions.Source: C. Mechanism Isolation · p. 34

D. Attitude Scaling and the Coupling Criterion

  • At high thrust, residuals scale as θ^1.94 on S1 and θ^1.38 on S5; at low thrust, they scale as θ^1.07 on S3 and θ^0.99 on S4.At high thrust, the larger attitude contribution to the departure impulse increases mid-course excursion and the second-order gravity remainder; at low thrust, attitude-induced gravity sets the accuracy floor.Source: D. Attitude Scaling and the Coupling… · pp. 34–35
  • The coupling correction exceeds 92% across tested attitude errors through 20°; the coupling overtakes the tidal term at 0.80° for chemical-thrust LEO and 0.34° for GEO electric propulsion.The criterion indicates whether coupling can dominate, but does not by itself predict the magnitude of the planner improvement.Source: D. Attitude Scaling and the Coupling… · p. 35

E. Conditioning and Computational Cost

  • Under thrust, degenerate transfer horizons shift from the coasting values by +1.8 × 10^-3π and −6.9 × 10^-3π on S1, and +6.7 × 10^-4π and −2.6 × 10^-3π on S3.At GEO, the tested 90-minute horizon has conditioning 1.1 versus 2.9 for the six-hour quarter orbit; terminal error is unchanged from 0.1 s to 60 s integration steps.Source: E. Conditioning and Computational Cost · pp. 35–36

F. Mismatched Control

  • On the geosynchronous scenario, the forced planner achieves 1.10 m terminal error versus 1436 m for the matched-control form applied naively.For chief thrusting with deputy coasting on S1, the forcing accounts for 9603 m of the naive variant’s 9892 m error.Source: F. Mismatched Control · p. 37
  • The mismatch residual bounds hold throughout the tested transfers, with two bounds attained near equality at interior times; a sustained angular-velocity mismatch of 1.45 × 10^-3 rad/s reaches the log-coordinate limit in 81 min.The bound coefficients must be evaluated along the trajectory in the slew case, where attitude-error magnitude changes during transfer.Source: F. Mismatched Control · p. 38

G. Gravity Linearization Bounds

The gravity bounds are uniformly tight across tested cases, while the applicability analysis identifies input knowledge and omitted perturbations as practical boundaries. Replanning substantially reduces terminal error, and the conclusions situate the framework as a generalization of HCW; the appendix supports the gravity-bound analysis with a singular-value derivation.

G. Gravity Linearization Bounds

  • All gravity bounds hold, with maximum ratios from 0.973 to 0.9995, showing they remain uniformly tight across the tested cases.The table reports transfer maxima; the near-saturated gravity bound on S1 corresponds to a radial worst case.Source: G. Gravity Linearization Bounds · p. 38, Table 5 · p. 39
  • At proximity-operation scales, the higher-order gravity error is two to four orders below the attitude-induced error; the errors become equal at 110 km in the stated low-Earth-orbit case at 5°.At 1000 km initial separation, the bound remains valid but is 5.5 times the tidal term, while the actual remainder is 0.97 times it.Source: G. Gravity Linearization Bounds · p. 39

H. Applicability Envelope and Replanning

  • Reference-thrust magnitude is the tightest knowledge requirement: retaining linearization residual as the dominant error requires thrust known to 1–2%.At 5° on the low-Earth-orbit scenario, terminal error grows 5.1 m per percent of thrust-magnitude error and vanishes at zero attitude error.Source: H. Applicability Envelope and Replanning · p. 39
  • Centered finite burns keep terminal error within 10% of the impulsive floor up to 8% of the transfer horizon at departure and 6% at arrival.Reference-frame rotation exceeds the finite-burn effect by a factor of 2.6 at every burn duration.Source: H. Applicability Envelope and Replanning · p. 40
  • Two interior replans reduce median terminal error from 19.6 m to 1.55 m and the 90th percentile from 36.0 m to 2.70 m, adding 4% to total impulse magnitude.The supplied figure shows the two error distributions; under perfect knowledge, replanning reduces error from 9.10 m to 0.57 m for a 1.4% impulse increase.Source: H. Applicability Envelope and Replanning · pp. 40–41, Figure 6 · p. 41
  • The model omits J2, differential drag, and differential solar radiation pressure; these can raise the error floor or become dominant on formation-keeping timescales.At 500 km, J2 contributes 0.59 m over a transfer and raises the zero-attitude-error floor from 2.2 mm to 0.50 m.Source: H. Applicability Envelope and Replanning · p. 41

VIII. Conclusions

  • Under circular coasting Keplerian assumptions, the framework recovers HCW exactly, while its two-impulse planner includes attitude–translation coupling for thrusting references.The coupling is represented by off-diagonal state-transition-matrix blocks absent from classical HCW-based planners and the Yamanaka–Ankersen matrix.Source: §VIII · pp. 41–42

A. Singular Values of the SO(3) Left Jacobian

  • The SO(3) left-Jacobian singular-value derivation separates the rotation-error axis from its orthogonal plane and analyzes the plane through a complex-number representation.The supplied derivation identifies the invariant axis and plane, then treats the plane map as a rotation and uniform scaling.Source: A. Singular Values of the SO(3) Left… · pp. 42–43

B. Commutator Norms for the Attitude-Induced Gravity Bound

The appendix derives commutator norms used in the attitude-induced gravity bound, establishes monotonicity of an inverse-Jacobian coefficient, and tests how attitude-axis and thrust-direction choices affect terminal error.

B. Commutator Norms for the Attitude-Induced Gravity Bound

  • The commutator-norm derivation uses z = S r̂, perpendicular to r̂ with norm θ sin φ, and resolves the second commutator using the angle between the attitude-error axis and radial direction.The symmetric matrix r̂zᵀ + z r̂ᵀ acts only on the span of r̂ and ẑ.Source: B. Commutator Norms for the… · p. 43

C. Monotonicity of the Inverse-Jacobian Coefficient

  • The coefficient c is strictly increasing on (0, π), and the inequality m_a > 2m_p becomes tight in the small-angle limit as their ratio approaches 2 from above.The appendix establishes the inequality by showing successive derivatives of h are nonnegative on [0, π].Source: C. Monotonicity of the… · pp. 43–44

D. Attitude Axis and Thrust Direction Dependence

  • The attitude-axis sweep gives S1 terminal errors from 1.88 to 39.8 m, while the bound ratio stays between 0.973 and 0.99996 across all 200 axes.The nominal axis produces 9.10 m, at the thirteenth percentile; the bound ratio has no correlation with terminal error.Source: D. Attitude Axis and Thrust Direction… · p. 44
  • At fixed attitude axis, sweeping thrust direction produces a 13.8-fold spread in terminal error, and neither angle to the radial direction nor angle to acceleration alone explains the pattern.The reported dependence is joint in the two directions.Source: D. Attitude Axis and Thrust Direction… · p. 44

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