Chance-Constrained Belief-Space Maneuver Planning for Autonomous Collision Avoidance Under Uncertainty

1. INTRODUCTION

• Operators must decide whether and when to maneuver as tracking updates refine risk, despite many encounters having fewer than ten CDMs before TCA.NASA CARA reports that conjunctions are generally first identified within seven days of TCA.Source: §1 · p. 1
• For an unmaneuverable secondary, waiting can bring informative tracking but reduces the time, fuel efficiency, and mitigation effectiveness available if intervention is needed.The primary spacecraft bears the mitigation burden when the secondary cannot coordinate or provide an authoritative maneuver plan.Source: §1 · p. 2
• Prior chance-constrained maneuver methods address risk in maneuver or trajectory optimization, while sequential collision-avoidance work considers evolving information without the terminal safety bound used here.The cited sequential work includes POMDP and MDP formulations [13] [3] [14] and analysis of delaying maneuvers [15].Source: §1 · p. 2
• The planner uses Gaussian orbital-state beliefs and chance-constrained MCTS to reason over future tracking and constrain the probability of an unsafe terminal belief at TCA.Its contributions include evaluation on NASA CARA scenarios and analysis of how measurement accuracy and tracking cadence affect decisions to wait or maneuver.Source: §1 · pp. 2–3

2. PROBLEM FORMULATION

• The central decision is whether an avoidance maneuver is necessary and, if so, whether to execute it now or wait for more tracking.The formulation treats this as a finite-horizon sequential decision problem under evolving state-estimate uncertainty.Source: §2 · p. 3
• Acting earlier preserves more time to change encounter geometry but commits before further tracking; waiting can refine the estimate but reduces intervention time.Source: §2 · p. 3
• The formulation represents the information–action tradeoff in belief space and uses probability of collision, Pc, as its probabilistic safety quantity.Source: §2 · p. 3

2.1 Conjunction Scenario and Scope

• The conjunction scenario comprises one maneuverable active spacecraft and one nonmaneuvering secondary resident space object.The secondary may be debris, a rocket body, or another object without a modeled avoidance action.Source: §2.1 · p. 3
• The planning horizon begins with the initial CDM and ends at TCA, with τ denoting time remaining until closest approach.τ = 0 corresponds to TCA.Source: §2.1 · p. 3
• Experiments cover short-duration encounters where the two-dimensional encounter-plane Pc model is appropriate; other encounters would need another risk metric.The framework is not restricted to that regime, but a higher-fidelity Pc method may increase computational cost.Source: §2.1 · p. 3

2.2 Belief-MDP Formulation

• A belief-MDP makes the current belief, rather than the unknown physical state, the state used for planning and risk evaluation.A POMDP allows noisy or incomplete observations; its belief-state is a distribution over possible physical states.Source: §2.2 · pp. 3–4
• At finite decision epochs, the belief transition represents a distribution over successor beliefs because future observations are not yet known.After an observation is realized, the corresponding posterior belief is obtained through a belief update.Source: §2.2 · p. 4

2.2.1 Belief Space

• The planner represents each object’s uncertain state with a multivariate Gaussian belief, using its state mean and covariance.Source: §2.2.1 · p. 4
• The joint belief includes both objects’ state estimates and uncertainty, while assuming independent estimation errors to omit cross-correlations.Source: §2.2.1 · p. 4

2.2.2 Action Space

• The action set is WAIT or MANEUVER; maneuvering applies a fixed 0.1 m/s impulsive change immediately at the chosen decision epoch.Fixing magnitude and direction isolates maneuver timing across epochs.Source: §2.2.2 · p. 4

2.2.3 Transition and Observation Model

• Beliefs evolve under the selected action and tracking information at prescribed decision times, with intermediate epochs set by tracking cadence.Source: §2.2.3 · p. 4
• At each decision epoch, the secondary belief is updated from its new measurement and the primary belief from assumed onboard navigation information.Source: §2.2.3 · p. 5
• During planning, future observations are sampled from the observation model and used in Kalman updates to generate possible successor beliefs.Different sampled observations let the planner assess how tracking may change encounter geometry and uncertainty before TCA.Source: §2.2.3 · p. 5

2.3.1 Probability of Collision

• At TCA, collision probability is the chance that the normally distributed relative position projected onto the encounter plane lies within the combined hard-body radius.The probability depends on both encounter geometry and its associated uncertainty.Source: §2.3.1 · p. 5

2.3.2 Chance-Constrained Objective

• The chance constraint limits the probability that a future terminal belief has Pc above δ; this work sets δ = 10−5 and α = 0.05.δ defines an unsafe terminal belief, while α caps the probability of reaching one.Source: §2.3.2 · pp. 5–6
• Because the constraint applies only at TCA, an intermediate belief with Pc above δ does not alone make an action infeasible.Later tracking observations may change the terminal belief and reduce the chance of exceeding the criterion.Source: §2.3.2 · p. 6
• For feasible actions, the objective balances expected terminal collision probability against whether a maneuver is executed.Expected risk measures average collision probability across terminal outcomes; violation probability instead measures how often risk exceeds δ.Source: §2.3.2 · p. 6

3. METHODOLOGY

• MCTS rollouts estimate each root action’s terminal risk-violation probability and expected collision probability; feasible actions are ranked by expected risk and maneuver cost.Figure 1 depicts the repeated process of selecting a root action, executing it, updating the belief, and planning again.Source: §3 · p. 6, Figure 1 · p. 7

3.1 Belief Propagation and Measurement Update

• Belief transitions propagate both objects’ Gaussian states and covariances, then update them with sampled full-state ECI tracking measurements.The linear measurement model uses H_j = I6; sampled future measurements create observation-dependent posterior beliefs.Source: §3.1 · pp. 6–7

3.2 Belief-Space Monte Carlo Tree Search

• MCTS samples future action-observation sequences to generate terminal beliefs, using rollout returns to guide search and exploration.The search value guides sampling; receding-horizon planning executes only the selected root action, incorporates the next observation, and replans.Source: §3.2 · pp. 7–8

3.3 Chance-Constrained Root Action Selection

• For each candidate root action, the fraction of terminal rollouts with Pc above δ estimates violation probability and determines chance-constraint feasibility.The same terminal outcomes estimate expected risk for ranking feasible actions; if none is feasible, the planner chooses the action with lowest estimated expected terminal collision probability.Source: §3.3 · pp. 8–9

4. EXPERIMENTS

• 96 evaluation configurations combine 8 conjunctions, 3 measurement-quality levels, and 4 tracking cadences.The evaluation varies tracking information to examine its effect on planner decisions.Source: §4 · p. 9

4.1 Conjunction Dataset and Scenario Selection

• Eight debris-secondary conjunctions near the operational maneuver-decision window are selected after a two-dimensional Pc validity assessment.The NASA CARA dataset provides historical conjunction messages with state estimates and covariances at TCA; these initialize beliefs propagated backward to the planning horizon.Source: §4.1 · p. 9

4.2 Tracking and Measurement Configurations

• Measurement quality and cadence vary independently; 2, 4, 8, and 24 h schedules set decision epochs for secondary updates and maneuver reconsideration.Quality levels represent favorable, typical, and degraded tracking conditions, derived from the 10th, 50th, and 90th percentiles of radar noise.Source: §4.2 · pp. 9–10

4.3 Baseline Policies and Ablations

• Rule-based policies maneuver only after a selected TCA-time threshold and when current Pc exceeds δ, while sharing measurements and belief updates with the planner.Decision thresholds span 28, 12, 6, and 3 h before TCA.Source: §4.3 · pp. 10–11
• The optimistic offline policy commits to WAIT when idealized future updates predict Pc < δ at TCA; an ablation replaces explicit terminal-feasibility enforcement with a soft Pc penalty.The offline forecast assumes measurements contract covariance without shifting the predicted trajectory; realized stochastic measurements may not confirm it.Source: §4.3 · p. 11

5.1 Safe Resolution Without Maneuver

• The 3 h rule resolves approximately 61–77% of encounters without maneuvering, while 10.6–11.2% of episodes terminate above δ.Later rule-based decisions allow more tracking before the fixed decision, but subsequent measurements can shift estimated geometry and risk.Source: §5.1 · p. 12
• The chance-constrained planner and soft-Pc ablation maintain zero observed terminal violations, while the chance-constrained planner resolves more encounters without maneuvering.The comparison shows that it improves no-maneuver resolution without increasing observed terminal risk.Source: Figure 2 · p. 12, §5.1 · pp. 12–13

5.2 Effect of Tracking Quality and Frequency

• No-maneuver resolution under best-quality sensors falls from 76% at a 2 h cadence to 38% at 24 h; under median quality it falls from 60% to 20%, while worst-quality resolution stays at or below 35%.All conjunctions resolved without maneuvering satisfy the terminal chance constraint.Source: Figure 3 · p. 13, §5.2 · pp. 13–14
• The chance-constrained planner outperforms the soft-Pc penalty ablation in eleven of twelve settings, with the largest gains under sparse measurements.The planners converge as measurements become more frequent.Source: Figure 3 · p. 13, §5.2 · p. 14

5.3 Information-Dependent Maneuver Timing

• For NORAD 38771 vs. 30802, the planner waits without maneuvering under best tracking, maneuvers later under median tracking, and intervenes immediately under worst tracking.The other illustrated cases either resolve through tracking alone or remain hazardous under continued waiting across tracking configurations.Source: §5.3 · pp. 14–15

5.4 Effect of Chance-Constrained Root Selection

• With best-quality tracking, WAIT remains feasible with one violating rollout among 27 (0.037 < α = 0.05), so the chance-constrained rule selects it while the soft-Pc penalty selects MANEUVER.The single violating rollout lowers the soft-penalty return, even though the estimated violation frequency remains below α.Source: §5.4 · p. 15
• Across 96 paired comparisons, the root rules disagree in 23 cases; each time, the chance-constrained rule selects WAIT and the soft-Pc penalty selects MANEUVER, with no observed increase in terminal violations.The disagreements occur near the constraint boundary.Source: §5.4 · p. 15
• Under worst-quality tracking, both root rules select MANEUVER when expected Pc is 1.1×10^-3, more than two orders of magnitude above δ.The chance-constrained rule treats WAIT as infeasible when available tracking information no longer supports it.Source: §5.4 · p. 15

• Approximately 40% of episodes reached TCA without maneuver with zero terminal violations, while the latest rule-based policy's threshold-violation rate rose to approximately 11%.Source: §6 · p. 15
• In one 325-node root search, WAIT remained feasible with 0.037 < α = 0.05, while the soft-Pc penalty selected MANEUVER after one violating rollout.Subsequent high-quality measurements resolved enough uncertainty for the encounter to reach TCA below the collision-risk threshold without maneuvering.Source: Figure 5 · p. 16
• Safe deferral ranged from 76% of episodes under the most accurate and frequent tracking to approximately 18–20% under the least informative conditions.For the same conjunction, the planner waited through TCA with high-quality tracking but maneuvered at the first decision epoch with low-quality tracking.Source: §6 · p. 17
• The terminal chance constraint identifies when uncertainty in future belief evolution makes continued deferral inadmissible.It evaluates whether future tracking is sufficiently likely to produce an acceptable terminal belief before allowing the planner to wait.Source: §6 · p. 17
• The formulation uses a fixed impulsive maneuver, prescribed tracking opportunities, and short-duration conjunctions within the two-dimensional encounter-plane model's validity regime.Continuous maneuver design and actively tasked observations are left for a more general joint planning problem.Source: §6 · p. 17

• The collision-risk calculation propagates each belief's mean state and covariance to TCA, retaining information gained from observations before the current time.It combines the propagated position covariances to form relative-position uncertainty.Source: APPENDIX · p. 19
• Short-duration, high-relative-velocity encounters permit an approximately linear relative-motion and static-covariance assumption for the two-dimensional reduction.Source: APPENDIX · p. 19
• An orthonormal basis normal to relative velocity projects the relative-position mean and covariance into the encounter plane, where collision probability is evaluated for each belief.The hard-body radius ρ is the sum of the primary and secondary hard-body radii.Source: APPENDIX · p. 19
• Fixed-order Gauss–Chebyshev quadrature evaluates the one-dimensional integral for the general two-dimensional Gaussian encounter-plane distribution.The method accommodates eccentric, potentially rotated covariance ellipses produced by the sensor model.Source: APPENDIX · p. 19