Safe Interval Path Planning (SIPP)

• Safe Interval Path Planning (SIPP) is a motion planning technique that identifies contiguous time windows during which a configuration remains collision-free in dynamic environments.
• It transforms exponential configuration-time search spaces into tractable augmented states by indexing nodes with spatial configurations and safe temporal intervals.
• Extensions of SIPP integrate kinematic, kinodynamic, and uncertainty-aware approaches, enhancing its scalability and applicability in multi-agent and high-dimensional scenarios.

Safe Interval Path Planning (SIPP) is a class of trajectory and motion planning algorithms that leverage the explicit computation and exploitation of maximal contiguous time windows—safe intervals—during which locations, configurations, or transitions are guaranteed to be free of collisions with dynamic obstacles. SIPP systematically transforms the exponential (configuration × time) search spaces of dynamic environments into tractable augmented spaces indexed by space and localized safe temporal sets, yielding both combinatorial efficiency and optimality guarantees. The methodology has become foundational in dynamic single- and multi-agent pathfinding, robotics under kinematic and kinodynamic constraints, uncertainty-aware planning, and high-dimensional manipulation.

1. Core Principles and Classical Formulation

At the heart of SIPP is the notion of a "safe interval": for a given configuration (e.g., a vertex in a roadmap, robot state, or joint configuration), a safe interval is a maximal time window [t_start, t_end] over which that configuration is guaranteed to remain free from collisions with dynamic obstacles. In the canonical SIPP (Ren et al., 2021, Yakovlev et al., 2020), a state is a tuple (v, I), where v is a configuration and I a distinct safe interval at v.

SIPP executes an A*-like search, using (configuration, interval) pairs as nodes, with g-values representing earliest feasible arrival times and h-values given by admissible lower bounds (e.g., Euclidean/Manhattan distance divided by maximum speed). Expansion of a state considers departures and successors only in temporal regions resulting in valid arrivals within the corresponding safe intervals of neighbor nodes.

By summarizing forbidden times at each configuration as a set of nonoverlapping intervals, SIPP achieves computational complexity O(n·k), where n is the configuration-space cardinality and k the maximum number of intervals per configuration—often several orders of magnitude smaller than explicit time discretization. For each expansion, only as many successors are generated as there are reachable safe intervals at neighbor nodes (Ren et al., 2021, Yakovlev et al., 2020, Huang et al., 16 Sep 2024).

2. Safe Interval Computation and State-Space Construction

Safe intervals are determined by taking the complement of the union of obstacle-occupancy intervals at each configuration:

$$\text{SI}(v) = \{ \text{maximal } [t_{\min}, t_{\max}] \subset [0, T] \mid v \notin \mathcal{O}_v(t) \ \forall t \in [t_{\min}, t_{\max}] \}$$

in which $\mathcal{O}_v(t)$ is an occupancy predicate that accounts for both static and dynamic obstacles.

Generalizations extend from pure configuration-vertex intervals to edge-safe intervals, as in the "Safe Interval Motion Planning for Quadrotors" framework, where for each edge, collision intervals $\text{CI}(e)$ are computed followed by aggregation of safe intervals on the edge as $SI(e) = T \setminus \text{CI}(e)$ under minimal traversal time constraints (Huang et al., 16 Sep 2024). In high-DoF manipulator settings, safe intervals are extracted per configuration by binarized collision checking across sampled time grids and collated via connected-component analysis (Kerimov et al., 27 Dec 2024).

For multi-agent prioritized and reservation-based schemes (e.g., SIPPwRT, TP-SIPPwRT), safe intervals are dynamically updated via reservation tables, using interval set operations and Offset functions to enforce collision avoidance across both space and time (Ma et al., 2018, Ali et al., 2021).

3. Planning with Kinematic and Kinodynamic Constraints

Classical SIPP assumes instantaneous velocity and turning, but this assumption fails for robots with acceleration, velocity, and non-holonomic constraints. Several extensions have been developed:

• Discrete Kinematic SIPP: States are augmented to include orientation and quantized speed; transitions are parameterized by feasible acceleration, deceleration, and turning primitives, with traversal times computed under kinematic constraints (Ali et al., 2021). Admissible heuristics precompute lower bounds on time-to-goal using DP on the configuration × speed graph.
• Kinodynamic SIPP-IP: For strictly-kinodynamic agents (bounded acceleration/deceleration), "waiting intervals" replace safe intervals: they represent sets of times and speeds for which feasible kinodynamic transitions can occur. Interval-projection ops and path-primitive tables allow complete and optimal planning despite the loss of instantaneous-wait properties (Ali et al., 2023).
• Non-Holonomic and Manipulation Domains: SIPPwRT enforces point-turn–then–forward-move primitives to represent the motion of car-like vehicles, and SI-RRT generalizes interval-based reasoning to the configuration space of high-DoF manipulators intertwined with randomized graph growth (Ma et al., 2018, Kerimov et al., 27 Dec 2024).

Completeness and optimality proofs for these variants are established by demonstrating that the expanded state space subsumes all A* time-configuration states reachable under the physical motion model (Ali et al., 2023).

4. Scalability, Multi-Agent, and Optimality Aspects

SIPP’s ability to compress the time dimension into intervals yields dramatic improvements in scalability compared to time-expanded A*; the pruning of wait states and dominance properties further reduce redundant exploration (Ren et al., 2021, Ma et al., 2018). For multi-agent settings, SIPP underpins a range of prioritized and coupled planners:

• Conflict-Based Search (CBS) with SIPP: Coupling SIPP at the low level with CBS at the high level admits sum-of-costs optimal, complete multi-agent plans in both discrete and continuous time, as in the CCBS algorithm (Andreychuk et al., 2019).
• Prioritized Multi-Agent SIPP: AA-SIPP(m) and TP-SIPPwRT plan for each agent sequentially, treating higher-priority agents' trajectories as dynamic obstacles. Completeness (in well-formed infrastructures) and empirical performance exceeding coupled approaches have been established (Yakovlev et al., 2017, Ma et al., 2018).
• Bounded Suboptimality: Weighted-SIPP and Focal SIPP variants allow explicit trade-off of optimality for computation via heuristic inflation or focal windows; formal suboptimality guarantees and experimental guidelines for parameter selection have been developed (Yakovlev et al., 2020).

5. Uncertainty-Aware and Real-Time Safe Interval Planning

Modern SIPP extensions address uncertainty in dynamic obstacle predictions and high-dimensional deployment environments:

• Conformal Prediction SIPP (CP-SIPP): CP quantifies uncertainty in predicted obstacle behaviors and scales SIPP states by confidence levels. Safe intervals become confidence-indexed, and an adaptive quantile mechanism provides finite-sample, distribution-free safety guarantees. Path risk can be explicitly weighted against travel time within the cost function (Liang et al., 22 Nov 2025). Under standard assumptions, the probability of trajectory-level collision is upper-bounded by the product of per-step miscoverage rates.
• Real-Time Quadrotor and Manipulation Planning: In "Safe Interval Motion Planning for Quadrotors," safe intervals are constructed for edges in dynamic visibility graphs, with trajectories parameterized as B-splines and optimized in 4D spatio-temporal corridors. Probabilistic completeness and real-time feasibility are demonstrated for dense and obstacle-rich settings, with >95% success rates and planning times below 300 ms (Huang et al., 16 Sep 2024). Similarly, SI-RRT for manipulators embeds safe-interval logic into RRT-Connect, achieving high success rates and lower solution costs across 1–300 moving obstacles (Kerimov et al., 27 Dec 2024).

6. Extensions, Theoretical Guarantees, and Limitations

Across its variants, SIPP and its derivatives maintain the following theoretical properties, subject to their model assumptions and completeness of safe interval extraction:

• Completeness: If a collision-free path exists (and the discretization is sufficiently fine), SIPP and its kinodynamic extensions will find it. This holds for prioritized and CBS-style multi-agent methods in well-formed instances (Andreychuk et al., 2019, Ali et al., 2021, Yakovlev et al., 2017).
• Optimality: The earliest arrival time at goal recorded during search is the minimum possible within the discretization and motion model (Ren et al., 2021, Ali et al., 2023).
• Complexity: State space growth with respect to time is controlled by the number of intervals, which is typically much smaller than explicit time expansion (Ren et al., 2021, Yakovlev et al., 2020).
• Safety under uncertainty: Confidence-indexed safe intervals allow explicit control of risk, with formal guarantees on trajectory-level collision probabilities (Liang et al., 22 Nov 2025).

Limitations of SIPP and its extensions arise in domains involving highly nonstationary/unpredictable obstacles, unknown environments, or situations where reachable safe intervals fragment excessively, leading to state explosion. For kinodynamic planning, safe-interval computation and the representation of high-dimensional intervals can become computational bottlenecks; batch collision checking and time adaptivity have been proposed to mitigate this (Kerimov et al., 27 Dec 2024). In non-convex environments or for highly-coupled multi-agent scenarios, additional coordination mechanisms (such as topological deformation classes or conflict-based branching) are required for practical solvability (Huang et al., 16 Sep 2024).

7. Practical Impact and Empirical Results

SIPP is deployed in a diverse array of robotic and multi-agent scenarios:

• Automated warehouses: SIPPwRT and TP-SIPPwRT yield planning subsecond for hundreds of AGVs and thousands of tasks, with linear scaling, outperforming post-processing approaches both in solution quality and throughput (Ma et al., 2018).
• High-DoF robotic manipulation: SI-RRT achieves high success and efficient planning in dynamic 3D workspaces with many moving obstacles (Kerimov et al., 27 Dec 2024).
• Quadrotor navigation: SIPP with dynamic visibility graphs and B-spline optimization achieves >95% success in highly dynamic environments and sub-second response (Huang et al., 16 Sep 2024).
• MAPF with kinematic/kinodynamic constraints: Prioritized SIPP with explicit speed and acceleration modeling allows for time-optimal, collision-free plans in large-scale settings, outperforming infinite-acceleration MAPF with post-processing (Ali et al., 2021).
• Uncertainty-aware planning: CP-SIPP maintains formal safety probabilities and increases the rate of collision-free plan generation under prediction uncertainty, at minor cost in solution length and time (Liang et al., 22 Nov 2025).

SIPP thus forms a methodological basis for tractable, optimal, and high-resilience motion planning in the presence of dynamic constraints, environment uncertainty, and complex robot capabilities.