Hybrid Online Trajectory Planning

• The paper presents a hybrid method that decomposes trajectory planning into discrete mode sequences and continuous optimizations for scalable, real-time performance.
• It leverages offline surrogate training and graph-based search to rapidly approximate edge costs while ensuring safety and dynamic feasibility.
• Empirical results across robotics applications demonstrate 1–3% optimality gaps and sub-100 ms planning times, validating its robust performance.

Hybrid methods for online trajectory planning represent a class of algorithms that decompose the motion planning problem into both discrete and continuous components, leveraging either structural hybrid models, computational decoupling, or algorithmic synergies (e.g., sampling + optimization, learning + MPC). These methods are distinguished by the integration of mode-switching logic, optimal transport over hybrid graphs, fast online feasibility via surrogate models, or fusion of learning and rule-based policies. This entry surveys the rigorous foundational models, algorithmic frameworks, representative instantiations, and computational trade-offs evidenced in current literature.

1. Hybrid System Formalism and Problem Decomposition

Hybrid online trajectory planning is fundamentally predicated on modeling the robot as a hybrid control system, which can be represented as $\mathrm{HL}=(FG, D, U, S, \Delta)$, where:

• $FG = \{(f_i, g_i)\}$ are the parametric continuous-dynamics vector fields for each locomotion mode $i$;
• $D = \{D_i\}$ are the domains in state space for each mode;
• $U = \{U_i\}$ are mode-specific control input sets;
• $S = \{S_{i,j}\}$ are guard surfaces (switching manifolds);
• $\Delta = \{\Delta_{i \rightarrow j}\}$ are the reset maps at mode transitions.

Trajectory planning within this formalism involves two coupled search problems: (i) finding (via discrete search) sequences of modes and switching events, and (ii) computing dynamically feasible continuous trajectories within each mode. The latter must respect domain constraints, guard conditions, and hybrid resets. The decoupling of these stages is critical for computational tractability, especially in systems with combinatorial mode diversity and high-dimensional continuous dynamics (Suh et al., 2019).

2. Algorithmic Architecture: Hybrid Graphs, Cost Surrogates, and Optimization

A unifying feature among hybrid trajectory planning methods is the representation of feasible motions as a state-lattice or graph $G=(V,E)$, whose nodes $V$ are sampled states partitioned by mode, and edges $E$ represent locally optimal motions within a mode or discrete edges for mode transitions. Edge costs are in general given by optimal control problems: $$c(s,s') = \min_{u(\cdot)} \int_0^T \ell_i(x(t),u(t))\,dt,\quad \text{subject to}\ \dot{x} = f_i(x)+g_i(x)u,\ x(0)=s,\ x(T)=s',\ u\in U_i.$$ Directly solving these boundary value problems for all $(s,s')$ is prohibitive at scale ($O(|V|^2)$), so a two-step surrogate approach is applied: Perform massive offline trajectory optimization between sample pairs to collect ground-truth cost data, then fit a regression model $\hat{W}_i(x^s, x^s')$ (e.g., SVR, neural net, nearest neighbor) as a cost surrogate. At runtime, this surrogate rapidly assigns edge weights for fast graph search (e.g., Dijkstra, A*) (Suh et al., 2019).

Subsequent to discrete path extraction, a final multi-phase optimal control smoothing step is performed—fixing the mode/guard sequence and re-optimizing the timings and continuous controls to further reduce suboptimality.

3. Representative Hybrid Methods: Instantiations and Variants

Several modern instantiations of hybrid online trajectory planning have been thoroughly validated:

• Multi-modal hybrid locomotion uses a graph over discretized states with mode-split nodes, edge costs learned via hundreds of thousands of offline optimal control solves, and fast graph search plus final smoothing. This methodology has been tested on hybrid double integrators, amphibious robots, and flying-driving drones, consistently finding plans within 1–3% of direct collocation optima and achieving 5–300 ms practical runtimes depending on state space size (Suh et al., 2019).
• Highway autonomous driving (learning-optimization hybrid): A two-layer architecture where a GNN predicts longitudinal velocity profiles, which inform safety-corridor construction for an MIQP-based lateral planner. Hard spatiotemporal non-overlap constraints guarantee collision avoidance at all times. The linearization of vehicle geometry via binary variables enables real-time feasibility (mean 54 ms total cycle), and the GNN speeds up and humanizes velocity selection, with achieved scenario-level success rates of 97% (Lu et al., 6 Aug 2025).
• Scenario-aware hybrid switching (rule vs. diffusion): Combines a fast, interpretable rule-based tracker for regular scenarios with a slower, highly generalizable diffusion-policy network, coordinated by a dual-timescale decision neuron that adaptively switches based on scenario score, frequency, and synaptic-like long-term adaptation. A proposal regulator modulates diffusion sample count for efficiency, and a fusion module ensures safety during transitions. Experiments confirm that the hybrid planner outperforms both constituents—achieving state-of-the-art results in urban "long-tail" driving challenges with no significant runtime penalty over mature rule-based systems (Fan et al., 30 May 2025).

4. Hybrid Planning in Different Robot Classes and Contexts

Hybrid online trajectory planning is not confined to a specific platform type but is broadly applicable where mode changes are fundamental:

• Articulated tracked robots: Modeled with a planar kinematic skeleton and reduced generalized coordinates, these systems switch between driving and traversing (flipper-assist) modes governed by geometric guards and velocity constraints. Trajectory optimization is decomposed into per-mode NLP subproblems solved in a receding horizon (single-switch) fashion, with map/terrain simplification and high-rate tracking control loops ensuring real-world viability at 5 Hz planning (Xu et al., 2023).
• Wheeled-legged quadrupeds: Decompose the high-dimensional hybrid trajectory optimization into per-wheel convex QPs and a base SQP, both receding-horizon. The full plan is synchronized and tracked by a hierarchical torque-level controller. Online replan rates reach 30–100 Hz, enabling robust, agile switching between walking and driving across rough terrain, confirmed in live DARPA SubT Challenge deployments (Bjelonic et al., 2019).
• Flying-crawling quadrotors: Leverage hybrid A* search with mode-labeled nodes for terrestrial vs. aerial primitives and minimum-snap polynomial smoothing. Transitions across the crawling/flying boundary are handled explicitly via trajectory pausing and local replanning. The method achieves sub-50 ms planning per cycle and smooth trajectory tracking across hybrid junctions (Hu et al., 2023).
• Cable-trailer quadruped systems: Exploit a two-stage pipeline—hybrid A* graph search for feasible mode sequences, followed by full nonlinear trajectory optimization with mode-indexed constraints and collision inequalities. Hybrid modes encode cable tautness, ensuring online applicability with typical solve times of 1–6 s (Zhang et al., 18 Apr 2024).
• Hybrid information in multi-agent scenarios: Planning under occlusion or partial observability is formulated as a dynamic hybrid game interleaving open-loop and feedback intervals. Efficient cubic-time solvers are applied for linear-quadratic games; iLQ-game iteration extends this to nonlinear, nonquadratic cases. This approach provides real-time capability for game-theoretic multi-agent planning in urban intersections and overtaking scenarios (Gupta et al., 2023).

5. Complexity, Runtime, and Empirical Validation

Hybrid trajectory planners are generally characterized by an upfront offline compute phase (optimal control problem datasets, regression fitting) amortized by highly efficient online execution:

• Offline cost: $O(N^2 T_\mathrm{opt})$ for surrogate training; $O(M d^3)$ for SVR regression; up to days for exhaustive mode-sequence enumeration where relevant.
• Online cost: $O(|V|^2)$ for edge cost assignments (typically $O(1)$–$O(\log M)$ per call); $O(|E|+|V|\log |V|)$ for shortest path search; $O(\mathrm{ms})$ to $O(100\,\mathrm{ms})$ for practical cycle times in representative machines.
• In all reviewed domains (locomotion, urban driving, cluttered flight), the hybrid method achieves (i) success and optimality guarantees (often to within 1–3% of the best known), (ii) real-time cycles ($<$1 s, frequently 10–300 ms), (iii) robust tracking and safety in both simulated and physical environments (Suh et al., 2019, Lu et al., 6 Aug 2025, Fan et al., 30 May 2025, Xu et al., 2023, Bjelonic et al., 2019, Hu et al., 2023, Zhang et al., 18 Apr 2024).

6. Limitations, Trade-offs, and Extensions

Empirical limitations include:

• Fixed or pre-enumerated mode sequences: Though relaxing this via learning (Fan et al., 30 May 2025) or hybrid A* (Zhang et al., 18 Apr 2024) reduces brittleness, globally optimal mode switching remains computationally intense.
• Static or deterministic environment assumptions are prevalent; robustification against stochastic predictions and dynamic obstacles is an active research area (Lu et al., 6 Aug 2025).
• Representation and surrogate modeling fidelity trade off sample complexity (offline) and online computation.
• Most methods target single-agent or well-structured environments; scaling to multi-agent or highly unstructured domains induces additional complexity.

Potential extensions include:

• Incorporation of probabilistic safety constraints (chance-constrained MIQP).
• Deployment of distributed hybrid games for multi-agent systems (Gupta et al., 2023).
• Fusion with model predictive control for disturbance rejection at the execution layer (Lu et al., 6 Aug 2025).
• Expansion of learned surrogates using active learning or adaptive sampling to handle nonstationary environments (Duburcq et al., 2019).

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In summary, hybrid online trajectory planning methods synthesize the discrete/continuous structure of robotic systems with computational surrogates and layered search/optimization, yielding scalable, robust, and real-time feasible solutions across diverse robotic and vehicle motion planning challenges (Suh et al., 2019, Lu et al., 6 Aug 2025, Fan et al., 30 May 2025).