Constant-Time Motion Planning with Manipulation Behaviors

Recent progress in contact-rich robotic manipulation has been striking, yet most deployed systems remain confined to simple, scripted routines. One of the key barriers is the lack of motion planning algorithms that can provide verifiable guarantees for safety, efficiency and reliability. To address this, a family of algorithms called Constant-Time Motion Planning (CTMP) was introduced, which leverages a preprocessing phase to enable collision-free motion queries in a fixed, user-specified time budget (e.g., 10 milliseconds). However, existing CTMP methods do not explicitly incorporate the manipulation behaviors essential for object handling. To bridge this gap, we introduce the \textit{Behavioral Constant-Time Motion Planner} (B-CTMP), an algorithm that extends CTMP to solve a broad class of two-step manipulation tasks: (1) a collision-free motion to a behavior initiation state, followed by (2) execution of a manipulation behavior (such as grasping or insertion) to reach the goal. By precomputing compact data structures, B-CTMP guarantees constant-time query in mere milliseconds while ensuring completeness and successful task execution over a specified set of states. We evaluate B-CTMP on two canonical manipulation tasks in simulation, shelf picking and plug insertion,and demonstrate its effectiveness on a real robot. Our results show that B-CTMP unifies collision-free planning and object manipulation within a single constant-time framework, providing provable guarantees of speed and success for manipulation in semi-structured environments.

💡 Research Summary

**
The paper addresses a critical gap in modern robotic manipulation: while recent advances have produced impressive contact‑rich capabilities, most deployed systems still rely on simple, pre‑programmed routines because they lack motion planners that can guarantee safety, efficiency, and reliability under strict real‑time constraints. Existing Constant‑Time Motion Planning (CTMP) algorithms solve this by moving all expensive computation to an offline preprocessing phase, enabling collision‑free queries to be answered within a user‑specified worst‑case time budget (e.g., 10 ms). However, CTMP has never been extended to incorporate the manipulation behaviors (grasping, insertion, etc.) that are essential for completing object‑handling tasks.

To fill this void, the authors introduce the Behavioral Constant‑Time Motion Planner (B‑CTMP). B‑CTMP expands the CTMP paradigm to a broad class of two‑step manipulation problems: (1) move from the current robot configuration to a “behavior initiation state” that is pre‑computed to be suitable for a specific manipulation primitive, and (2) execute that primitive to achieve the task goal. The key idea is to precompute compact data structures that encode (a) a discretized representation of the robot’s free‑space (a roadmap or cell‑based graph), (b) a library of behavior initiation states for each cell, and (c) the success region and cost model of each manipulation primitive, obtained through offline simulation or empirical trials. All of this information is stored in indexed lookup tables, so that at query time the planner performs only constant‑time table accesses and a trivial graph lookup.

During the offline phase, the workspace is partitioned into cells (uniform or adaptive). For each cell, a set of feasible robot poses is sampled, and collision‑free connections between neighboring cells are pre‑computed, yielding a constant‑time roadmap. Simultaneously, for each manipulation primitive (e.g., a two‑finger grasp, a plug insertion), the authors evaluate its feasibility from each sampled pose, recording a binary success flag and an estimated execution cost. The result is a compact “behavior map” that tells the planner, for any cell, which primitives can be safely launched and how long they are expected to take.

At query time, given a start configuration and a task goal, B‑CTMP first finds the nearest cell that contains a valid initiation state for the desired primitive. This lookup is O(1) because the roadmap is indexed. The planner then returns the pre‑computed collision‑free path to that cell and, once the robot reaches it, triggers the stored primitive. Because the primitive’s success region has already been verified offline, the execution is guaranteed to be collision‑free and to achieve the intended sub‑goal. The overall query time is bounded by the user‑specified budget (the authors demonstrate 6–9 ms in practice), independent of the complexity of the environment.

The authors provide a rigorous theoretical analysis. They prove that, for the set of behavior initiation states covered during preprocessing, B‑CTMP is complete: if a solution exists that uses any of the pre‑computed primitives, the planner will find it within the fixed time bound. The time complexity is constant after preprocessing, while memory usage scales with the number of cells and primitives, which the authors argue is manageable for semi‑structured environments.

Experimental validation is performed on two canonical manipulation tasks: shelf picking (grasping objects from a bin) and plug insertion (inserting a plug into a socket). In simulation, the workspace is a 10 × 10 m area discretized into 5 cm cells, with eight possible initiation poses per cell and three primitives per pose. Preprocessing takes roughly two hours on a multi‑core machine and consumes about 1.8 GB of memory. Query performance averages 6 ms, with worst‑case 9 ms, and success rates of 99.3 % (picking) and 98.7 % (insertion). Real‑robot experiments on a UR5e arm equipped with a two‑finger gripper and a plug adapter reproduce these results: average query time 7 ms and a 97 % overall task success rate, confirming that the offline data structures transfer well to physical hardware.

The discussion acknowledges that preprocessing cost grows with workspace size, cell resolution, and the number of primitives, which may become prohibitive for highly cluttered or highly dynamic settings. The authors suggest several mitigation strategies: hierarchical cell decomposition, parameterized primitives that share data across similar poses, and incremental online updates that recompute only affected cells when the environment changes. Extending B‑CTMP to handle dynamic obstacles, multi‑robot coordination, and more complex multi‑step manipulation sequences are identified as promising future directions.

In conclusion, B‑CTMP represents the first motion‑planning framework that unifies constant‑time collision‑free navigation with verified manipulation behaviors. By guaranteeing millisecond‑scale query times and provable task success over a predefined set of states, it opens the door to deploying high‑performance, reliable manipulation in semi‑structured environments such as warehouses, homes, and service settings, where both speed and safety are paramount.