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.

Robotic manipulators are essential for future autonomous systems, yet limited trust in their autonomy has confined them to rigid, task-specific systems. The intricate configuration space of manipulators, coupled with the challenges of obstacle avoidance and constraint satisfaction, often makes motion planning the bottleneck for achieving reliable and adaptable autonomy. Recently, a class of constant-time motion planners (CTMP) was introduced. These planners employ a preprocessing phase to compute data structures that enable online planning provably guarantee the ability to generate motion plans, potentially sub-optimal, within a user defined time bound. This framework has been demonstrated to be effective in a number of time-critical tasks. However, robotic systems often have more time allotted for planning than the online portion of CTMP requires, time that can be used to improve the solution. To this end, we propose an anytime refinement approach that works in combination with CTMP algorithms. Our proposed framework, as it operates as a constant time algorithm, rapidly generates an initial solution within a user-defined time threshold. Furthermore, functioning as an anytime algorithm, it iteratively refines the solution's quality within the allocated time budget. This enables our approach to strike a balance between guaranteed fast plan generation and the pursuit of optimization over time. We support our approach by elucidating its analytical properties, showing the convergence of the anytime component towards optimal solutions. Additionally, we provide empirical validation through simulation and real-world demonstrations on a 6 degree-of-freedom robot manipulator, applied to an assembly domain.