NAMO: Navigation Among Movable Obstacles

• NAMO is a robotics planning problem where robots compute paths that either bypass or manipulate obstacles to reach a target while adhering to physical constraints.
• Algorithmic strategies include sampling-based path and push planning, physics-driven push modeling, and optimization-based trajectory planning with explicit contact and friction modeling.
• Recent advances integrate uncertainty quantification, online learning, and socially-aware constraints to enhance performance in dynamic and cluttered real-world environments.

Navigation Among Movable Obstacles (NAMO) is a classically defined robotic planning problem in which a robot must traverse an environment containing obstacles that can either be circumvented or actively manipulated—most often by pushing or relocating the obstacles—en route to a goal. While originally rooted in 2D navigation and nonprehensile manipulation, NAMO’s scope now encompasses dynamic perception, decision-making under uncertainty, high-dimensional multi-contact planning, and even extensions to 3D disconnected environments and machine learning–driven motion generation.

1. Problem Formalization and Mathematical Foundations

NAMO is typically posed as a hybrid navigation-manipulation problem: the robot must compute a sequence of actions to reach a specified goal, via either collision-free motion or through the manipulation (push, pick, or rearrange) of some subset of obstacles. The formal model consists of:

• Configuration Space: $C \subseteq SE(2)$ for planar rigid body motion, possibly $SE(3)$ or higher for manipulators or 3D settings.
• Obstacle Set: $O = \{O_1, \ldots, O_m\}$, where each $O_i$ may be static (fixed) or movable (with current pose $q_i$).
• Robot Actions: Partitioned into motion actions (continuous or discrete moves in $C$) and manipulation actions (e.g., push $u_j = (O_i, \phi, d)$, grasp, or pick-and-place).
• Objective Function: A combined cost

$$\min_{P,S} \;\; \alpha \cdot \text{Length}(P) + \beta \cdot \text{PushCost}(S)$$

where $P$ is the path, $S$ the sequence of manipulations, and $SE(3)$0 > 0, subject to physics-based, collision, and reachability constraints (Emeli et al., 2020).

• Constraints: Collision-free motion at all times, proper physical evolution of objects (e.g., Newton’s laws, Coulomb friction), and feasibility of the final configuration.

NP-hardness holds for the most general NAMO, with special cases ($SE(3)$1 class: only single-object 'keyhole' manipulations required) being more tractable via greedy and backtracking methods (Muguira-Iturralde et al., 2022).

2. Algorithmic Approaches for NAMO

Algorithmic strategies for NAMO are diverse, reflecting the hybrid and combinatorially rich nature of the task. Common approaches include:

• Sampling-based Joint Path and Push Planning: Alternating between candidate path generation (e.g., using RRT-Connect with a shrunken robot approximation) and push planning using dynamic physics simulation. The best plan is selected by minimizing a combined cost over path length and manipulation effort (Emeli et al., 2020).
• Physics-Based Push Planning: Push actions are computed using contact models (quasi-static or dynamic) and simulated with engines such as Box2D, with push cost proportional to the displacement and the complexity of multi-object interactions (Emeli et al., 2020).
• Optimization-Based Trajectory Planning: Contact-Aware Motion Planning (CAMP) formulates the NAMO problem as a trajectory optimization problem with complementarity constraints for all robot–object and object–object contacts. Trajectories are computed by solving an augmented Lagrangian with quasi-Newton steps, explicitly modeling complementarity for contacts and friction (Wang et al., 5 Feb 2025).
• Affordance-Driven Planning: Obstacle movability is estimated online via geometric and force-based affordance extraction from perceptual data, guiding a contact-implicit motion planner that can decide whether to bypass, push, or lift obstacles (Wang et al., 2021).

Recent methods extend to hierarchical reinforcement learning, dynamic property estimation, and contact-aware planning in high-clutter or 3D environments (Yang et al., 18 Jun 2025, Kalluraya et al., 6 Jan 2026).

3. Explicit Uncertainty, Partial Observability, and Visibility Constraints

Advanced NAMO formulations move beyond deterministic, fully observable settings:

• Uncertainty Intervals: NAMOUnc models action failures, observation noise, model approximation, and unknown blockages as cost intervals. Decision-making is carried out by interval-aware criteria, comparing expected time and success probability of path bypass and object removal (Zhang et al., 16 Sep 2025).
• Visibility-Aware Planning: VANAMO (Visibility-Aware NAMO) augments the classic problem with constraints such that robot actions may only traverse regions already viewed by the robot’s sensors. The LaMB (Look and Manipulate Backchaining) algorithm recursively plans observations and object manipulations, explicitly maintaining accumulative visibility and dynamically switching between look and move subgoals (Muguira-Iturralde et al., 2022).
• Adaptive Object Models: Model-Predictive Path Integral (MPPI) control employing learned SE(2) object dynamics and online adaptation enables robust decision-making for unknown objects—determining on-the-fly whether to attempt manipulation or reroute (Dai et al., 2024).

These advances formally integrate probabilistic state estimation, online learning of physical parameters, and real-time adaptation to non-deterministic environments into the NAMO framework.

4. Extensions to Multi-Object, 3D, and Socially-Aware NAMO

NAMO has been generalized in several significant directions:

• 3D Environments and Bridge-Building: In disconnected 3D environments, such as separate planes at differing elevations, NAMO includes placing movable blocks to construct traversable connections, requiring sampling-based planners (such as BRiDGE) to search joint robot–object configuration spaces under placement and manipulation constraints (Kalluraya et al., 6 Jan 2026).
• High-Dimensional and Multi-Contact NAMO: Multi-object rearrangement and coordinated contact sequence planning are encoded via optimization over robot and object waypoints, time, and contact mode schedules (Wang et al., 5 Feb 2025).
• Socially-Aware NAMO (S-NAMO): Adding constraints relevant to human-robot interaction—evaluating the “social movability” (e.g., white-listing which objects are acceptable to manipulate), “social placement” (avoiding taboo zones such as doorways, carpets), and integrating social cost into the overall plan cost. S-NAMO further supports explicit reasoning about human feedback and social comfort (Renault et al., 2019).

5. Learning-Based and Hierarchical NAMO Solutions

Recent NAMO frameworks leverage both hierarchical policy learning and interaction-driven property estimation:

• Hierarchical Reinforcement Learning (HRL): The high-level policy generates base or arm pushing commands conditioned on path tracking and inferred object properties, while the low-level controller executes the whole-body motion necessary for stable, precise non-prehensile interaction (Yang et al., 18 Jun 2025). The estimator continuously updates obstacle mass, center-of-mass, and friction properties as the robot interacts.
• Structured Pushing and Curriculum Design: Restricting the action space to parametrized push commands simplifies learning while enabling the discovery of efficient, interpretable pushing strategies for clearing paths in clutter.

These approaches have demonstrated substantial improvements in success rate, path optimality, and time-to-goal over classical collision-avoidance or non-adaptive pushing baselines, particularly in densely cluttered settings.

6. Quantitative Performance and Empirical Validation

Controlled simulation and real-world benchmarks reveal NAMO’s advantages in high-clutter and dynamic scenarios:

Planner / Approach	Max Clutter% Solved	Success Rate (%)	Context
RRT only	18	100	2D, baseline (Emeli et al., 2020)
Straight-line push	37	40	2D, baseline
Joint NAMO planner	49	70–100	2D, RRT+push (Emeli et al., 2020)
CAMP (contact-aware)	n/a	95–100	up to 7 obstacles (Wang et al., 5 Feb 2025)
SVG-MPPI (movability)	n/a	54.5	real robot, clutter room (Weeda et al., 27 Feb 2025)
HRL + Estimation	n/a	85.3–92.7	Map1/Map2, 6 obs (Yang et al., 18 Jun 2025)
NAMOUnc (uncertainty)	n/a	100 (warehouse)	1–3 MOs (Zhang et al., 16 Sep 2025)

Empirical results consistently demonstrate that NAMO frameworks substantially outperform pure collision-avoidance planners, particularly when leveraging contact-awareness, uncertainty quantification, or property adaptation.

7. Current Limitations and Future Directions

• Computational Efficiency: Solving optimization-based or sampling-based NAMO planners in high-dimensional or highly-cluttered environments remains computationally intensive (e.g., CAMP back-end versus avoidance).
• Model Uncertainty and Adaptation: While learning-based object models and property estimators improve robustness, further advances are required for fast, generalizable adaptation—especially in 3D, dynamic, or human-populated spaces.
• Integration of Perception and Semantics: Many frameworks assume perfect or near-perfect perception; integrating full-stack visual detection, semantic understanding, and uncertainty modeling remains open.
• Extending to Social and Multi-Agent Contexts: S-NAMO highlights the need for rich social cost models, real-time human presence reasoning, and dynamic interaction policies.
• Bridging Algorithmic and Learned Planning: Unifying the strengths of sampling-based completeness, optimization-based smoothness, and HRL-driven adaptability is a principal research direction.

NAMO is thus a central problem at the interface of geometric motion planning, adaptive control, learning, and human-robot interaction, with active research spanning exact theoretical formulations, scalable algorithms, and embodied real-world systems (Emeli et al., 2020, Wang et al., 2021, Wang et al., 5 Feb 2025, Muguira-Iturralde et al., 2022, Dai et al., 2024, Zhang et al., 16 Sep 2025, Weeda et al., 27 Feb 2025, Yang et al., 18 Jun 2025, Kalluraya et al., 6 Jan 2026, Renault et al., 2019).