Geometric Planning

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1 paper in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

Jul 6Week of Sep 21

Latest papers 17

Sep 1, 2026cs.RO

SG-AMP: Scene-Graph-Guided Active Perception and Semantics-Aware Motion Planning for Pepper Plants

We present SG-AMP, integrating robust depth completion with input-conditioned uncertainty, persistent panoptic mapping, plant scene-graph reasoning, and semantics-aware active view-motion planning. Beyond inspecting uncertain observed regions, the scene graph explicitly hypothesizes unobserved pepper--peduncle attachments and directs close-range sensing toward them. Candidate views are selected according to expected information gain, while class-dependent motion costs distinguish protected peppers, peduncles, and stems from conditionally traversable foliage. On pepper data, the perception network achieves

55.27\%

semantic mIoU,

38.67\%

PQ, and

40.62\,\mathrm{mm}

depth RMSE, while input-conditioned uncertainty improves NYUv2 NLL from

-1.6518

to

-1.6925

and AUSE from

0.0102

to

0.0087

.

Rohit Menon, Shiva Rudra Lolla, Niklas Mueller-Goldingen +3

Humanoid Robots Lab, University of Bonn, Germany · Center for Robotics, Bonn, Germany · Institute of Geodesy and Geoinformation, University of Bonn +1

Depth Estimation 3D Perception Geometric Planning Planning

Aug 6, 2026cs.CV

Iterative Hybrid Discrete-Continuous Viewpoint Planning for UAV Photogrammetry

Unmanned aerial vehicle (UAV) photogrammetry requires camera networks that provide sufficient surface coverage, image overlap, parallax, and resolution, yet conventional flight patterns are often poorly adapted to scene geometry resulting in local reconstruction errors. This paper proposes an iterative hybrid discrete-continuous viewpoint planning method for targeted UAV photogrammetry from a proxy reconstruction. The method scores sampled surface points using photogrammetric heuristics based on frontality, imaging distance, parallax, and multi-view observation count, while also evaluating the full viewpoint set in terms of visibility, pairwise overlap, and graph connectivity. Candidate viewpoints are generated around weakly observed regions, refined using clustered Covariance matrix adaptation evolution strategy (CMA-ES) optimisation, and removed when redundant. The final flight path combines close-range detail viewpoints with wider model-coverage viewpoints, balancing local reconstruction quality with global image-network robustness. Evaluation on three synthetic scenes shows that the proposed method improves both reconstruction accuracy and completeness compared with prior UAV path-planning methods.

Alan Grech, Daniel Pisani, Andre Grima +3

University of Malta Msida, Malta · Stargate Studios Malta Mosta, Malta

Unmanned Aerial Vehicles Aerial Imagery Geometric Planning Reconstruction Fidelity Planning Iterative

Aug 2, 2026cs.RO

Sampling-Based Visibility Task Planning

Robot Task and Motion Planning (TAMP) algorithms enable autonomous operation by incorporating the specific functions and constraints of end-effector tools, such as grippers or soldering irons, directly into the planning process. In this paper, we explore sampling-based TAMP algorithms specifically designed for a critical subset of devices whose unique properties make traditional planning methods ineffective. Visibility-based instruments, such as exteroceptive sensors, cameras, flashlights and directional antennas, are essential across a vast array of human activities. The unique properties of these devices, and particularly, their field-of-view, render many widely used heuristics and distance metrics less effective. We introduce two new sampling-based algorithms, FOV-PRM and FOV-RRT, designed to tackle visibility-based tasks. FOV-PRM employs a hierarchical decomposition of the environment, leveraging the concept of visibility integrity, to efficiently sample configurations with a clear line-of-sight to the target. A specialized Inverse Kinematics solver enables FOV-RRT to "glance" in the direction of the target at opportune moments, facilitating the rapid discovery of key configurations. We show that FOV-PRM and FOV-RRT achieve a higher success rate and faster runtimes compared to adaptations of RRT, PRM and VIR, through both simulated and physical experiments.

Stav Ashur, Avishai Sintov

School of Mechanical Engineering, Tel-Aviv University, Israel

Geometric Planning Motion Planning Visibility Planning

Jul 21, 2026cs.RO

Stochastic Multi-Objective Kinodynamic Planning Against Adversaries

This paper addresses multi-objective kinodynamic planning in environments with stochastic hybrid adversaries that probabilistically transition to adversarial modes based on the ego state. The goal is to construct the Pareto-front of paths that trade off execution cost and the probability of safety constraint violation (risk). Existing chance-constrained planners evaluate risk over open-loop trajectories, yielding overly conservative solutions that fail to account for ego-agent reactivity. To address this limitation, we shift the planning space to sequences of closed-loop policies, and integrate sample-based risk evaluation directly into tree construction via Monte-Carlo particle rollouts. We first introduce Stochastic Multi-Objective RRT (SMO-RRT), for which we prove probabilistic completeness, followed by Stochastic Multi-Objective Stable Sparse RRT (SMO-SST), which leverages selective pruning to improve numerical performance at the cost of completeness. For both algorithms, we derive a finite-sample bound on the probability of chance constraint violation for systems with non-Gaussian, state-dependent uncertainty, enabling probabilistically safe planning in a broad class of environments applicable to multi-agent systems, social navigation, and autonomous driving.

Thomas Marshall Vielmetti, Daniel Cherenson, Dimitra Panagou

Geometric Planning Planning Stochastic Bounded Adversary Reactive Adversarial Robustness

Jul 8, 2026cs.RO

Shift & Drift: A Zero-Shot Benchmark for Generalizable and Robust Autonomous Driving Motion Planning

While closed-loop motion planners trained on large-scale, object-level datasets, e.g., nuPlan, demonstrate strong in-distribution (ID) performance, their generalization to novel urban topologies and recovery mechanisms following execution perturbations remain under-explored. To address this, we present Shift & Drift, a novel dual-track benchmark designed to rigorously stress-test motion planners across two critical axes of distribution shift: (1) The Semantic Shift Track leverages a novel conversion pipeline that transforms the aerial, DeepScenario Open 3D dataset into the nuPlan simulation framework. This enables zero-shot evaluation of planners trained on North American and Singaporean data against 1,182 scenarios spanning four German cities and the US city of San Francisco featuring dense pedestrian-cyclist interactions. (2) The State-Distribution Drift Track injects stochastic perturbations into the ego vehicle's dynamics to quantify robustness against compounding execution errors. Based on this, we systematically evaluate the failure modes of diverse planning paradigms under semantic and state-distribution shifts. While imitation learning methods achieve high scores in ID benchmarks, they exhibit significant failures under semantic shift, particularly in pedestrian-dense environments, and suffer from persistent drift when subjected to temporally correlated actuation noise. In contrast, the evaluated reinforcement-learning-based planner demonstrates more graceful degradation, maintaining higher safety and progress metrics across both tracks. Our findings reveal an empirical trade-off between imitation fidelity and closed-loop resilience, providing the community with a rigorous benchmark to evaluate progress toward reliable deployment.

Alessandro Canevaro, Hang Yu, Julian Schmidt +5

Mercedes-Benz AG, Research & Development, Sindelfingen, Germany · University of Tübingen, Tübingen, Germany · Karlsruhe Institute of Technology, ITIV, Karlsruhe, Germany

Autonomous Driving Simulation Geometric Planning Distribution Shifts Pedestrian Detection Motion Imitation

Jul 4, 2026cs.RO

Fast Asymptotically Optimal Kinodynamic Planning via Vectorization

Sampling-based motion planners have been shown to be effective for systems with complex kinodynamic constraints and high dimensionality. However, these algorithms struggle to achieve real-time performance, leading to recent efforts to parallelize planning. While GPU-accelerated planners have achieved significant speedups, existing approaches require specialized CUDA programming that limits accessibility and portability. We present Parallel Asymptotically Optimal Kinodynamic RRT (PAKR), a massively parallel kinodynamic planner leveraging JAX and the XLA compiler to achieve GPU acceleration through standard Python tooling. By combining our parallel planner with the AO-x meta-algorithm, we achieve asymptotic optimality through fast iterative replanning. We provide a theoretical analysis of probabilistic completeness, analyze the effects of batch size and branching factor on convergence, and demonstrate scalability to complex dynamics using the MuJoCo-XLA simulator. Experiments show competitive runtimes with state-of-the-art GPU planners and superior solution quality.

Yitian Gao, Andrew Lu, Zachary Kingston

Department of Computer Science, Purdue University

Geometric Planning Iterative Re-Planning Planning Kinematics Vectorization

Jun 30, 2026cs.RO

MVP-Nav: Multi-layer Value Map Planner Navigator

Zero-shot Object Goal Navigation (ZSON) with RGB-only perception poses a fundamental challenge for embodied agents, as the absence of explicit depth information introduces severe physical uncertainty and semantic-physical misalignment. Existing approaches either rely on high-level semantic reasoning without geometric grounding or learn end-to-end policies that lack explicit physical constraints, often resulting in semantically plausible but physically unsafe behaviors. In this paper, we propose MVP-Nav, a physical-aware RGB-only navigation framework that aligns perception, planning, and control with the real 3D world. MVP-Nav reconstructs explicit physical occupancy from monocular observations by leveraging 3D foundation models to project 2D semantic instances into 3D oriented bounding boxes, forming a global spatial semantic representation. To unify high-level semantic reasoning and low-level physical constraints, we introduce a Multi-layer Value Map (MVM) that integrates semantic priorities and reconstructed geometry into a shared cost space, enabling physically grounded geometric planning. Extensive experiments on zero-shot object navigation benchmarks demonstrate that MVP-Nav significantly outperforms existing depth-free methods, achieving state-of-the-art performance and validating that structured physical priors can effectively compensate for the absence of active depth sensors.

Wenyuan Xie, Shaokai Wu, Yijin Zhou +7

Shanghai Jiao Tong University

Object Goal Navigation Vision-Language Navigation Geometric Planning Menu Embodied Agents Semantic Geometry+1

Jun 29, 2026cs.RO

ReactiveBFM: Reactive Closed-Loop Motion Planning Towards Universal Humanoid Whole-Body Control

While current Behavior Foundation Models (BFMs) provide robust control priors for humanoids, they only execute pre-defined reference motions. As a result, they are vulnerable to environmental shifts and incapable of reactive whole-body coordination. Naively cascading them with generative motion planners fails to achieve true reactivity, as inevitable tracking discrepancies induce fatal cumulative exposure bias. To bridge this gap, we propose ReactiveBFM, a real-time closed-loop planning-control framework. At its core, we effectively mitigate exposure bias via a scheduled prefix sampling curriculum, forcing the generative planner to actively learn error-recovery behaviors from imperfect physical states rather than ground-truth trajectories. Systematically, to reconcile the severe latency mismatch between auto-regressive planning and high-frequency tracking, we introduce an asynchronous replanning mechanism. Combined with trajectory chunking to temporally ensemble spatial references, our system guarantees spatio-temporally fluid execution without physical jitter. Deployed on the Unitree G1 humanoid, ReactiveBFM demonstrates unprecedented physical agility across a vast repertoire of text-conditioned closed-loop motions. Notably, ReactiveBFM achieves zero-shot moving target reaching, showcasing intricate whole-body coordination and on-the-fly replanning. In sim-to-sim benchmarking under severe perturbations, ReactiveBFM achieves a 93.1% success rate, significantly outperforming cascaded open-loop baselines by 28.6%.

Xiao Chen, Weishuai Zeng, Xiaojie Niu +12

The Chinese University of Hong Kong · Shanghai AI Laboratory

Whole-Body Control Reactive Geometric Planning Wireless Foundation Models

Jun 29, 2026cs.RO

Multi-UAV Formation Cooperative Obstacle Avoidance and Adaptive Shape Deformation Control in Complex Environments Based on BI-APF-RRT and Affine Transformation

Aiming at the problem that obstacle avoidance flexibility and formation integrity are difficult to coexist in multi-UAV formation motion in complex obstacle environments , and that the traditional artificial potential field (APF) method easily falls into local optima, a cooperative obstacle avoidance algorithm for multi-UAV formations integrating BI-APF-RRT and affine transformation is proposed. First, abandoning the traditional APF centroid path planning method , a goal-biased Bidirectional Artificial Potential Field method RRT (BI-APF-RRT) algorithm is adopted to conduct global collision-free path planning for the centroid of the leader formation. By introducing an improved artificial potential field and cubic B-spline interpolation, the smoothness and rapid convergence of the global path are ensured. Secondly, using the generated global path as the guiding trajectory for the formation's centroid , combined with an affine transformation matrix (including non-uniform scaling and rotation) , the formation can adaptively deform based on the distance to obstacles while moving along the optimal path. Finally, the followers track the leaders through a distributed control law , enabling the entire formation to safely cross complex obstacle areas without disassembling.

Yiliang Wu, Weican Chen, Yendo Hu

College of Marine Information Engineering, Jimei University, Xiamen, China

Multi-Uav Multi-Robot Systems Geometric Planning Obstacle Avoidance Dynamic Environments Deformable Cnn+1

Jun 15, 2026cs.RO

SemGeoNav:A Safety-Guided Visual Navigation Approach with Semantic Reasoning and Geometric Planning

Learning-based visual navigation has enhanced semantic goal-reaching capabilities. However, due to their black-box nature, purely end-to-end models often lack explicit geometric constraints, leading to unpredictable and unreliable obstacle avoidance in open environments. Conversely, traditional geometric planners ensure safety but struggle with high-dimensional visual targets. To address these limitations, we propose SemGeoNav, a novel hierarchical visual navigation framework.It tightly integrates the high-level semantic reasoning of end-to-end models with the reliable local planning ability of geometry-based methods, achieving robust image-based navigation while significantly improving obstacle avoidance. Furthermore, we introduce a temporal trajectory smoothing mechanism to ensure continuous and stable robot motion. We evaluated SemGeoNav on a Unitree Go2 quadruped robot in real-world environments. The results demonstrate that SemGeoNav outperforms existing representative methods, including ViNT and NoMaD, achieving higher success rates and shorter navigation times.

Yu Liu, Zongyang Chen, Yan Guo +2

College of Intelligence Science and Technology, National University of Defense Technology, Changsha 410073, China.

Geometric Planning Obstacle Avoidance

Jun 10, 2026cs.RO

Fibration Trees: A Unified Approach to Multi-Robot Motion Planning

State space projections and decompositions have emerged as powerful tools to tackle the curse of dimensionality in high-dimensional, multi-robot motion planning problems. However, existing methods lack a unified framework which seamlessly handles combinations of projections (prioritization or task-space) and decompositions (parallel or decoupled subspaces). To fill this gap, we introduce fibration trees, which are trees consisting of state spaces as nodes and fibrations as edges, whereby a fibration models a projection from a higher-dimensional space to a lower-dimensional (or simplified) space. By modeling projections as fibrations, we unify sequential prioritization, parallel decomposition, and task-space projections under a single, coherent formalism. Building on this, we develop the rapidly-exploring random fibration trees (Fibration-RRT) planner, a sampling-based motion planner that generalizes strategies from quotient-space RRT (for sequential prioritizations) and discrete RRT (for parallel decompositions), while allowing the inclusion of task-space projections. Fibration-RRT operates on user-defined fibration trees and is proven to be probabilistically complete. To test the generality and efficiency of Fibration-RRT, we provide an open-source implementation and conduct experiments on 32 scenarios using multi robot teams with up to 96 degrees of freedom. Our results indicate that Fibration-RRT efficiently solves high-dimensional problems by exploiting user-defined fibration trees, thereby establishing fibration trees as a powerful, unified framework for multi-robot motion planning.

Andreas Orthey, Florian T. Pokorny, Lydia E. Kavraki

Technical University of Berlin, Germany · KTH Royal Institute of Technology, Stockholm, Sweden · Rice University and the Ken Kennedy Institute, Houston, TX, US

Multi-Robot Motion Planning Geometric Planning Motion Planning Planning

Jun 9, 2026cs.LG

Geometry-Aware Reinforcement Learning for 2D Irregular Nesting

Traditional heuristic solvers for the 2D irregular nesting problem share a fundamental limitation: they are blind to polygon geometry, relying on guided brute-force to navigate the continuous placement space with minimal geometrical guidance. In this paper, we argue that Reinforcement Learning is uniquely positioned to overcome this bottleneck. By pairing an optimization policy with a geometry-aware neural encoder, an agent can automatically discover rich geometric priors directly from data, utilizing these learned intuitions to strategically guide exploration. To realize this, we introduce the Polygons Transformer (PoT), a novel architecture that encodes 2D continuous vector geometries while allowing cross-polygons attention. We couple this novel architecture with a Combinatorial Optimization Reinforcement Learning (CORL) training framework to find optimal solutions. To support this paradigm, we release an open-source training dataset derived from complex geographic contours alongside a dedicated evaluation benchmark. Our empirical validation demonstrates that our trained agent achieves area utilization performance highly competitive with Sparrow, the state-of-the-art heuristic solver, proving that reinforcement learning can successfully discover and exploit geometric awareness for precise spatial tasks.

Auguste Lehuger, Guillaume Henon-Just

Valeo Brain

Geometry-Aware Geometric Planning Geometric Guidance

May 24, 2026cs.RO

Performance Comparison of Classical and Neural Sampling Algorithms for Robotic Navigation

Integrating artificial intelligence (AI) into sampling-based motion planning provides new possibilities for improving autonomous navigation efficiency. In this paper, three algorithms, namely RRT*, Neural RRT*, and Neural Informed RRT*, are implemented and evaluated on environments containing convex and concave obstacles with different obstacle densities. The obtained results indicate that neural-guided planners improve path quality, producing up to 14% shorter paths and 55--75% smoother trajectories compared with the conventional RRT* algorithm. Among the evaluated methods, Neural Informed RRT* achieves the best overall performance in terms of path length and trajectory smoothness. These results demonstrate the effectiveness of AI-guided sampling strategies for improving reliability and trajectory efficiency in robotic and UAV navigation, despite a slight increase in computation time. Overall, the study highlights the growing importance of artificial intelligence in real-time robotic path planning applications.

Hichem Cheriet, Badra Khellat Kihel, Samira Chouraqui

dept. of Computer Science Universit´e d’USTO Mohamed Boudiaf Oran, Algeria · dept. of Economics Oran2 Mohamed BenAhmed University Oran, Algeria

Geometric Planning Robot Planning Autonomous Navigation Navigation

May 24, 2026cs.RO

Convex-Neural RRT*: Fast and Reliable Learning-Guided Sampling for High-Quality Robot Path Planning

Sampling-based algorithms for robot path planning offer probabilistic completeness and strong empirical convergence properties across environments with diverse obstacle configurations. However, in practice, these methods often require many iterations to obtain high-quality solutions. This paper proposes Convex-Neural RRT*, an enhanced RRT* variant that incorporates neural guidance to predict informative waypoint regions near high-quality paths. Convex candidate regions are extracted from these predictions, enabling the planner to concentrate exploration on geometrically relevant areas while preserving global exploration. The proposed algorithm is evaluated against Neural RRT*, Neural Informed RRT*, classical RRT*, and LTA* across three environment types and 18 benchmark maps. Experimental results show that Convex-Neural RRT* reduces computation time by 30-75% compared to neural-guided variants and up to 88-98% relative to LTA*, while achieving an average path length reduction of approximately 5% compared to classical RRT*, with larger improvements observed in complex environments. The method also maintains an overall success rate above 99% across varying obstacle densities. These findings indicate that convex-guided neural sampling provides an effective balance between computational efficiency and solution quality, supporting its applicability to time-sensitive robotic navigation tasks.

Hichem Cheriet, Badra Khellat Kihel, Samira Chouraqui +1

Geometric Planning Robot Planning Path Planning Obstacle Avoidance Geometric Guidance Diffusion Sampling

Apr 25, 2026cs.RO

An Efficient Beam Search Algorithm for Active Perception in Mobile Robotics

Active perception is a fundamental problem in autonomous robotics in which the robot must decide where to move and what to sense in order to obtain the most informative observations for accomplishing its mission. Existing approaches either solve a computationally expensive traveling salesman problem over heuristically selected informative nodes, or adopt a more efficient but overly constrained shortest path tree formulation. To address these limitations, we explore beam search algorithms as scalable alternatives. While the standard beam search provides scalability by preserving the top-B paths at each depth level, it is prone to local optima and exhibits parameter sensitivity. Our first contribution is a node-wise beam search (NBS) algorithm, which maintains top-B candidates per node to enable more effective exploration of the solution space. Systematic benchmarking on graphs shows that NBS consistently outperforms other baselines and maintains strong performance even at low beam widths. As a second contribution, we integrate the concept of frontiers into the path selection criterion, introducing the expected gain metric, which better balances exploration and exploitation compared to existing alternatives. Our third contribution proposes the rapidly-exploring random annulus graph (RRAG), a novel graph construction method that preserves full orientation sampling and ensures connectivity in cluttered environments through a fallback local sampling-based planner. Extensive experiments demonstrate that NBS combined with RRAG achieves the highest performance across all three representative active perception tasks, outperforming state-of-the-art algorithms by at least 20% in one or more tasks. We further validate the approach on real robotic platforms in different scenarios.

Kaixian Qu, Han Wang, Victor Klemm +2

Robotic Systems Lab, ETH Zurich

Viewpoint-Dependent Active Perception Geometric Planning Beam Search Efficient Exploration Inhomogeneous Random Graphs Device

Aug 26, 2025cs.RO

Real-Time Model Checking for Closed-Loop Robot Reactive Planning

Reactive obstacle avoidance methods often cause agents to become trapped in local minima, because they can often only reason one step ahead (i.e., the next action based on the current state). In this paper, we use model checking to achieve reactive multi-step planning and obstacle avoidance on an autonomous robot. Our small, purpose-built model checking algorithm generates plans in situ (within the robot's code) based on ``core'' knowledge and attention as found in biological agents. This is achieved in real-time using no pre-computed data on a low-powered device. Our approach is based on chaining temporary control systems that are spawned to counteract disturbances in the local environment which disrupt an autonomous agent from its preferred action (or resting state). We mitigate state-space explosion by relying on temporary snapshots of the immediate environment, restricting the number of states. Multi-step planning using counter-examples generated by depth-first search and a negated LTL path property is applied to scenarios involving a cul-de-sac and a free-standing obstacle. Empirical results and informal proofs of two fundamental properties demonstrate the effectiveness of our approach for the creation of efficient multi-step plans for local obstacle avoidance. We significantly improve performance compared to a purely reactive agent that can only plan one step ahead. Our approach is an instructional case study for the development of safe and reliable navigation in the context of autonomous vehicles. We believe it also has general application in navigation for mission-critical mobile robots.

Christopher Chandler, Bernd Porr, Giulia Lafratta +1

Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow, Scotland, UK · School of Engineering, University of Glasgow, Glasgow, Scotland, UK · School of Computing Science, University of Glasgow, Glasgow, Scotland, UK

Geometric Planning Obstacle Avoidance Probabilistic Model Checking Mobile Robots Robot Systems Linear Temporal Logics+2

Date pendingcs.RO

OA-NBV: Occlusion-Aware Next-Best-View Planning for Human-Centered Active Perception on Mobile Robots

We naturally step sideways or lean to see around the obstacle when our view is blocked, and recover a more informative observation. Enabling robots to make the same kind of viewpoint choice is critical for human-centered operations, including search, triage, and disaster response, where cluttered environments and partial visibility frequently degrade downstream perception. However, many Next-Best-View (NBV) methods primarily optimize generic exploration or long-horizon coverage, and do not explicitly target the immediate goal of obtaining a single usable observation of a partially occluded person under real motion constraints. We present Occlusion-Aware Next-Best-View Planning for Human-Centered Active Perception on Mobile Robots (OA-NBV), an occlusion-aware NBV pipeline that autonomously selects the next traversable viewpoint to obtain a more complete view of an occluded human. OA-NBV integrates perception and motion planning by scoring candidate viewpoints using a target-centric visibility model that accounts for occlusion, target scale, and target completeness, while restricting candidates to feasible robot poses. OA-NBV achieves over 90% success rate in both simulation and real-world trials, while baseline NBV methods degrade sharply under occlusion. Beyond success rate, OA-NBV improves observation quality: compared to the strongest baseline, it increases normalized target area by at least 81% and keypoint visibility by at least 58% across settings, making it a drop-in view-selection module for diverse human-centered downstream tasks.

Boxun Hu, Chang Chang, Jiawei Ge +4

Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218, USA · Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA · Laboratory for Computational Sensing and Robotics, Johns Hopkins University, Baltimore, MD 21218, USA +1

Viewpoint-Dependent Active Perception Diverse Occlusion-And-Revelation Scenarios Geometric Planning Mobile Robots Planning