1. Bibliographic Information

1.1. Title

NoMaD: Goal Masked Diffusion Policies for Navigation and Exploration

1.2. Authors

Ajay Sridhar, Dhruv Shah, Catherine Glossop, and Sergey Levine. The authors are affiliated with the University of California, Berkeley. Sergey Levine is a prominent professor and researcher in the fields of robotics, machine learning, and control, known for his significant contributions to deep reinforcement learning and robotic learning.

1.3. Journal/Conference

The paper was published in the 7th Annual Conference on Robot Learning (CoRL) in 2023. CoRL is a highly respected and selective international conference that focuses on the intersection of robotics and machine learning, making it a premier venue for this type of research.

1.4. Publication Year

2023

1.5. Abstract

The paper addresses the challenge of creating a single robotic policy that can handle both task-oriented navigation (moving towards a known goal) and task-agnostic exploration (searching for a goal in a new environment). Traditionally, these two functions are managed by separate models. The authors propose NoMaD (Navigation with Goal Masked Diffusion), a unified policy that accomplishes both tasks. The architecture uses a large-scale Transformer to process visual data and a diffusion model decoder to generate actions. A key innovation is "goal masking," which allows the policy to flexibly switch between being conditioned on a goal image (for navigation) and ignoring it (for exploration). The authors demonstrate through real-world experiments on a mobile robot that NoMaD outperforms alternative approaches, including those using generative models for subgoal proposals, while being significantly more computationally efficient and achieving lower collision rates.

1.6. Original Source Link

• Official Source: https://arxiv.org/abs/2310.07896
• PDF Link: https://arxiv.org/pdf/2310.07896v1.pdf
• Publication Status: The paper is available as a preprint on arXiv and was officially published at the CoRL 2023 conference.

2. Executive Summary

2.1. Background & Motivation

• Core Problem: In real-world robotics, particularly for navigation in unfamiliar areas, a robot needs two fundamental capabilities:
  1. Exploration: When a goal's location is unknown, the robot must intelligently and safely search its environment. This is a task-agnostic behavior, as the immediate actions are not directed at a specific, visible goal.
  2. Navigation: Once a goal is located (or specified), the robot must efficiently and safely travel to it. This is a task-oriented or goal-directed behavior.
• Existing Gaps: Prior research typically treats these as separate problems, requiring distinct systems. For example, a robot might use a high-level planner or a generative model to propose "subgoals" for exploration, and a separate navigation policy to reach those subgoals. This approach introduces complexity, increases computational overhead, and may not allow the two behaviors to share learned knowledge effectively.
• Innovative Idea: The paper's central idea is to question this separation. Can a single, highly expressive policy be trained to seamlessly perform both exploration and navigation? The authors hypothesize that exploration can be framed as navigation without a specific goal, and that a powerful enough model can learn this duality.

2.2. Main Contributions / Findings

The paper makes several key contributions:

1. NoMaD Architecture: The authors propose a novel architecture that unifies exploration and navigation into a single policy. Its core components are:
   • A Transformer-based encoder to process sequences of visual observations.
   • An attention-based goal masking mechanism that allows the policy to dynamically "turn off" its conditioning on a goal image, switching it from navigation mode to exploration mode.
   • A diffusion model decoder that generates sequences of future actions. This is crucial for modeling the complex, often multi-choice (multimodal) action distributions required for exploration (e.g., turn left or right at an intersection).
2. State-of-the-Art Performance: Through extensive real-world experiments, NoMaD is shown to significantly outperform existing methods. In exploration tasks, it improves the success rate by over 25% compared to the previous state-of-the-art while drastically reducing collisions.
3. Computational Efficiency: NoMaD is remarkably efficient. It achieves its superior performance using a model that is 15 times smaller (19M vs. 335M parameters) than the leading competitor (Subgoal Diffusion). This allows it to run directly on the robot's onboard computer (e.g., NVIDIA Jetson Orin), a major practical advantage.
4. First Real-World Goal-Conditioned Diffusion Policy: To the best of the authors' knowledge, NoMaD is the first successful implementation of a goal-conditioned action diffusion model deployed and validated on a physical robot for navigation.

3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

3.1.1. Diffusion Models

Diffusion Models, formally known as Denoising Diffusion Probabilistic Models (DDPMs), are a class of generative models that have become state-of-the-art in generating high-quality data, especially images. They work in two stages:

1. Forward Process (Noise Addition): This is a fixed process where Gaussian noise is gradually added to a real data sample (e.g., an image or, in this paper, a sequence of robot actions) over a series of timesteps, $T$. At each step, a small amount of noise is added, until the original data is transformed into pure, random noise.

2. Reverse Process (Denoising): This is the learned part. The model, typically a neural network like a U-Net, is trained to reverse the noising process. At each timestep $t$, it takes the noisy data from that step, $x_t$, and learns to predict the small amount of noise that was added between step t-1 and $t$. By repeatedly subtracting this predicted noise, the model can start from pure random noise ($x_T$) and gradually reconstruct a clean, realistic data sample ($x_0$).

   For NoMaD, the "data" being generated is a sequence of future actions. The key advantage is that diffusion models can naturally represent multimodal distributions. For a robot at an intersection, there might be two good options (turn left, turn right) and many bad ones (go straight into a wall). A simple regression model might predict the average of the good options (e.g., a slight turn), which is useless. A diffusion model can assign high probability to both turning left and turning right, making it ideal for exploration.

3.1.2. Transformer Architecture

The Transformer is a neural network architecture originally designed for natural language processing tasks. Its core innovation is the attention mechanism, which allows the model to weigh the importance of different parts of the input sequence when making a prediction.

The most common form is scaled dot-product attention, which is calculated using three vectors for each input token: a Query ($Q$), a Key ($K$), and a Value ($V$). The formula is: $$\mathrm{Attention}(Q, K, V) = \mathrm{softmax}\left(\frac{QK^T}{\sqrt{d_k}}\right)V$$

• Conceptual Definition: Imagine you are translating a sentence. To translate the word "it," you need to pay "attention" to which noun "it" refers to. The attention mechanism does this automatically.
• Mathematical Formula:

  • $Q$: A representation of the current token, asking a "query" about other tokens.

  • $K$: A representation of all tokens in the sequence, acting as "keys" that can be matched with the query.

  • $V$: A representation of all tokens, holding the actual "values" or information.

  • $QK^T$: The dot product between queries and keys calculates a similarity score, indicating how much attention each token should pay to every other token.

  • $\sqrt{d_k}$: A scaling factor to stabilize gradients, where $d_k$ is the dimension of the key vectors.

  • $\mathrm{softmax}$: This function converts the similarity scores into probabilities (attention weights) that sum to 1.

  • The final output is a weighted sum of the values, where the weights are determined by the attention scores.

    In NoMaD, the Transformer processes a sequence of visual embeddings from past observations and a potential goal image, allowing it to reason about temporal context and goal relevance.

3.1.3. Topological Maps

In robotics, a topological map is a simplified representation of an environment, structured as a graph.

• Nodes: Represent significant places or viewpoints in the environment (e.g., a specific visual observation at a hallway intersection).

• Edges: Represent navigable paths between these places. An edge exists between two nodes if the robot can successfully travel from one to the other.

  This is different from a geometric map (like a grid map), which captures detailed spatial information. Topological maps are more abstract and are excellent for long-horizon planning. To get from point A to a distant point B, the robot doesn't need to plan every single motor command; instead, it can find a path of nodes in the graph (A -> C -> D -> B) and use its navigation policy to travel between consecutive nodes. NoMaD uses this framework for navigating large environments.

3.2. Previous Works

• ViNT (Visual Navigation Transformer): This is the direct predecessor and backbone of NoMaD. ViNT is a goal-conditioned policy that uses an EfficientNet to encode a history of observations and a goal image, and a Transformer to fuse this information and predict a sequence of actions. Its key limitation is that it can only perform goal-directed navigation. For exploration, it requires a completely separate, large-scale model to propose subgoals.
• Subgoal Diffusion (from the ViNT paper): This is the state-of-the-art system that NoMaD is compared against. It is a two-part system:
  1. A massive (300M parameter) image diffusion model is trained to generate plausible subgoal images from the robot's current viewpoint.
  2. The ViNT policy is then used to navigate to these generated subgoals. This approach is powerful but computationally expensive and complex.
• Diffusion Policy (Chi et al., 2023): This work demonstrated that diffusion models can be used to learn visuomotor policies by directly generating action sequences from visual inputs. However, it was designed for task-agnostic or implicitly conditioned behaviors, not explicitly goal-conditioned navigation. NoMaD extends this concept to be explicitly goal-conditioned (or not) via masking.
• VIB (Variational Information Bottleneck): This is a baseline method for exploration that uses a latent variable model. It tries to model a distribution of possible actions by encoding the observation into a compressed latent space, from which actions are decoded. This is another way to tackle multimodal action distributions but, as results show, is less effective than diffusion models in this context.

3.3. Technological Evolution

The field of robotic navigation has evolved from classical methods to learned policies:

1. Classical Methods: Separate modules for mapping (e.g., SLAM), localization, planning (e.g., A*), and control. These are often robust but can be brittle in unstructured environments and don't learn from experience.
2. End-to-End Learning: Early learned policies used CNNs to map pixels directly to actions. These worked for simple tasks but struggled with long-horizon reasoning.
3. High-Capacity Policies: The introduction of Transformers (e.g., ViNT) allowed policies to reason over longer histories of observations and better integrate goal information, leading to state-of-the-art performance in goal-conditioned navigation.
4. Generative Models for Exploration: To handle exploration, the field moved towards using powerful generative models (like image diffusion models) to propose exploratory subgoals for the high-capacity navigation policies to follow.
5. Unified Generative Policies (NoMaD): NoMaD represents the next step in this evolution. Instead of using a generative model for goals and a separate policy for actions, NoMaD proposes a single generative model for actions that can operate in both goal-agnostic (exploration) and goal-oriented (navigation) modes.

3.4. Differentiation Analysis

Compared to its main competitor, Subgoal Diffusion, NoMaD's innovation is its unification and directness.

• Subgoal Diffusion: Operates in a high-dimensional, indirect space. It first generates a subgoal image, then uses a policy to infer actions to reach that image. This is a two-step, computationally heavy process.

• NoMaD: Operates directly in the action space. It uses a single model to generate actions directly from observations. The diffusion process models the multimodality of exploration (e.g., multiple possible paths) within the action space itself. This is more direct, efficient, and, as the experiments show, more effective.

  The core difference is what is being generated: subgoal images vs. action sequences.

4. Methodology

4.1. Principles

The central principle of NoMaD is that exploration and goal-reaching are two facets of a single, competent navigation behavior. Exploration can be viewed as goal-reaching without a specific goal. Therefore, a single, powerful policy should be able to learn both.

To achieve this, NoMaD relies on two key ideas:

1. A Flexible Conditioning Mechanism: The policy needs a "switch" to toggle between goal-conditioned and goal-agnostic behavior. The paper introduces goal masking within the Transformer's attention mechanism as an elegant and effective way to implement this switch.
2. An Expressive Action Distribution Model: For exploration, the policy must represent complex, multimodal action distributions (e.g., at a fork in the road, both left and right are valid options). The paper leverages a diffusion model as the policy's decoder, as it is exceptionally well-suited for this task.

4.2. Core Methodology In-depth

The NoMaD architecture, shown in Figure 2 of the paper, can be broken down into a step-by-step data flow from input to action output.

该图像是示意图，展示了NoMaD模型架构。图中包含两个EfficientNet编码器$\psi$和$\phi$，用于生成输入标记，随后输入到一个四层的Transformer解码器。通过目标掩码（goal masking），模型能够联合推理任务无关和任务导向的行为。该架构利用上下文$c_t$实现动作扩散，以获得可用于目标条件和无方向导航的多样性策略。

4.2.1. Step 1: Input Processing and Visual Encoding

The policy takes as input a history of $P+1$ recent RGB observations, $\mathbf{o}_t := o_{t-P:t}$, and an optional RGB goal image, $o_g$. These images are first processed by visual encoders:

• Observation Encoder ($\psi$): Each observation image $o_i$ (for $i \in \{t-P, \dots, t\}$) is passed through an EfficientNet-B0 encoder, $\psi(o_i)$, to produce a sequence of observation tokens.
• Goal Fusion Encoder ($\phi$): The current observation $o_t$ and the goal image $o_g$ are processed together by a second EfficientNet-B0 encoder, $\phi(o_t, o_g)$, to produce a single goal token. This token encapsulates information about the goal relative to the current view.

4.2.2. Step 2: Goal Masking in the Transformer

The observation tokens and the goal token are fed into a Transformer decoder. This is where the core mechanism of goal masking comes into play. A binary mask, $m \in \{0, 1\}$, controls whether the policy attends to the goal token.

• For Exploration ($m=1$): The goal mask is set to 1. The attention mechanism within the Transformer is modified to prevent all observation tokens from attending to the goal token. The goal information is effectively "blocked," forcing the policy to rely only on the observation history.

• For Navigation ($m=0$): The goal mask is set to 0. The attention mechanism operates normally, allowing all tokens to attend to each other. The policy can use the goal token to inform its actions.

  The Transformer processes these inputs and produces a final context vector, $c_t = f(\psi(o_i), \phi(o_t, o_g), m)$, which summarizes all relevant information from the observations and, if unmasked, the goal.

4.2.3. Step 3: Action Generation with the Diffusion Policy

The context vector $c_t$ is used to condition the diffusion model decoder, which generates a sequence of $H$ future actions, $\mathbf{a}_t := a_{t:t+H}$. This is an iterative denoising process:

1. Initialization: The process starts with a sequence of actions sampled from pure Gaussian noise: $\mathbf{a}_t^K \sim \mathcal{N}(0, I)$, where $K$ is the total number of diffusion steps.
2. Iterative Denoising: For each step $k$ from $K$ down to 1, the model refines the action sequence. The update rule is given by the following formula from the paper: $$\mathbf { a } _ { t } ^ { k - 1 } = { \boldsymbol { \alpha } } \cdot \left( a _ { t } ^ { k } - \gamma \epsilon _ { \theta } ( c _ { t } , \mathbf { a } _ { t } ^ { k } , k ) + { \mathcal { N } } ( 0 , \sigma ^ { 2 } I ) \right)$$

   • $\mathbf{a}_t^k$: The noisy action sequence at denoising step $k$.

   • $\epsilon_\theta$: The core of the diffusion model, a noise prediction network (implemented as a 1D conditional U-Net) with parameters $\theta$. It takes the noisy action $\mathbf{a}_t^k$, the current denoising step $k$, and the conditioning context $c_t$ as input, and predicts the noise that was added to the clean action to produce $\mathbf{a}_t^k$.

   • $c_t$: The context vector from the Transformer, which provides all the visual and goal information.

   • $\alpha, \gamma, \sigma$: These are scalar functions derived from a predefined "noise schedule," which controls how much noise is removed and added at each step. After $K$ iterations, this process yields a clean, noise-free action sequence $\mathbf{a}_t^0$. This is the final output of the policy. The ability of $\epsilon_\theta$ to predict noise conditioned on $c_t$ is what allows the diffusion model to generate actions that are consistent with the robot's observations and goals. Figure 3 from the paper visualizes this beautifully, showing how the distribution of predicted actions is multimodal for exploration (yellow) and becomes unimodal when conditioned on a goal (green or blue).

     该图像是一个示意图，展示了NoMaD在两种目标图像（绿色、蓝色）下学习的任务无关（黄色）和目标导向路径。NoMaD在没有目标时预测出双峰的无碰撞动作分布，而在条件化于这两个不同目标图像后则收缩到更窄的分布。

4.2.4. Step 4: Training

The entire NoMaD model is trained end-to-end using supervised learning on a large dataset of robot trajectories. The training objective consists of two parts, captured by the following loss function: $$\begin{array} { r } { \mathcal { L } _ { \mathrm { N oM aD } } ( \phi , \psi , f , \theta , f _ { d } ) = M S E ( \varepsilon ^ { k } , \varepsilon _ { \theta } ( c _ { t } , \mathbf { a } _ { t } ^ { 0 } + \varepsilon ^ { k } , k ) ) } \\ { + \lambda \cdot M S E ( d ( \mathbf { o } _ { t } , o _ { g } ) , f _ { d } ( c _ { t } ) ) } \end{array}$$

1. Diffusion Loss: The first term is the core loss for training the diffusion model. For a given ground-truth action sequence $\mathbf{a}_t^0$, a random noise level $k$ is chosen, and corresponding noise $\varepsilon^k$ is added to create a noisy sample. The network $\epsilon_\theta$ predicts the noise from this sample, and the loss is the Mean Squared Error (MSE) between the predicted noise and the actual noise $\varepsilon^k$.
2. Temporal Distance Loss: The second term trains an auxiliary prediction head, $f_d$, which takes the context $c_t$ and predicts the temporal distance (number of timesteps) to the goal, $d(\mathbf{o}_t, o_g)$. This is also an MSE loss against the ground-truth distance. This distance prediction is useful for the high-level topological planner.

   • $\lambda$: A hyperparameter (set to $10^{-4}$) that balances the two loss terms.

     During training, for each sample, the goal mask $m$ is randomly set to 1 with a probability of $p_m = 0.5$. This ensures the model is trained equally on both exploration (masked) and navigation (unmasked) behaviors.

5. Experimental Setup

5.1. Datasets

• Datasets Used: The model was trained on a combination of two large-scale, real-world robotics datasets:
  1. GNM (General Navigation Model) Dataset: A heterogeneous dataset collected from multiple different robot platforms across a wide variety of environments.
  2. SACSoN (Scalable Autonomous Data Collection for Social Navigation) Dataset: A dataset focused on navigation in environments with many pedestrians.
• Scale and Characteristics: Together, these datasets comprise over 100 hours of real-world robot trajectories, providing a rich and diverse source of data for learning generalizable navigation behaviors.
• Reason for Choice: Using large, diverse, real-world datasets is crucial for training policies that can generalize to new, unseen environments, which is a core focus of this paper.

5.2. Evaluation Metrics

• Success Rate:

  1. Conceptual Definition: This metric measures the reliability of the navigation or exploration policy. It is the fraction of experimental runs in which the robot successfully reaches its designated goal region. A higher success rate indicates a more effective and robust policy.
  2. Mathematical Formula: $$\text{Success Rate} = \frac{\text{Number of Successful Trials}}{\text{Total Number of Trials}}$$
  3. Symbol Explanation:
     • Number of Successful Trials: The count of experiments where the robot reached the goal.
     • Total Number of Trials: The total number of experiments conducted.

• Collisions (Coll.):

  1. Conceptual Definition: This metric measures the safety of the policy. It is the average number of times the robot collides with an obstacle during an experimental run. A lower value is better, indicating a safer policy that is better at obstacle avoidance.
  2. Mathematical Formula: $$\text{Collisions per Trial} = \frac{\text{Total Number of Collisions}}{\text{Total Number of Trials}}$$
  3. Symbol Explanation:
     • Total Number of Collisions: The sum of all collision events across all trials.
     • Total Number of Trials: The total number of experiments conducted.

5.3. Baselines

NoMaD was compared against five alternative methods, covering different approaches to exploration and navigation:

• VIB [17]: An exploration method based on a latent goal model that uses a variational information bottleneck to model diverse actions.
• Masked ViNT: A direct ablation of NoMaD. It uses the same ViNT backbone and goal masking technique but replaces the diffusion decoder with a simple regression head that predicts a single action sequence. This baseline tests the importance of the diffusion model for handling multimodal action distributions.
• Autoregressive: A baseline that models multimodal action distributions by discretizing the action space and predicting actions autoregressively (one step at a time).
• Random Subgoals [3]: A variant of the ViNT system where, for exploration, subgoals are not generated but are randomly sampled from the training dataset. This tests whether a sophisticated subgoal generator is necessary at all.
• Subgoal Diffusion [3]: The previous state-of-the-art. This is the full ViNT system combined with a large (300M parameter) image diffusion model that generates subgoal images for the ViNT policy to navigate towards. This is the strongest and most direct competitor.

6. Results & Analysis

6.1. Core Results Analysis

The main experimental results, comparing NoMaD to the baselines on exploration and navigation tasks, are summarized in Table I.

The following are the results from Table I of the original paper:

• Exploration Performance: NoMaD achieves a 98% success rate with an extremely low collision rate of 0.2. This is a dramatic improvement over the state-of-the-art Subgoal Diffusion, which only achieves 77% success with 1.7 collisions. This result strongly validates the paper's central hypothesis: a unified policy that directly models multimodal actions is superior to a complex, two-stage system that generates subgoal images.
• Navigation Performance: In the goal-conditioned navigation task, NoMaD achieves a 90% success rate, matching the performance of the best baselines (Subgoal Diffusion and Random Subgoals). This demonstrates that unifying the policy for exploration did not compromise its goal-reaching capabilities.
• Model Efficiency: NoMaD (19M parameters) is over 15 times smaller than Subgoal Diffusion (335M parameters). This is a massive advantage, making NoMaD practical for deployment on resource-constrained onboard robot hardware.
• Importance of Diffusion Decoder: The Masked ViNT baseline, which lacks the diffusion decoder, performs poorly (50% exploration success). This shows that simply masking the goal is insufficient; the ability of the diffusion model to represent complex action distributions is critical for effective exploration.

6.2. Unified v/s Dedicated Policies

To understand if the unified model makes compromises, the authors compared NoMaD to specialized policies trained only for one task.

The following are the results from Table II of the original paper:

• No Compromise: The results are striking. NoMaD, with a comparable parameter count (19M), performs identically to the specialized Diffusion Policy (15M) on undirected exploration (98% success) and the specialized ViNT Policy (16M) on goal-conditioned navigation (92% success). This shows that a single policy can indeed master both behaviors without any performance degradation, effectively learning shared representations that benefit both tasks.

6.3. Ablation Studies / Parameter Analysis

The authors investigated the impact of the visual encoder and goal masking strategy on performance (Table III).

The following are the results from Table III of the original paper:

• Encoder Architecture is Crucial: The NoMaD architecture, which uses an EfficientNet + Transformer design (the ViNT backbone), vastly outperforms alternatives. Both CNN-based architectures struggle, and a standard Vision Transformer (ViT) performs very poorly, likely due to optimization difficulties when trained end-to-end with a diffusion model. This confirms that the specific visual processing architecture from ViNT is a key component of NoMaD's success.
• Masking Strategy: The paper's use of attention-based masking within the Transformer is shown to be superior to simpler strategies like using dropout on CNN embeddings.

7. Conclusion & Reflections

7.1. Conclusion Summary

The paper successfully presents NoMaD, a novel and highly effective policy that unifies task-agnostic exploration and task-oriented navigation for mobile robots. By combining a powerful Transformer encoder with an innovative goal masking technique and a multimodal diffusion decoder for actions, NoMaD sets a new state-of-the-art in navigating unseen environments. It significantly improves exploration success rates and safety while being drastically more computationally efficient than previous methods. The key finding is that a single, unified policy can not only match but exceed the performance of complex, multi-model systems, paving the way for more capable and practical robotic navigation systems.

7.2. Limitations & Future Work

The authors acknowledge several limitations and suggest future research directions:

• Limited Goal Modalities: The system currently relies on goal images to specify tasks. While general, this is not always the most intuitive modality for a human user. Future work could extend NoMaD to accept goals specified via language instructions or spatial coordinates (e.g., GPS, clicks on a map).
• Simple High-Level Planning: NoMaD is integrated with a standard frontier-based exploration strategy for its high-level planner. More sophisticated strategies that leverage semantics (e.g., "search for the kitchen first") or prior knowledge about environment layouts could further enhance exploration efficiency.

7.3. Personal Insights & Critique

• Elegance in Simplicity: The core innovation of "goal masking" is impressively simple yet powerful. It avoids complex architectural changes, instead using a simple switch to unlock dual functionality from a single powerful model. This highlights a valuable principle in model design: sometimes the most effective solutions are the most elegant ones.
• Directness Pays Off: The paper's success reinforces the idea that solving problems directly in the most relevant space (in this case, the action space) can be more effective than indirect, multi-stage approaches (like generating subgoals in image space). The diffusion model's ability to directly capture the multimodality of actions is key.
• Potential for Generalization: The concept of a conditionally masked diffusion policy is highly generalizable. It could be applied to other robotics domains, such as manipulation. For example, a robot arm policy could be trained to perform both "explore possible grasps on this object" (unconditional/partially conditioned) and "execute this specific top-down grasp" (fully conditioned).
• Unanswered Questions: While the ViT backbone performed poorly, the authors attribute this to "optimization challenges." It would be interesting to see a deeper analysis of why this is the case, as ViTs are typically very powerful vision models. Is it an issue with the dataset size, the specific training regime, or a more fundamental incompatibility with action diffusion models? This could be an area for future investigation. Overall, NoMaD is a significant step forward, presenting a clean, efficient, and high-performing solution to a long-standing problem in robot navigation.