SanD-Planner: Diffusion & Sand Physics

• SanD-Planner is a unified planning framework that integrates diffusion-based trajectory planning for robot navigation with physics-driven sand management tools.
• It employs a hybrid architecture combining CNNs and Transformers to encode sensor data and context, achieving high success rates (e.g., 93% SR) using minimal expert episodes.
• Physics modules leverage CFD simulations and deep RL to optimize sand fence design and sand–tool interactions, ensuring robust, sample-efficient planning in uncertain environments.

SanD-Planner refers to two distinct classes of algorithms and tools within the academic literature: (1) a state-of-the-art diffusion-based trajectory planning framework for local robot navigation in complex environments, and (2) a family of physics-based and data-driven planning modules for predictive modeling and optimization in aeolian sand management and sand–tool interaction contexts. This article synthesizes these technical lines under the unifying theme of sample-efficient, physically-aware planning under uncertainty, highlighting algorithmic advances, mathematical formulations, benchmarking results, and integration with domain-specific simulators.

1. Diffusion-Based SanD-Planner for Robust Local Navigation

The framework introduced in "SanD-Planner: Sample-Efficient Diffusion Planner in B-Spline Space for Robust Local Navigation" (Wang et al., 31 Jan 2026) is formulated for mapless, receding-horizon planning, targeting robot navigation in cluttered and dynamic environments. The approach operates directly on a compact representation: a sequence of $N$ clamped B-spline control points $$\mathcal Q_t = \{Q_{t,i}\}_{i=0}^{N-1} \subset \mathbb R^3$$, with the initial point $Q_0$ fixed at the robot origin.

At each timestep $t$, the system processes:

• Four recent depth images, $\mathcal O_t = \{D_{t-3}, D_{t-2}, D_{t-1}, D_t\}$, from an onboard sensor.
• The relative goal vector $g_t = [x, y, z]^\top$.
• A previous-trajectory velocity token $v_t^{\rm prev}$.

Context encoding is performed via a hybrid architecture: ResNet-18 for image features, MLPs for $g_t$ and $v_t^{\rm prev}$, and a two-layer Transformer encoder composing the feature tokens into the planning context $\mathcal C_t$.

A conditional Denoising Diffusion Probabilistic Model (DDPM) $p_\theta(\mathcal Q_t \mid \mathcal C_t)$ is trained to match the empirical distribution of expert-generated B-spline control points. Sampling is performed in latent space; $K$ candidates are generated per replanning iteration, mapped to continuous candidate trajectories $\tau(u)$ via standard clamped-cubic B-spline basis:

$$\tau(u) = \sum_{i=0}^{N-1} B_{i,3}(u) Q_i, \quad u \in [u_3, u_N],$$

where $B_{i,3}(u)$ are Cox–de Boor basis functions, ensuring $C^2$ continuity and the convex-hull property.

2. Objective Function, Safety Critic, and Execution

Candidate trajectories are discretized into $M$ arc-length–equidistant waypoints and scored by an analytic, non-learned safety critic based on a Euclidean Signed Distance Field (ESDF) computed from the current depth map. The cost functional comprises weighted terms:

• Discounted safety clearance:

$$J_{\rm safe}(\tau) = \sum_{j=0}^{M-1} \gamma^j \max(0, d_{\rm safe} - E(x_j)),$$

with $\gamma \in (0,1)$ emphasizing near-horizon safety.

• Path length:

$$J_{\rm len}(\tau) = \sum_{j=0}^{M-2} \|x_{j+1} - x_j\|_2,$$

• Goal proximity:

$J_{\rm goal}(\tau) = \|x_{M-1} - g_t\|_2,$

• Total cost:

$$J(\tau) = \lambda_1 J_{\rm safe} + \lambda_2 J_{\rm len} + \lambda_3 J_{\rm goal}.$$

The trajectory with minimal $J(\tau)$ is selected; only the first segment is executed before the process repeats with updated observations (Wang et al., 31 Jan 2026).

3. Training Efficiency and Representation Analysis

The system demonstrates high sample efficiency: only 500 expert episodes (0.25% of the scale used by baselines such as NavDP) are required for state-of-the-art performance. Sub-trajectory resampling over these episodes amplifies effective training samples. Ablations confirm B-spline superiority: for equivalent control-point counts, B-spline representation achieves 93.0% success rate (SR) and 83.8% success path length (SPL) in cluttered environments, significantly outperforming waypoint-based or cubic-spline planners.

Temporal conditioning via $v_t^{\rm prev}$ yields further gains; its removal causes SR to drop from $\sim93\%$ to $\sim86\%$ and induces oscillatory behavior.

4. SanD-Planner for Predictive Sand Physics and Tool Interaction

The term SanD-Planner also encapsulates a physics-based planning toolkit for wind-driven sand transport, sand manipulation, and optimal environmental design (Lima et al., 2017, Kim et al., 2019, Preziosi et al., 2015). In (Lima et al., 2017), SanD-Planner denotes an embedded planning module leveraging 2D CFD (RANS+$k{-}\varepsilon$) wind flow simulations to optimize linear arrays of porous sand fences, minimizing material cost $C$ while guaranteeing a user-specified protected-area fraction $A_p$. The algorithmic skeleton performs parameter sweeps over fence height $H$, porosity $\phi$, and spacing $L$, referencing pre-tabulated maximal safe-spacing functions $L_t(\phi, H, u_{*t})$ derived from CFD.

For sand–tool interaction, (Kim et al., 2019) introduces a hybrid planning system:

• State: Sand surface modeled as $H_t \in \mathbb{R}^{m\times n}$, with post-action steady-state achieved after each "tool push" through iterative erosion relaxation.
• Planning: Discrete trenching tasks solved via classical A* on binary thresholded maps; complex shapes approached via deep RL (DQN for discrete, DDPG+HER for continuous actions) with shape-matching rewards.

5. Physics Modules and Simulation Integration

The self-contained physics module in (Preziosi et al., 2015) extends SanD-Planner with a two-phase model of air and dilute sand, governed by coupled RANS+$k$–$\omega$ and sand advection–diffusion equations, including sedimentation, collision-diffusion, and erosion boundary layers. Boundary inputs include surface roughness, friction velocity thresholds, and adjustable mass-flux closures, supporting physically-consistent benchmarking (e.g., saltation layer growth, non-equilibrium sand transport in erodible/ non-erodible beds).

These physics modules are referenced directly by SanD-Planner optimization kernels to predict wind–sand–obstacle dynamics or tool–sand reshaping outcomes, enabling closed-loop or open-loop planning in both environmental engineering and autonomous robotics contexts.

6. Experimental Results and Benchmarks

In diffusion-based local navigation (Wang et al., 31 Jan 2026), SanD-Planner surpasses prior approaches on the InternNav ClutteredEnv benchmark ($90.1\%$ SR, $84.0\%$ SPL) and demonstrates robust zero-shot sim-to-real transfer on a Unitree Go2 with RealSense D435. Sand manipulation planning achieves global A* optimality for small trench shapes (up to $7 \times 7$ grid), while deep RL methods reproduce complex multi-stroke "alphabet" patterns with variable fidelity, reporting typical convergence after $10^4$ episodes (Kim et al., 2019).

For sand fence optimization, SanD-Planner predicts the minimal-cost design at $H \approx 0.5\,\mathrm{m}$ and $\phi \approx 0.4$, with spacing $L=L_t(0.4,0.5,u_{*t})$, outperforming traditional (taller) fence arrays for equivalent protection levels (Lima et al., 2017). The CFD-driven planner accommodates user constraints (partial protection, wind exceedance probability) and provides rapid lookup of $L_t$ and $C$ for field-scale deployment.

7. Integration Strategies, Limitations, and Extensions

SanD-Planner methodologies, whether diffusion-based or physics-driven, are designed for modular integration with simulation environments (e.g., Gazebo, Matterport3D (Wang et al., 31 Jan 2026), in-house sand physics engines (Kim et al., 2019, Preziosi et al., 2015)). While diffusion planners enjoy high sample efficiency and analytic cost functional evaluation, their accuracy is contingent on ESDF fidelity and sufficient domain randomization. In sand physical planning, the simplifications (steady-state, kinematic tools, omission of force/dynamic feedback) restrict realism for rapid or transient interactions. In large-scale or high-dimensional planning, A* search exhibits exponential node growth; RL approaches suffer from sparse rewards and slow convergence as task complexity increases.

Plausible implications include the benefit of hierarchical decomposition, more aggressive curriculum learning, and improved reward shaping in RL sand manipulation, as well as the potential for 3D or non-perpendicular-wind extensions in environmental planning. Parameter uncertainties (e.g., grain size, threshold velocity) should be addressed via in situ calibration.

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References:

• SanD-Planner: Sample-Efficient Diffusion Planner in B-Spline Space for Robust Local Navigation (Wang et al., 31 Jan 2026)
• Optimal array of sand fences (Lima et al., 2017)
• Developing a Simple Model for Sand-Tool Interaction and Autonomously Shaping Sand (Kim et al., 2019)
• A Multiphase First Order Model for Non-Equilibrium Sand Erosion, Transport and Sedimentation (Preziosi et al., 2015)