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+---
+title: "Distribution Estimation for Global Data Association via Approximate Bayesian Inference"
+
+date: 2026-09-04
+
+description: Leverage approximate Bayesian inference to estimate the distribution of potential loop closures.
+
+featured_image: '/images/projects/mmda.png'
+
+authors:
+- yixuany
+- masonbp
+- andyli
+
+active: true
+---
+
+### About 
+
+Global data association is an essential prerequisite
+for robot operation in environments seen at different times
+or by different robots. Repetitive or symmetric data creates
+significant challenges for existing methods, which typically rely
+on maximum likelihood estimation or maximum consensus
+to produce a single set of associations. However, in these
+ambiguous scenarios, the distribution of solutions to global
+data association problems is often highly multimodal, and such
+single-solution approaches frequently fail. In this work, we
+introduce a data association framework that leverages approx-
+imate Bayesian inference to capture multiple solution modes
+to the data association problem, thereby avoiding premature
+commitment to a single solution under ambiguity. Our approach
+represents hypothetical solutions as particles that evolve via
+deterministic or randomized updates, naturally parallelizable
+on GPUs, to cover the modes of the underlying solution
+distribution. Simulated and real-world experiments with highly
+ambiguous data show that our method correctly estimates the
+distribution over transformations when registering point clouds
+or object maps. Code is available at: [Link](https://github.com/mit-acl/mmda).
+
+![](/images/projects/mmda.png)
+*Symmetric and repetitive structures are common in human
+environments and induce perceptual aliasing. To localize, robots
+must explicitly model the uncertainty from ambiguities inherent
+in those environments. The top row shows several such scenarios,
+where the visualized object associations are perceptually similar
+but incorrect, necessitating a multimodal representation. The bottom
+row shows the multimodal distribution generated by our proposed
+method. The orange mode is incorrect and corresponds to the
+ambiguous associations shown in the top row.*
diff --git a/_projects/trail.md b/_projects/trail.md
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+---
+title: "Off-Road Navigation via Implicit Neural Representation of Terrain Traversability"
+
+date: 2026-09-04
+
+description: Leverage implicit representation for gradient-based trajectory optimization that modulates path geometry and speed profile simultaneously.
+
+featured_image: '/images/projects/trail.png'
+
+authors:
+- yixuany
+- andyli
+
+active: true
+---
+
+### About 
+
+Autonomous off-road navigation requires robots to estimate terrain traversability from onboard sensors and plan motion accordingly. 
+Conventional approaches typically rely on sampling-based planners such as MPPI to generate short-term control actions that aim to minimize traversal time and risk measures derived from the traversability estimates. These planners can react quickly but optimize only over a short look-ahead window, limiting their ability to reason about the full path geometry, which is important for navigating in challenging off-road environments. Moreover, they lack the ability to adjust speed based on the terrain-induced vibrations, which is important for smooth navigation on challenging terrains.
+In this paper, we introduce TRAIL (Traversability with an Implicit Learned Representation), an off-road navigation framework that leverages an implicit neural representation to model terrain properties as a continuous field that can be queried at arbitrary locations. This representation yields spatial gradients that enable integration with a novel gradient-based trajectory optimization method that adapts the path geometry and speed profile based on terrain traversability.
+
+![](/images/projects/trail.png)
+***Top**: Optimized trajectories overlaid on the blended cost map, generated by combining two cost maps with 50% transparency (sampled at fixed grid resolution for visualization). The larger dark blocks represent higher geometric risk inflated using vehicle radius (e.g. the highlighted tree trunks) while finer greyscale variations correspond to predicted terrain bumpiness (e.g. grey regions around trees from tree roots). Redder trajectory segments indicate higher speed.
+**Bottom**: Corresponding onboard camera images. 
+The optimized trajectories avoid hard obstacles, slow down when approaching bumpy regions, and speed up in smoother areas.*
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