V-MORALS: Visual Morse Graph-Aided Estimation of Regions of Attraction in a Learned Latent Space

TL;DR

Visual Morse Graph-Aided Estimation of Regions of Attraction in a Learned Latent Space (V- MORALS) provides capabilities similar to the original MORALS architecture without relying on state knowledge, and using only high-level sensor data.

Abstract

Reachability analysis has become increasingly important in robotics to distinguish safe from unsafe states. Unfortunately, existing reachability and safety analysis methods often fall short, as they typically require known system dynamics or large datasets to estimate accurate system models, are computationally expensive, and assume full state information. A recent method, called MORALS, aims to address these shortcomings by using topological tools to estimate Regions of Attraction (ROA) in a low-dimensional latent space. However, MORALS still relies on full state knowledge and has not been studied when only sensor measurements are available. This paper presents Visual Morse Graph-Aided Estimation of Regions of Attraction in a Learned Latent Space (V-MORALS). V-MORALS takes in a dataset of image-based trajectories of a system under a given controller, and learns a latent space for reachability analysis. Using this learned latent space, our method is able to generate well-defined Morse Graphs, from which we can compute ROAs for various systems and controllers. V-MORALS provides capabilities similar to the original MORALS architecture without relying on state knowledge, and using only high-level sensor data. Our project website is at: https://v-morals.onrender.com.

Figures (6)

• Figure 1: Regions of Attraction (ROA) of the GetUp controller 10.1145/3528233.3530697 on a Humanoid. Given a trajectory of images, we learn a latent space to generate a Morse Graph and ROA. $I1$ and $I2$ are examples of image sequences (with simplified notation to suppress details). Here, $E(\cdot)$ refers to the encoded latent vector of an image sequence. The $t_0$ and $t_f$ represent, respectively, the initial and final images in the trajectory. Each colored region corresponds to a different attractor, allowing the system to predict the long-term outcome of a trajectory, such as success $I1 \rightarrow F1$ or failure $I2 \rightarrow F2$, and providing safety analysis without access to the system's state information.
• Figure 2: The process for constructing the directed graph F, which serves as a combinatorial map of the system's dynamics within the learned latent space $\mathcal{Z}$. In MORALS (Section \ref{['sec:morals_background']}), state sequences are first projected into the latent space by the encoder $E$. V-MORALS (Section \ref{['sec:proposed_method']}) builds upon this by projecting image sequences into the latent space as shown above. This space is then discretized into a grid of cells $C$. To determine the flow between cells, the corner points of a given cell ($c_i$) are propagated through the learned dynamics network $LD$. A safety bubble with radius $\delta$ is created around each predicted point to account for prediction uncertainty. A directed edge is drawn from $c_i$ to another cell $c_j$ in the graph $F(C)$ if the union of these safety bubbles intersects with $c_j$. This process is repeated for all valid cells to build a complete map of the latent dynamics.
• Figure 3: The V-MORALS architecture and training pipeline. To the left, we show representative binary images from our dataset $\mathcal{M}_I$ used to form training samples (from the Humanoid task). Using $\mathcal{M}_I$, we randomly sample a sequence of input images to form a training sample, $\bar{I}_{k-h:k}$. This is mapped to a low-dimensional latent vector $z_k$ by an Encoder ($E$). The dotted arrow lines represent a visualization of the operations in latent space $\mathcal{Z}$. A latent dynamics network ($LD$) is trained to predict the future latent state $\hat{z}_{k+1}$. A Decoder ($D$) reconstructs the image sequence $\bar{I}_{k-h:k}^{\prime}$ from the latent state $z_k$.
• Figure 4: Morse Graph and ROA of the Get Up controller applied to a Humanoid, with a latent space dimension of 3. The colors for each Morse Node describe how the set of states changes between each node. The dark blue node represents the attractor of the success region while the dark purple node represents the attractor of the failure. V-MORALS can successfully analyze a complex, high-degree-of-freedom system using only high-dimensional image data, providing an interpretable, low-dimensional map to predict bistable tasks. Additionally, V-MORALS further extends MORALS by being able to visualize ROAs in a 3 dimensional space.
• Figure 5: A comparison of Morse Graphs generated for Pendulum using different latent space dimensions. (Left) A 2-dimensional latent space yields a complex graph with multiple attractors (leaf nodes), failing to capture the true bistable nature of the task. (Right) Increasing the latent dimension to 3 results in a simpler and more accurate topological representation. The dynamics correctly converge to two distinct attractors (dark purple and green), which correspond to the task's success and failure modes.