${\tt MORALS}$: Analysis of High-Dimensional Robot Controllers via Topological Tools in a Latent Space

TL;DR

MORALS integrates autoencoder-based latent representations with Morse Graphs to enable data-efficient RoA estimation for high-dimensional, data-driven robot controllers. By projecting dynamics into a learned latent space and constructing a reduced bistable Morse Graph, the method identifies attractors and their basins while vastly reducing discretization complexity. Evaluations on a 67-dim humanoid, a 96-dim 3-finger manipulator, and additional benchmarks show accurate RoA estimation with substantially fewer trajectories than prior approaches. This approach offers a scalable, explainable tool for safe controller design and planning in complex robotic systems.

Abstract

Estimating the region of attraction (${\tt RoA}$) for a robot controller is essential for safe application and controller composition. Many existing methods require a closed-form expression that limit applicability to data-driven controllers. Methods that operate only over trajectory rollouts tend to be data-hungry. In prior work, we have demonstrated that topological tools based on ${\it Morse Graphs}$ (directed acyclic graphs that combinatorially represent the underlying nonlinear dynamics) offer data-efficient ${\tt RoA}$ estimation without needing an analytical model. They struggle, however, with high-dimensional systems as they operate over a state-space discretization. This paper presents ${\it Mo}$rse Graph-aided discovery of ${\it R}$egions of ${\it A}$ttraction in a learned ${\it L}$atent ${\it S}$pace (${\tt MORALS}$). The approach combines auto-encoding neural networks with Morse Graphs. ${\tt MORALS}$ shows promising predictive capabilities in estimating attractors and their ${\tt RoA}$s for data-driven controllers operating over high-dimensional systems, including a 67-dim humanoid robot and a 96-dim 3-fingered manipulator. It first projects the dynamics of the controlled system into a learned latent space. Then, it constructs a reduced form of Morse Graphs representing the bistability of the underlying dynamics, i.e., detecting when the controller results in a desired versus an undesired behavior. The evaluation on high-dimensional robotic datasets indicates data efficiency in ${\tt RoA}$ estimation.

Figures (5)

• Figure 1: The 67-dim. state space of a bipedal humanoid robot controlled by a Soft Actor-Critic (SAC) controller is encoded to a 2-dim. learned latent space by MORALS, which then discovers two attractors and their corresponding Regions of Attraction (RoA s) in the latent space. Encoded final states A2, B2 are mapped to a desired (dark green) and undesired attractor (dark purple) respectively. Encoded initial states A1, B1 lie respectively in the RoA s (light green and light purple) of the desired and undesired attractors. The yellow region contains the separatrix (undecidable region), indicating initial states may go to A2 (node 2 $\rightarrow$ node 1) or B2 (node 2 $\rightarrow$ node 0). Best viewed in color.
• Figure 2: Example of $N$-dim. bistability (left and middle) and the learned dynamics on the 2-dim. encoded space (right). (left & middle) The state space is $X=X_1 \times \prod_{i=2}^N X_i$ where $X_i=[-3,3]\times[-2,2]^{N-1}$ and the dynamics $\phi : X \to X$ is given by $\phi_1(x) = \arctan(4x)$ plotted in black and $\phi_i(x) = x/2$ plotted in red, where $i=2, \ldots, N$. The domains $X_1 = [-3,3]$ and $X_i=[-2,2]$ are decomposed into intervals $a$ through $e$ and $f_i, g_i, h_i$, respectively. Forward propagation of $B=b\times \prod g_i$ is depicted by the lines from the boundary of $b$ and $g_i$'s. ${\mathcal{F}}$ is a directed graph capturing reachable vertices from other vertices, (regions in $\{a,b,c,d,e\}\times\prod_{i=2}^N\{f_i,g_i,h_i\}$). Strongly connected components of ${\mathcal{F}}$ result in $\mathsf{CG}({\mathcal{F}})$. Finally, the Morse Graph ${\mathsf{ MG}}({\mathcal{F}})$ (nodes B, C, D) contains the attractors and expresses their RoA s. (right) The $N$-dim. bistability dynamics are encoded into a 2-dim. latent space represented by a bistable Morse Graph ${\mathsf R}_{{\mathsf{ MG}}}$.
• Figure 3: (Left) The autoencoding neural network and loss functions used for training the encoder $h_\text{enc}$, decoder $h_\text{dec},$ and the latent dynamics $h_\text{dyn}$. (Middle/Right) Visualizing the learned latent dynamics for a 4-dim. version of a pendulum $(x,\dot{x},y,\dot{y})$ controlled by LQR. (Middle) the ground-truth trajectory for the same initial conditions (circles). (Right) iteratively calling $h_\text{dyn}$ for a fixed number of timesteps. Both plots capture the 3 true attractors ($\times$). The right plot does not contain regions where the trajectories move from one RoA to another.
• Figure 4: a) A Morse graph ${\mathsf{ MG}}({\mathcal{F}})$ for the humanoid of Fig. \ref{['fig:overview']}; b) bistable Morse graph ${\mathsf R}_{{\mathsf{ MG}}}$ from Th. \ref{['thm:order_retraction']}.
• Figure 5: (L-R) Analytical Pendulum, Cartpole simulated using MuJoCo, real robot dataset collected using a TriFinger guertler2023benchmarking.

Theorems & Definitions (3)

• Theorem IV.1
• proof
• Theorem IV.2