Online Hybrid-Belief POMDP with Coupled Semantic-Geometric Models

TL;DR

This work tackles planning under uncertainty for autonomous robots that must reason about both geometry and semantics by introducing a hybrid semantic-geometric belief within a POMDP framework. It develops a factorized belief form that enables planning-time sampling and explicit computation over semantic mappings under structured rewards, addressing the exponential growth of semantic hypotheses. The authors show that the value function and probability of safety can be estimated efficiently with representative samples, using planning-time sampling (MH/SNIS) and Rao-Blackwellization to reduce MSE, under certain reward structures. Experiments in a synthetic 2D setting demonstrate that these methods achieve accuracy comparable to exhaustive methods but with polynomial rather than exponential complexity, supporting safer and more reliable planning in complex semantic-geometric environments.

Abstract

Robots operating in complex and unknown environments frequently require geometric-semantic representations of the environment to safely perform their tasks. While inferring the environment, they must account for many possible scenarios when planning future actions. Since objects' class types are discrete and the robot's self-pose and the objects' poses are continuous, the environment can be represented by a hybrid discrete-continuous belief which is updated according to models and incoming data. Prior probabilities and observation models representing the environment can be learned from data using deep learning algorithms. Such models often couple environmental semantic and geometric properties. As a result, semantic variables are interconnected, causing semantic state space dimensionality to increase exponentially. In this paper, we consider planning under uncertainty using partially observable Markov decision processes (POMDPs) with hybrid semantic-geometric beliefs. The models and priors consider the coupling between semantic and geometric variables. Within POMDP, we introduce the concept of semantically aware safety. Obtaining representative samples of the theoretical hybrid belief, required for estimating the value function, is very challenging. As a key contribution, we develop a novel form of the hybrid belief and leverage it to sample representative samples. We show that under certain conditions, the value function and probability of safety can be calculated efficiently with an explicit expectation over all possible semantic mappings. Our simulations show that our estimates of the objective function and probability of safety achieve similar levels of accuracy compared to estimators that run exhaustively on the entire semantic state-space using samples from the theoretical hybrid belief. Nevertheless, the complexity of our estimators is polynomial rather than exponential.

Figures (8)

• Figure 1: An illustration of a viewpoint-dependent observation model. Sensors receive different information from different viewpoints. Consider the semantic observations are the output of a classifier. The blue bars represent classifier output. From each viewpoint the semantic observation, in this case the distribution over classes obtained from a classifier, can vary due to change in visual appearance or detected features.
• Figure 2: A factor graph diagram, representing the belief $b_k[C,X_k]$. The region outlined by the blue dotted line highlights the factor graph of the geometric belief $b_k^g[X_k]$, which corresponds to the geometric observations $z^g_{1:3}$, the actions $a_{1:2}$, and the robot prior probability $\mathbb{P}_0(X_0)$. The semantic variables $c_1$ and $c_2$, in orange, are incorporated into the graph through semantic observations $z^s_{2, n}$ and semantic priors $\mathbb{P}_0$. Given the state $X_k$, the semantic variables ($c_1, c_2$) are independent.
• Figure 3: (a) The true trajectory and objects' unsafe area. Robot starts at (0,0) and reaches (9,9). The class of an object specifies the unsafe area around it. Each time step, $\mathbb{P}_{\text{safe}}$ is calculated assuming the robot will continue to move in a straight line. (b) One trial probability of safety versus time, compared between different methods. In the case of the theoretical-all-hyp , Hoffding's inequality holds, ensuring this result. For the rest of the estimators, MCMC-Ours and PF-all-hyp are aligned with the theoretical-all-hyp , SNIS-Ours is slightly further away, and theoretical-pruned , PF-pruned and GS-MAP are more biased. The bias can be positive or negative, and there is no indication as to which it will be. In the case of GS-MAP we get the least accurate results.
• Figure 4: Expected reward error versus number of samples. The error decreases as the sample size increases for the theoretical-all-hyp , PF-all-hyp , MCMC-Ours and SNIS-Ours estimators. Theoretical-pruned , PF-pruned and GS-MAP estimators contain a bias that is not affected by sample size.
• Figure 5: RMSE of $\mathbb{P}_{\text{safe}}$ estimations versus time-step.

Theorems & Definitions (15)

• Lemma 3.1
• Definition 3.2
• Theorem 5.1
• Theorem 5.5
• Theorem 5.6: Rao-Blackwellization for Sampled Based Estimations
• Lemma 5.7
• Theorem 5.8
• Proposition 5.9
• proof
• proof