Many hybrid problems in robotics involve both continuous and discrete components, and modeling them together for estimation tasks has been a long standing and difficult problem. Hybrid Factor Graphs give us a mathematical framework to model these types of problems, however existing approaches for solving them are based on approximations. In this work, we propose an efficient Hybrid Factor Graph framework alongwith a variable elimination algorithm to produce a hybrid Bayes network, which can then be used for exact Maximum A Posteriori estimation and marginalization over both sets of variables. Our approach first develops a novel hybrid Gaussian factor which can connect to both discrete and continuous variables, and a hybrid conditional which can represent multiple continuous hypotheses conditioned on the discrete variables. Using these representations, we derive the process of hybrid variable elimination under the Conditional Linear Gaussian scheme, giving us exact posteriors as hybrid Bayes network. To bound the number of discrete hypotheses, we use a tree-structured representation of the factors coupled with a simple pruning and probabilistic assignment scheme, which allows for tractable inference. We demonstrate the applicability of our framework on a SLAM dataset with ambiguous measurements, where discrete choices for the most likely measurement have to be made. Our demonstrated results showcase the accuracy, generality, and simplicity of our hybrid factor graph framework.

翻译：机器人学中的许多混合问题同时涉及连续和离散分量，为估计任务对它们进行联合建模长期以来一直是一个困难问题。混合因子图为这类问题的建模提供了数学框架，然而现有的求解方法均基于近似。本文提出一种高效的混合因子图框架及相应的变量消元算法，可生成混合贝叶斯网络，进而用于对两组变量进行精确的最大后验估计与边缘化。我们的方法首先构建了新型混合高斯因子——该因子可同时连接离散与连续变量，以及混合条件分布——该分布能表征以离散变量为条件的多个连续假设。基于这些表示，我们在条件线性高斯框架下推导出混合变量消元过程，从而获得以混合贝叶斯网络形式呈现的精确后验分布。为约束离散假设数量，我们采用树状结构表示因子，并结合剪枝与概率分配方案，从而实现可处理的推理。我们在具有模糊观测的SLAM数据集上验证了所提框架的适用性，该场景需要为最可能观测做出离散选择。实验结果表明了本混合因子图框架的精确性、通用性与简洁性。