Fig. 3: DR-FREE evaluation.
From: Distributionally robust free energy principle for decision-making
a Unicycle robots (from the Robotarium) of 11cm × 8.5cm × 7.5cm (width, length, height) that need to achieve the goal destination, xd, avoiding obstacles. The Robotarium work area is 3m × 2m, the robot position is the state, and actions are vertical/horizontal speeds; \({q}_{k}({{{{\bf{x}}}}}_{k}| {{{{\bf{x}}}}}_{k-1},{{{{\bf{u}}}}}_{k})\) is a Gaussian centered in xd and \({q}_{k}({{{{\bf{u}}}}}_{k}| {{{{\bf{x}}}}}_{k-1})\) is uniform. See Methods for the settings. b The non-convex state cost for the navigation task. See Methods for the expression. c Comparison between DR-FREE and a free-energy minimizing agent that makes optimal decisions but is unaware of the ambiguity. DR-FREE enables the robot to successfully complete the task at each training stage. The ambiguity-unaware agent fails, except when the shortest path is obstacle-free. Training details are in “Methods”. d Screenshots from Supplementary Movie 1. This is from the Robotarium platform. DR-FREE allows the robot (starting top-right) to complete the task (trained model from stage 3 used). e How DR-FREE policy changes as a function of ambiguity. By increasing the radius of ambiguity by 50%, DR-FREE policy (left) becomes a policy dominated by ambiguity (right). As a result, actions with low ambiguity are assigned higher probability. Screenshot of the robot policy when this is in position [0.2, 0.9], i.e., near the middle obstacle. The ambiguity increase deterministically drives the robot bottom-left (note the higher probability) regardless of the presence of the obstacle. f Belief update. Speeds/positions from the top-right experiments in (c) are used together with F = 16 state/action features, \({\varphi }_{i}({{{{\bf{x}}}}}_{k-1},{{{{\bf{u}}}}}_{k})={{\mathbb{E}}}_{{\overline{p}}_{k}({{{{\bf{x}}}}}_{k}| {{{{\bf{x}}}}}_{k-1},{{{{\bf{u}}}}}_{k})}[{\phi }_{i}({{{{\bf{X}}}}}_{k})]\) in Supplementary Fig. 1b. Once the optimal weights, \({w}_{i}^{\star }\), are obtained, the reconstructed cost is \(-{E}_{{\overline{p}}_{k}({{{{\bf{x}}}}}_{k}| {{{{\bf{x}}}}}_{k-1},{{{{\bf{u}}}}}_{k})}[{\sum }_{i=1}^{16}{w}_{i}^{\star }{\phi }_{i}({{{{\bf{X}}}}}_{k})]\). Since this lives in a 4-dimensional space, we show \(-{\sum }_{i=1}^{16}{w}_{i}^{\star }{\phi }_{i}({{{{\bf{x}}}}}_{k})\), which can be conveniently plotted.
