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@@ -25,7 +25,7 @@ Whether learned, simulated, or analytical, approximations of a robot's dynamics
 
 # Problem Statement
 
-**Problem.** Given an approximation `$\tilde f$` of the system's *unknown* stochastic dynamics `$f$`, a goal region `$\mathcal G \subseteq \mathcal S$`, a safe set `$\mathscr C \subseteq \mathcal S$`, a calibration dataset `$D_{cal}$` and an acceptable failure-rate `$\alpha \in (0,1)$`, we aim to recursively solve the following stochastic optimization problem with planning horizon `$H \in \mathbb N$`:
+**Problem.** Given an approximation `$\tilde f$` of the system's *unknown* stochastic dynamics `$f$`, a goal region `$\mathcal G \subseteq \mathcal S$`, a safe set `$\mathscr C \subseteq \mathcal S$`, a calibration dataset of state transitions `$D_{cal}$` and an acceptable failure-rate `$\alpha \in (0,1)$`, we aim to recursively solve the following stochastic optimization problem with planning horizon `$H \in \mathbb N$`:
 ```
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 \begin{alignat}{3}
@@ -43,13 +43,13 @@ $$
 
 # LUCCa
 
-Given a dynamics predictor and a small calibration dataset, LUCCa provides probabilistically valid prediction regions for the robot's future states accounting for both aleatoric and epistemic uncertainty. We prove its validity for any finite set of calibration data, predictors outputting a multivariate normal uncertainty, any unknown dynamics function, and uncharacterized aleatoric perturbations. LUCCa calibrates the uncertainty locally relative to the system's state-action space, leading to prediction regions that are representative of predictive uncertainty. For the first planning step, the dynamics predictor can take any form. However, for subsequent planning steps, exact calibration. We use Loewner's partial order to prove that linear dynamics approximations (true dynamics are unconstrained), LUCCa provides the desired coverage guarantee for all planning steps (see Appendix `$A$` for discussion and proof).
+Given a dynamics predictor and a small calibration dataset, LUCCa provides probabilistically valid prediction regions for the robot's future states accounting for both aleatoric and epistemic uncertainty. We prove its validity for any finite set of calibration data, predictors outputting a multivariate normal uncertainty, any unknown true dynamics function, and uncharacterized aleatoric perturbations. LUCCa calibrates the uncertainty locally relative to the system's state-action space, leading to prediction regions that are representative of predictive uncertainty and therefore useful for planning. For the first planning step, LUCCa satisfies the safety condition (3) above for any predictor form. For subsequent planning steps, this becomes more complex, but we use Loewner orders to prove that, if the dynamics approximation is linear (actual dynamics are still unconstrained), then LUCCa satisfies the safety condition (3) for all planning steps no matter the horizon length (see Appendix `$A$` for a discussion and the proof).
 
 {{ figure(alt=["LUCCA Algorithm", "Calibrated Rollout"] src=["lucca_algo.png", "calibrated_rollout.png"] dark_invert=[true, true]) }}
 
 # Experiments
 
-We conducted experiments on an MPC controller that uses LUCCa to plan short-horizon trajectories to reach a goal. We compared LUCCa with a baseline on a double-integrator system over four environments shown below. In the white regions the dynamics predictor corresponds to the ground truth dynamics, but in the yellow regions there is a significant mismatch (actually dynamics become lower-friction). In both areas there is aleatoric uncertainty.
+We conducted experiments with an MPC controller that uses LUCCa to plan short-horizon trajectories to reach a goal. We compared LUCCa with a baseline on a double-integrator system over four environments shown below. In the white regions the dynamics predictor corresponds to the ground truth dynamics, but in the yellow regions there is a significant mismatch (actual dynamics become lower-friction). In both areas there is aleatoric uncertainty.
 
 {{ figure(src = ["./Corridor_Uncalibrated Baseline.mp4","./Corridor_LUCCa.mp4"], subcaption = ["**Uncalibrated Baseline** (Corridor Map)","**LUCCa** (Corridor Map)"], dark_invert=[false,false]) }}