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Towards Certified Sim-to-Real Transfer via Stochastic Simulation-Gap Functions

P Sangeerth, Abolfazl Lavaei, Pushpak Jagtap

Abstract

This paper introduces the notion of stochastic simulation-gap function, which formally quantifies the gap between an approximate mathematical model and a high-fidelity stochastic simulator. Since controllers designed for the mathematical model may fail in practice due to unmodeled gaps, the stochastic simulation-gap function enables the simulator to be interpreted as the nominal model with bounded state- and input-dependent disturbances. We propose a data-driven approach and establish a formal guarantee on the quantification of this gap. Leveraging the stochastic simulation-gap function, we design a controller for the mathematical model that ensures the desired specification is satisfied in the high-fidelity simulator with high confidence, taking a step toward bridging the sim-to-real gap. We demonstrate the effectiveness of the proposed method using a TurtleBot model and a pendulum system in stochastic simulators.

Towards Certified Sim-to-Real Transfer via Stochastic Simulation-Gap Functions

Abstract

This paper introduces the notion of stochastic simulation-gap function, which formally quantifies the gap between an approximate mathematical model and a high-fidelity stochastic simulator. Since controllers designed for the mathematical model may fail in practice due to unmodeled gaps, the stochastic simulation-gap function enables the simulator to be interpreted as the nominal model with bounded state- and input-dependent disturbances. We propose a data-driven approach and establish a formal guarantee on the quantification of this gap. Leveraging the stochastic simulation-gap function, we design a controller for the mathematical model that ensures the desired specification is satisfied in the high-fidelity simulator with high confidence, taking a step toward bridging the sim-to-real gap. We demonstrate the effectiveness of the proposed method using a TurtleBot model and a pendulum system in stochastic simulators.
Paper Structure (7 sections, 4 theorems, 18 equations, 5 figures)

This paper contains 7 sections, 4 theorems, 18 equations, 5 figures.

Table of Contents

  1. Introduction
  2. System Description
  3. Data-driven Framework
  4. Formal Quantification of Stochastic Simulation-Gap Function
  5. Controller Synthesis
  6. Case Studies
  7. Conclusion and Future Work

Key Result

Lemma III.2

Let $[\eta_i^*;q^{(1)}_i;\cdots;q^{(z_i)}_i]$ be a feasible solution of SCP-$\hat{N}_1$eq: SCP_for_one_state_n. Suppose that Assumption variance assumption holds with a given $\hat{M}^{(i)}$. Then, for some arbitrary $\delta_{1,i} > 0$, one has for some $\beta_{1,i} \geq \frac{\hat{M}^{(i)}}{\delta_{1,i}^2 \hat{N}_{1}}$.

Figures (5)

  • Figure 1: Simulation showing that out of 1000 realizations of TurtleBot from $(x,y,\theta) = (3.5,2,1.5)$ using the controller and deterministic simulation gap $\gamma(x,u)$ from sangeerth2024towards, 44 resulted in collisions in the Gazebo simulator due to unmodeled simulator stochasticity. Obstacles are shown in black.
  • Figure 2: TurtleBot model in the Gazebo simulator.
  • Figure 3: The realization of 1000 state trajectories from the initial state $(x,y,\theta) = (3.5,2,1.5)$. Black regions indicate obstacles. The controller designed using the mathematical model, along with the stochastic simulation gap, ensures the satisfaction of the specification in the stochastic simulator for all 1000 trajectories, even though the theorem guaranteed satisfaction in the first-moment sense. The dark black line shows the average trajectory.
  • Figure 4: Pendulum model in the PyBullet simulator.
  • Figure 5: The mathematical system trajectory is shown in red, and the stochastic simulator trajectory in blue. In PyBullet, the controller synthesized using $\gamma(x,u)$ from sangeerth2024towards violates the invariance specification (top). In contrast, the controller synthesized using the proposed approach ensures invariance in PyBullet in the first-moment sense (bottom), based on an average of 1000 trajectories. The $x_1$ trajectory, which remained within the safe set in both cases, is omitted for brevity. Multiple runs showing stochasticity are also omitted for brevity.

Theorems & Definitions (11)

  • Remark II.2
  • Lemma III.2
  • proof
  • Lemma III.3
  • proof
  • Remark IV.2
  • Theorem IV.3
  • proof
  • Theorem IV.4
  • proof
  • ...and 1 more