Control Barrier Functions With Real-Time Gaussian Process Modeling

TL;DR

A recursive formulation for the model update is presented, which reduces time complexity of the update from O(p3) to O(p2), where p is the number of data used and O(p3) is the number of data used.

Abstract

We present an approach for satisfying state constraints in systems with nonparametric uncertainty by estimating this uncertainty with a real-time-update Gaussian process (GP) model. Notably, new data is incorporated into the model in real time as it is obtained and select old data is removed from the model. This update process helps improve the model estimate while keeping the model size (memory required) and computational complexity fixed. We present a recursive formulation for the model update, which reduces time complexity of the update from O(p3) to O(p2), where p is the number of data used. The GP model includes a computable upper bound on the model error. Together, the model and upper bound are used to construct a control-barrier-function (CBF) constraint that guarantees state constraints are satisfied.

Figures (11)

• Figure 1: $\gamma$, $\dot{\gamma}$ and $u$ for Cases 1, 2 and 3, Note that $\gamma_{\rm{d}}$, $\dot{\gamma}_{\rm{d}}$ and $u_{\rm{d}}$ are shown with dashed lines.
• Figure 2: $\psi_{0}$ and $\psi_{1}$ for Cases 1, 2 and 3.
• Figure 3: $\psi$ for Cases 1, 2 and 3.
• Figure 4: $\mu_{[2]}$ for Cases 1, 2 and 3. Note that $w_{[2]}$ is shown using dashed line.
• Figure 5: $(\varphi)_{[2]}$ for Cases 1, 2 and 3. Note that $|(\mu-w)_{[2]}|$ is shown with dashed lines.