Reachable Predictive Control: A Novel Control Algorithm for Nonlinear Systems with Unknown Dynamics and its Practical Applications

TL;DR

It is demonstrated that it is possible to follow a set of waypoints comprised of states analytically proven to be reachable despite not knowing the system dynamics.

Abstract

This paper proposes an algorithm capable of driving a system to follow a piecewise linear trajectory without prior knowledge of the system dynamics. Motivated by a critical failure scenario in which a system can experience an abrupt change in its dynamics, we demonstrate that it is possible to follow a set of waypoints comprised of states analytically proven to be reachable despite not knowing the system dynamics. The proposed algorithm first applies small perturbations to locally learn the system dynamics around the current state, then computes the set of states that are provably reachable using the locally learned dynamics and their corresponding maximum growth-rate bounds, and finally synthesizes a control action that navigates the system to a guaranteed reachable state.

Figures (6)

• Figure 1: Trajectory following for unknown systems where the line (red) represents the desired reachable linear path contained within the GRS (gray). The $\mathfrak{x}_{n} = \tilde{x}(\tau_n)$ denotes the begining point of each learning cycle, while $z_n$ represents the sequence of points automatically generated by Algorithm \ref{['alg: one_time']}, each located an $r$-distance apart and progressively approaching $y:=\tilde{x}_f$, toward which the system state converges within each learning cycle.
• Figure 2: An optimal path determined by solving \ref{['eq: optimal_path_problem']}. Path obstructing obstacles are shown in green, brown, and blue with the tolerance radius $\mathbb{B}^2(c_k, r_k + \Delta)$ for all $k$ obstacles.
• Figure 3: On the left, the true reachable set (green) of actuated states $\theta,v$ of system \ref{['eq:kinematic_bicycle_model']} calculated knowing the dynamics, and an underapproximation (red) calculated using solely the knowledge from Assumption \ref{['ass: growth_rate_bounds']} for system \ref{['eq:simple_bicycle_model']}. On the right, we apply Algorithm \ref{['alg: one_time']} to learn a path (blue) designed to follow a reachable path (black). The learned path (blue) is calculated without knowing the dynamics, but instead using $\tilde{x}_0 = 01^T$, $L_f^\mathcal{N} = 0$, $L_G^\mathcal{N} = 3$, $\tilde{f}(\tilde{x}_0) = 0$, $\tilde{G}(\tilde{x}_0) = I$.
• Figure 4: On the left, we have the reachable set (blue) of states $\overline{x} = xy^T$ of system \ref{['eq:simple_bicycle_model']}, calculated knowing the dynamics, and its guaranteed underapproximation (red). On the right, we calculate $\mathcal{R}_{\overline{x}}^{\tilde{n}}(T,x_0)$. The sets in red are calculated without knowledge of the dynamics, using solely the knowledge of $x(0) = 0002.0^T$ for $T = 0.1$ seconds, $\tilde{f}(\tilde{x}_0) = 0$, $\tilde{G}(\tilde{x}_0) = \operatorname{diag}(1,2)$, $L_f^{\hat{n}} = 0$, and $L_G^{\hat{n}} = 3$ for $\hat{n} \in \{1,\hdots,5\}$.
• Figure 5: Autonomous path following of the desired path (green) by informed decision making using calculated guaranteed reachable positions (red) over a total of $0.5$ seconds.

Theorems & Definitions (3)

• proof
• proof
• proof