A hopping rover is a robot that can move in low gravity planets by the characteristic motion called the hopping motion. For its autonomous explorations, the so-called SLAM (Simultaneous Localization and Mapping) is a basic function. SLAM is the combination of estimating the position of a robot and creating a map of an unknown environment. Most conventional methods of SLAM are based on odometry to estimate the position of the robot. However, in the case of the hopping rover, the error of odometry becomes considerably large because its hopping motion involves unpredictable bounce on the rough ground on an unexplored planet. Motivated by the above discussion, this paper addresses a problem of finding an optimal movement of the hopping rover for the estimation performance of the SLAM. For the problem, we first set the model of the SLAM system for the hopping rover. The problem is formulated as minimizing the expectation of the estimation error at a pre-specified time with respect to the sequence of control inputs. We show that the optimal input sequence tends to force the final position to be not at the landmark but in front of the landmark, and furthermore, the optimal input sequence is constant on the time interval for optimization.

When a mathematical model is not available for a dynamical system, it is reasonable to use a data-driven approach for analysis and control of the system. With this motivation, the authors have recently developed a data-driven solution to Lyapunov equations, which uses not the model but the data of several state trajectories of the system. However, the number of state trajectories to uniquely determine the solution is O(n2) for the dimension n of the system. This prevents us from applying the method to a case with a large n. Thus, this paper proposes a novel class of data-driven Lyapunov equations, which requires a smaller amount of data. Although the previous method constructs one scalar equation from one state trajectory, the proposed method constructs three scalar equations from any combination of two state trajectories. Based on this idea, we derive data-driven Lyapunov equations such that the number of state trajectories to uniquely determine the solution is O(n).