Figure 5
From: Combined perception and control for timing in robotic music performances
A trajectory with using optimal control u τ for κ = 150. Here, the arrows denote the magnitude and the direction of the optimal control action u τ as a function of the state. Since the control action is defined as acceleration or deceleration only, actions can only affect the velocity. Here, the robot was initially both faster, and its position was ahead of the performer. Hence, using the corresponding optimal control action, it tends to decelerate. However, it cannot directly catch the performer, since both the deceleration would affect the position in time, and there is also an associated penalty with large controls. Hence, the tempo cannot change quickly and the robot follows a non-trivial trajectory until convergence.