for the ball to bounce at in the $(x,y)$ plane, $ {a}_{min}$ and $ {a}_{max}$ are constraints on the magnitudes of the action, and $P_{obstacle}$ is a set of locations for which the ball is not permitted to bounce at. The goal of this optimization problem is to get the ball as close to the target while maintaining action and state constraints. More details of this controller are given in the supplementary material.

All forward models are multilayer perceptrons with two hidden layers of 128 and 64 neurons. Leaky rectified linear activations are used for the hidden layers.

![Ball-Paddle Simulator](images/simulator.png){width="30.00000%"}

Sufficiently Accurate Models
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To investigate the performance of our model, we first consider the effect of learning a model with an objective described by Eq. \[eq:constrained\_model\_learning\_example\] rather than Eq. \[eq:model\_learning\]. Does the normalization and constraints make a difference in the quality and performance of the model? In Fig. \[fig:model\_errs\], the magnitude of the errors is plotted against the magnitude of the output state.

![**Model errors vs. state magnitude**. On the left side, data from a model trained with Eq. \[eq:model\_learning\] is shown. The right side shows data from a model using Eq. \[eq:constrained\_model\_learning\]. The unconstrained problem leads to a 