This assignment explores optimizing a one-shot task (basketball free throw) to maximize the probability of success.
We crudely approximate a basketball free throw (also known as a foul shot). The ball is a point. The release point of the ball is 4m horizontally away from and 1m below the midpoint of the basket opening. Our basket radius is 0.15m (actual hoop radius minus fudge factor). Gravity is 9.81 m/s^2.
1. Find the minimum velocity and corresponding launch angle to send the point ball through the basket. Note that the ball must enter the basket coming down.
Note that the ball trajectory should skim the basket's edge on the optimal trajectory, and any uncertainty might cause the shot to miss. The rest of this assignment addresses ways to reduce the effect of uncertainty. Before we start, can you think of a quick hack to reduce the effect of uncertainty? What is the trajectory that minimizes the effect of uncertainty?
2. Let's assume the launch angle has additive zero-mean Gaussian noise with variance 0.01 (N(0,0.01)), and the velocity magnitude has multiplicative zero-mean Gaussian noise with variance 0.0001 (N(0,0.0001)). What is launch angle and velocity maximize the probablity of success?
2A. Calculate this by propagating the state uncertainty forward in time and linarizing the dynamics. For extra credit try calculating this by propagating the state uncertainty forward in time using the techniques used by unscented filters.
2B. Calculate this by evaluating many launches. How many launches are needed to get reliable results?
2C. Combine the approaches in 2A and 2B. For each thow, propagate uncertainty along that trajectory. This is similar to the approach of particle filters.