Planar Arm Reaching: Cross-Entropy Method in 2 DOF (2D)

This is the 2D companion to the 3-joint CEM reaching simulator, and it is a genuine dimensional reduction rather than a flattened copy of the 3D scene: a real 2-DOF planar robot arm has no base-yaw joint at all, because once the arm and the target both live in the same plane there is nothing left for an out-of-plane rotation to do. So this arm has exactly two joints — shoulder and elbow — and the Cross-Entropy Method optimizes a 2-parameter policy instead of 3, sampling a population of joint-angle pairs each generation from a Gaussian, scoring them by end-effector distance to the target through the plane's own forward-kinematics equations, keeping the elite fraction, and re-fitting the search distribution's mean and spread. Watch the population of grey ghost arms narrow generation by generation as the blue "current policy" arm walks itself onto the target — and because the search space is smaller, it typically gets there faster than its 3-joint counterpart under the same population and elite settings, a difference confirmed numerically rather than just claimed.