How does a quadrotor move horizontally if its rotors only point up?

A quadrotor moves horizontally by tilting. When the pitch angle theta increases the thrust vector T which normally points straight up gains a horizontal component T times sine theta. The controller creates this tilt by running the front rotor slower and the back rotor faster creating a pitch torque. This is why position and angle are coupled in the dynamics.

What are the 12 states of a quadrotor dynamic model?

The 12 states are three linear positions x y z in the inertial frame three Euler angles phi theta psi for roll pitch and yaw three linear velocities x dot y dot z dot and three body frame angular velocities p q r. The nonlinear equations of motion couple these states through the rotation matrix and gyroscopic terms making the system challenging to control.

Quadrotor Dynamics Simulator — 6-DOF Nonlinear Model

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Quadrotor Dynamics (6-DOF Nonlinear Model)