This project chronicles the design, analytical modeling, and control synthesis of increasingly complex, underactuated dynamical systems, demonstrating a progression from a standard benchmark to a highly chaotic multi-body system. Beginning with a 1D Inverted Pendulum on a Cart (IPL), Euler-Lagrange mechanics were utilized to map the kinetic and potential energies into equations of motion for the generalized coordinates $q = [x, \theta]^T$. This 4-state nonlinear system was linearized around its unstable upright equilibrium to synthesize a Linear Quadratic Regulator (LQR) that minimizes state deviation against control effort. The architecture was then extended to 2D planar cart motion, expanding the state-space to 8 states ($x, y, \theta, \phi$ and their derivatives) and introducing complex gyroscopic coupling as the cart's orthogonal $X$-$Y$ forces were leveraged to regulate the spherical pendulum's pitch and roll. The final, most demanding phase culminated in the Double Inverted Pendulum on a Cart (DIPL)—a 3-degree-of-freedom system with 6 states, controlled by a single actuator force ($F$). The dynamics of this phase conform to the rigorous robotic manipulator structure $M(q)\ddot{q} + C(q, \dot{q})\dot{q} + G(q) = \tau$, where the unactuated upper link relies entirely on the precise inertial forces transferred through the lower link. By successfully linearizing this highly non-minimum phase system and implementing advanced MIMO state-space controllers, a robust basin of attraction was established to stabilize both links vertically, showcasing a deep mastery of analytical mechanics, state-space control theory, and advanced system decoupling.