![]() With j = 2 and q represents generalized coordinate variables of x and theta. The nonlinear dynamics are then found by finding the Euler-Lagrange Equations using: The potential energy of the system is represented by the potential energy of the pole. The kinetic energy of the system is represented by: The Lagrangian of the cart pole system uses the general energy equation L = T – V where L is the Lagrangian term generalizing the total energy in the system, T is the kinetic energy of the system, and V is the potential energy of the system. The dynamics of the cart pole system is determined by evaluating the system using Lagrangian Mechanics. The concepts explored here can be extended to control any dynamic system, though not all systems are so straightforward to implement. In this case, there is only 1 actuated DOF, but 2 system DOF. This means there are fewer actuated degrees of freedom than there are degrees of freedom of the system. ![]() This is a classic control problem because deriving the dynamics is relatively straightforward, but it still belies some inherent complexity due to its underactuated nature. ![]() ![]() Here we will explore dynamics, modern control methods, and trajectory optimization by implementing various methods to control the canonical underactuated system, the cart-pole example. Dynamics, Control, and “Learning”: Cart-Pole ![]()
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