This paper describes the kinematics and dynamics modelling of a mechanical system consisting of a spherical inverted pendulum whose base is mounted on a parallel planar mechanism. better known as a five-bar mechanism. The whole mechanism has four degrees of freedom. but it has only two actuators and so it is an under-actuated system. https://www.aitsglobal.com/product-category/%d7%97%d7%a4%d7%a6%d7%99-%d7%90%d7%95%d7%9e%d7%a0%d7%95%d7%aa/