## 1 Introduction  
Underactuated systems are a crucial area of research in nonlinear control theory and robotics [1, 2]. Underactuated systems have fewer independent inputs than degrees of freedom [3, 4], such as balance bikes [5], flexible robots [6], crane systems [7], surface vessels [8], and quadcopter drones [9]. Underactuated systems can occur for a number of reasons:  
1. Active reduction of brakes to make the manipulator lighter and more energy-efficient [10], such as space station robotic arms [11] and snake robots [12];
2. An experimental system designed for the research of nonlinear control theory [13], such as the ball and beam system [14], the rotary pendulum [15], and underactuated manipulators [16];
3. Partial manipulator actuator damage [17]. With the continuous development of automation technology, scholars have focused on the motion control problem of mechanical systems [18, 19]. As a lot of work in space exploration and ocean engineering is carried out in a weightless or microgravity environment, the planar manipulator--without considering the effect of gravity--is the main actuator inside. When part of the drive of the planar fully driven mechanical manipulator is damaged, it can be switched to the control method of an underactuated system, which greatly enhances the fault tolerance of the entire system [20, 21]. Exploring the position control problem of the planar underactuated manipulator holds significant practical value for rapidly developing fields such as space exploration and ocean engineering [22]. For a gravity-unconstrained planar underactuated manipulator, it can be stabilized in any position on the plane [23; 24]. In addition, the approximate model of the planar underactuated manipulator is difficult to control because the approximate model does not satisfy the linear controllability condition of the equilibrium point [25; 26]. According to the integration characteristics of planar underactuated manipulators, they can be categorized into three different types:  
1. The holonomic system that is fully integrable [27; 28];
2. The first-order non-holonomic system that is partially integrable [29; 30];
3. The second-order non-holonomic system that is non-integrable [31; 32]. The planar Acrobot is a typical holonomic system with angular constraints. [PERSON] et al. [33] analyzed the integrability of planar Acrobot, and deduced the constraint relationship between the angle and angular velocity of each link. According to this relationship, a control strategy is designed by using the Lyapunov function method to realize the stable control of the planar Acrobot. [PERSON] et al. [34] analyzed the controllability, minimum phase characteristics, small-time local controllability, differential flatness, and nilpotentizable properties of the linear approximation model of the planar Acrobot. They used parametric polynomial input to stabilize the system to a nonsingular equilibrium configuration. The PMLUM with the first passive link is a non-holonomic system with angular velocity limitations. [PERSON] et al. [35] constructed the coupling relationship between the control torques, transformed the stability of the 3-DOF underactuated manipulator into the stability of the 2-DOF underactuated manipulator, and designed the torque coupling control strategy according to the Lyapunov stability. [PERSON] et al. [36] proposed a continuous control strategy based on a differential evolution algorithm for the planar four-link passive first-joint underactuated manipulator according to the constraint relationship between the active and passive joints. Meanwhile, in order to overcome the sudden initial torque change, a step PD controller is designed, and the controller parameters are optimized by a differential evolution algorithm. The PMLUM containing the final or middle PL belongs to the second-order non-holonomic system. [PERSON] et al. [37] applied the principle of energy decay to a target value to simplify the planar APAA system into a planar PAA system.