## 1 Introduction  
With the rapid development of robot technology, various types of robots are being widely used in various fields. There are many kinds of robots, such as industrial robots [1; 2], medical robots [3], handling robots [4], and so on. Most of these robots are fully actuated robots. When a joint of the robot is damaged, the actuated robot becomes an underactuated system. Underactuated systems are one typical kind of nonlinear systems, whose number of the control inputs is less than the degrees of freedom of the system motion space [5; 6; 7]. Such systems are widespread in daily life and include crane [8; 9], quadrotor [10; 11], helicopter [12; 13], unmanned ship [14; 15], soft robot [16; 17], and underactuated robot [18; 19], among others. The study and development of the underactuated systems have been receiving more attention due to their outstanding features of lightweight quality, moderate price, and high work efficiency [20; 21; 22]. Therefore, many experts and scholars have concentrated on the control problems of the underactuated systems [23; 24; 25]. Moreover, the underactuated systems are frequently applied in aerospace [26] and deep-water exploration [27]. One of typical underactuated systems is the planar underactuated robot [28; 29; 30]. Usually, the problem with these systems is achieving stable control. For an underactuated robot, the linear approximation model at every equilibrium point in motion space is not directly controllable [31; 32; 33]. Therefore, it is very challenging to achieve the control of underactuated systems [34; 35; 36]. Among the underactuated systems, planar \(n\)R (\(n>3\)) underactuated robots with multiple degrees of freedom have different nonholonomic constraints. For the system with a passive first joint, [PERSON] [37] proposes a control method via iterative correction steering to achieve its stable control, and [PERSON] [38] has developed a method to realize the control objective of this system via model reduction and intelligent optimization algorithm. For the system with a passive second joint, [PERSON] [39] presents a control method based on the energy attenuation to achieve its control objective, whose control program is divided into three stages. For the system with a passive third joint, [PERSON] [40] presents a position control method based on model reduction. For the planar \(n\)R underactuated robot with a passive last joint, [PERSON] [41] presents a stable control method based on bidirectional trajectory planning to achieve its stable control, and [PERSON] [42] proposes a control approach to solve the problem of the position control of this system. The planar 3R underactuated robots have different situations. The first situation is the system with a passive first joint, which has the first nonholonomic characteristic. [PERSON] [43] presents a continuous state feedback control method to realize its control objective, and [PERSON] [44] solves the control problem of this system by using the hybrid control method and particle swarm optimization. The second situation is the system with a passive middle joint, which has the second nonholonomic characteristic. [PERSON] [45] presents a control method based on model reduction to achieve its stable control, and [PERSON] [46] proposes a phased control method based on the structural characteristics of this system. The third situation is the system with a passive last joint. [PERSON] [47] proposes a position control method by using motion planning. The planar 2R underactuated robots have two situations based on the different location of the passive joint. The first one is planar Acrobot [48]. [PERSON] presents a stable control method based on its holonomic constraint [49]. The other one is the planar 2R underactuated manipulator [50]. The intelligent optimization method [51] and double integrator method [52] have been developed to achieve its stable control.