## 1 Introduction  
With the continuous in-depth exploration of the deep sea and deep space, the role of robots in such environments is becoming increasingly important. In fact, in extreme environments, such as weightlessness or microgravity, people cannot directly perform manual operations, and these basic operation tasks must be realized indirectly by robots [1, 2, 3, 4, 5]. When a part of the robot fails or is destroyed, the robot has underactuated characteristics in a certain sense. Analyzing the matching between the quantity of control inputs and the quantity of robot degrees of freedom, they can be divided into fully actuated robots with the quantity of control inputs equal to the quantity of degrees of freedom and underactuated robots with the quantity of control inputs less than the quantity of degrees of freedom [6, 7, 8, 9, 10]. The fully actuated robot has linearized control characteristics, and each control input can control the robot's movement in one degree-of-freedom direction [11, 12, 13, 14]. The number of control inputs of underactuated robots is insufficient. To complete a motion in the direction of more degrees of freedom, it is indispensable to conduct in-depth research and analysis on the complex nonlinear characteristics and specific conditions of underactuated robots. In the same practical engineering applications, underactuated robots only need a small number of controllers to complete the task, which has advantages such as energy saving, lightness, and flexibility compared with fully actuated robots under the same conditions [15; 16; 17; 18; 19; 20]. More complex planar underactuated robots can be constructed based on prismatic linkage and rotational underactuated linkage, and their control strategies are more complicated. The prismatic-rotational (PR) underactuated robot is the simplest form of construction, consisting of these two most basic linkages. From the perspective of motion stability, [PERSON] et al. [21] proved that there are irregularity and strong instability in the motion process of PR underactuated robots. Ref. [22] measured the absolute position of the prism-actuated link rod and the relative angle of the rotating link rod and designed a nonlinear observer utilizing the lower-order theory to verify the asymptotic convergence of the observation error. [PERSON] [23] eliminated the angular acceleration of the actuated link from the dynamics equation of a PR underactuated robot and developed a unified controller with a brake to address the point-to-point control challenge in the motion space. Planar prismatic-prismatic-rotational (PPR) underactuated robots have a wider working range for extra-pearular exploration in special environments. In Ref. [24], the Jacobian matrix of the planar PPR underactuated robot was obtained by using spiral theory, and a controller was crafted based on this matrix to achieve stability control. [PERSON] [25] converted the dynamic model of the planar PPR underactuated robot into an extended non-holonomic double-integral form and designed a controller to achieve the robot's position control. In Ref. [26], the end effector of the planar PPR underactuated robot achieved the desired position with arbitrary attitude. Ref. [27] analyzed the passivity of a planar PPR underactuated robot, and a controller was developed utilizing the energy strategy to meet specific control objectives. In Ref. [28], a time-varying feedback controller was constructed using a direct method and backstepping technique to bring the planar PPR underactuated robot convergence to the target state. Most of the above research on planar PPR underactuated robots is based on model construction and controller design, and some achievements have been realized. However, the control methods for this structure are relatively simple and need to be improved in terms of stability time and control performance indicators. The stability control methods for planar PPR underactuated robots need to be further expanded.