## 2 System Model and Analysis  
### Dynamic Model  
The PPR underactuated robot moves in horizontal space without considering the gravitational potential energy, and the prismatic connecting rod is independently controlled by the actuated joint. The first and second joints of the robot are mutually orthogonal moving joints, the third joint is a rotating joint, the two moving joints are actuated joints, and the rotating joints are underactuated joints. The translational motion of the prismatic link is controlled under the action of the control torque, and the rotating link is connected with the underactuated joint without a driving device, which is not directly controlled by any controller and can rotate freely. It is an important control strategy for the planar PPR underactuated robot to propel the rotating underactuated link indirectly to reach the target state by controlling the prism-actuated link motion. The end point of the first actuated joint is taken as the origin, and a planar PPR underactuated robot model is established in the \(O\)-\(xy\) plane, as shown in Figure 1. There are three links in the model \((r=1,2,3)\), and each link has a number of symbolic parameters, whose meanings and units are shown in Table 1.  
The dynamics model is established utilizing the Euler-Lagrange approach. Firstly, the Lagrange function \(L(q,\dot{q})\) of the system is determined as  
\[L(q,\dot{q})=K(q,\dot{q})-P(q) \tag{1}\]  
\(q=[q_{1}q_{2}q_{3}]^{T}\in\mathbb{R}^{3\times 1}\) is the state vector of the system link and \(\dot{q}=[q_{1}\dot{q}_{2}q_{3}]^{T}\in\mathbb{R}^{3\times 1}\) is the velocity vector of the system link. Since the system is moving in the horizontal plane, the potential energy \(P(q)\) is zero, and the kinetic energy \(K(q,\dot{q})\) is expressed as  
\[K(q,\dot{q})=\frac{1}{2}\dot{q}^{T}M(q)\dot{q} \tag{2}\]  
The Lagrange equation is defined as  
\[\frac{d}{dt}\frac{\partial L(q,\dot{q})}{\partial\dot{q}}-\frac{\partial L(q, \dot{q})}{\partial q}=\tau \tag{3}\]  
By differentiating the Lagrange function of the system with the velocity quantity, respectively, for the state quantity and substituting it into Equation (3), we can obtain  
\[M(q)\dot{q}+\dot{M}(q)\dot{q}-\frac{1}{2}\frac{\partial}{\partial q}\Big{(} \dot{q}^{T}M(q)\dot{q}\Big{)}=\tau \tag{4}\]  
In the above equation, \(M(q)\in\mathbb{R}^{3\times 3}\) is a symmetric matrix with positive-definite characteristics, and \(\tau=[\tau_{1}\tau_{2}\tau_{3}]\in\mathbb{R}^{3\times 1}\) is the control torque. \(\dot{q}=[\dot{q}_{1}\dot{q}_{2}q_{3}]^{T}\in\mathbb{R}^{3\times 1}\) is the system acceleration vector, and \(H(q,\dot{q})\) is the combined vector of Coriolis force and centrifugal force. In the dynamic Equation (4), we let  
\[H(q,\dot{q})=\dot{M}(q)\dot{q}-\frac{1}{2}\frac{\partial}{\partial\dot{q}} \Big{(}\dot{q}^{T}M(q)\dot{q}\Big{)} \tag{5}\]  
where \(M(q)\) and \(H(q,\dot{q})\) are expressed as follows:  
\[M(q)= \begin{bmatrix}M_{11}&M_{12}&M_{13}\\ M_{12}&M_{22}&M_{23}\\ M_{13}&M_{23}&M_{33}\end{bmatrix} = \begin{bmatrix}m_{1}+m_{2}+m_{3}&0&-m_{3}l_ {3}\sin q_{3}\\ 0&m_{2}+m_{3}&m_{3}l_{3}\cos q_{3}\\ -m_{3}l_{3}\sin q_{3}&m_{3}l_{3}\cos q_{3}&l_{3}+m_{3}l_{3}^{2}\end{bmatrix}  \tag{6}\]  
\[H(q,\dot{q})= \begin{bmatrix}H_{1}\\ H_{2}\\ H_{3}\end{bmatrix} = \begin{bmatrix}-m_{3}l_{3}q_{3}^{2}\cos q_{3}\\ -m_{3}l_{3}q_{3}^{2}\sin q_{3}\\ 0\end{bmatrix}  \tag{7}\]  
\[\tau= \begin{bmatrix}\tau_{1}\\ \tau_{2}\\ 0\end{bmatrix}  \tag{8}\]  
The nonlinearity of planar PPR underactuated robots is mainly reflected in the following aspects:  
1. In terms of dynamic equations, the dynamic equation of the planar PPR underactuated robot is established based on the Euler-Lagrange equation. The inertia matrix \(M(q)\) is a nonlinear function of the generalized coordinate \(q\), and the Coriolis force and centrifugal force terms \(H(q,\dot{q})\) are nonlinear functions of the generalized coordinates \(q\) and \(\dot{q}\). 2.