## 4 The Second-Phase Control  
### Model Reduction  
When the planar PPR system drives all links to maintain their respective target states, the underactuated links rotate freely. After the first-phase control, the planar PPR system will be degraded to a planar virtual PR system, and the second and third linkages of the planar PPR system are the actuated links and underactuated links of the planar virtual PRsystem, respectively. Meanwhile, the first actuated link will stop moving, and the control torque is applied alone to the second actuated link, thus driving the third underactuated link to reach the target state. The order-reduction model is shown in Figure 3.  
Supplementary: After the end of the first phase, the first joint link has already stabilized to the target state and stopped moving, and the open-loop iterative control strategy acts solely on the second link. At this time, it enters the second phase, the PPR model is reduced to PR model, and the first link stops participating in the second phase.  
The planar PR underactuated robot model after order reduction is as follows:  
\[M(q)^{*}\ddot{q}+H^{*}(q,\dot{q})=\tau^{*} \tag{26}\]  
where \(M(q)^{*}\) and \(H^{*}(q,\dot{q})\) are as follows, and the control torque is \(\tau_{2}^{*}\).  
\[M(q)^{*}=\begin{bmatrix}m_{2}+m_{3}&-m_{3}l_{3}\sin q_{3}\\ -m_{3}l_{3}\sin q_{3}&m_{3}l_{3}^{3}+l_{3}\end{bmatrix} \tag{27}\]  
\[H^{*}(q,\dot{q})=\begin{bmatrix}-m_{3}l_{3}l_{3}^{2}\cos q_{3}\\ 0\end{bmatrix} \tag{28}\]  
According to Equations (26)-(28), the underactuated part of the reduced-order dynamics equation of planar PR robot can be obtained as follows:  
\[\left(-m_{3}l_{3}\sin q_{3}\right)\ddot{q}_{2}+\left(m_{3}l_{3}^{2}+l_{3} \right)\ddot{q}_{3}=0 \tag{29}\]  
\[\ddot{q}_{3}=\ddot{q}_{2}\frac{m_{3}l_{3}}{m_{3}l_{3}^{2}+l_{3}}\sin q_{3} \tag{30}\]  
By combining Equation (30), a new state-space expression can be obtained as follows:  
\[\dot{x}=h(x)+k(x)A(t) \tag{31}\]  
where \(A(t)\) represents the newly developed control input, and  
Figure 3: Virtual planar PR underactuated robot model.  
\[\begin{cases}h(x)=&\left[\begin{matrix}q_{2}&q_{3}&0&0\end{matrix}\right]^{\text{T}} \\ k(x)=&\left[\begin{matrix}0&0&1&C\sin q_{3}\end{matrix}\right]^{\text{T}}\\ \\ \end{cases}\\ C=\frac{m_{3}l_{3}}{m_{3}l_{3}^{2}+l_{3}}\end{cases} \tag{33}\]  
Equation (31) has non-holonomic integral property, which makes it difficult to obtain the constraint relation directly through integration. Therefore, we construct the accessibility matrix in the initial state \(x_{0}=[q_{2}^{0},q_{3}^{0},q_{2}^{0},q_{3}^{0}]^{\text{T}}\) based on Equation (31) and calculate the local coordinates at \(x_{0}\) and perform the privileged coordinate transformation [31]. The obtained privileged coordinate \(z=[z_{1},z_{2},z_{3},z_{4}]^{\text{T}}\) preserves the reachability of the original system. By processing the privileged coordinates with Taylor expansion method, a nilpotent approximation model is obtained as follows:  
\[\dot{z}=\begin{bmatrix}1\\ 0\\ -z_{2}\\ 0\end{bmatrix}-A(t)\begin{bmatrix}0\\ -1\\ 0\\ \left(q_{3}^{1}z_{1}\right)^{2}/4C\cos q_{3}^{1}+\frac{1}{2}z_{3}\end{bmatrix} \tag{34}\]  
at the beginning of each cycle, the angle and angular velocity of the underactuated link are \(q_{3}^{1}\) and \(q_{3}^{1}\), respectively. Equation (34) has exactly the same characteristics as Equation (31). The planar PPR underactuated robot can be shrunk to a stable target state by designing and using the auxiliary control input \(A(t)\).