## 2 Preliminaries
### Dynamic Model  
The planar A\({}^{m}\)P manipulator model is depicted in Figure 1 as the research object of PMLUM. This system has one PL at the end and \(m\) ALs in the front. The model parameters for the \(i\)-th \((i=1,2,\ldots,m+1)\) link are as follows: the mass is \(m_{i}\), the length is \(L_{i}\), the moment of inertia is \(l_{i}\), the distance between the mass center and the \(i\)-th joint is \(l_{i}\), and the driving torque imparted to the \(i\)-th joint is \(\tau_{i}\). Additionally, the endpoint coordinate is \((x,y)\). Figure 1: Model structure. The planar A\({}^{m}\)P manipulator's dynamic model is as follows:  
\[M(q)\ddot{q}+H(q,\dot{q})=\tau \tag{1}\]  
where \(q\in\mathbb{R}^{(m+1)\times 1}\) is the angle vector, and \(\tau\in\mathbb{R}^{(m+1)\times 1}\) is the input torque vector. The \((m+1)\)-th link is the PL; hence, \(\tau_{m+1}=0\). The inertia matrix \(M(q)\in\mathbb{R}^{(m+1)\times(m+1)}\) is symmetric and positive definite, and \(H(q,\dot{q})=\dot{M}(q)\dot{q}-(1/2)\partial\big{(}\dot{q}^{T}M(q)\dot{q} \big{)}/\partial q\in\mathbb{R}^{(m+1)\times 1}\) is the combination of Coriolis and centrifugal forces. Specific expressions for M and H are given in [41]. Let \(X= \begin{bmatrix}X_{1}^{T}&X_{2}^{T}\end{bmatrix} ^{T}\), where \(X_{1}=[q_{1},\,q_{2},\cdots,q_{m+1}]^{T}\) and \(X_{2}=[q_{m+2},\,q_{m+3},\cdots,\,q_{2m+2}]^{T}\). The planar A\({}^{m}\)P manipulator's state-space equations are as follows:  
\[\left\{ \begin{array}{l}\dot{X}_{1}=X_{2}\\ \dot{X}_{2}=F(X)+G(X)\tau\end{array} \right. \tag{2}\]  
where \(\tau\) is the torque vector  
\[ \begin{array}{l}F(X)=\left[\begin{array}{cccc}f_{1}&f_{2}&\cdots&f_{m+1} \end{array}\right]^{T}=-M^{-1}(q)H(q,\dot{q})\\ G(X)=\left[\begin{array}{cccc}g_{11}&&\cdots&g_{1(m+1)}\\ g_{21}&&\cdots&g_{2(m+1)}\\ \vdots&&\ddots&\vdots\\ g_{(m+1)1}&\cdots&g_{(m+1)(m+1)}\end{array}\right]=M^{-1}(q)\end{array}  \tag{3}\]  
The system's control goal is to make the last link transform from the original coordinates to the desired coordinates. The relationship between the coordinate position of the endpoint and the angles of all links is obtained by using the coordinate projection method, as follows:  
\[\left\{ \begin{array}{l}x=-\sum\limits_{i=1}^{m+1}\left(L_{i}\sin\left( \sum\limits_{j=1}^{i}q_{j}\right)\right)\\ y=\sum\limits_{i=1}^{m+1}\left(L_{i}\cos\left(\sum\limits_{j=1}^{i}q_{j} \right)\right)\end{array} \right. \tag{4}\]  
In accordance with Equation (4), the system's control objective is changed to shift every link to the relevant target angles. According to the control target of controlling the endpoint of the manipulator to the specified position, the angle of each link needs to be obtained first. Considering that--for the same endpoint position--the angle of each link is multi-solvable, we can easily acquire a set of the desired angles using the DE algorithm. The distance between the endpoint's real-time position \((x,y)\) and the target position \((x_{d},y_{d})\), expressed as an absolute value, serves as the DE algorithm's evaluation function, which is described as follows:  
\[h=|x-x_{d}|+|y-y_{d}| \tag{5}\]  
Algorithm 1 illustrates the steps taken by the DE algorithm to determine each link's target angle.