## 3 Equivalent Order Reduction Method  
In matrix calculation, especially in the process of robotic arm control calculation, \(X_{M_{k}\times N_{k}}\) is often sparse. Considering that the dimension of the matrix in the calculation process of robotic arm control is generally less than eight, the matrix is only divided into four blocks, which are expressed as follows:  
\[X_{M_{k}\times N_{k}}=[ \begin{matrix}X_{(M_{k}/2\times N_{k}/2,LU)}&X_{(M_{k}/ 2\times N_{k}/2,RU)}\\ X_{(M_{k}/2\times N_{k}/2,LD)}&X_{(M_{k}/2\times N_{k}/2,RD)}\end{matrix} ] \tag{3}\]  
Then the calculation result of matrix \(X_{M_{k-1}\times N_{k-1}}\) and \(X_{M_{k}\times N_{k}}\) multiplication is  
\[Y_{M_{k-1}\times N_{k}}(M_{k-1},M_{k})= \begin{matrix}Y_{(M_{k-1}/2\times N_{k} /2,LU)}(M_{k-1},M_{k})&Y_{(M_{k-1}/2\times N_{k}/2,RU)}(M_{k-1},M_{k})\\ Y_{(M_{k-1}/2\times N_{k}/2,LD)}(M_{k-1},M_{k})&Y_{(M_{k-1}/2\times N_{k}/2,RD)} (M_{k-1},M_{k})\end{matrix}  \tag{4}\]where  
\[\left\{ \begin{array}{l}Y_{(M_{k-1}/2\times N_{k}/2,LU)}(M_{k-1},M_{k})\\ =X_{(M_{k-1}/2\times N_{k-1}/2,LU)}X_{(M_{k}/2\times N_{k}/2,LU)}+X_{(M_{k-1}/2 \times N_{k-1}/2,RU)}X_{(M_{k}/2\times N_{k}/2,LD)}\\ Y_{(M_{k-1}/2\times N_{k}/2,RU)}(M_{k-1},M_{k})\\ =X_{(M_{k-1}/2\times N_{k-1}/2,LU)}X_{(M_{k}/2\times N_{k}/2,RU)}+X_{(M_{k-1}/2 \times N_{k-1}/2,RU)}X_{(M_{k}/2\times N_{k}/2,RD)}\\ Y_{(M_{k-1}/2\times N_{k}/2,LD)}(M_{k-1},M_{k})\\ =X_{(M_{k-1}/2\times N_{k-1}/2,LD)}X_{(M_{k}/2\times N_{k}/2,LU)}+X_{(M_{k-1}/2 \times N_{k-1}/2,RD)}X_{(M_{k}/2\times N_{k}/2,LD)}\\ Y_{(M_{k-1}/2\times N_{k}/2,RD)}(M_{k-1},M_{k})\\ =X_{(M_{k-1}/2\times N_{k-1}/2,LD)}X_{(M_{k}/2\times N_{k}/2,RU)}+X_{(M_{k-1}/2 \times N_{k}/2,RD)}X_{(M_{k}/2\times N_{k}/2,RD)}\end{array} \right.\]  
Considering the partitioned matrix \(X_{(M_{k}/2\times N_{k}/2,v)}(v\in[LU,RU,LD,RD])\), when it is zero matrix or identity matrix, the calculation amount of matrix multiplication is zero, and let the proportion of partial columns and columns be \(\beta_{(w,p)}(\beta_{(w,p)}\in[0,1],w=k-1,k)\), then the calculation amount of adjacent matrix multiplication of four partitioned blocks is  
\[T_{quad}(k-1,k)=0.125M_{k-1}N_{k-1}N_{k}\sum_{v,w}\beta_{(k-1,k,v,w)}=0.125 \prod_{u=k-1}^{k+1}M_{u}\sum_{v}\sum_{w=k-1}\beta_{(w,p)} \tag{5}\]  
If the zero value of the multiplied matrix partially occupies \(\lambda\) rows (\(\lambda\in Z^{+}\), \(\lambda\leq M_{k-1}/2\)) and the zero value of the multiplied matrix partially occupies \(\gamma\) columns (\(\gamma\in Z^{+}\), \(\gamma\leq N_{k}/2\)), the value of \(\beta_{(w,p)}\) is as follows:  
\[\left\{ \begin{array}{c}X_{(M_{k-1}/2\times N_{k-1}/2,v)}=[\overset{\rightharpoonup}{0 }_{M_{k-1}/2\times\lambda_{(k-1,p)}},X_{M_{k-1}/2\times(N_{k-1}/2-\lambda_{(k -1,p)})}]\\ X_{(M_{k}/2\times N_{k}/2,v)}=[\overset{\rightharpoonup}{0}_{\gamma_{(k,p)} \times N_{k}/2}]\\ \beta_{(k-1,v)}=1-2\lambda_{(k-1,v)}/M_{k-1}\\ \beta_{(k,v)}=1-2\gamma_{(k,v)}/N_{k}\end{array} \right. \tag{6}\]  
Substitute Formula (6) into Formula (5), and obtain  
\[T_{quad}(k-1,k)=0.125\prod_{u=k-1}^{k+1}M_{u}\sum_{v}(2\lambda_{(k-1,v)}/M_{k -1}+2\gamma_{(k,v)}/N_{k}) \tag{7}\]  
According to Formula (7), after the sparse matrix zero vector is counted, the column number of the multiplied matrix and the row number of the multiplied matrix can be equivalent to  
\[N_{k-1}^{\prime}=M_{k}^{\prime}=0.125M_{k}\sqrt{\sum_{v}\left(1-\lambda_{(k-1,v)}/M_{k-1}  
Figure 1: Equivalent reduced-order calculation based on sparse features of block matrix. Step 1: The matrix \(X_{M_{k-1}\times N_{k-1}}\) of row \(M_{k-1}\) and column \(N_{k-1}\) and the matrix \(X_{M_{k}\times N_{k}}\) of row \(M_{k}\) and column \(N_{k}\) are divided into four blocks according to Formula (3) to obtain \(X_{(M_{k-1}/2\times N_{k-1}/2,LU)}\), \(X_{(M_{k-1}/2\times N_{k-1}/2,RU)}\), \(X_{(M_{k-1}/2\times N_{k-1}/2,LD)}\), \(X_{(M_{k-1}/2\times N_{k-1}/2,RD)}\), \(X_{(M_{k}/2\times N_{k}/2,LU)}\), \(X_{(M_{k}/2\times N_{k}/2,RU)}\), \(X_{(M_{k}/2\times N_{k}/2,LD)}\), and \(X_{(M_{k}/2\times N_{k}/2,RD)}\).