# Model-Based Adaptive Collaboration of Multi-Terminal Internal Force Tracking
## 3 Multi-Terminal Adaptive Motion
### Unknown Motion Estimation of the Internal Command Force Direction  
Next, we will use the internal command force \(F_{rj}\), (\(j=b,c,d\)) to estimate the relative position vector \({}^{hj}P_{rj}\) shown in Figure 5, i.e., the unknown time-varying position \(p_{rjo}\) of the direction \(\overrightarrow{O_{hj}O_{ha}}\), which is critical for \(p_{rj}^{r}\) presented in Section 3.3. In general, we model \(p_{rjo}\) with the following time-varying trajectories:  
\[ \begin{split} p_{r00}(t)&=b_{0}+b_{1}t+b_{2}t^{2}+ \ldots+b_{n-1}t^{n-1}\\ p_{r00}(t)&=c_{0}+c_{1}t+c_{2}t^{2}+\ldots+c_{n-1}t ^{n-1}\\ p_{r00}(t)&=d_{0}+d_{1}t+d_{2}t^{2}+\ldots+d_{n-1}t ^{n-1}\end{split}  \tag{31}\]  
where \(t\) is time \(b_{n-1}\), \(c_{n-1}\), \(d_{n-1}\), where \(n=1,2,3\cdots\), which is considered a constant for a certain period. For the conciseness of the subsequent formulas, we will set  
Figure 5: Time-varying force tracking converted into virtual spring two-stage motion. \(g_{n-1}=b_{n-1}\), \(c_{n-1}\), \(d_{n-1}\). Thus, in Equation (31), the corresponding velocity and acceleration will be:  
\[ \begin{split} p_{rjo}(t)&=g_{0}+2g_{1}t+\ldots+g_{n- 1}t^{n-1}\;,\\ \dot{p}_{rjo}(t)&=g_{1}+2g_{2}t+\ldots+(n-1)g_{n-1}t ^{n-2}\;,\;(g=b,c,d),(j=b,c,d)\\ \dddot{p}_{rjo}(t)&=2g_{2}+\ldots+(n-1)(n-2)g_{n-1}t ^{n-3}\;.\end{split}  \tag{32}\]  
Based on the model in Equation (32), the estimate of \(p_{rjo}\) is described as follows:  
\[ \begin{split}\dot{p}_{rjo}^{\wedge}&=\dot{\delta}_{ 0}^{\wedge}+\dot{\delta}_{1}^{\wedge}t+\dot{\delta}_{2}^{\wedge}t^{2}\\ \dot{\hat{p}}_{rjo}&=\dot{\delta}_{1}+2\dot{\delta} _{2}^{\wedge}t\\ \dddot{\hat{p}}_{rjo}&=2\dot{\delta}_{2}^{\wedge} \end{split}  \tag{33}\]  
where \(\hat{\ast}\) is an estimated value of \(\ast\), defined according to Equation (33). \[\dot{\hat{F}}_{rj}^{\wedge}=\dot{k}(p_{rjo}^{\wedge}-p_{rje}) \tag{34}\]  
where \(\dot{\hat{F}}_{rj}^{\wedge}\) and \(\dot{k}\) are the estimated command internal forces and the Hooke coefficient of the object, respectively. By subtracting Equations (34) and (29), we obtain:  
\[ \begin{split}\dot{\hat{F}}_{rj}^{\wedge}-F_{rj}&= \dot{k}\;\left(\dot{\delta}_{0}^{\wedge}+\dot{\delta}_{1}t+\dot{\delta}_{2}t^ {2}-p_{rje}\right)-k\left(g_{0}+g_{1}t+g_{2}t^{2}-p_{rje}\right)\\ &=\left[p_{rje}\quad 1\quad t\quad t^{2}\right]\;H_{j}.\;(g=b,c,d),\; (\;j=b,c,d\;)\end{split}  \tag{35}\]  
In this equation,  
\[H_{j}= \begin{bmatrix}\dot{\wedge}\\ -\dot{k}+k\\ \dot{\delta}_{0}^{\wedge}-\;kg_{0}\\ \dot{\delta}_{1}^{\wedge}-\;kg_{1}\\ \dot{k}\dot{\delta}_{2}^{\wedge}-\;kg_{2}\end{bmatrix}  \tag{36}\]  
Equation (36) represents the estimated error for the unknown total kinetic parameters set. Then, we design the adaptive renewal law of \(H_{j}\) based on [PERSON] theory:  
\[\dot{H_{j}}= \begin{bmatrix}\dot{\cdot}\\ \vdots\\ \dot{k}\;\dot{\delta}_{0}+\dot{k}\;\dot{\delta}_{0}\\ \dot{k}\;\dot{\delta}_{1}+\dot{k}\;\dot{\delta}_{1}\\ \dot{k}\;\dot{\delta}_{2}+\dot{k}\;\dot{\delta}_{2}\end{bmatrix} =-\;\Gamma_{j} ^{-1} \begin{bmatrix}p_{rje}\\ 1\\ t\\ t^{2}\end{bmatrix} (\hat{F}_{rj}-F_{rj}),\;(\;j=b,c,d\;),(\;g=b,c,d\;) \tag{37}\]  
Thus, the adaptive renewal law of \(\dot{k}\) and \(\dot{\delta}_{i}\) in Equation (37) can be determined as  
\[\dot{\dot{k}}=\gamma_{j1}^{-1}p_{rje}(\hat{F}_{rj}-F_{rj})\;,j=b,c,d. \tag{38}\]  
\[\dot{\dot{g}}_{i-1}=-\frac{1}{\dot{k}}(\gamma_{j1,i+1}^{-1}t^{-1}(\hat{F}_{rj}- F_{rj})+k_{e}\dot{\delta}_{i-1})\;,g=b,c,d.\;i=1,2,3,j=b,c,d. \tag{39}\]  
where the renewal rate in the equation is \(\gamma_{j,i}\). In Equation (37), \(\Gamma_{j}\)\(=\) diag\(\left[\gamma_{j,1},\;\gamma_{j,2},\;\gamma_{j,3},\;\gamma_{j,4}\right]\), \(j=b,c,d\) is a positive definite matrix.