# Nonlinear Robust Adaptive Control of Universal Manipulators Based on Desired Trajectory
## 2 Dynamic Model and Preliminary
### Properties and Assumptions  
The dynamics are first reconstructed into a linear form with respect to the dynamic parameters. **Theorem 1**. : _(Parameter linearization) The dynamic model in (1) can be linearized by parameters as follows:_  
\[M(q)\ddot{q_{r}}+C\big{(}q,\dot{q}\big{)}\dot{q_{r}}+G(q)+F\big{(}\dot{q} \big{)}=Y_{l}\big{(}q,\dot{q},\dot{q_{r}},\ddot{q_{r}}\big{)}\theta_{l}+Y_{f} \big{(}\dot{q}\big{)}\theta_{f} \tag{3}\]  
_where \(\dot{q_{r}},\ddot{q_{r}}\in R^{6}\) are arbitrary reference vectors, which will be defined as model reference state variables in the controller design; and \(Y_{l}\) is the regression matrix of the dynamic term of the links with respect to the inertial parameters \(\theta_{l}\). Specifically:_\[\gamma_{I}\big{(}q,\dot{q},\dot{q}_{r},\ddot{q}_{r}\big{)}\theta_{l}= \begin{bmatrix} W_{11}&\cdots&W_{16}\\ \vdots&\ddots&\vdots\\ 0&\cdots&W_{66}\end{bmatrix}\begin{bmatrix}p_{1}\\ p_{2}\\ \vdots\\ p_{6}\end{bmatrix}  \tag{4}\]  
_where \(W_{ij}\in R^{1\times 10}\) represents the regression matrix of link \(i\) with respect to the inertial parameters \(p_{j}\) of link \(j\), which can be shown as follows:_  
\[p_{j}= \begin{bmatrix}iI_{xx},\,^{j}I_{xy},\,^{j}I_{xz},\,^{j}I_{yy},\,^{j}I_{ yz},\,^{j}I_{zz},\,m_{j}\,^{j}r_{cx},m_{j}\,^{j}r_{cy},m_{j}\,^{j}r_{cz}\end{bmatrix} ^{T} \tag{5}\]  
_Respectively, there are six independent parameters in the inertia matrix \(\,^{j}I\), the mass moment in three directions, and the mass of the link. Therefore, \(Y_{I}\in R^{6\times 60}\), \(\theta_{l}\in R^{60}\)._  
_Correspondingly, \(Y_{f}\) is the regression matrix of the joint friction term with respect to the Coulomb friction coefficients and viscous friction coefficients. Specifically:_  
\[Y_{f}\big{(}\dot{q}\big{)}\theta_{f}=\begin{bmatrix}sign\big{(}\dot{q_{1}} \big{)},\dot{q_{1}}&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&sign\big{(}\dot{q_{6}}\big{)},\dot{q_{6}}\big{]} \begin{bmatrix}c_{1} \\ v_{1}\\ \vdots\\ c_{6}\\ v_{6}\end{bmatrix}  \tag{6}\]  
_where the diagonal elements are the regression terms of the friction moment of joint \(i\) with respect to the Coulombic friction coefficient \(c_{i}\) and the viscous friction coefficient \(v_{i}\). Therefore, \(Y_{f}\in R^{6\times 12}\), \(\theta_{f}\in R^{12}\)._  
**Theorem 2**. : _(skew-symmetry) When the Coriolis and centrifugal matrix \(C(q)\) follow the definition of Christoffel symbols (of the first kind), the matrix obtained by the following expression is a skew-symmetric matrix:_  
\[N\big{(}q,\dot{q}\big{)}=\dot{M}(q)-2C\big{(}q,\dot{q}\big{)} \tag{7}\]  
_where the element of \(N\big{(}q,\dot{q}\big{)}\in R^{6\times 6}\) satisfies \(n_{jk}=-n_{kj}\)._  
**Theorem 3**. : _(Boundedness of inertial matrix) The inertia matrix \(M(q)\) is symmetric positive definite. For a robot containing only rotational joints, the elements of the \(M(q)\) contain only terms involving the sine and cosine functions, so that the constants \(\lambda_{m}\) and \(\lambda_{M}\) can be found, providing consistent bounds in the inertia matrix, i.e.,_  
\[\lambda_{m}I_{6\times 6}\leq M(q)\leq\lambda_{M}I_{6\times 6} \tag{8}\]  
_The above inequality is interpreted in the sense of the matrix inequality, i.e., if \(A\leq B\), it means that the matrix \(B-A\) is positive semidefinite._  
Next, the uncertainties are analyzed. In view of the linearization of the robot dynamic model with respect to the link inertia parameters \(\theta_{l}\) and the joint friction coefficients \(\theta_{f}\) in Equation (3), the following reasonable assumptions are made:  
**Assumption 1**. : _After long-term operation of the robot, the inertial parameters \(\theta_{l}\) of the links will be perturbed within a certain range, and the boundary can be estimated in advance as follows:_  
\[\theta_{ml}\leq\theta_{l}\leq\theta_{Ml} \tag{9}\]  
_where \(\theta_{ml}\) and \(\theta_{Ml}\) are both known constant vectors, which are the lower and upper bounds of \(\theta_{l}\), respectively.