# Inertial Parameter Identification in Robotics: A Survey
## 2 Robot Modeling and Control in the Context of Parameter Identification
### Base Parameters and Identification Model  
As explained in [2], the rank of the sampled observation system is function of two factors, namely the nature of the collected data samples, and the internal structure of the regressor matrix \(\Upsilon_{\chi}\). Data rank deficiency of the observation system is a direct consequence of noisy or improper data samples that do not adequately _excite_ the model parameters (i.e., that do not sufficiently reveal their influence on the system dynamics). This issue can most of the time be solved by having the robot track a trajectory that is specifically designed to excite the model parameters [38]. The other aspect of the problem lies in the fact that the intrinsic geometrical properties of robot manipulators naturally imply that some of the dynamic parameters within the \(\chi\) vector simply cannot be identified, for having rigorously no influence on the actual robot motions whatever the followed trajectory. In the same manner, some parameters can only be identified jointly because of their combined influence on the system. From a formal point of view, these issues can be characterized as a _structural rank deficiency_ of the regression matrix \(\Upsilon_{\chi}\). To solve this issue, it is necessary to perform a set of column rearrangements within \(\Upsilon_{\chi}\), eventually leading to the set of \(b\leq p\)_base parameters_ that are actually linear combinations of the standard parameters \(\chi\). Base inertial parameters are defined by [39; 40] as the set of dynamic parameters which is sufficient to completely describe the-intrinsically constrained-dynamics of a robot mechanism. As exposed in [37], the base parameter vector \(\beta\in\mathbb{R}^{b}\) can be obtained by performing a set of rearrangements in the symbolic expressions of (4) (note that in this context, some of the performed symbolic simplifications are only made possible using proximal DH convention). In [41], an alternative computation method was proposed based on Fourier series decomposition of the robot dynamic equations. Dedicated numerical methods based, for example, on QR decomposition can also be used. Consider the observation matrix \(W_{\chi}(\tilde{\mathbf{q}},\tilde{\mathbf{q}},\mathbf{q})\in\mathbb{R}^{(n\cdot N)\times p}\) constructed by stacking the samples of \(\Upsilon_{\chi}\) obtained from a randomly generated set of \(N>>1\) distinct joint values:  
\[W_{\chi}P=Q \begin{bmatrix}\mathbf{R}\\ \mathbf{0}_{(r-b)\times p}\end{bmatrix}  \tag{5}\]  
where for \(r=nN\), \(\mathbf{Q}\in\mathbf{O}(r)\), \(\mathbf{R}\in\mathbb{R}^{b\times p}\) is an upper-triangular matrix of rank \(b\leq p\) and \(\mathbf{P}\in\mathbf{O}(p)\) is a permutation matrix chosen - _by default_ - so that the diagonal values of \(\mathbf{R}\) are arranged in decreasing order. Denoting by \(\overline{\mathbf{P}}\in\mathbb{R}^{p\times b}\) and \(\underline{\mathbf{P}}\in\mathbb{R}^{p\times(p-b)}\) the first \(b\) and last \(p-b\) columns of \(\mathbf{P}\) leads to  
\[\underbrace{\big{[}W_{\chi}\overline{\mathbf{P}}\ \ \ \ \ \ \  
and hence the corresponding _non-bijective_ mapping between the base parameter vector \(\beta\) and the standard parameter vector \(\chi\)  
\[\beta=\Big{[}\overline{P}^{\top}\quad\quad\overline{R}^{-1}\underline{R}\ \underline{P}^{\top}\Big{]}\chi \tag{13}\]  
Note that--as explained in [20]--a bijective map \(\mathfrak{m}\) can still be defined between \((\beta,\underline{\chi})\) and \(\chi\) as  
\[\mathfrak{m}(\chi)=\underbrace{ \begin{bmatrix}\mathbf{1}_{b\times b}&\overline {R}^{-1}\underline{R}\\ \mathbf{0}_{(p-b)\times b}&\mathbf{1}_{(p-b)\times(p-b)}\end{bmatrix} P^{\top} }_{G}\chi \tag{14}\]  
in which case the inverse mapping is  
\[\mathfrak{m}^{-1}(\beta,\underline{\chi})=\underbrace{ \begin{bmatrix}\mathbf{ 1}_{b\times b}&-\overline{R}^{-1}\underline{R}\\ \mathbf{0}_{(p-b)\times b}&\mathbf{1}_{(p-b)\times(p-b)}\end{bmatrix} }_{G^{-1} } \begin{bmatrix}\beta\\ \underline{\chi}\end{bmatrix}