# Inertial Parameter Identification in Robotics: A Survey
## 7 Enforcing Physical Consistency within Inertial Parameter Identification  
Mathematical Formulation of the Physical Consistency Constraints within a Parameter Identification Process  
The possibility of enforcing physical consistency as part of a parameter identification process has been the subject of extensive research over the past two decades, in robotics. Early contributions, such as [15; 16; 20], formulated the parameter physicality of a given robot link \(j\), as a set of _positivity constraints_ on its mass \(M_{j}\), viscous-Coulomb friction parameters \(Fv_{j},Fc_{j}\) and on its transmission-chain inertia \(la_{j}\), as well as a _positivity-definition constraint_ on the inertia tensor \(\boldsymbol{I}_{j}\) expressed at the link's CoM, namely  
\[\forall\,\text{link}\,j: \begin{cases}M_{j}>0,Fv_{j}>0,Fc_{j}>0,Ia_{j}>0\\ \boldsymbol{I}_{j}\succ 0.\end{cases}  \tag{56}\]  
It is worth noting that prior [20], approaches to physically consistent inertial parameter identification were usually leveraging the equivalence between the positivity-definition constraint on the inertia matrix \(\boldsymbol{I}_{j}\) a set of strict positivity constraints on its eigenvalues \(\tilde{I}_{xxj},\tilde{I}_{yyj}\) and \(\tilde{I}_{zzj}\). These eigenvalues are in fact the moments of inertia along the so-called link's _principal axes of inertia_, yielding  
\[\boldsymbol{I}_{j}=\boldsymbol{R}_{j}\tilde{\boldsymbol{I}}_{j}\boldsymbol{R} _{j}^{\top}\succ 0\equiv\tilde{I}_{xxj}>0,\tilde{I}_{yyj}>0,\tilde{I}_{zzj}>0, \tag{57}\]  
where \(\tilde{\boldsymbol{I}}_{j}=\text{diag}(\tilde{I}_{xxj},\tilde{I}_{yyj},\tilde{ I}_{zzj})\) and \(\boldsymbol{R}_{j}\in SO(3)\) is the rotation matrix relating the frame \(\mathcal{I}_{j}\) of the principal axes of inertia, to the link's Center-of-Mass frame \(\mathcal{C}_{j}\). In this context, ref. [16] proposed a reformulation of IDIM-OLS as a nonlinearly constrained optimization problem, solved using Sequential Quadratic Programming (SQP), while [79] suggested to directly identify the elements of \(\tilde{\boldsymbol{I}}_{j}\) instead of \(\boldsymbol{L}_{j}\), using sigmoid functions within an EKF identification loop to smoothly bound the parameter estimates to physicality. It is worth pointing that such an approach could also possibly be applied to the HTRNN identification process by modifying the hyperbolic tangent activation function, although, to the best of our knowledge, this was never done. Conversely, ref. [20] proposed reformulating IDIM-LS under physicality constraints in a Semi-Definite Programming (SDP) perspective,writing (56) in the form of a Linear Matrix inequality (LMI). Provided that the reference frame \(\mathcal{L}_{j}\) of link \(j\) and its CoM frame \(\mathcal{C}_{j}\) have the same orientation, \(\mathbf{I}_{j}\) can indeed be related to the inertia tensor \(\mathbf{L}_{j}\) expressed in \(\mathcal{L}_{j}\), using the _Huygens-Steiner_ theorem  
\[\mathbf{I}_{j}=\mathbf{L}_{j}-M_{j}^{-1}\mathbf{S}(\mathbf{I}_{j})^{\top}\mathbf{S}(\mathbf{I}_{j}), \tag{58}\]  
where \(\mathbf{I}_{j}= \begin{bmatrix}MX_{j},&M\dot{Y}_{j},&MZ_{j}\end{bmatrix} ^{\top}\in \mathbb{R}^{3}\) denotes the vector of link's first moments and \(\forall\mathbf{u},\mathbf{v}\in\mathbb{R}^{3}\ \mathbf{S}(\mathbf{u})\mathbf{v}=\mathbf{u}\times\mathbf{v}\). Observing that (58) is in fact the Schur complement of a matrix \(\mathbf{D}_{Lj}(\mathbf{\chi})\in\mathbb{S}^{6\times 6}\) allows rewriting (56) as  
\[\mathbf{D}_{j}(\mathbf{\chi})= \begin{bmatrix}\mathbf{D}_{Lj}(\mathbf{\chi})&\mathbf{0}_{6\times 3 }\\ \mathbf{0}_{3\times 6}&\mathbf{D}_{Aj}(\mathbf{\chi})\end{bmatrix} \succ 0,\text{where:} \tag{59}\]  
where \(\mathbf{0}_{3\times 3},\mathbf{1}_{3\times 3}\) respectively denote the \(3\times 3\) zero and identity matrices. In [21, 22, 24], the authors went one step further by noticing that (56) and (59) could still potentially lead to inconsistent link mass distributions.