# Inertial Parameter Identification in Robotics: A Survey
## 7 Enforcing Physical Consistency within Inertial Parameter Identification
They consequently defined a _density-realizability_ criterion as an additional triangle-inequality condition to be fulfilled alongside with (56), namely  
\[ \begin{cases}I_{xxj}+I_{yyj}>I_{zzj}\\ I_{yyj}+I_{zzj}>I_{xxj}\\ I_{zzj}+I_{xxj}>I_{yyj}\end{cases} \equiv\text{tr}(\mathbf{I}_{j})>0\equiv\text{tr }(\mathbf{I}_{j})>0 \tag{60}\]  
where \(\text{tr}\left(\cdot\right)\) denotes the trace operator. A reformulation \(\mathbf{D}^{\prime}_{Lj}(\mathbf{\chi})\in\mathbb{S}^{4\times 4}\) of the term \(\mathbf{D}_{Lj}(\mathbf{\chi})\) within the constraint matrix \(\mathbf{D}_{j}(\mathbf{\chi})\) was eventually proposed to account for density realizability constraints alongside with (56)  
\[\mathbf{D}^{\prime}_{Lj}(\mathbf{\chi})= \begin{bmatrix}\dfrac{1}{2}\ \text{tr}\left(\mathbf{L}_{j}\right)\mathbf{1}_{3\times 3 }-\mathbf{L}_{j}&\mathbf{I}_{j}\\ \mathbf{I}_{j}^{\top}&M_{j}\end{bmatrix} \succ 0, \tag{61}\]  
naturally leading to the following SDP reformulation of IDIM-OLS and -WLS, that will be here referred to as Physically Consistent (PC-IDIM-OLS and -WLS)  
\[ \begin{split}\hat{\mathbf{\beta}}_{WLS}& =\underset{\mathbf{\beta},\mathbf{\chi}}{\arg\min}(\mathbf{W}\mathbf{\beta}-\mathbf{y}_{\tau})^{ \top}\mathbf{\Sigma}^{-1}(\mathbf{W}\mathbf{\beta}-\mathbf{y}_{\tau})\\ & s.t.\ \mathbf{\chi}=\mathbf{G}^{-1}\begin{bmatrix}\mathbf{\beta}^{\top}&\mathbf{\chi}^{ \top}\end{bmatrix}^{\top}\\ &\mathbf{D}_{j}(\mathbf{\chi})\succ 0\ \forall j=\{1\cdots n\},\end{split}  \tag{62}\]  
where \(\mathbf{G}^{-1}\) denotes the inverse mapping \(\mathbf{\Sigma}^{-1}(\mathbf{\beta},\mathbf{\chi})\) defined in (15). It is worth mentioning that the very structure of IDIM-IRLS (c.f. [23]), IDIM-IV and DIDIM (c.f. [92]) also makes it possible to seamlessly integrate the LMI physicality constraints by actually solving an SDP at each iteration of the corresponding algorithms. For instance, the PC-DIDIM algorithm (proposed in [92]) can be implemented by solving at each iteration \(i\)  
\[ \begin{split}\hat{\mathbf{\beta}}^{i}_{DIDIM}& =\underset{\mathbf{\beta}^{i},\mathbf{\chi}}{\arg\min}\Big{(}\mathbf{W}^{i}_{s} \mathbf{\beta}^{i}-\mathbf{y}_{\tau}\Big{)}^{\top}\mathbf{\Sigma}^{-1}\Big{(}\mathbf{W}^{i}_{s }\mathbf{\beta}^{i}-\mathbf{y}_{\tau}\Big{)}\\ & s.t.\ \mathbf{\chi}=\mathbf{G}^{-1}\begin{bmatrix}\mathbf{\beta}^{\top}&\mathbf{\chi}^{ \top}\end{bmatrix}^{\top}\\ &\mathbf{D}_{j}(\mathbf{\chi})\succ 0\ \forall j=\{1\cdots n\}.\end{split}  \tag{63}\]Please note that [19] recently proposed a set of alternative parametrizations, allowing direct enforcement of physicality of the reconstructed dynamic parameters without need for constrained optimization techniques. Following their method, one could for instance enforce the constraints in (61) by adopting the parametrization \(\chi^{\prime}\), based on the Cholesky decomposition of \(\mathbf{D}^{\prime}_{Lj}(\mathbf{\chi})\)  
\[ \begin{split}\mathbf{D}^{\prime}_{Lj}(\mathbf{\chi})=\mathbf{H}\mathbf{H}^{\top}+ \varepsilon\mathbf{1}_{4\times 4},\\ \chi^{\prime}=&[H_{11},H_{22},H_{33},H_{44},H_{21},H_{31},H_{32},H_{41},H_{42},H_{43}]^{\top}\end{split}  \tag{64}\]  
where \(\forall i,j\in[1,\cdots,4]\), \(H_{ij}\in\mathbb{R}\) is the element of row \(i\) and column \(j\) of the lower triangular matrix \(\mathbf{H}\in\mathbb{R}^{4\times 4}\) and \(\varepsilon<<1\) is a regularization term. It is also worth mentioning that [93] independently proposed a linearization of both the positivity-definition and density realizability constraints applied to \(\mathbf{I}_{j}\), in the form of a mass-positivity constraint applied to a distribution of point-masses located within each robot link (this is not without reminding the linearized friction-cone constraints applied to walking robots to ensure stable contacts with the environment). Provided that the number of point-masses is \"high enough\", PC-IDIM-OLS and -WLS can be reformulated as Quadratic Programs (QP) with a good precision.