# Inertial Parameter Identification in Robotics: A Survey
## 4 The Instrumental Variables (IV) and Maximum Likelihood (ML) Approaches
### Maximum Likelihood (ML) Identification Method  
Investigated in [9, 10, 11], the Maximum Likelihood (ML) parameter identification algorithms aims at tackling the issue of noisy torque and joint angle measurements. These works are the first ones actually addressing the issue of noisy observation matrices. In [10], the authors present an improvement of the original ML approach presented in [9]. The ML criterion they adopt can be formulated as  
\[\mathbf{\hat{\beta}}_{ML}=\operatorname*{arg\,min}_{\mathbf{\beta}}\frac{1}{2}\sum_{k= 1}^{N}\mathbf{\epsilon}^{\top}(t_{k})\Big{(}\mathbf{G}_{k}\sigma_{k}^{2}\mathbf{G}_{k}^{ \top}\Big{)}^{-2}\mathbf{\epsilon}(t_{k}), \tag{36}\]  
where \(\sigma_{k}^{2}\in\mathbb{R}^{4n}\) denotes the diagonal variance matrix of the \(k^{th}\) sample (this allows in particular to account for the effects of time differentiation in the joint signal in terms of noise amplification), \(\mathbf{\epsilon}(t_{k})\) is the error at the \(k^{th}\) torque sample defined in (19); and \(\mathbf{G}_{k}=\partial\mathbf{\epsilon}(t_{k})/\partial\mathbf{s}_{k}\in\mathbb{R}^{n \times 4\cdot n}\) is the Jacobian matrix of this error relative to the measurement vector \(\mathbf{s}_{k}=\big{[}\mathbf{q}(t_{k})^{\top}\mathbf{\dot{q}}(t_{k})^{\top}\mathbf{\ddot{q}}( t_{k})^{\top}\mathbf{\tau}(t_{k})^{\top}\big{]}^{\top}\in\mathbb{R}^{4\cdot n}\). Because \(\mathbf{G}_{k}\) involves the IDM, this ML approach can be called IDIM-ML. In the IDIM-ML approach, the authors suggest constructing the vector \(\mathbf{s}_{k}\) by averaging the original measurements of \(\mathbf{q}(t_{k}),\ \mathbf{\dot{q}}(t_{k}),\ \mathbf{\ddot{q}}(t_{k})\) and \(\mathbf{\tau}(t_{k})\) over \(N_{real}\) realizations. Then, \(\mathbf{s}_{k}\) is used to construct the following observation matrix denoted \(\mathbf{\overline{W}}\). It must be noticed that provided \(N_{real}\) is big enough and the variances of the original measurements are finite, the Lyapunov criterion ensures that \(\mathbf{s}_{k}\) is close to a  
Figure 1: Block diagram of the IDIM-IV identification method.  
Gaussian distribution. Note also that it can be even further assumed that \(\mathbf{s}_{k}\) tends to the noise-free original measurements as \(N_{real}\) grows and the noise level is _reasonable_ which means that one has \(\overline{\mathbf{W}}\to\mathbf{W}_{nf}\). Interestingly, this ML approach is somehow related with the IDIM-IV method developed in [6]. Indeed, the authors suggest building an instrumental matrix such that \(\mathbf{Z}=\mathbf{W}_{nf}\) using simulated data and the IDM while in [10] the authors suggest constructing \(\overline{\mathbf{W}}\) with \(\overline{\mathbf{W}}\to\mathbf{W}_{nf}\). Furthermore, in [71], the author establishes some interesting relationships between IV and ML approaches. It is thus expected that IDIM-IV and ML approaches give similar results. However, in [9] a _prewhitening_ process has been carried out to remove all the coloring and correlation of the measurement noise. Without this _prewhitening_ process which can be connected with the use of the _decimate_ filter, the statistical assumptions made on the noise may be violated making the ML approach potentially unable to provide consistent estimates. Furthermore, it should also be stressed that the joint torques in these approaches are measured via _current-shunt monitors_ instead of being the outputs of low-level controllers as usually done in [42]. Therefore, if the ML method does not require the simulation of the DDM, it could prove more sensitive to noises and is more time-consuming than IDIM-IV is.