# Inertial Parameter Identification in Robotics: A Survey
## 10 Results, Discussion and Perspectives
### Analysis and Discussion of the Results
#### 10.1.1 Noise Immunity  
The Monte Carlo simulation results in terms of the different figures of merit shown in the tables of the supplementary material clearly indicate that the AdaNN, HTRNN, ML, IDIM-OLS, -WLS, -IRLS and -TLS methods are not robust to the presence of noise in the joint-position signal and its temporal derivatives, especially when the sampling rate is \"high\", as is for instance the case in _MCS-TX40-1-1-MCS-TX40-4-5_. On a theoretical point of view, this stems from the noise-induced correlation between the observation matrix \(\mathbf{W}\) and the vector of sampled torque errors \(\mathbf{\varepsilon}\), yielding \(\mathrm{E}(\mathbf{W}^{\top}\mathbf{\varepsilon})\
eq\mathbf{0}\). It should be emphasized that since the torque noise is purely additive, it does not affect the statistical consistency of the estimates, but rather their statistical efficiency. This can be demonstrated by noticing the similarity between the results of _MCS-TX40-1-4, MCS-TX40-4-4_ and _MCS-RV2 SQ-1-4_, (c.f. supplementary material, Section 11), where the joint torque noise level is set to \(\mathbf{\sigma}_{\mathbf{\tau}}=10^{-1}\) N\(\cdot\)m, with that of _MCS-TX40-1-2, MCS-TX40-4-2_ and _MCS-RV2 SQ-1-2_ (c.f. supplementary material, Section 11), where the torque noise is set to be twice as low. It should also be noted in practice that a low torque signal-to-noise ratio (SNR) could also imply that the trajectories are not exciting enough or that the hardware is not well designed. The most critical noise is, actually, that corrupting the joint angle readings as it significantly contributes to the bias of the IDIM-LS methods. The influence of noise in the joint angle measurements is made visible within the experiments described in the first three rows of Table 1, namely in [_MCS-TX40-1-1, MCS-TX40-1-2, MCS-TX40-1-3_], [_MCS-TX40-4-1, MCS-TX40-4-2, MCS-TX40-4-3_], and [_MCS-RV2 SQ-1-1, MCS-RV2 SQ-1-2, MCS-RV2 SQ-1-3_]. Within these experiments, the robots are actually tracking the same excitation trajectories, with the same control laws and initial parameter estimates, but with three different orders of magnitude in terms of joint position noise, namely \(\mathbf{\sigma}_{\mathbf{q}}=10^{-4}\) rad, \(\mathbf{\sigma}_{\mathbf{q}}=10^{-3}\) rad and \(\mathbf{\sigma}_{\mathbf{q}}=10^{-2}\) rad. Although the first noise level within these experiments is consistent with observations made on actual robots, it should be emphasized that the last two noise levels, are here only provided for indicative purposes. In practice, re-injecting such signals into the low-level control loop of a robot would in turn result in substantial noise in the control signals--as shown by the degraded values of the \(d_{\mathbf{\tau}}\) figure of merit obtained for \(\mathbf{\sigma}_{\mathbf{q}}=10^{-3}\) rad and \(\mathbf{\sigma}_{\mathbf{q}}=10^{-2}\) rad--eventually leading to poor control performance. The experimental results suggest that without data-filtering, the AdaNN, HTRNN, ML, IDIM-OLS, -WLS, -IRLS and -TLS methods provide biased estimates that poorly - or at least improperly - match the objective values, although the enhanced robustness of ML, IDIM-WLS and IDIM-IRLS compared to AdaNN, HTRNN, IDIM-OLS and -TLS should be noted. In any case, it appears that performing a tailor-made data-filtering based on a zero-shift forward-backward Butterworth filter significantly improves the results. This can be explained by the fact that filtering turns the noisy observation matrix \(\mathbf{W}\) into a noise-free matrix denoted by \(\mathbf{W}_{nf}\). Loosely speaking, this breaks the correlation between \(\mathbf{W}\) and the vector \(\mathbf{\varepsilon}\) of sampled errors. It is here worth pointing out that the use of simple forward low-pass filters generally leads to strongly biased estimates. This is justified in [7] by the fact that the phase shift induced by such filters is not accounted for in the IDM and is, therefore, considered to be a modeling error yielding \(\mathbf{W}\
eq\mathbf{W}_{nf}\).