# State Estimation of Over-Sensored Systems Applied to a Low-Cost Robotic Manipulator
## 4 State Estimation
### Joint Model Estimation
#### 4.2.4 Full Model  
Finally, the results of the full model for the same test are presented in Figure 20. The gyroscope problem of the initially incorrect steady-state estimates of \(\theta_{2}\) and \(\theta_{3}\) is removed, and the overshoots are detected.  
A summary of the errors observed when using individual sensors is displayed in Table 6.  
Figure 19: Evolution of estimates of the joints’ angles when testing the joint model using encoder measurements.  
Figure 20: Evolution of estimates of the joints’ angles when testing the joint model using all measurements.  
A notable difference is the overall performance of this filter relative to the other. The maximum error was around 5.7 cm for the link model, whereas the error for the joint model always stays below 3.6 cm. This improvement is not drastic. However, if the effects of drift are set aside, the vertical projection of the error goes from a maximum of 13.6 mm in the link model to 4.2 mm in the joint model. Apparently, the freedom given to the link model of tilting the links independently allows it to tilt to far off the real orientations. These extra degrees of freedom cause the drift to project to the other axes, allowing the maximum error to be higher.  
For the sake of completeness, Table 8 presents the errors of the joints' angles. Overall, the error of this model, when not considering drift, is always below 1 cm, which, given the limited precision with which the structure was assembled, means this model might have even better performance when tested with a different ground truth.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Sensors Used** & **Problem** & **Cause** \\ \hline \multirow{3}{*}{Gyroscopes} & Incorrect amount of rotation estimated & Misalignment of the rotation axes caused by a misalignment of the links’ orientations \\ \cline{2-3}  & & A small acceleration projected into the horizontal plane is obtained from the tilt in the links and justified as centrifugal acceleration \\ \cline{2-3}  & (After correction) inability & Gravity acceleration vector parallel to rotation axis of joint 1 \\ \cline{2-3}  & (After correction) significant noise on & Large distance from the base of the manipulator \\ \cline{2-3}  & \(\theta_{3 est}\) during transitions & Fast angular velocity of a joint \\ \cline{2-3} Encoders & Overshoot not detected & causes the encoder to miss pulses \\ \hline \hline \end{tabular}
\end{table}
Table 6: Summary of the errors of the joint model using individual sensors.  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Point** & **B** & **C** & **D** & **E** & **F** & **G** & **A** \\ \hline Norm of the Error (mm) & 1.50 & 29.83 & 7.59 & 4.45 & 24.30 & 35.60 & 2.12 \\ \hline Norm of the Error & & & & & & \\ projected on the \(xy\)-plane (mm) & 1.06 & 29.80 & 6.44 & 3.67 & 23.94 & 35.55 & 2.12 \\ \hline Norm of the Error & & & & & & & \\ projected on the \(z\) axis (mm) & 1.07 & 1.37 & 4.02 & 2.52 & 4.19 & 1.90 & 0.01 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Norm of the errors of the accuracy test performed on the joint model.  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Point** & **B** & **C** & **D** & **E** & **F** & **G** & **A** \\ \hline Error in \(\theta_{1}\) (\({}^{\circ}\)) & 0.16 & 4.63 & 0.93 & 0.85 & 5.55 & 8.33 & 10.01 \\ \hline Error in \(\theta_{2}\) (\({}^{\circ}\)) & 0.83 & 0.61 & 0.31 & 0.08 & 1.69 & 0.01 & 0.02 \\ \hline Error in \(\theta_{3}\) (\({}^{\circ}\)) & 1.94 & 1.60 & 2.22 & 1.03 & 4.56 & 0.75 & 0.68 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Error of the angles of the joints during the accuracy test for the joint model.