# State Estimation of Over-Sensored Systems Applied to a Low-Cost Robotic Manipulator
## 4 State Estimation
### Link Model Estimation
#### 4.1.5 Accuracy Test  
The accuracy test was performed using the link model under the previously specified conditions.  
The norm of the error vectors was collected for each point and is displayed in Table 4. All the values from the table are presented in millimeters. Even though the tip of the manipulator was initially at Point A, its error is not displayed. This is because, at the starting point, the error is always zero since the initial conditions provided to the estimation methods correspond to the expectations of the position of the tip.  
The maximum error has a norm of 5.7 cm, which is quite large in the scale of the manipulator. Note that most of the error is located in the horizontal projection. Furthermore, this error has a tendency to increase, except when reaching Points E and A. This large error is caused by a drift of \(\theta_{1}\). The accelerometers are the only sensors capable of correcting for sources of drift. Given that the acceleration of gravity is almost parallel to the rotation axis of \(\theta_{1}\), drift on this angle cannot be compensated. The reason this error is decreased when reaching Points E and A is that the error of the tip's position becomes less dependent on the value of \(\theta_{1}\) as the manipulator approaches a vertical pose.  
A confirmation of the cause of the problem can be observed in Table 5. From it, it becomes even more evident that the error in \(\theta_{1}\) increases and that it is the most responsible joint for the accumulation of error.  
\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) & 9.50 & 12.69 & 21.40 & 20.25 & 43.79 & 56.64 & 40.75 \\ \hline Norm of the Error & 9.49 & 10.60 & 19.32 & 16.19 & 42.48 & 54.98 & 40.04 \\ projected on the \(xy\)-plane (mm) & & & & & & & \\ \hline Norm of the Error & 0.28 & 6.98 & 9.20 & 12.16 & 10.64 & 13.62 & 7.57 \\ projected on the \(z\) axis (mm) & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Norm of the errors of the accuracy test performed on the link model.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Sensors Used** & **Problem** & **Cause** \\ \hline \multirow{4}{*}{Gyroscopes} & Static \(\theta_{1}\) rotation vector & Missing IMU in Link 1 \\ \cline{2-2}  & No direct relation of the components & Static orientation of Link 1 causes the orientation of Link 2 to have 2 DOF \\ \hline \multirow{4}{*}{Accelerometers} & Static \(\theta_{1}\) rotation vector & Missing IMU in Link 1 \\ \cline{2-2}  & Significant values for the components & Initial misalignment of \\ \cline{2-2}  & \(\theta_{2x}\) and \(\theta_{2y}\) & Link 2 relative to Link 3 \\ \cline{2-2}  & Significant noise on \(\theta_{3z}\) & Large distance from the \\ \cline{2-2}  & during transitions & base of the manipulator \\ \hline \multirow{4}{*}{Encoders} & Low noise on \(\theta_{2b}\) & “Inertia” of the model causes the orientations of the links to fall behind \\ \cline{2-2}  & Overshoot not detected & Fast angular velocity of a joint \\ \cline{1-1} \cline{2-2}  & & causes the encoder to miss pulses \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary of the errors of the link model using individual sensors.