# Using Full Pose Measurement for Serial Robot Calibration
## 8 Results  
Fitted DH parameters revealed different amounts of variations for different simulated conditions. The twist angles \(\mathbf{\hat{\alpha}}\) estimated from 3D data generated for the CPA procedure showed moderate variations. The largest absolute deviation \(\delta\alpha_{max}\) from the GT value over all \(N\) joints and all simulated conditions (\(M_{n}\) noise levels, \(N_{rep}\) repeats and both deviations \(\Delta\mathbf{\kappa}_{1,2}\) from the default values \(\mathbf{\kappa}_{0}\)) was \(0.3^{\circ}\). Zero offsets \(\mathbf{\xi}\) revealed larger  
Figure 1: Characteristics of simulated small random rotations \(\mathcal{N}_{k}\) in (22): (**a**) histograms of \(x\) component of angle vectors \(\xi_{k^{\prime}}\); (**b**) histograms of angle of rotation \(\beta\) of rotation matrix \(\mathcal{N}_{k}\). Blue lines correspond to weak noise with \(\sigma_{a}=0.05^{\circ}\) and black lines correspond to strong noise with \(\sigma_{a}=0.15^{\circ}\). deviations: the largest absolute deviation \(\delta\epsilon_{max}=1.18^{\circ}\). The largest link length deviation was \(\delta d_{max}=4.7\) mm and the largest link offset deviation was \(\delta r_{max}=4.2\) mm. Such large differences between the fitted and the GT parameters were observed mostly for large noise levels \(\sigma_{p}\) and \(\sigma_{a}\). Figure 2 shows an example of robot end-effector errors at \(J=50\) test poses. Position errors \(q_{j}\) and orientation errors \(\eta_{j}\) were calculated in (24) for robot DH parameters calibrated with Method 1 and Method 2. Presented errors were calculated for simulated sensor poses perturbed by \(i=14\) noise realization (selected arbitrary from \(N_{rep}\) repeats) and \(m=7\) noise levels \(\left(\sigma_{a},\sigma_{p}\right)_{m}\). These \(\left(q_{j},\eta_{j}\right)_{m,i}\) errors were then used to calculate \(\left(v_{m,i},\rho_{m,i}\right)\) and then, mean errors \(\overline{v}_{m}\) and \(\overline{p}_{m}\) in (25) and the corresponding standard deviations \(\delta v_{m}\) and \(\delta\rho_{m}\) for each \(m\)-th noise level. These means and standard deviations were then used to create the plots in the remaining Figures 3-6. Figure 3 shows the outcomes of two registration transformations \(\left(\mathbf{\Omega}_{1},\mathbf{\tau}_{1}\right)\) and \(\left(\mathbf{\Omega}_{2},\mathbf{\tau}_{2}\right)\) described in Section 6. In both cases, robot was calibrated with Method 2. GT parameters used in simulation of 6D data, i.e., end-effector poses and noisy poses as measured by sensor, were obtained by modifying the default DH parameters with deviations shown in Table 2. For both registrations, mean errors were calculated at the same values of sensor noise (\(\sigma_{p}\) in Figure 3a,c and \(\sigma_{a}\) in Figure 3b,d). In each subplot, two graphs are slightly shifted horizontally only for better visualization. Error bars \(\delta v_{m}\) in Figure 3a,c and \(\delta\rho_{m}\) in Figure 3b,d are the corresponding standard deviations calculated in (25) from \(N_{rep}\) repeated simulations of noisy sensor data. Figure 4 shows the outcomes of two registration procedures applied after robot was calibrated using Method 3 and the error function \(Err_{full}\) defined in (18) with the scaling factor \(w=1\). Presented results were obtained for 6D data generated with GT values of DH parameters deviating from their default values by \(\left(\Delta\alpha_{2},\Delta\tau_{2},\Delta d_{2}\right)\) shown in Table 2.  
Figure 2: Robot end-effector errors calculated at \(J=50\) test poses for fixed sensor noise \(\sigma_{p}=0.055\) mm, \(\sigma_{a}=0.055^{\circ}\) and one, arbitrary selected noise realization: (**a**) positional errors \(q_{j}\); (**b**) orientation errors \(\eta_{j}\). Robot was calibrated with Method 1 (black lines) and Method 2 (blue lines).