Alan You
AYEngineering
Back to Research

Mar 2025 – May 2025

5 DoF Kinematic Calibration

Calibration of a 5 DoF surgical robot using an optical tracker and polynomial fitting. The worst single-axis position error on the test set fell from 8.58 mm to 0.30 mm.

Grid of 3D scatter plots over five iterations for the R, T and HE data sets, with optical tracker points in red and forward kinematics points in blue. The red and blue points overlap more closely after the first iteration

Approach

This was a two-month kinematic and hand-eye (HE) calibration. An Atracsys infrared optical tracker took the sensor readings, the robot’s internal forward kinematics (FK) supplied the other side of the comparison, and the calibration itself used polynomial fitting.

What makes a 5 DoF delta-plus-rotation robot unusual is that translation and rotation can be separated, so I calibrated them separately.

Results

On the test HE data set, mean position error fell from 4.80, 2.61 and 0.76 mm to 0.08, 0.12 and 0.16 mm in X, Y and Z. The largest single-axis error fell from 8.58 mm to 0.30 mm.

Position error (mm)

Data set State Mean X Mean Y Mean Z Std X Std Y Std Z Max X Max Y Max Z
Test HE Uncalibrated 4.803 2.613 0.755 3.167 1.654 0.536 8.576 5.084 2.126
Test HE Calibrated 0.077 0.116 0.155 0.056 0.078 0.077 0.238 0.282 0.302
T-only Uncalibrated 0.363 0.409 0.403 0.268 0.296 0.290 1.264 1.146 1.267
T-only Calibrated 0.044 0.053 0.041 0.032 0.035 0.024 0.148 0.149 0.088
R-only Uncalibrated 3.740 1.931 0.514 2.167 1.097 0.360 7.365 4.023 1.464
R-only Calibrated 0.050 0.097 0.107 0.032 0.064 0.050 0.129 0.223 0.222

Rotation error (deg)

Data set State Mean X Mean Y Mean Z Std X Std Y Std Z Max X Max Y Max Z
Test HE Uncalibrated 0.077 0.126 0.063 0.059 0.084 0.043 0.209 0.297 0.167
Test HE Calibrated 0.012 0.077 0.045 0.008 0.025 0.016 0.026 0.108 0.070
T-only Uncalibrated 0.015 0.03 0.04 0.011 0.024 0.030 0.046 0.111 0.125
T-only Calibrated 0.004 0.077 0.042 0.000 0.000 0.000 0.004 0.077 0.042
R-only Uncalibrated 0.050 0.069 0.038 0.042 0.061 0.031 0.161 0.257 0.128
R-only Calibrated 0.008 0.077 0.044 0.006 0.019 0.012 0.026 0.108 0.070

Residual error

Each arrow is the residual at one point, drawn at scale 10. The color scales differ between plots, so compare them using the captions.

3D arrow plot of initial residual error on the R data. Long arrows cover the surface, colored up to about 1.9 mm
R data, initial: up to about 1.9 mm
3D arrow plot of final residual error on the R data, on the same axes. The arrows are much shorter, colored up to about 0.3 mm
R data, final: up to about 0.3 mm
3D arrow plot of initial residual error on the T data, with short arrows colored up to about 0.47 mm
T data, initial: up to about 0.47 mm
3D arrow plot of final residual error on the T data, on the same axes. The arrows are barely visible, colored up to about 0.18 mm
T data, final: up to about 0.18 mm