GBEC: Geometry-Based Hand-Eye Calibration

TL;DR

Geometry-Based End-Effector Calibration (GBEC) is proposed that enhances the repeatability and accuracy of the derived transformation compared to traditional hand-eye calibrations, making it independent of workspace constraints or robot accuracy, without sacrificing the orthogonality of the rotation matrix.

Abstract

Hand-eye calibration is the problem of solving the transformation from the end-effector of a robot to the sensor attached to it. Commonly employed techniques, such as AXXB or AXZB formulations, rely on regression methods that require collecting pose data from different robot configurations, which can produce low accuracy and repeatability. However, the derived transformation should solely depend on the geometry of the end-effector and the sensor attachment. We propose Geometry-Based End-Effector Calibration (GBEC) that enhances the repeatability and accuracy of the derived transformation compared to traditional hand-eye calibrations. To demonstrate improvements, we apply the approach to two different robot-assisted procedures: Transcranial Magnetic Stimulation (TMS) and femoroplasty. We also discuss the generalizability of GBEC for camera-in-hand and marker-in-hand sensor mounting methods. In the experiments, we perform GBEC between the robot end-effector and an optical tracker's rigid body marker attached to the TMS coil or femoroplasty drill guide. Previous research documents low repeatability and accuracy of the conventional methods for robot-assisted TMS hand-eye calibration. When compared to some existing methods, the proposed method relies solely on the geometry of the flange and the pose of the rigid-body marker, making it independent of workspace constraints or robot accuracy, without sacrificing the orthogonality of the rotation matrix. Our results validate the accuracy and applicability of the approach, providing a new and generalizable methodology for obtaining the transformation from the end-effector to a sensor.

Figures (13)

• Figure 1: Kinematic chains of robot-assisted TMS. The numbers in Fig. 1(a) and 1(b) refer to: 1 the optical tracking camera, 2 the RBM affixed to the subject's head, enabling the acquisition of spatial information, 3 the RBM affixed on the TMS coil, 4 the TMS coil, 5 the robotic arm (depicted in semi-transparent) at its initial pose, and 6 the robotic arm at the aligned pose. The dashed line arrows labeled $^{eff}T_{coilRef}$ represent the transformation between the end-effector and the RBM affixed to the TMS coil. This transformation is the derived result of the proposed GBEC.
• Figure 2: The "camera-in-hand" and "marker-in-hand" setups. The numbers 1-5 match the numbering in Fig. \ref{['fig:alignkinematics']}. Here, 6 is the Portable Projection Mapping Device used in liu2022inside, and 7 is a rigid body probe used to digitize the position of a point.
• Figure 3: The formulation of $AX=ZB$ and $AX=XB$ problems. To simplify the equations, here, the superscripts and subscripts from Fig. \ref{['fig:alignkinematics']} are simplified to the initial letters. In this figure, unknown transformations are in red and orange backgrounds. In (b), the black arrows on the equations indicate rearrangement of the transformations to formulate $AX=XB$.
• Figure 4: End-effector attachment (TMS coil holder) and landmarks. Fig. (a) is the top view of the end-effector attachment, where green dots represent the landmarks used for paired-point registration. The coordinates of these landmarks with respect to $^{eff}F$ can be determined using $\theta_i$ and $h_i$, as shown in Equation \ref{['eq:landmarkcoor']}. Fig. (b) provides an isometric view of the end-effector attachment. Fig. (c) shows a side view, illustrating the reference frame $^{eff}F$ represented by blue arrows. The coordinates of the landmarks (green dots) with respect to $^{eff}F$ can be derived by the geometry of the end-effector attachment design. Fig. (d) shows the fitted lines of the grooves. The fitted lines with respect to $^{eff}F$ can also be derived by the geometry of the end-effector attachment design. Fig. (e) shows the digitization of landmarks or grooves.
• Figure 5: Line fitting using digitized point cloud. (a) shows the digitized point cloud $\mathcal{P}_{dig}$ in blue crosses and the regression lines $\mathcal{L}_{dig}$ in red lines, with respect to the reference frame $^{coilRef}F$. (b) and (c) show the point cloud $\mathcal{P}_{eff}$ (green dots) defined in Fig. \ref{['fig:coilholder']} and the sampled cloud point $\mathcal{P}_{sam}$ (red crosses) from regression lines $\mathcal{L}_{dig}$. The point clouds are with respect to $^{eff}F$ and $^{coilRef}F$ with overlapped origins (Fig. \ref{['fig:coilholder']}e). (b) is the side view and (c) is the top view. The result of the paired-point registration is the transformation $^{eff}T_{coilRef}$ (Fig. \ref{['fig:alignkinematics']}(b)), and the result transformation is denoted in red arrows in (b) and (c).