The standard description of a serial arm assigns four parameters to each link (Denavit and Hartenberg, 1955). It serves well for nominal kinematics but poorly for calibration when two consecutive joint axes are nearly parallel, because a small misalignment then produces a very large change in one parameter. Hayati and Mirmirani (1985) introduced an additional rotation about the y axis for such pairs, and this modification is routinely adopted. The point is of direct relevance here, since many six-axis collaborative arms have three consecutive parallel axes at the shoulder, elbow and first wrist joint. A model used for identification should be complete, continuous and free of redundant parameters (Mooring et al., 1991).

A calibrated camera that sees a pattern of known geometry can recover its own pose relative to the pattern. The intrinsic parameters and lens distortion are obtained from several views of a planar target (Zhang, 2000), and the pose from each later image by solving the perspective-n-point problem (Lepetit et al., 2009); the underlying geometry is set out by Hartley and Zisserman (2004). Square fiducial markers make the pattern self-identifying, so that each detected corner is matched to its known position without ambiguity even when only a fraction of the target is in view (Garrido-Jurado et al., 2014; Olson, 2011). Open implementations of all of these steps are available in the OpenCV library (Bradski, 2000).

The camera measures in its own frame, not the flange frame, and the fixed transform between the two is unknown. Finding it is the hand-eye problem, classically written as AX = XB and solved in closed form by Tsai and Lenz (1989), with later formulations on the Euclidean group (Park and Martin, 1994) and with dual quaternions (Daniilidis, 1999). These solutions assume that the robot poses are exact, which is precisely what is in doubt during calibration. They are therefore used here only for initial values.

A board of fiducial markers is printed on matt paper, matt because glossy stock produces specular highlights that disturb corner detection. Office printers do not reproduce dimensions exactly and may scale differently along and across the feed direction, so the printed board is measured in both directions with a steel rule or caliper and the measured pitch, not the nominal one, is entered in the software. The sheet is bonded without bubbles to a flat rigid backing such as float glass or a machined plate. Two or three boards placed at different heights and orientations in the workspace give better conditioning than one, and their relative positions need not be known in advance.

The camera is clamped to the flange with a bracket stiff enough that it does not shift during motion. Focus, aperture and zoom are set for the working distance and then locked mechanically; autofocus must be disabled, because any change of focus alters the intrinsic parameters. Intrinsics and distortion are estimated by the planar method from a few dozen views that tilt the target strongly and cover the whole image, including the corners. The reprojection residual should be inspected as a pattern across the image and not only as a single number, since a low average can conceal a poorly modelled lens edge.

The model takes the nominal parameters from the manufacturer and adds a small correction to each, with the Hayati term on the parallel-axis pairs. Not every correction can be identified. Some are indistinguishable from the unknown base and tool transforms, and others are nearly dependent for a given arm geometry. The practical test is to compute the identification Jacobian for the planned poses and examine its singular values; parameters associated with very small singular values are removed or fixed. On a first attempt it is prudent to identify only the joint zero offsets, which usually account for the larger part of the error, and to add link parameters only if validation shows a benefit.

Identification needs poses that excite every parameter, which means that each joint must move through a wide range while a target stays in view. The poses are generated in advance, checked for visibility and collisions in simulation, and split into an identification set and a validation set before any data are seen. At each pose the arm is allowed to settle, and the image is stored with the joint angles reported by the controller, not the commanded Cartesian pose. Approaching every pose from the same joint direction keeps backlash from appearing as random noise. Images in which markers are blurred, heavily foreshortened or fewer than a set minimum are discarded by rule.

The unknowns are the kinematic corrections, the hand-eye transform and the pose of each board in the robot base frame. Initial values for the last two come from a closed-form hand-eye solution on the nominal kinematics. All unknowns are then refined together by minimising the reprojection error, the distance in pixels between each detected marker corner and its position predicted through the chain of base, arm, camera and board. Working in pixels weights each observation by what the camera actually measured and avoids treating the poorly observed depth direction as equal to the others. The minimisation is a standard nonlinear least-squares problem handled by the Levenberg-Marquardt method (Marquardt, 1963), preferably with a robust loss so that a few bad detections do not dominate.

The identified model is judged only on the validation poses. For each, the board pose predicted from the joint angles is compared with the pose measured by the camera, before and after calibration, and the position and orientation errors are summarised. Residuals should also be plotted against each joint angle; a remaining trend indicates an unmodelled effect, and a model that fits the identification set well but the validation set badly has too many parameters. In either case the procedure returns to Step 3 or Step 4. An independent check with a different instrument, even a dial gauge against a reference block, is worth making because the camera cannot reveal its own bias.

Some controllers accept modified kinematic parameters directly. Where they do not, the correction is applied outside the controller: the desired Cartesian pose is passed through the inverse kinematics of the identified model, and the resulting joint angles are sent as a joint-space target (Siciliano et al., 2009). The identified values, the pose sets, the images and the software versions are archived so that the calibration can be repeated after a collision or a joint replacement.