Robot calibration poses move on a sphere: predict the motion

Run a camera calibration on a robot and the arm starts moving in arcs nobody taught it. That is by design: the pose list is computed from geometry centered on the calibration grid, not hand-taught, and the wrist is driven through every point of it. New operators read those sweeps as faults, and a cell risk-assessed for taught paths suddenly contains motion nobody scripted. This explainer gives you the geometry these routines use — a sphere around the grid, its radius set by arm size, its surface sampled by rotations and small radius offsets — so you can predict each move, check reach and clearances, and size the camera lens before the first sweep.

The poses are computed, not taught

The pose set a calibration routine commands is relative to the camera position after a centering step, which makes it indirectly relative to the grid. One published routine generates 18 poses that way; hand-eye calibration in general works through 10 to 20 postures, each pairing a registered end-effector pose with one image of the plate. Expect one untaught move before the first pose: the centering step swings the camera until its optical axis points at the grid center. Everything after that is arithmetic on a sphere.

Which positions make the solve accurate is a separate question — eye in hand calibration pose selection: robot positions that work covers that side. This article is about the motion itself, and robot mounted camera calibration: capture and solve walks the full workflow around the images.

The sphere the camera rides on

Picture a sphere whose center is the middle of the calibration grid. The routine assumes your snapshot position placed the camera on that sphere, optical axis through the center. It then generates poses by rotating the sphere about x, y and z, and by varying the radius with an offset of ±0.05 m: rotations slide the camera across the surface at constant distance, radius-offset poses dip 0.05 m closer and 0.05 m farther. At every pose the camera still looks approximately at the grid center, so the plate stays framed while its apparent size and skew change. The radius itself is set by arm size; one published routine uses these values:

If your arm sits between classes, lay out clearances for the larger radius.

What the motion looks like pose to pose

Consecutive poses differ by a rotation about the sphere center, so between them the wrist travels an arc of a circle around the grid. The tool does not just translate: the wrist reorients at every pose to keep the plate near the middle of the frame, and those reorientations — not the travel — are the moves that surprise new operators. A sweep can put the wrist through orientations the production program never visits, so cable and hose dress on the arm gets exercised in both directions. The camera never comes closer to the plate than r - 0.05 m.

Predict the swept volume before you run

The sweep occupies a spherical shell: inner radius r - 0.05 m, outer radius r + 0.05 m, both centered on the plate. On a mid-size arm that is the shell between 0.50 m and 0.60 m around the grid center. Check four things against that shell before the routine runs:

• Reach. Every sampled point of the shell must sit inside the working envelope at the wrist orientations the routine commands, not only at the snapshot orientation.
• Fixtures. Nothing may intersect the shell. A bracket 0.45 m from the plate center clears the mid-size shell; the same bracket at 0.55 m is squarely inside it and will be struck.
• Cables. The orbit plus the reorientations swing the wrist through its full range; check dress packages and connector strain relief at the extreme poses.
• Preview. Jog to the first and last commanded pose at low override before releasing the routine at speed.

Worked example: sizing the lens for the 0.55 m sphere

Inputs, each with its source:

• Sphere radius r = 0.55 m — the mid-size row of the table above.
• Radius offset ±0.05 m — part of the same pose generator, so the plate distance spans 0.50 m to 0.60 m.
• Calibration plate 80 mm square with 5 mm checker squares — a common board for wrist cameras; substitute your own numbers here.
• Sensor format 1/1.8-inch, about 7.2 mm × 5.3 mm — call it 3000 pixels across the width, which is 2.4 µm pixels.

Rule in words: the plate should fill 60 to 85 percent of the sensor width at both radius extremes. Below 60 percent the measured corners crowd together and pose noise rises; above 85 percent a tilted near pose clips the plate corners out of the frame.

Far constraint, WD = 0.60 m with fill at least 60 percent: FOV ≤ 80 / 0.60 = 133 mm, so f ≥ 7.2 × 600 / (133 + 7.2) = 4320 / 140.2 = 30.8 mm. Near constraint, WD = 0.50 m with fill at most 85 percent: FOV ≥ 80 / 0.85 = 94 mm, so f ≤ 7.2 × 500 / (94 + 7.2) = 3600 / 101.2 = 35.6 mm. The standard focal length inside 30.8-35.6 mm is 35 mm.

Check the 35 mm lens at both extremes:

• FOV at 0.50 m: 7.2 × (500 - 35) / 35 = 7.2 × 465 / 35 = 95.7 mm; fill = 80 / 95.7 = 84 percent.
• FOV at 0.60 m: 7.2 × (600 - 35) / 35 = 7.2 × 565 / 35 = 116.2 mm; fill = 80 / 116.2 = 69 percent.
• Resolution at the worst case: 3000 px / 116.2 mm = 25.8 px/mm, so a 5 mm square spans 129 px — corner detection is comfortable.

The neighbors fail in opposite directions. A 25 mm lens gives FOV = 7.2 × 575 / 25 = 165.6 mm at the far radius, a 48 percent fill. A 50 mm lens gives FOV = 7.2 × 450 / 50 = 64.8 mm at the near radius, which crops the 80 mm plate.

Focus is the second constraint: focused at 0.55 m, the lens must stay sharp across ±50 mm of radius offset.

At WD = 550 mm, f = 35 mm, N = 11 and a 0.02 mm circle of confusion: DOF ≈ 2 × 302500 × 11 × 0.02 / 1225 ≈ 109 mm, about ±54 mm, which covers the sweep with margin. The same sum at f/2.8 gives ±14 mm, so the offset poses soften. The depth of field calculator runs your own numbers.

Exposure is unconstrained by motion: the arm settles at each pose before the image, so set 2-10 ms for the lighting and let the controller dwell. Rolling shutter costs nothing on a static plate.

Decision: 35 mm lens, focus at 0.55 m, f/11, any 1/1.8-inch camera of 5 MP or more resolves the plate across the whole sweep. Run your own case in the lens focal length calculator.

At these magnifications the sensor format drives the camera choice, not frame rate — one image per pose is enough. Two wrist-friendly cable options in that sensor format:

Both put a 6 MP, 1/1.8-inch C-mount sensor behind a rolling shutter; one runs GigE, the other USB3. USB3 keeps wrist wiring short, while GigE tolerates longer runs back to the cabinet. Match the shutter and interface to the production task the camera does after calibration, then browse area scan cameras or the camera finder and start a build at the lens step.

Keep the sphere center still: mounting the grid

Every pose is computed about the grid center, so the plate must not move between the centering step and the last pose. A plate on a rubber-footed bench or a magnetic base shifts a fraction of a millimeter under wrist vibration, and the sphere the routine assumed no longer matches the physical one; the error surfaces later in the hand-eye calibration accuracy verification test. Bolt the plate to a machined stand:

The 700 mm column and 400 × 450 mm base hold the plate steady through wrist reorientations, and the slide table lets you center the plate under the snapshot position without touching the robot. For bigger targets, the construction notes in a large calibration grid for a 1 m field of view scale down to any plate. Choose the pattern with the grid target guide, then pick the support from camera stands, the accessory finder or start a build at the stand step.

When the sphere model no longer holds

• Snapshot position off the sphere. The centering step corrects small errors, but a start far from the assumed radius produces an extra untaught move or a reposition prompt. Teach the snapshot at the radius your arm class uses.
• Plate touched after centering. The whole pose set is then wrong by the same offset; re-run the centering step after any plate adjustment.
• Fast lens. At f/2.8 the ±0.05 m offsets exceed depth of field, corners soften exactly where pose diversity is highest, and the solve absorbs the bias. f/11 at this radius holds the sweep sharp.
• Reach limits. Where the shell leaves the envelope, part of the sweep is unreachable and the solve runs on whatever was captured. Teach a smaller-radius snapshot or a manual, diverse pose set instead.
• Thermal drift. In hot or outdoor cells, focus can drift between calibration and production — camera focus drift with temperature covers the diagnosis.

Full sweeps earn their keep at commissioning. For routine in-cell checks, robot cell vision recalibration with one reference target is faster than repeating the sphere.

Checklist before the first sweep

• Confirm r for your arm class — 0.30, 0.55 or 0.65 m — and compute the shell r ± 0.05 m around the plate center.
• Clear the shell of fixtures and confirm every sampled point is inside the working envelope.
• Jog the first and last pose at low override; check cable dress at both.
• Bolt the plate to a rigid stand; re-run centering if the plate is touched.
• Lens: plate held at 60-85 percent fill at r - 0.05 m and r + 0.05 m, focus at r, f/11 or slower.
• People out, speed override down, interlock gated on actual axis motion for the whole sweep.

With the geometry fixed, start a system build from the camera, or send the arm class and plate size and we will lay out the sweep volumes with you.