The question

Short answer

What “mathematical coupling” needs (verified)

An external kinematic system can move in two ways. The distinction is the whole problem here:

Definitions: External Axes — Setup & Programming (KSS 5.5), §2.3, p.12. Coupling vs. static base in programming: §9.2, p.93–94.

Where the data sits: the table’s transformation chain

Transformation variables and chain: External Axes — Setup & Programming (KSS 5.5), §7.3.1, p.61–63; worked DKP 400 example with $ET1_TPINFL, Fig. 10‑2, p.102. The green items are what the cell move / tool swap invalidate; the orange items describe the table’s own geometry, which did not change.

Step 0 — preconditions before you calibrate

Calibration is a start‑up task: T1, no program selected, Expert user group. If the external axis itself was disturbed mechanically, master it first via Setup > Master > Dial gauge; otherwise leave mastering alone. The manual is explicit that calibration is what makes coupling possible at all: “Calibration of a kinematic system is necessary to enable the motion of the axes… to be synchronized and mathematically coupled with the robot axes.”

Mastering with the dial gauge: §6.1.5, p.45–48. Calibration purpose: §6.2.1, p.49.

Step 1 — re-measure the new tool (TCP) first

Every calibration step below moves a previously calibrated tool to reference points — the root‑point method’s precondition is literally “A previously calibrated tool is mounted on the mounting flange.” Because you changed the tool, its TCP must be re‑measured before anything else, or every later step inherits the error. Re‑measure with the standard tool calibration (XYZ 4‑point + ABC), or enter it numerically via Setup > Measure > Tool > Numeric Input if the values are known.

Step 2 — re-calibrate the table’s root point

The root point is the controller’s knowledge of where the table is: “In order to be able to move the robot with a mathematical coupling to a kinematic system, the robot must know the precise location of the kinematic system. This location is determined by means of root point calibration.” Moving the robot and table relative to each other is exactly what invalidates it. Two sub‑steps:

• Assign a reference point as a TOOL. A fixed measuring point (the reference pin) on the table is entered as a TOOL coordinate system, relative to the table’s flange: Setup > Measure > Tool > Numeric Input. The manual requires that “Reference point and flange center point must be sufficiently far apart” — this is the documented form of the field rule “this value cannot be zero.”
• Touch the reference point in 4 different table positions. Setup > Measure > External kinematic > Root point; jog the table to a new angle and drive the TCP onto the reference point, Measure, repeat 4×, Save. The controller computes the root point from the four positions. If the location is known from CAD, enter it numerically via … > Root point (numeric) instead.

Root point purpose & steps: §6.2.2–6.2.5, p.50–52. “Sufficiently far apart”: §6.2.3, p.51.

Step 3 — re-calibrate the offset base (this is “base 17”)

The moving base that rides with the table is the offset base: “a moving base that moves in the same way as the kinematic system… its reference point refers to the flange center point of the kinematic system and not the WORLD coordinate system.” Its coordinates live in BASE_DATA[17…22], and only one offset base exists per kinematic system. Teach it by touching three points with the calibrated TCP:

• Setup > Measure > External kinematic > Offset; name the kinematic system, enter the mounted tool number.
• Drive the TCP to the offset‑base origin (Continue), then a point on the +X axis (Continue), then a point in the XY plane with +Y (Continue), then Save. Those three points define the base.

Offset base definition, BASE_DATA[17…22], 3‑point procedure: §6.2.6, p.52–53.

Step 4 — activate the coupling and test

Coupling is not a separate “mode” switch — it is which base the motion is relative to. The manual states it directly: “To activate mathematical coupling, select the offset base in the option window Frames as the base to which the robot motion is relative. The coordinates of an offset base are saved as BASE_DATA[17…22].” So program the motion with that base selected:

; coupled — robot tracks the rotating table
PTP P10 VEL=100% PDAT50 Tool[1]:Pen Base[17]:TABLE
LIN P11 VEL=0.2m/s CPDAT1 Tool[1]:Pen Base[17]:TABLE

Test by jogging E1 (the table) a few degrees with the TCP held near the reference point: with the offset base selected, the TCP must stay on the point as the table turns. If it does, coupling is restored.

Should you edit $ETx_TPINFL in $MACHINE.DAT?

This is where the common field shortcut is half‑right and worth pinning down. $ETx_TPINFL is the offset from the table’s flange center point to the reference pin (in the manual’s DKP 400 example, $ET1_TPINFL={X 210.0,…}). The pin offset is real and must be non‑trivial — but on your KSS it is no longer a machine‑data value:

Sources: §7.3.2, p.64; worked example lines & note, p.103–104.

Open points / to verify

• Version drift on menu labels. The corpus is KSS 5.5; you run KSS 5.4.14. The variables and method are the same, but exact softkey/menu wording on your pendant may differ slightly — confirm Setup > Measure > External kinematic on the controller.
• We can’t see your machine data. This answer assumes the table’s mechanical geometry was unchanged by the move. If the table itself was rebuilt or the axis re‑geared, the $ETx_* transformation would also need revisiting — not the case for a simple reposition.
• Mastering. If the external axis lost mastering (encoder/brake work, mechanical knock), redo it (§6.1.5, p.45–48) before Step 2; otherwise skip it.
• Single‑axis vs. multi‑axis table. Steps are written for a one‑axis rotary (E1). A 2‑axis tilt‑rotate positioner adds $ET1_TA2A1 to the chain but the calibration sequence is identical.