Drape-Compatible Tool-Tip Localization for Hand-Held Laparoscopic Instruments via UWB Carrier-Phase Ranging and Trocar-Constrained Geometry

A draped instrument hides every optical path on itself, so we estimate the tool tips from distances and an IMU alone. Ultra-wideband carrier phase between nodes on the handle side of the instruments gives three distances; the trocar reduces each instrument to a single insertion depth, leaving one equation to spare that exposes a wrong integer instead of absorbing it. In an 11-hole phantom among metal instruments the three distances hold σ = 0.10–0.33 mm at 35–46 Hz, and a sterile drape pressed onto the antennas leaves the link unchanged.

A surgical robot policy consumes geometry at each timestep: where each instrument's working end sits relative to the camera and to the other instrument. A conventional laparoscopic operation records none of it. Optical trackers need line of sight to markers while the part to be localized is inside the patient; electromagnetic trackers degrade among the metal instruments that fill a surgical field; markerless endoscopic estimation degrades under occlusion. The operating room compounds all three, because an instrument that enters it must be draped unless all of its electronics are sterilized — and the drape seals any optical path on the instrument inside the barrier.

Distance is the one quantity the drape leaves intact. We therefore clip an ultra-wideband node near the handle of each hand-held instrument and one on the endoscope, all outside the sterile barrier, and let them range against each other. Radio crosses the drape, and a distance, unlike a marker or an image, needs no line of sight.

A clip-on node on an unmodified laparoscopic instrument — The assembled node — a Qorvo DWM3001CDK UWB board, a BNO085 IMU and a 1000 mAh cell on a 3D-printed mount — clipped to a standard 5 mm instrument between the handle and the trocar. Nothing is added to the part that enters the patient.

Two problems, one radio and one geometry

The exchange whose pulses give a coarse time of flight also carries the carrier's phase of arrival. At channel 5 one cycle spans 46.2 mm, so a few degrees of phase noise is tens of micrometres of path. Between rounds we accumulate the wrapped phase increment,

\delta_k = \operatorname{wrap}(\phi_k - \phi_{k-1})\,\frac{\Delta}{2\pi}, \qquad \tilde d_k = \tilde d_{k-1} + \delta_k,

which is exact as long as the instrument moves less than $\Delta/2 = 11.5$ mm between rounds — up to 400 mm/s at 35 Hz. Time of flight is not fused into that chain: the antenna delay of these development boards is uncalibrated and its per-boot bias re-randomizes by ±70–100 mm, so ToF only seeds where each episode starts.

That leaves two difficulties. Keeping the phase chain continuous is the radio problem; turning three distances into two tool tips is the geometric one. Three scalar distances cannot place two free-floating instruments — the two instrument antennas carry six coordinates against three measurements — and phase fixes distance only modulo $\lambda/2 = 23.1$ mm, so a lost round slips the registration by a whole cell.

Keeping the chain unbroken

Four mechanisms that keep the carrier phase continuous — (a) Ranging rate per pair: the packet path raised the two-node link from 34 to 73 rounds/s, and the single-channel three-node schedule delivers 35–46 rounds/s per pair where the dual-channel schedule left each chain only 10–17. (b) Per-round phase noise against the crystal trim. (c) A static link should trace a flat line: without the post-final packet the chain drifts. (d) Median phase error by SNR bin; the rounds left of the gate carry the largest errors and are discarded.

Four mechanisms do the work. A post-final packet measures each round's residual carrier-frequency term, which holds block-mean phase coherence at 0.964 even under a forced PLL re-lock every 2 s (0.885 uncorrected) and suppresses static drift from −3.3 to +0.1 mm/min. Crystal trimming halves per-round phase noise, from 24.3° to 10.8° on channel 5. A per-round quality gate on SNR rejects rounds at multipath amplitude nulls, whose phase is not the path's; fewer than 0.2% of rounds fall below it. And the three nodes stay on one channel: the usual two-channel synthetic wavelength admits near-coincident false fixes, cannot be sampled simultaneously, and re-randomizes the carrier phase at every hop — our two-channel version slipped a cell every 30–60 s.

The three-node single-channel schedule sustained 35.1, 45.5 and 35.2 rounds/s on the three pairs with no failed exchange in 125,176 rounds.

The trocar supplies what distance lacks

From three distances to two tool tips — Each instrument pivots at its port, so its antenna and its tip lie on one line whose direction the IMU measures, and the insertion coordinate is the instrument's only unknown.

Because the shaft passes through the port, the antenna of instrument $i$ lies on the line through the port $\mathbf{P}_i$ with direction $\hat{\mathbf{f}}_i$ from the on-board IMU,

\mathbf{a}_i(s_i) = \mathbf{P}_i - \hat{\mathbf{f}}_i s_i + \mathbf{R}_i \mathbf{o}_i,

and with the antenna-to-tip shaft length $D_i$ the tool tip is $\mathbf{p}_i = \mathbf{P}_i + \hat{\mathbf{f}}_i (D_i - s_i)$ . The insertion coordinate $s_i$ is the instrument's only unknown. A two-way carrier-phase observation on pair $ij$ then reads

d_{ij} = \lVert \mathbf{a}_i - \mathbf{a}_j \rVert - n_{ij}\Delta + \varepsilon_{ij}, \qquad n_{ij} \in \mathbb{Z},

so three pairs give three equations in two continuous unknowns and three integers. Unlike the standard treatment of carrier-phase ambiguity, which buys the integer with more measurements, we buy it with kinematics the operating room already provides — and the spare equation is what makes a wrong integer visible instead of absorbed. The integers are enumerated over 31 hypotheses covering ±116 mm, and the arbiter commits a correction only when the residual margin exceeds 3 mm for five consecutive solves.

What the measurements show

Ranging test setup — (a) Two DWM3001CDK nodes on 3D-printed mounts face each other on an aluminium-profile rail, with a digital caliper of 0.01 mm resolution as ground truth; the drape experiments use the same rail. (b) A laparoscopic box trainer with an 11-hole trocar plate: the endoscope node in the central port, and a gripper and a cutter — each carrying a node — hand-held through two further ports.

Rail experiments — (a) A 75 s freehand sequence, 2,757 raw rounds. (b) Raw static σ at 250.00 mm, bare, with the drape laid loosely over both nodes and pressed onto both antennas. (c) Blind displacements with the drape pressed on. (d) Endpoint error of each chain and of ToF on a symmetric-log axis; the red lines mark one cell.

The drape changes nothing. Bare, loosely draped and with the drape pressed onto both chip antennas, the round rate was 73.1, 73.0 and 73.0 rounds/s, first-path SNR stayed within 0.93–1.07 of its bare value, and raw static σ was 0.83/0.73/0.89 mm on channel 5 — the spread follows the channel, not drape contact. With the drape pressed on, three blind displacements were tracked at 0.81 mm mean absolute error, while ToF on the very same rounds erred by 6.7–149 mm.

Static precision in the phantom — (a) Static σ of the tracked distance per pair, on a log axis, against the rule stack without the Kalman filter and against ToF of the same rounds. (b) Deviation against averaging window: the phase sits at its floor from 0.1 s while ToF stays an order of magnitude above it at every window. (c) The ToF bias re-draws at every boot, which is why it only seeds the offset.

In the 11-hole phantom, among metal instruments and with oblique antennas, the tracked distance held a median σ of 0.22, 0.33 and 0.10 mm on the three pairs. Averaging does not rescue ToF: across two power cycles of the same static bench pair the ToF median moved by 35.2 mm and the single-shot σ grew from 31 to 105 mm. A bias that re-draws at every boot cannot be averaged away.

Grid integrity — Candidate integer-slip events across the freehand episodes, with time-of-flight from the same rounds used as an independent witness.

Over 22 freehand episodes — 30.7 pair-minutes of motion — the window detector raised 31 candidates, of which 20 carried a ToF witness: three were real motion and none fell in the slip band, so no slip was confirmed, with a one-sided 95% bound of 0.29 slips per pair-minute. The absorption gate is what buys this: lowering it from 250 to 25 mm/s produced six confirmed slips over a comparable 31.4 pair-minutes.

Trocar-constrained solve — (a) Residual RMS of the null solve, of an uncorrected one-cell slip on each pair, and of a wrong mounting-offset sign; the red line is the 3 mm commit margin. (b) Outcome against injected slip size: correct correction, silence, or a wrong cell. (c, d) Tip error against range noise and against IMU attitude error.

The spare equation behaves as a self-check. With everything correct the solve closed at a median residual RMS of 0.17 mm — the level of the range noise itself. An uncorrected one-cell slip raised it to 1.5, 2.4 and 4.3 mm depending on the pair, visible but often below the commit margin, which is why the arbiter stays silent on many single-cell slips. A wrong mounting-offset sign raised it to 26.9 mm, so a setup error of that kind cannot pass unnoticed.

Over 200 slip-free poses the ±5-cell hypothesis set committed no false fix. Over 200 injected slips it committed the correct correction in 126 (63%), stayed silent in 61 and picked a wrong cell in 13 — so 82% of its failures are silence rather than a wrong answer. With exact attitude, 0.5 mm range noise gives a median tip error of 0.92 mm; an IMU attitude error of 1° and 2° raises that to 3.6 and 7.4 mm, because the error enters through both the lever arm and the solved depth. Attitude, not range, sets tip accuracy, and that is where the next gain lies.

In-vivo deployment — Appendectomy on a live rabbit with a node on each hand-held instrument and a third on the endoscope, all outside the trocars. Inset: the endoscope view of the same episode.

The logger was then carried into a live rabbit appendectomy with the same mounts and logging chain as in the phantom, and logged 11 episodes — 133 s of three-pair ranging and 6.0 pair-minutes of motion exposure — with the working ends inside the animal.

Limitations

In-vivo tip localization is not yet validated: those recordings need more episodes, and one instrument's IMU often produced incomplete attitude, so the records hold distances rather than solved tips. Ranging is relative — each episode is seeded by a ToF whose bias re-draws at every power cycle, so absolute accuracy against surveyed fiducials was not measured — and the integer is bounded rather than proven. The pivot is not perfectly fixed: the abdominal wall is viscoelastic, so a port that shifts during manipulation biases the solved depth. The nodes are development boards on 3D-printed mounts and need sterilizable packaging before clinical use, and the endoscope node is treated as fixed, so a moving endoscope would add a third insertion depth.

#UWB#carrier phase#tool tracking#trocar#ICRA 2027