Sensorless Probe Motion Estimation

• Sensorless probe motion estimation is the indirect recovery of an instrument’s movement from non-traditional signals such as ultrasound images, proprioceptive data, or inertial measurements.
• Techniques address different inverse problems by leveraging local texture cues, global anatomical context, or mechanical residuals to enhance motion observability.
• Advanced methods, including deep learning models and constraint-based optimizations, mitigate challenges like drift, out-of-plane ambiguity, and low signal excitation.

Sensorless probe motion estimation is the recovery of an instrument, probe, or payload motion state from indirect measurements rather than from external position sensors, tracking hardware, or dedicated tip-localization devices. In current arXiv literature, the term spans at least three technically distinct regimes: image-only recovery of freehand ultrasound probe trajectories from sequences of 2D B-mode frames, inference of constrained laparoscopic pivot misalignment from robot proprioception and residual force estimation, and estimation of cable-suspended payload motion from onboard inertial and odometric sensing without any load-side sensor (Wilson et al., 11 Sep 2025). A broader adjacent literature formulates the same estimation principle through active probing: a known excitation is injected, a configuration-dependent response is demodulated, and the hidden motion state is inferred from that response (Jebai et al., 2012).

1. Problem classes and state representations

Across the literature, sensorless probe motion estimation is not a single state-estimation problem but a family of inverse problems with different hidden states, different sensing channels, and different constraints. In freehand 3D ultrasound, the hidden state is the probe trajectory or relative inter-frame motion. In robotic laparoscopy, the hidden state is the offset between a nominal remote center of motion and the actual trocar/incision pivot. In cable-suspended transport, the hidden state is the payload position, velocity, cable direction, and cable angular velocity (Wilson et al., 11 Sep 2025).

The state dimensionality changes accordingly. DualTrack formulates ultrasound tracking with an ordered sequence $(X_i)_{i=1}^{n}$ of 2D ultrasound frames and represents pose either as $T_i\in\mathbb{R}^{4\times 4}$ or as a 6-DoF vector $\mathbf{p}_i\in\mathbb{R}^6$, with the task cast in terms of relative transforms $T_{j\leftarrow i}=T_i^{-1}T_j$ and composed at inference as $T_i=T_0\prod_{k=0}^{i-1}T_{k+1\leftarrow k}$, where $T_0=\mathrm{Id}$ (Wilson et al., 11 Sep 2025). DCL-Net uses the related formulation $M'_i=M_{i+1}M_i^{-1}$, decomposed into $$\theta_i=\{t_x,t_y,t_z,\alpha_x,\alpha_y,\alpha_z\}_i$$, and predicts segment-level mean relative motion before sliding-window aggregation (Guo et al., 2020). By contrast, the laparoscopic RCM paper defines the main unknown as a scalar misalignment $D$, the Euclidean distance from the robot’s true RCM to the misaligned incision port location $I_c$, rather than a full 6-DoF pose (Yang et al., 17 Mar 2025).

A concise comparison is useful because the same label “sensorless” hides materially different estimation targets.

Application	Hidden state	Primary inputs
Freehand 3D ultrasound	Probe pose or relative 6-DoF motion	Sequence of 2D B-mode ultrasound frames
Robotic laparoscopy	Scalar RCM-incision misalignment $T_i\in\mathbb{R}^{4\times 4}$0	Joint positions, joint torques, kinematics
Cable-suspended transport	$T_i\in\mathbb{R}^{4\times 4}$1	UAV IMU acceleration and odometry

These formulations also imply different reconstruction operators. Ultrasound methods typically estimate relative motion and rely on cumulative composition, so local error can drift into large global error. Constraint-based robotic methods often solve for a lower-dimensional latent geometric variable, such as $T_i\in\mathbb{R}^{4\times 4}$2, and therefore emphasize observability of that variable under contact or pivot excitation. Cable-suspended estimation lies between these extremes: the hidden state is high-dimensional, but geometry sharply constrains it through a fixed-length cable relation (Yang et al., 17 Mar 2025).

2. Information channels and observability

The central technical issue is not merely estimation accuracy but observability: what in the available signals actually encodes motion? The ultrasound literature makes this explicit by separating local and global visual context. Local information comprises fine-grained speckle patterns, decorrelation, and short-term frame-to-frame appearance change; global information comprises higher-level anatomical structure, coarse shape, scan progression, and long-range temporal dependencies. DualTrack’s main claim is that accurate sensorless ultrasound tracking requires both, because local cues alone do not reliably situate the probe in 3D over long freehand sweeps, particularly in turns or out-of-plane ambiguous segments (Wilson et al., 11 Sep 2025).

DCL-Net identifies a related observability mechanism at the level of local texture. Its attention maps align with speckle-rich areas, motivated by prior decorrelation theory stating that correlation between neighboring ultrasound frames reflects relative separation and orientation. This makes the signal informative but also fragile: homogeneous tissue, acoustic shadowing, weak texture, or severe out-of-plane ambiguity reduce the discriminative content available to a local image model (Guo et al., 2020). A direct implication is that “sensorless” does not mean “free of latent sensing assumptions”; it means that image content itself is the sensing medium.

In robotic laparoscopy, observability comes from mechanically induced residuals rather than visual content. The RCM misalignment framework infers $T_i\in\mathbb{R}^{4\times 4}$3 from the reaction forces that appear when a shaft pivots about a trocar that is misaligned with the robot’s nominal RCM. The key relation is $T_i\in\mathbb{R}^{4\times 4}$4, with external force estimated from residual torque after subtracting a learned free-space torque model. The paper is explicit that small misalignments are difficult to recover because low contact forces make the residual comparable to friction, modeling error, and torque-estimation error; the approach is effective mainly when the external force is large enough, and the authors state that any sensorless force estimation scheme they tried produces poor RCM estimates when the external force is small, e.g. $T_i\in\mathbb{R}^{4\times 4}$5 N (Yang et al., 17 Mar 2025).

The cable-suspended-load estimator states its observability condition even more directly: persistent excitation requires $T_i\in\mathbb{R}^{4\times 4}$6. In hover, where $T_i\in\mathbb{R}^{4\times 4}$7, the transverse excitation vanishes and the load direction becomes weakly observable (Nascimento et al., 5 May 2026). This is structurally analogous to low-speed electromechanical sensorless estimation, where the natural dynamics become uninformative and an auxiliary excitation must reveal the hidden state. The active-probing PMSM literature makes this equivalence explicit: low-speed unobservability is overcome by injecting a known high-frequency signal and using the resulting anisotropic response as a virtual measurement of position (Jebai et al., 2012).

3. Image-only ultrasound trajectory estimation

The best-developed probe-specific branch of the literature is freehand 3D ultrasound reconstruction. The practical motivation is consistent across papers: conventional 3D ultrasound either requires matrix-array probes or external tracking systems, both of which increase cost, complexity, calibration burden, and workflow friction. Sensorless reconstruction instead seeks to infer probe motion directly from ultrasound image sequences and then place the 2D B-scans into 3D space (Wilson et al., 11 Sep 2025).

DCL-Net represents an early deep contextual formulation of this problem. It operates on TRUS video segments, uses a 3D ResNeXt backbone with residual blocks, and predicts a segment-level mean relative 6-DoF motion vector $T_i\in\mathbb{R}^{4\times 4}$8. At test time, a window of length $T_i\in\mathbb{R}^{4\times 4}$9 is slid along the sequence and neighboring-frame motion is obtained by averaging all window predictions that cover a frame pair. Two design choices are central. First, 3D convolutions treat time as the third axis and thereby learn spatiotemporal filters over neighboring frames rather than isolated 2D features. Second, a self-attention block focuses the model on speckle-rich regions judged informative for motion estimation. The paper adds a case-wise correlation loss,

$\mathbf{p}_i\in\mathbb{R}^6$0

to counter regression toward the average trajectory style and to increase within-case correlation between prediction and true motion (Guo et al., 2020).

DualTrack reframes the same problem around scale separation rather than only local temporal context. Its architecture contains a local encoder, a global encoder, and a fusion module. The local branch is a 3D ResNet18 with small temporal windows, trained on short contiguous full-resolution subsequences so that it remains texture-biased and sensitive to speckle-level frame-to-frame cues. The global branch uses an image backbone $\mathbf{p}_i\in\mathbb{R}^6$1 followed by temporal self-attention $\mathbf{p}_i\in\mathbb{R}^6$2, and is explicitly trained on randomly sampled non-contiguous subsequences with images downsampled to $\mathbf{p}_i\in\mathbb{R}^6$3 so that it cannot exploit adjacent-frame speckle motion and is forced toward coarse anatomy, landmarks, structural continuity, and sequence-level positional awareness. A transformer-decoder-style fusion module then uses self-attention over local embeddings and cross-attention from local embeddings to global embeddings (Wilson et al., 11 Sep 2025).

This explicit decoupling is the conceptual departure. Prior sensorless ultrasound methods either emphasized local frame-to-frame motion or appended a global temporal model on top of locally extracted features, thereby forcing a single representation to serve incompatible purposes. DualTrack instead treats local and global evidence as separate regimes and fuses them only at the regression stage. The paper argues that this is especially important for out-of-plane direction ambiguities and scan turns, where local appearance changes alone do not resolve direction (Wilson et al., 11 Sep 2025).

Quantitatively, the two papers mark a clear progression in benchmarked ultrasound sensorless reconstruction. On 70 TRUS test cases, DCL-Net reported average distance error $\mathbf{p}_i\in\mathbb{R}^6$4 mm and average final drift $\mathbf{p}_i\in\mathbb{R}^6$5 mm, improving over linear motion, classical decorrelation, a prior 2D CNN, and a vanilla 3D CNN, with paired $\mathbf{p}_i\in\mathbb{R}^6$6-test significance at $\mathbf{p}_i\in\mathbb{R}^6$7 (Guo et al., 2020). On the TUS-REC benchmark, DualTrack reported on 72 public test scans a GPE of $\mathbf{p}_i\in\mathbb{R}^6$8 mm, LPE of $\mathbf{p}_i\in\mathbb{R}^6$9, FDR of $T_{j\leftarrow i}=T_i^{-1}T_j$0, and maximum drift of $T_{j\leftarrow i}=T_i^{-1}T_j$1 mm, outperforming DCL-net, 2-Frame CNN, MoNet, and Hybrid Transformer; the gains over Hybrid Transformer were reported significant by Wilcoxon signed-rank test with $T_{j\leftarrow i}=T_i^{-1}T_j$2 (Wilson et al., 11 Sep 2025).

4. Proprioceptive and constraint-based estimation in robotic systems

A second major branch replaces image evidence with proprioceptive residuals and geometric constraints. In robotic laparoscopy, the problem is not free-space pose recovery but estimation of a hidden constrained motion parameter: misalignment between the robot’s nominal remote center of motion and the true trocar/incision pivot. The paper models the dVRK Patient Side Manipulator with the base frame at the robot RCM and modifies the Jacobian to be evaluated at the incision rather than the tip. The core residual relation is

$T_{j\leftarrow i}=T_i^{-1}T_j$3

where $T_{j\leftarrow i}=T_i^{-1}T_j$4 is a learned estimate of free-space torque. Misalignment is then estimated through a constrained nonlinear least-squares problem over $T_{j\leftarrow i}=T_i^{-1}T_j$5 and $T_{j\leftarrow i}=T_i^{-1}T_j$6, with coaxial displacement $T_{j\leftarrow i}=T_i^{-1}T_j$7 and $T_{j\leftarrow i}=T_i^{-1}T_j$8 (Yang et al., 17 Mar 2025).

The physical interpretation is direct: if the robot’s assumed pivot is misaligned with the actual incision, rotational motion of the shaft generates lateral tissue loading. The paper reports that if the incision port is more than $T_{j\leftarrow i}=T_i^{-1}T_j$9 mm from the RCM, interaction force can exceed $T_i=T_0\prod_{k=0}^{i-1}T_{k+1\leftarrow k}$0 N; at $T_i=T_0\prod_{k=0}^{i-1}T_{k+1\leftarrow k}$1, the force reaches $T_i=T_0\prod_{k=0}^{i-1}T_{k+1\leftarrow k}$2 N; and the maximum reported force is about $T_i=T_0\prod_{k=0}^{i-1}T_{k+1\leftarrow k}$3 N. For $T_i=T_0\prod_{k=0}^{i-1}T_{k+1\leftarrow k}$4 mm, the optimization achieved absolute error within $T_i=T_0\prod_{k=0}^{i-1}T_{k+1\leftarrow k}$5 mm, whereas small misalignments remained difficult under sensorless force estimation because low residual forces are insufficiently informative (Yang et al., 17 Mar 2025). This is sensorless probe motion estimation in a constrained rather than free-space sense: the unknown is not pose but hidden geometric inconsistency of the probe-environment constraint.

The cable-suspended payload paper moves in the opposite direction: it estimates a high-dimensional load state without any load-side sensing. The model imposes the rigid cable constraint $T_i=T_0\prod_{k=0}^{i-1}T_{k+1\leftarrow k}$6 and derives the tension force via the Udwadia–Kalaba construction. The resulting constrained dynamics introduce a cable force $T_i=T_0\prod_{k=0}^{i-1}T_{k+1\leftarrow k}$7, with tension magnitude

$T_i=T_0\prod_{k=0}^{i-1}T_{k+1\leftarrow k}$8

The EKF state is $T_i=T_0\prod_{k=0}^{i-1}T_{k+1\leftarrow k}$9, the filter input is world-frame UAV acceleration $T_0=\mathrm{Id}$0, and the measurement is the onboard UAV odometry state. The measurement model is geometric rather than load-direct: $T_0=\mathrm{Id}$1 The estimator also includes a taut/slack switch based on

$T_0=\mathrm{Id}$2

In real-robot experiments, the estimated load state was then fed into NMPC, and the reported tracking metrics improved from RMSE Drone $T_0=\mathrm{Id}$3 to $T_0=\mathrm{Id}$4, STD Drone $T_0=\mathrm{Id}$5 to $T_0=\mathrm{Id}$6, RMSE Payload $T_0=\mathrm{Id}$7 to $T_0=\mathrm{Id}$8, and STD Payload $T_0=\mathrm{Id}$9 to $M'_i=M_{i+1}M_i^{-1}$0 (Nascimento et al., 5 May 2026).

A plausible unifying view is that these robotic methods replace direct probe localization with physically structured latent-variable estimation. The state can be scalar, as in $M'_i=M_{i+1}M_i^{-1}$1, or vector-valued, as in $M'_i=M_{i+1}M_i^{-1}$2, but in both cases observability is created by geometry, excitation, and internal consistency rather than by an external tracker.

5. Active probing, virtual measurements, and adjacent sensorless paradigms

The electromechanical literature is not about biomedical or surgical probes, yet it supplies a rigorous general paradigm for sensorless motion estimation: inject a known excitation, measure the induced response, and recover configuration from the system’s anisotropic local transfer characteristics. The 2012 PMSM signal-injection paper gives the clearest formulation. A high-frequency voltage perturbation

$M'_i=M_{i+1}M_i^{-1}$3

induces a current decomposition

$M'_i=M_{i+1}M_i^{-1}$4

with the demodulated high-frequency component

$M'_i=M_{i+1}M_i^{-1}$5

where $M'_i=M_{i+1}M_i^{-1}$6 is an angle-dependent saliency matrix determined by the differential magnetic model. The angle is then estimated by

$M'_i=M_{i+1}M_i^{-1}$7

The paper’s point is general: observability lost in the nominal low-frequency model can be restored through high-frequency probing and synchronous demodulation of a virtual output (Jebai et al., 2012).

The PWM-induced PMSM paper pushes the same idea further by exploiting excitation already present in the system. Instead of adding a dedicated probing signal, it uses the switching ripple created by PWM and shows that the demodulated current ripple yields a virtual measurement $M'_i=M_{i+1}M_i^{-1}$8 of the saliency matrix. For interleaved PWM, one reconstructs $M'_i=M_{i+1}M_i^{-1}$9 and estimates

$$\theta_i=\{t_x,t_y,t_z,\alpha_x,\alpha_y,\alpha_z\}_i$$0

The broader relevance is methodological rather than domain-specific: intrinsic actuation ripple can function as a probing carrier if the response is sufficiently configuration dependent (Surroop et al., 2020).

A learned counterpart appears in the BLDC paper, which estimates rotor electrical position from conditioned terminal phase voltages and timestamps using a small MLP, then estimates speed from temporal differences of inferred position. The reported performance was about $$\theta_i=\{t_x,t_y,t_z,\alpha_x,\alpha_y,\alpha_z\}_i$$1 electrical degrees MAE for position and $$\theta_i=\{t_x,t_y,t_z,\alpha_x,\alpha_y,\alpha_z\}_i$$2 rpm MAE for speed over $$\theta_i=\{t_x,t_y,t_z,\alpha_x,\alpha_y,\alpha_z\}_i$$3 to $$\theta_i=\{t_x,t_y,t_z,\alpha_x,\alpha_y,\alpha_z\}_i$$4 rpm, with training sensor-based and testing sensorless (Gamazo-Real et al., 2024). This suggests a second general template for sensorless probe motion estimation: when a direct constitutive inversion is difficult, a supervised nonlinear map from indirect waveform features to latent motion state can replace analytic observers.

These works support a general editor’s term, “virtual-measurement sensorless estimation.” In that regime, the hidden state is not observed directly; instead, one constructs or learns a surrogate output whose dependence on state is sharper than that of the raw measurements. In ultrasound this surrogate can be a fused local-global representation; in residual-based robotics it can be an external torque or force proxy; in signal injection it is the demodulated response envelope. The papers suggest convergence of these ideas even though the sensing media differ.

6. Evaluation criteria, misconceptions, and open directions

Evaluation protocols reflect the structure of each hidden-state problem. Ultrasound work distinguishes local motion quality from long-horizon consistency. DualTrack evaluates Final Drift Rate, Maximum Drift, Local Point Error between adjacent frames, and Global Point Error against the fully reconstructed trajectory, with particular emphasis on GPE and FDR because cumulative composition can turn small local bias into global failure (Wilson et al., 11 Sep 2025). DCL-Net uses average distance between corresponding frame corner-points throughout a video and final drift between the center points of transformed end frames, again separating trajectory-shape error from accumulated end-point drift (Guo et al., 2020). The robotic RCM paper uses force RMSE and absolute error in $$\theta_i=\{t_x,t_y,t_z,\alpha_x,\alpha_y,\alpha_z\}_i$$5, while the cable-suspended work reports closed-loop trajectory RMSE and standard deviation rather than a dedicated load-state RMSE table (Yang et al., 17 Mar 2025).

Several common misconceptions are corrected by the literature. One is that sensorless estimation must be label-free or unsupervised. The opposite is common: DCL-Net trains with EM tracker labels, DualTrack uses ground-truth relative transforms from TUS-REC, and the BLDC ANN trains with encoder-provided labels while remaining sensorless only at runtime (Guo et al., 2020). Another misconception is that good adjacent-frame prediction suffices. The ultrasound papers show that locally plausible relative motions can still produce globally inconsistent reconstructions after composition, which is why DualTrack’s strongest gains appear precisely on global metrics (Wilson et al., 11 Sep 2025).

A third misconception is that the absence of external tracking eliminates calibration. In fact, several systems shift calibration rather than remove it. The RCM framework uses ATI force sensing only in calibration and validation, then deploys sensorlessly (Yang et al., 17 Mar 2025). Adjacent residual-based manipulator work similarly shows that changing payloads corrupt sensorless contact estimation unless a payload residual model is recalibrated; the proposed pre-trained calibration scheme reduces online calibration to a 4-second trajectory and substantially lowers joint-current RMSE across joints, although that work estimates contact/wrench rather than full probe trajectory (Shan et al., 2024). Likewise, a hybrid manipulator dynamics paper builds a virtual force sensor from estimated external torque and demonstrates peg-in-hole interaction-state inference, but it does not directly recover probe pose (Yang et al., 2024). These adjacent results indicate that “sensorless” often means “without a dedicated external motion sensor,” not “without model maintenance.”

Current limitations are remarkably consistent across modalities. Ultrasound methods remain vulnerable to out-of-plane ambiguity, texture-poor regions, and dataset shifts. The RCM method estimates magnitude more reliably than sign and degrades when external forces are small. Cable-suspended estimation becomes weakly observable in hover and assumes a rigid, inextensible, massless cable with a point-mass payload. Active-probing electromechanical methods depend on saliency or saturation-induced anisotropy and on sufficient measurement bandwidth. Across all of them, hidden-state observability is conditional rather than automatic.

The research direction emerging from these papers is therefore not a single universal estimator but a structured design doctrine. Local evidence and long-range context may need to be decoupled rather than entangled; residual-based methods require an explicit account of internal dynamics and payload variation; and when nominal dynamics are weakly informative, deliberately or intrinsically generated excitation can be turned into a virtual measurement. Sensorless probe motion estimation, in this technical sense, is best understood as constrained latent-state inference from indirect but state-dependent signatures rather than as the mere omission of a tracker.