Continuum Body Odometry Overview

• Continuum body odometry is a framework that models state continuously—via arclength, time, or contact constraints—to estimate both deformable shapes and rigid trajectories.
• It leverages diverse estimation architectures such as batch MAP, sliding-window optimization, and continuous observers to achieve real-time, accurate state reconstruction.
• The approach integrates heterogeneous sensor data, including IMUs, cameras, lidar, radar, and strain sensors, to address estimation challenges like process noise and odometric drift.

Searching arXiv for the cited papers to ground the article in current preprints. Continuum body odometry denotes a class of state-estimation problems in which body state is represented continuously rather than only as a sequence of isolated poses. In the literature summarized here, that continuity appears in more than one technically distinct form: as a continuum robot state distributed along arclength $s$, as a rigid-body trajectory evolving continuously in time $t$, and as odometry constrained by continuous support geometry or intermittent contact anchors. The common theme is that estimation is organized around a continuous state or continuous constraint structure, typically on $SE(3)$, and fused with heterogeneous sensing such as pose sensors, FBG strain sensors, cameras, IMUs, radar, lidar, wheel encoders, or proprioceptive contact measurements (Lilge et al., 2024, Mo et al., 2021, Burnett et al., 2024, Jiang et al., 2024, Sun et al., 19 Feb 2026).

1. Terminological scope

The literature does not use the phrase in a single uniform sense. One usage is literal continuum-body state estimation: reconstructing the shape and strain of a deformable or continuum robot body along its backbone. A second usage is continuous-time odometry of a rigid sensor platform, where the body remains rigid but the estimated motion is a smooth trajectory $t \mapsto (R(t),p(t))$ or $$\mathbf{x}(t)=\{\mathbf{T}(t),\boldsymbol{\varpi}(t),\mathbf{b}(t)\}$$. A third, related usage concerns rigid-body odometry under continuous environmental or contact structure, such as a globally continuous ground manifold or contact anchors that intermittently tie the body to the world (Lilge et al., 2024, Burnett et al., 2024).

Usage	Continuous variable	Representative papers
Continuum-robot state reconstruction	Arclength $s$ along the body	(Lilge et al., 2024)
Continuous-time rigid-body odometry	Time $t$	(Mo et al., 2021, Burnett et al., 2024, Bouazza et al., 28 Aug 2025)
Odometry under continuous support/contact structure	Ground manifold or contact-anchor history	(Jiang et al., 2024, Sun et al., 19 Feb 2026)

A common misconception is to treat all of these as equivalent. They are not. The temporally continuous formulations in visual-inertial, lidar-inertial, radar-inertial, and optical-flow/IMU odometry estimate a single rigid-body trajectory, whereas the continuum-robot formulation estimates distributed body shape and strain. Conversely, methods based on ground manifolds or contact anchors use continuity in the environment or in body-environment constraints rather than in a deformable body model itself. This suggests that “continuum body odometry” is best read as an umbrella term whose precise meaning depends on whether the continuity is spatial, temporal, or geometric.

2. Spatial continuum-body reconstruction on $SE(3)$

A direct formulation of continuum body odometry appears in "State Estimation for Continuum Multi-Robot Systems on $SE(3)$" (Lilge et al., 2024). That work formulates shape-and-strain estimation for multiple coupled continuum robots as a batch MAP estimation problem on $SE(3)$ with a Gaussian-process prior along arclength, explicit coupling constraints, and support for heterogeneous sensing such as pose sensors and FBG strain sensors. The estimator is not restricted to a single serial continuum arm: it estimates the full state of a system of $t$0 coupled continuum bodies, plus rigid bodies such as a shared end-effector, under arbitrary coupling topology, while preserving computational efficiency through sparse factor-graph structure.

The paper is careful about the meaning of odometry in this setting. It is not a temporal recursive odometry filter in the usual sense. The state is reconstructed quasi-statically at each instant, using a prior defined over robot arclength $t$1, not over time $t$2. Temporal propagation is explicitly not part of the model and is identified as a limitation. For that reason, the contribution is more precisely characterized as continuum body state reconstruction with uncertainty than as temporal odometry in the conventional inertial-navigation sense (Lilge et al., 2024).

Within that formulation, the substantive advance is support for multiple coupled continuum robots with arbitrary topology. Simulations and experiments show average end-effector errors of $t$3 mm and $t$4 depending on the sensor setup. The reported computation time is below $t$5 ms, enabling quasi-static real-time scenarios with average update rates of $t$6-$t$7 Hz, and an open-source C++ implementation is provided (Lilge et al., 2024). In practical terms, this places continuum-robot state estimation inside the same algorithmic regime as modern sparse geometric estimation, but with the continuous variable indexed by body arclength rather than by time.

3. Continuous-time rigid-body odometry

A second major line of work treats the body as rigid but represents its motion continuously in time. "Continuous-Time Spline Visual-Inertial Odometry" (Mo et al., 2021) replaces the standard discrete-time motion model of monocular VIO with a continuous-time cubic spline trajectory inside a sliding keyframe window. The method models translation with a cubic polynomial and rotation through a Lie-algebra interpolation anchored at each keyframe, then uses spline derivatives to synthesize accelerometer and gyroscope predictions at exact IMU timestamps. Visual information is inherited from Direct Sparse Odometry through direct photometric error, so the estimator jointly optimizes camera intrinsics, scale, gravity roll/pitch, keyframe poses, affine brightness parameters, inverse depths, IMU biases, and spline coefficients. The full problem is posed as a constrained nonlinear optimization solved with Lagrange multipliers, and First Estimate Jacobians are used to improve consistency and efficiency (Mo et al., 2021).

The principal odometric significance of this formulation is temporal flexibility. Because the trajectory is defined for every $t$8 in the active window, IMU samples can be compared directly to analytically differentiated motion rather than being forced into preintegrated constraints between sparse states. The paper explicitly states that continuous-time pose representation “makes it possible to address many VIO challenges, e.g. rolling shutter distortion and sensors that may lack synchronization,” although those advantages remain mostly latent in the reported system: no explicit rolling-shutter projection model and no temporal offset parameter are introduced. Empirically, SplineVIO is reported as state-of-the-art in accuracy and real-time computational efficiency; on EuRoC Machine Hall sequence MH1, average keyframe optimization time is $t$9 ms and overall framerate is $SE(3)$0 FPS (Mo et al., 2021).

"Continuous-Time Radar-Inertial and Lidar-Inertial Odometry using a Gaussian Process Motion Prior" (Burnett et al., 2024) adopts a different continuous-time backbone. Its state is $SE(3)$1, where $SE(3)$2, $SE(3)$3 is body-centric velocity, and $SE(3)$4 contains IMU biases. The motion prior is white-noise-on-acceleration, yielding an exact sparse GP with block-tridiagonal inverse kernel. Support states are maintained in a fixed-lag sliding window of about $SE(3)$5 ms, interpolation depends only on the two bracketing support states, and every lidar point, radar return, and IMU sample is attached to its own timestamp. A notable modeling choice is to use the gyroscope as a direct measurement of the state while preintegrating accelerometer readings only into a relative velocity factor. The result is a sliding-window batch nonlinear least-squares estimator solved with Gauss-Newton and designed for real-time continuous-time lidar-inertial, radar-inertial, and radar-only operation (Burnett et al., 2024).

The continuous-time state is particularly consequential for rolling-acquisition sensors. In lidar, each point is transformed using $SE(3)$6 at its own acquisition time, so motion distortion is removed through the trajectory representation itself. In radar, continuous-time body velocity appears directly in the Doppler-compensated measurement model. The paper reports real-time operation across datasets, and notes that radar-inertial odometry benefits especially from IMU fusion, whereas lidar-inertial odometry is only marginally better than lidar-only on slow-vehicle data because the GP continuous-time model already compensates scan distortion well (Burnett et al., 2024).

A third continuous-time rigid-body formulation is "Observer Design for Optical Flow-Based Visual-Inertial Odometry with Almost-Global Convergence" (Bouazza et al., 28 Aug 2025). That paper uses monocular optical flow only to estimate velocity direction, feeds it into a globally exponentially stable Riccati observer for body-frame velocity and gravity direction, and then applies a complementary observer on $SE(3)$7 for attitude. The architecture is explicitly continuous-time and observer-theoretic, with global exponential convergence for the translational observer under persistent excitation and almost-global asymptotic stability for the cascaded interconnection. At the same time, it is not a full modern VIO pipeline: it does not estimate landmarks or depth, and position is obtained only by integration. It is therefore directly relevant to continuous rigid-body odometry, but only indirectly relevant to full continuum-body state estimation (Bouazza et al., 28 Aug 2025).

4. Continuous geometric and contact priors

Continuity can also enter odometry through constraints imposed by the support geometry rather than through a continuous trajectory model. "WING: Wheel-Inertial Neural Odometry with Ground Manifold Constraints" (Jiang et al., 2024) is an interoceptive-only odometry system for ground robots that fuses IMU and wheel encoders with learned sensor correction and a globally continuous ground manifold. The ground is modeled as a $SE(3)$8 dual cubic B-spline manifold, and the estimator introduces soft constraints requiring the wheel-ground contact point to lie on the manifold and the wheel/contact-frame up vector to align with the surface normal. The filter is a space-based sliding-window EKF whose window grows when the robot travels more than a fixed distance $SE(3)$9, not after a fixed time interval. This design is meant to exploit the spatial continuity of the manifold rather than only the recency of measurements (Jiang et al., 2024).

The continuity assumption is not in the robot body, which remains rigid, but in the environmental support surface. That distinction is crucial. The estimator maintains current IMU state, a sliding window of cloned pose states, and an active manifold control vector $t \mapsto (R(t),p(t))$0, while neural networks estimate IMU de-biasing, wheel-velocity correction, and associated covariances. The manifold ablation reported in the paper shows the effect of continuity explicitly: on KAIST urban17, ATE is $t \mapsto (R(t),p(t))$1 for a segmented 3rd-order manifold, $t \mapsto (R(t),p(t))$2 for the continuous cubic B-spline manifold, $t \mapsto (R(t),p(t))$3 for a continuous quadratic B-spline, $t \mapsto (R(t),p(t))$4 for a continuous linear B-spline, and $t \mapsto (R(t),p(t))$5 with no manifold; on NCLT 12/08/20, ATE is $t \mapsto (R(t),p(t))$6 for segmented, $t \mapsto (R(t),p(t))$7 for continuous cubic, and $t \mapsto (R(t),p(t))$8 with no manifold (Jiang et al., 2024).

A different continuity mechanism appears in "Contact-Anchored Proprioceptive Odometry for Quadruped Robots" (Sun et al., 19 Feb 2026). That work addresses purely proprioceptive body odometry using IMU and motor measurements only. Its key idea is to treat each contacting leg as a kinematic anchor: at touchdown, the estimator records the foot’s world-frame location, and during stance that stored footfall acts as a world-frame geometric constraint on body position; a corresponding zero-velocity or wheel-propagated contact model constrains body velocity. The system further includes joint-torque-based foot wrench estimation for contact selection, a height clustering and time-decay correction to suppress elevation drift, an inverse-kinematics cubature Kalman filter for foot-end velocity estimation, and a multi-contact geometric heading reference for yaw stabilization (Sun et al., 19 Feb 2026).

Although this method is not a continuum-robot estimator, it is conceptually relevant wherever body odometry depends on intermittent attachment to the environment. The empirical results emphasize long-horizon drift suppression through anchors: Astrall point-foot robot A closes a $t \mapsto (R(t),p(t))$9 m horizontal loop with $$\mathbf{x}(t)=\{\mathbf{T}(t),\boldsymbol{\varpi}(t),\mathbf{b}(t)\}$$0 m error and a $$\mathbf{x}(t)=\{\mathbf{T}(t),\boldsymbol{\varpi}(t),\mathbf{b}(t)\}$$1 m vertical loop with $$\mathbf{x}(t)=\{\mathbf{T}(t),\boldsymbol{\varpi}(t),\mathbf{b}(t)\}$$2 m error; wheel-legged robot C closes a $$\mathbf{x}(t)=\{\mathbf{T}(t),\boldsymbol{\varpi}(t),\mathbf{b}(t)\}$$3 m horizontal loop with $$\mathbf{x}(t)=\{\mathbf{T}(t),\boldsymbol{\varpi}(t),\mathbf{b}(t)\}$$4 m error and a $$\mathbf{x}(t)=\{\mathbf{T}(t),\boldsymbol{\varpi}(t),\mathbf{b}(t)\}$$5 m vertical loop with $$\mathbf{x}(t)=\{\mathbf{T}(t),\boldsymbol{\varpi}(t),\mathbf{b}(t)\}$$6 m error; Unitree Go2 EDU closes a $$\mathbf{x}(t)=\{\mathbf{T}(t),\boldsymbol{\varpi}(t),\mathbf{b}(t)\}$$7 m horizontal loop with $$\mathbf{x}(t)=\{\mathbf{T}(t),\boldsymbol{\varpi}(t),\mathbf{b}(t)\}$$8 m error and a $$\mathbf{x}(t)=\{\mathbf{T}(t),\boldsymbol{\varpi}(t),\mathbf{b}(t)\}$$9 m vertical loop with less than $s$0 m vertical error (Sun et al., 19 Feb 2026). A plausible implication is that, in contact-rich systems, persistent odometric drift can be controlled not only by better propagation models but also by explicit memory of trusted body-environment attachments.

5. Estimation architectures and computational structure

Across these works, several estimator architectures recur. The continuum multi-robot formulation uses batch MAP estimation on $s$1 with a Gaussian-process prior along arclength and a sparse factor-graph structure (Lilge et al., 2024). SplineVIO uses a sliding-window constrained nonlinear optimization with cubic spline boundary conditions, Lagrange multipliers, and First Estimate Jacobians (Mo et al., 2021). The radar/lidar-inertial GP approach uses sliding-window batch nonlinear least squares with Gauss-Newton, Schur-complement marginalization, and a sparse GP prior whose inverse kernel is block tridiagonal (Burnett et al., 2024). WING uses a space-based sliding-window EKF with cloned states and manifold control variables (Jiang et al., 2024). The optical-flow method replaces batch optimization with a cascaded continuous-time observer composed of a Riccati observer and a complementary observer on $s$2 (Bouazza et al., 28 Aug 2025). CAPO is modular rather than monolithic, combining translational fusion with an inverse-kinematics cubature Kalman filter and separate yaw-correction logic (Sun et al., 19 Feb 2026).

These design choices produce different computational tradeoffs. The continuum multi-robot estimator reports computation below $s$3 ms and $s$4-$s$5 Hz updates in quasi-static scenarios (Lilge et al., 2024). SplineVIO remains real-time despite recomputing many IMU residuals, with MH1 backend optimization slower than DSO or VI-DSO but still online (Mo et al., 2021). The GP lidar/radar framework emphasizes that interpolation at a query time is $s$6 because it depends only on the two bracketing support states, and reports runtime examples of $s$7 ms for STEAM-LO on KITTI-raw, $s$8 ms for STEAM-LIO on Newer College, and $s$9 ms for STEAM-RIO on Boreas (Burnett et al., 2024). WING reports $t$0 ms for the IMU De-Bias Net, $t$1 ms for the Wheel Encoder Net, and $t$2 ms for the space-based sliding-window fusion at a $t$3 Hz sensor rate (Jiang et al., 2024).

The methodological contrast is therefore not merely between “continuous” and “discrete.” It is also between batch smoothing, constrained windowed optimization, sparse GP regression, EKF-style recursive filtering, and continuous-time observer design. This suggests that continuity does not determine a unique estimator family; rather, it determines the state representation and residual structure, after which multiple inference strategies remain viable.

6. Limitations, misconceptions, and conceptual boundaries

The most important conceptual boundary is that continuous-time rigid-body odometry is not the same problem as continuum-body shape estimation. The optical-flow/IMU observer explicitly assumes a single rigid body with fixed camera-IMU extrinsics, static landmarks, and one global $t$4; it is therefore suitable for a rigid sensor head on a continuum robot, but not for full-body deformation state estimation (Bouazza et al., 28 Aug 2025). The lidar-inertial and radar-inertial GP framework likewise models a rigid body on $t$5, with continuity in time rather than in material configuration (Burnett et al., 2024). WING assumes a rigid ground vehicle and a globally continuous support surface, not a deformable body (Jiang et al., 2024). CAPO assumes discrete rigid contacts, known low-dimensional kinematics, and indexed end-effectors, which limits direct transfer to distributed or deformable contact (Sun et al., 19 Feb 2026).

The second boundary concerns what counts as odometry. The continuum multi-robot estimator is explicit that temporal propagation is not part of the model; the prior is over arclength $t$6, not time $t$7, and the state is reconstructed quasi-statically at each instant (Lilge et al., 2024). For that reason, describing it as “odometry” requires care. By contrast, the continuous-time VIO and lidar/radar-inertial systems estimate time-indexed motion trajectories and are odometry systems in the usual navigation sense (Mo et al., 2021, Burnett et al., 2024).

Several practical limitations are also recurrent. In SplineVIO, rolling shutter and unsynchronized sensing are motivations rather than fully realized model components, since no explicit rolling-shutter projection model or sensor time-offset estimation is introduced; robustness also degrades when there is insufficient fast motion early in the sequence to initialize the IMU (Mo et al., 2021). In the GP lidar/radar framework, the local $t$8 GP approximation assumes small process noise and small rotational motion between adjacent support states, and posterior interpolation at intermediate times is an approximation rather than the result of explicitly including all measurement-time states (Burnett et al., 2024). The optical-flow observer is validated only in simulation, does not estimate a map or landmark depths, and without magnetometer cannot recover yaw (Bouazza et al., 28 Aug 2025). WING depends on terrain that is reasonably approximable by a globally continuous height manifold, and its experimental initialization uses ground truth for the IMU state (Jiang et al., 2024). CAPO does not explicitly model slip, assumes horizontal ground in wheel-contact propagation, and treats heading correction through practical modules rather than a fully unified probabilistic state-space model (Sun et al., 19 Feb 2026).

Taken together, these works indicate that continuum body odometry is not a single canonical algorithmic object. A more precise reading is that it is a family of estimation problems unified by continuous representations: body shape indexed by arclength, rigid-body trajectory indexed by time, or odometric constraints indexed by continuous support geometry or contact history. The exact meaning depends on which quantity is continuous and which physical assumptions remain rigid.