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State estimation for a hybrid rigid-continuum space manipulator system

Lorenzo Ticozzi, Patricio A. Vela, Panagiotis Tsiotras

arXiv:2610.37043v1cs.RO

TL;DR

State estimation for continuum manipulators commonly assumes a fixed base and specialized sensing, unlike space platforms with uncertain free-floating bases and vision/encoder measurements. This paper formulates hybrid soft-rigid manipulator reconstruction as factor-graph MAP inference using Geometric Variable Strain variables. In simulation, it reports a mean end-effector position error of 2.3% of system characteristic length, with millimeter-level relative position accuracy to the base.

  • Problem

    Continuum-robot state estimation largely assumes a fixed base and motion-capture or embedded-strain sensing, whereas space manipulators have uncertain free-floating bases and may be limited to vision and encoders.

    Source: Related work · p. 2

  • Method

    The framework uses factor-graph MAP optimization with analytic residual Jacobians and Geometric Variable Strain coefficients among the estimation variables.

    Source: Factor graph optimization · p. 5, Conclusion · p. 8

  • Results

    The 60-second simulated maneuver yielded a mean end-effector position error of 2.3% of system characteristic length, millimeter-level relative position accuracy to the base, and an average solution time of 1.23 ms.

    Source: Conclusion · p. 8

  • Takeaways & Limitations

    The relative position estimate is accurate to the millimeter, compared with centimeter-level accuracy in the inertial frame, with absolute-position accuracy limited by relative navigation rather than the visual frontend.

    Source: Results · p. 7

    The visual pipeline can lose the continuum link through occlusion or field-of-view exit, and the simulated trajectory therefore kept it continuously visible.

    Source: Visual frontend · p. 6

Abstract

from arXiv · show

Spacecraft-mounted robotic manipulators can support long-term operability of orbiting assets by conducting tasks such as on-orbit assembly, repair, upgrade, and refueling. In this context, the integration of continuum/soft robotic mechanisms into space manipulator designs promises to reduce stowage volume and mass, as well as to improve operational safety through the inherent mechanical compliance of soft structures. This work presents a novel state reconstruction pipeline for a hybrid soft-rigid space manipulator, in which the distal link is a tendon-driven continuum robot. The estimation problem is formulated as a maximum a posteriori (MAP) inference problem over the hybrid manipulator's configuration, conditioned on discrete, noisy measurements from joint encoders and a vision-based sensing pipeline. The resulting soft-constrained optimization problem is transcribed as a factor graph, providing a compact representation for fusing heterogeneous measurements with the rigid-soft kinematic model. Furthermore, the adoption of the Geometric Variable Strain (GVS) approach yields a minimum-order parameterization of the configuration, a property expected to facilitate model-based feedback control design in future work. The proposed implementation is validated in simulation on a closed-loop trajectory where, for better realism, position measurements for the continuum link are obtained by means of a simulated visual pipeline leveraging synthetic photorealistic images. The state reconstruction accuracy and computational performance confirm the potential of the proposed framework for hybrid soft-rigid robotic applications in orbit.

Introduction

Space manipulator research has begun incorporating tendon-driven continuum links, but state-estimation methods for continuum robots largely assume a fixed base and specialized sensing. The paper proposes a factor-graph framework for hybrid soft-rigid manipulators and uses Geometric Variable Strain kinematics to represent their configuration.

Related work

  • Existing continuum-robot estimators largely assume a fixed base and motion-capture or embedded-strain sensing, while space platforms have uncertain free-floating bases and may rely on vision and encoders.The related-work discussion identifies joint probabilistic reconstruction of base and appendage uncertainty within a consistent kinematic and sensing model as uninvestigated.Source: Related work · p. 2

Contribution

  • The proposed framework combines hybrid soft-rigid kinematics with factor-graph state estimation and is evaluated using stereo-triangulated continuum-link positions from photorealistic synthetic images.The contribution passage describes the approach as generalizable to soft-rigid chains and sensor configurations, with system-level information propagation and computational efficiency as design aims.Source: Contribution · p. 2

Hybrid soft-rigid kinematics

  • Geometric Variable Strain reduces the continuum-link configuration problem to a finite-dimensional one by projecting its strain state.The considered manipulator has two revolute-joint rigid links and a tendon-driven continuum link; the continuum configuration is parameterized along arc length.Source: Figure 1 · p. 2, Hybrid soft-rigid kinematics · p. 3

Factor graph inference

The section presents factor graphs as a representation for factored joint posteriors and MAP estimation. Their variable and factor nodes provide a way to express the estimation problem's dependencies.

Factor graph inference

  • Factor graphs are bipartite graphs whose variable and factor nodes connect across node types, matching the factored form of joint posterior distributions.The example depicts variable nodes as circles and factor nodes as squares.Source: Factor graph inference · p. 3, Figure 2 · p. 3
  • The graph's two node types provide a compact visual distinction between estimated variables and the factors connecting them.Circles denote variables and squares denote factors.Source: Figure 2 · p. 3
  • The unknown state and its observations are represented through a maximum a posteriori estimate whose posterior factorization maps to a factor graph.The section frames the graph as a representation of the right-hand side of the MAP formulation.Source: Factor graph inference · p. 3

Problem formulation

The formulation estimates a collection of absolute poses, continuum-backbone poses, and configuration variables. It defines these unknowns as the state for MAP estimation conditioned on sensor measurements.

Problem formulation

  • The endpoint continuum poses correspond by construction to frames F and E, respectively.The pose samples span the appendage from s0 = 0 to sN = L.Source: Problem formulation · p. 3
  • The unknown vector includes absolute poses for the base and rigid-chain frames, N + 1 poses along the continuum appendage, and the configuration vector q.The continuum poses are indexed from i = 0 to N, with endpoints at s0 = 0 and sN = L.Source: Problem formulation · p. 3
  • The MAP estimate is conditioned on the collected sensor measurements, denoted z.The text gives star trackers, joint encoders, and strain sensors as examples of possible measurement sources.Source: Problem formulation · p. 3

Hybrid space manipulator factor graph

The factor graph exposes local dependencies among the hybrid manipulator’s estimation variables and represents the kinematic and measurement relationships as factors. Its pose-offset, joint, camera, and continuum factors connect the rigid and soft portions of the system.

Hybrid space manipulator factor graph

Pose offset factors:

  • Pose-offset factors encode known frame relationships while keeping the formulation sensor-configuration agnostic and providing confidence measures for estimated variables.The offset factors also support adding visual information on a rigid link without altering the rest of the graph.Source: Pose offset factors · p. 4
  • Revolute-joint factors encode parent-to-child pose transformations, and the same kinematic-factor family extends to other joint types such as prismatic joints.For the first joint, the relationship uses its home configuration and unit joint twist.Source: Pose offset factors · pp. 4–5
  • Stereo cameras mounted on the first joint’s frame provide triangulated 3D position measurements of the continuum link in the left-camera frame.The camera point factors relate backbone-node poses to those measurements using the fixed joint-to-camera offset.Source: Pose offset factors · p. 5
  • Continuum-specific Magnus factors encode kinematic relations between consecutive backbone poses under the strain parameterization, alongside a prior factor on the GVS coordinates.The graph also includes a corresponding factor centered on the strain coefficients.Source: Pose offset factors · p. 5

Factor graph optimization

The MAP problem is converted into nonlinear least squares and solved iteratively with Levenberg–Marquardt. For trajectory estimation, the fixed-time formulation is solved repeatedly, using each solution to initialize the next.

Factor graph optimization

  • The negative logarithm of the factor product converts the MAP problem into an equivalent nonlinear least-squares problem.The factor graph is optimized using Levenberg–Marquardt with analytic Jacobians for each residual with respect to adjacent variables.Source: Factor graph optimization · p. 5
  • The MAP solution estimates the configuration at a fixed time instant because the formulation contains no notion of time variation.Along a trajectory, the problem is solved repeatedly, seeding each step with the preceding solution.Source: Factor graph optimization · pp. 5–6
  • Figure 5 depicts the manipulator configuration along the 60-second simulated maneuver, with the end-effector trajectory shown in white.Source: Figure 5 · p. 6

Data generation pipeline

The evaluation uses simulated HSMS data over a 60-second trajectory, with sensor measurements generated from noisy ground truth. Continuum-link position measurements come from a stereo visual pipeline using rendered images and colored backbone markers.

Data generation pipeline

  • Ground truth for a 60-second maneuver tracking a slowly evolving Cartesian position reference was generated with SoRoSim V6.3, and synthetic sensor data was corrupted with Gaussian noise.Source: Data generation pipeline · p. 6
  • The stereo cameras are rigidly attached to the first joint frame, so their geometry relative to the manipulator is independent of the first joint angle.The camera model uses ideal pinhole sensors; the available passages also specify a nominal camera-to-appendage distance of approximately 3 m.Source: Data generation pipeline · p. 6
  • Five distinctly colored markers at predefined, equally spaced arclength positions on the continuum backbone are detected in stereo views and triangulated into position measurements.The rendered left-camera view shows the five bands used by the visual pipeline.Source: Figure 6 · p. 6

Visual frontend

The visual frontend uses color-coded backbone markers to associate stereo-triangulated points with graph factors. Estimation uses a spline-based strain representation, factor covariances, and sequential warm starts.

Visual frontend

  • Five differently colored backbone markers encode predefined arclengths, enabling stereo correspondences to be assigned directly to camera point factors.The markers support color-based matching between the left and right images.Source: Visual frontend · p. 6, Figure 7 · p. 7
  • Occlusion or field-of-view exit limits the visual pipeline, so the simulated trajectory keeps the continuum link continuously visible.Additional cameras could improve workspace coverage, but sensor-placement analysis is outside the work's scope.Source: Visual frontend · p. 6

Estimator configuration

  • The continuum strain field uses cubic spline bases with four control points for the x- and y- components.The factor graph was constructed and optimized using GTSAM.Source: Estimator configuration · p. 6
  • Table 1 lists the covariance values applied to the HSMS factors, including an isotropic noise model for camera point factors.The covariance values are applied in Eq. (28).Source: Estimator configuration · p. 6, Table 1 · p. 7
  • The estimator runs at 30 Hz over 1,801 trajectory timesteps, warm-starting each optimization from the preceding solution.The initial Levenberg–Marquardt solve was seeded with the reference configuration.Source: Estimator configuration · p. 7

Results

The tip position error averages 90.03 mm, or 2.3% of the system's characteristic length, while relative position is accurate to the millimeter. The results also show uncertainty accumulating along the position chain, near-deterministic rigid-chain attitude propagation, and calibrated predicted uncertainty.

Results

  • 90.03 mm is the tip-frame trajectory mean position error, equal to 2.3% of the 3.905 m characteristic length.Position error decreases monotonically toward the base, consistent with uncertainty accumulation along the kinematic chain.Source: Results · p. 7
  • Encoder and rigid-link factors add no appreciable attitude uncertainty, so base attitude error propagates unchanged along the rigid links.Orientation error statistics are almost identical from B to F, as expected for a near-deterministic rigid chain.Source: Results · p. 7
  • Predicted position uncertainty is calibrated: the empirical-to-predicted standard-deviation ratio stays close to 0.577 at every frame, matching the applied variance inflation factor.The passage also reports the same consistency for orientation uncertainty of rigid-body frames.Source: Results · p. 7
  • Relative position is accurate to the millimeter, compared with centimeter-level accuracy in the inertial frame, because the base pose cancels in the relative composition.Stereo measurements constrain the relative base-to-tip pose independently of the base estimate; absolute-position accuracy is limited by relative navigation rather than the visual frontend.Source: Results · p. 7
  • Orientation estimates for local backbone frames M show a different behavior, with increasingly conservative uncertainty bands in the last column of Table 2.The reported orientation error statistics include 4.20°; the supplied passage does not preserve the full associated comparison.Source: Results · p. 7

3 reference

Simulation on a 60-second closed-loop maneuver yielded millimeter-level end-effector position accuracy relative to the base, calibrated predicted position uncertainty, and an average solution time of 1.23 ms. The conclusion identifies missing direct attitude information, sensor configuration, and hardware testing as areas for future work.

Conclusion

  • Across a 60-sec closed-loop simulation, mean end-effector position error was 2.3% of system characteristic length, reaching millimeter level relative to the base, with calibrated predicted position uncertainty.The optimization returned a marginal covariance for every variable, and predicted position uncertainty was accurately calibrated across the chain.Source: Conclusion · p. 8
  • Average solution time was 1.23 ms; Table 3 reports the computational performance data.Source: Conclusion · p. 8, Table 3 · p. 9
  • The method lacks direct attitude information for the continuum appendage, which the authors identify as a target for future work.Source: Conclusion · pp. 8–9
  • Future work should investigate optimal sensor configurations and test the formulation on representative hardware.Source: Conclusion · p. 9
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