Observability-Informed Optimal Sensor Placement for Soft Robots

I. INTRODUCTION

• Soft robots’ continuous deformability and many passive degrees of freedom complicate sensing and precise open-loop control, while exhaustive placement search grows computationally intractable with candidate locations.Sensor integration is also constrained by electronics, sensor size, limited space, and the risk that placements compromise softness or movement.Source: §I · p. 2
• Morphology-general placement methods may neglect dynamics, while a dynamics-aware neural-network approach is limited to strain sensors and entails material-point modeling and extensive data collection.Observability-based placement methods incorporating dynamics had not yet been evaluated on soft robots.Source: §I · p. 2
• The work implements and experimentally evaluates the Gramian-based optimization formulation in on a physical soft robot, using a continuum robot as a representative example.Placements yield lower or comparable shape-estimation RMSE to evenly spaced sensors.Source: §I · pp. 2–3

A. Observability Metrics

• The framework designs the measurement matrix to maximize a Gramian-based observability metric, which quantifies how observable system states are.Common metrics include λmin(Wo), tr(Wo), and log(det(Wo)).Source: §II · p. 3, A. Observability Metrics · p. 3

B. Optimization Problem

• The method constructs candidate measurement rows and selects p of them with a binary selector α so that C* = diag(α)C maximizes a chosen observability-Gramian metric.The measurement matrix is designed so the pair (A, C) yields a Gramian that maximizes observability.Source: B. Optimization Problem · p. 3
• The original selection problem is nonconvex, so the method rewrites its sensor contribution as a sum over candidate locations and relaxes the binary constraint to obtain a convex optimization.The rewrite is equivalent when selector entries are binary; a relaxed solution satisfying the original integer constraint is optimal for the integer problem, with tolerance 0.15 used to assess that condition.Source: B. Optimization Problem · p. 3
• The formulation accommodates concurrent sensing modalities by assigning each modality its own candidate matrix and selector vector.The optimization imposes the corresponding sensor-count constraints for the selectors.Source: B. Optimization Problem · pp. 3–4
• The workflow models robot dynamics, gathers candidate locations, solves the optimization, and forms the selected measurement matrix; nonlinear systems can first be linearized at one operating point or multiple trajectory points.With multiple linearization points, the optimization can maximize observability across them.Source: B. Optimization Problem · p. 4

III. APPLICATION TO A CONTINUUM ROBOT ARM

• The evaluation used a pneumatically actuated continuum arm with two segments, each containing three silicone-cast chambers, and did not use the replica’s proprioceptive backbone.Fibers around each chamber induce bending toward the opposing chambers when pressurized.Source: §III · p. 4

A. Modelling

• The segment is modelled as a cantilever Euler-Bernoulli beam, a linear approximation that can enter the optimization directly.The model is considered reasonable because the arm does not undergo torsion or shear; nonlinear alternatives such as the Cosserat rod model would require linearization.Source: A. Modelling · p. 4
• Rayleigh-Ritz approximates beam deflection with fixed-free mode shapes, and the Euler-Lagrange equation yields the unforced, undamped modal equation of motion.The modal coordinates express deflection through spatial basis functions and time-dependent coefficients.Source: A. Modelling · p. 4
• The modal model is converted to state-space form after mass normalization and addition of damping and generalized forces.The modal coordinates are transformed into the state vector used for the system dynamics.Source: A. Modelling · pp. 4–5
• Position sensors measure displacement from the neutral axis, while strain sensors measure strain at a beam location; their measurement rows form matrix C.Matrix C and the state-space dynamics together form the complete system model.Source: A. Modelling · p. 5

B. Experimental Setup

• The experiments used the robot’s 12 cm second segment, actuated through a FESTO valve manifold with pressure signals generated in Simulink and run on a Speedgoat real-time machine.The manifold measured valve-port pressure at 100 Hz.Source: B. Experimental Setup · p. 5
• A Vicon system tracked seven equally spaced markers along each chamber of the second segment at 100 Hz.Only one chamber was actuated at a time, producing planar motion.Source: B. Experimental Setup · p. 5
• Markers at corresponding heights were grouped into centroids, then projected from 3D onto a plane of best fit for the planar arm model.The projection is illustrated in Fig. 2.Source: B. Experimental Setup · p. 5

C. System Identification

• The first two natural frequencies were estimated by averaging frequency peaks from impulse responses across trials with chamber pressure set from 0 to 100 kPa.Pressure settings were tested in 10 kPa increments, and tip-centroid motion was analyzed with a Fourier transform.Source: C. System Identification · p. 5
• The identified beam parameters were ϱ = 2.001 kg/m and EI = 0.022 N m2, fitted to match measured natural frequencies with model eigenvalue roots.Only two natural frequencies and mode shapes were retained for the beam model.Source: C. System Identification · p. 5
• Stiffness-proportional damping was assumed, yielding identified damping ratios ζ1 = 0.181 and ζ2 = 1.385.The first-mode ratio was estimated from logarithmic decrement, then used with the natural frequencies to determine the second-mode ratio.Source: C. System Identification · p. 5
• ±97%, ±45%, and ±9% variation in ϱ, EI, and ζi, respectively, left the optimal placement and objective function value unchanged.These are the reported tolerances for the physical parameters and damping ratios in the sensitivity analysis.Source: C. System Identification · p. 5
• The pressure-to-force coefficient was fitted from static positions measured over 18 trials, giving average b = 0.231.The model treated pressure actuation as a uniformly distributed force.Source: C. System Identification · p. 5

D. Experimental Procedures

• The evaluation used 21 step-input trials and 30 sinusoidal-input trials across the stated pressure amplitudes and frequencies.Step pressures ranged from 0 to 100 kPa in 5 kPa increments; sinusoidal amplitudes ranged from 10 to 50 kPa and frequencies from 1 to 6 rad/s.Source: D. Experimental Procedures · p. 5
• Position-sensor candidates were seven arm centroids spaced 2 cm apart, while strain candidates were 13 locations at 1 cm intervals.Strain profiles were approximated by fitting beam deflection to mocap positions; placements were optimized for p = 1, 2, 3 sensors using all three objectives.Source: D. Experimental Procedures · p. 6
• A Kalman filter estimated shape using only measurements at the selected placement, with noise covariances set from pressure variance and inflated position- and strain-noise estimates.Position noise began with 0.5 mm mocap calibration error, inflated tenfold; strain noise was based on position noise and inflated by two orders of magnitude.Source: D. Experimental Procedures · p. 6
• Estimated deflections were evaluated at marker locations against a ground-truth Kalman-filter estimate using all seven mocap centroid positions, with RMSE.The ground truth avoids direct reliance on noisy mocap measurements affected by interpolation during occluded frames.Source: D. Experimental Procedures · p. 6
• The combined case used one position sensor and three strain sensors, excluding candidate locations near the arm end to emulate possible occlusion or end-effector space limits.The last two position candidates and last four strain candidates were excluded.Source: D. Experimental Procedures · p. 6

A. Position Sensor Placement

• Among position-sensor objectives, λmin(Wo) produces a nonbinary selector vector, so it does not identify an optimal sensor configuration.For a single position sensor, the baseline is deliberately placed at the middle location to compare against the intuitively optimal tip placement.Source: A. Position Sensor Placement · p. 6, Table I · p. 6
• For position sensors under step inputs, p = 1 tip placement has lower RMSE than the middle-position baseline, while p = 2 and 3 optimized placements generally have higher RMSE.The p = 2 log(det(Wo)) placement matches baseline performance.Source: A. Position Sensor Placement · p. 6, Table II · p. 7
• Under sinusoidal excitation, position placements outperform the baseline for p = 1 and p = 3, while the p = 2 log(det(Wo)) placement matches baseline performance.These trials provide broader frequency excitation than step inputs.Source: A. Position Sensor Placement · p. 6
• For position sensors, log(det(Wo)) with p = 3 has lower RMSE than the baseline at higher actuation amplitudes and frequencies.The analysis uses sinusoidal trials; the log(det(Wo)) configuration was the best-performing p = 3 setup under those inputs.Source: A. Position Sensor Placement · p. 7, Figure 4 · p. 7

B. Strain Sensor Placement

• For strain sensors, tr(Wo) placements with p = 2 or 3 outperform the evenly spaced baseline for both step and sinusoidal inputs.Across input types, the relative ranking of placements is consistent.Source: B. Strain Sensor Placement · p. 7, Table IV · p. 8
• For strain sensors, tr(Wo) with p = 3 maintains lower RMSE than the baseline across tested frequencies, with a larger advantage at higher amplitudes.This amplitude-frequency analysis uses sinusoidal inputs.Source: B. Strain Sensor Placement · p. 7, Table IV · p. 8

C. Combined Sensor Placement

• The combined test used one position and three strain sensors; no optimal position-sensor placement was found for λmin(Wo), while baseline sensors were evenly spaced.The baseline position sensor was at the arm’s middle, and the strain sensors were evenly distributed.Source: C. Combined Sensor Placement · p. 8, Table V · p. 8
• The tr(Wo) placement maintained lower RMSE than baseline at higher amplitudes and across tested frequencies.This is the combined-sensor amplitude and frequency analysis.Source: C. Combined Sensor Placement · p. 8
• Both tr(Wo) and log(det(Wo)) placements achieved lower RMSE than baseline across all trials, with tr(Wo) performing best.Table VI reports total trajectory RMSE grouped by actuation input type.Source: Table VI · p. 8, C. Combined Sensor Placement · p. 8

V. DISCUSSION

• Tip-position and base-strain placements beat middle baselines; trace maximization can leave weakly observable modes, while log(det(Wo)) mitigates model error at higher amplitudes and frequencies.For p = 2 and p = 3, trace placements showed higher RMSE where a weakly observable second mode remained; log(det(Wo)) performed slightly worse for p = 3 under step inputs but better under sinusoidal inputs.Source: §V · p. 8
• Optimization (5) accommodates excluded candidate locations, while automated placement is especially useful for multiple sensors or sensor types whose layouts are less intuitive.In the combined experiment, trace and log-determinant placements outperformed baseline, possibly due to position sensors’ stronger contribution to system observability.Source: §V · p. 8
• Linear-observability placements performed well under increased nonlinearity at higher sinusoidal amplitudes, suggesting broader applicability to periodic actuation.The paper notes that sinusoids form a basis for periodic signals via Fourier series.Source: §V · pp. 8–9
• No observability metric consistently wins: for p = 2 and p = 3, the best choice alternates between trace and log-determinant depending on sensor type.All metrics recover the optimal placement for p = 1 cases.Source: §V · p. 9
• Extension is framed for linear state-space models, but requires a suitable state/dynamics representation and may yield nonbinary selectors or no positive-definite-Gramian placement.The discussion says a simplified linear model may suffice for placement, even if it is not accurate enough for estimation or control.Source: §V · p. 9

• Convex optimization maximizes system observability using different Gramian-based metrics in a sensor-placement framework that accounts for dynamics.Source: §VI · p. 9
• In most cases, optimal placements matched or outperformed baseline RMSE for shape estimation, with strong performance under larger deflections.The conclusion identifies this as potential robustness to modelling errors.Source: §VI · p. 9
• Automated placement is useful where many candidate locations make intuitive choices unclear; the continuum-arm results suggest possible extension to other nonlinear soft robots.Source: §VI · p. 9