Physics-Informed Identification of Tensegrity Prestress across Taut-Slack Transitions
Authors: Sandesh Lamsal, Sumit Kumar, Rubi Bhandari, Subhamoy Sen, Andrzej Rutkiewicz, Leszek Małyszko, Landolf Rhode-Barbarigos
Organizations: Department of Civil and Architectural Engineering, University of Miami, Coral Gables, FL 33146, USA · Department of Civil and Environmental Engineering, Florida International University, Miami, FL 33174, USA · i4S Laboratory, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh, India · Faculty of Geoengineering, University of Warmia and Mazury in Olsztyn, Prawoche´nskiego 15, 10-720 Olsztyn, Poland · Institute of Technology, University of Applied Sciences in Elbl ˛ag, Al. Grunwaldzka 137, 82-300 Elbl ˛ag, Poland
Prestress governs the stability, stiffness, and dynamic behavior of tensegrity structures, but it is not directly observable from sparse structural measurements, and it may decrease during service life. Under strong excitation, the cables may slacken and the dynamics become nonsmooth. Existing identification methods are limited to the taut regime, and physics-informed networks for tensegrities solve only the forward problem of predicting the response from known member parameters. This study formulates the inverse problem. Neural networks are trained on the equation-of-motion residual in constrained Cartesian coordinates, where the mass matrix is constant and the force densities enter linearly. The force densities, damping, and axial stiffness are trainable parameters, and the tension-only cable law is kept. A Fisher-information analysis shows, before inversion, which member parameters a record can determine. The method is tested on simulated planar tensegrity frames with up to five modules and a three-dimensional prism, and on measured records of a biomimetic leg and a full-scale tensegrity simplex. On the one-module frame, the identifiable force densities are recovered within 1.5% from noisy accelerations at half the free coordinates, on records with and without taut-slack transitions. Records with transitions determine members that taut records leave undetermined. On the five-module frame, a 16-cable two-span bridge, the median force-density error rises moderately, to 4.2%. On measured records, an identified spring stiffness is within 0.1% of its independently measured value on average, and the identified simplex prestress at 13 levels agrees with its cable force transducers to 2.9% root-mean-square.
Figures & tables
Figure 1: The benchmark structures, drawn from the simulator geometry: (a) the planar X n staircase frames, (b) the X2 instance, (c) the triplex prism. Blue arrows mark the loaded coordinates and green squares a representative accelerometer placement; properties are given in Table 1 .
Figure 2: Physics-informed identification framework: (a) pipeline from sparse, noisy accelerometer records to identified member parameters and the structural assessment; (b) a single record, with sparse sensor samples feeding the data loss and dense collocation points enforcing the physics residual through the taut–slack transitions; (c) the computational graph, with the member parameters entering the residual as trainable inputs.
Structure
Nodes
Bars
Cables
Mech. DOFs
Prestress source
X1 frame
4
2
4
2
LP, 25 N/m min. force density
X2 frame
6
4
7
4
LP, 25 N/m min. force density
X3 frame
8
6
10
6
LP, 25 N/m min. force density
X5 bridge (two-span)
12
10
16
8
LP, 25 N/m; intermediate support
Triplex prism ( 45∘ , gravity)
6
3
9
6
uniquely fixed by equilibrium
Prism series ( 30∘ , zero- g )
6
3
9
6
self-stress scale 1–15 N/m
Table 1: Benchmark structures used for identification. The classic tensegrity forms assessed but not identified are listed in Section S3 of the Supplementary Material.
Figure 3: Model-based identifiability analysis of the X1 frame. (a, b) Root-mean-square sensitivity of the sensor accelerations to each parameter on the taut and slack-transition records. (c) Cramér–Rao lower bounds at 2 % noise (floor lines) against the multi-seed scatter and the single-run achieved error on the taut record.
Check
Level
Residual against simulator state derivatives, slack included
10−15
Residual at exact parameters, exact cable law
10−15
Smoothed-cable-law peak force-density bias at β=800
<1 %
Simulator against classical-limit truss frequencies
3×10−9
Frame generator against earlier X2 frequencies
10−6
Flexible-bar residual against flexible-bar simulator
10−9
Table 2: Verification checks of the numerical implementation, as relative figures. The checks of Section 2.5 bound the error of the simulated records themselves.
Figure 4: The X1 inversions on (a) the taut record and (b) the slack-transition record with 20 taut–slack transitions: parameter convergence normalized by the exact values (top, ±5 % band shaded) and the training histories (bottom). The inert base cable does not converge to its value, as expected for a member that cannot be identified.
Figure 5: Posterior-predictive validation for (a) the taut record and (b) the slack-transition record: the structure re-simulated with the recovered parameters under the same forcing, compared with the original record at a coordinate that was never measured. Shading marks the intervals with a slack cable; the strip below each trace is the signed error.
Structure
Record
Forcing
Transitions
max σ^ err (%)
c^ err (%)
recon. (%)
held-out ratio
Outcome
X1 frame
taut
1-pt
0
3.5
1.3
4.1
2.8
identified
X1 frame
slack-transition
1-pt
20
18.3
3.2
11.5
9.5
partial
X2 frame
taut
1-pt
0
79.8
7.0
5.3
4.6
silent failure
X2 frame
taut
2-pt
0
2.1
0.1
1.3
6.3
identified
X2 frame
slack-transition
1-pt
20
20.8
5.5
1.7
6.0
partial
X2 frame
slack-transition
2-pt
61
1.8
2.7
0.7
6.9
identified
Table 3: Identification across the benchmark structures at 2 % noise. Errors are over the identifiable members; reconstruction is the root-mean-square error at unmeasured coordinates; the held-out ratio and the Outcome verdicts are defined in Section 4.3 . 1-pt rows use single-point forcing and 2-pt rows are repeated runs with two-point forcing; step counts are given in the text.
Intervention
max σ^ err (%)
c^ err (%)
Baseline (one-point taut, four sensors)
79.8
7.0
+ sensors (all eight coordinates)
76.7
1.9
+ two-point off-resonance forcing
12.4
0.4
+ three pooled taut records
90.4
1.8
One-point slack-transition record
20.8
5.5
Combined (transitions, two-point, eight sensors)
9.2
3.3
Table 4: Removal of the silent failure on the X2 taut record. One factor is changed at a time from the baseline record, at a fixed 12,000 -step run; the first error column is the maximum over the identifiable force densities and the second is the damping-coefficient error.
Structure
Treatment
Steps
max σ^ err (%)
c^ err (%)
recon. (%)
transitions
X1
smoothed cable law
12,000
18.82
3.8
11.6
20
X1
Fischer–Burmeister NCP
12,000
26.97
42.6
15.5
20
X1
event-aware collocation
12,000
17.90
2.4
10.9
20
X1
smoothed cable law
25,000
0.77
6.5
0.5
20
X1
Fischer–Burmeister NCP
25,000
2.55
24.4
1.7
20
X1
event-aware collocation
25,000
0.82
6.3
0.6
20
Table 5: Comparison of the transition treatments on identical slack-transition records, at both training lengths. Every entry is the median over four seeds of the same protocol.
Figure 6: Recovered against exact force densities for the zero-gravity prism prestress series, structurally inert members excluded, with the ±10 % band shaded. Levels 5 and 15 N/m lie on the 1:1 line; the two misses at level 1 N/m are an optimization limit (Section 4.6 ).
Figure 7: The X5 two-span tensegrity bridge, drawn from the simulator geometry, with the two end base nodes and one intermediate base node pinned. Blue arrows mark the four driven coordinates; the green squares sketch the accelerometer layout.
Taut record
Slack-transition record
Cost and tuning
Method
maxσ^
c^
maxσ^
c^
wall
hyper-
err. (%)
err. (%)
err. (%)
err. (%)
(s)
parameters
Physics-informed identification
1.0
0.3
1.3
6.2
2819 / 4643
15
Levenberg–Marquardt shooting
97.4
6504
0.07
0.2
10700 / 83
2
Unscented Kalman filter
3.1
1.5
182.7
133.5
63 / 47
11
Particle filter with EnKF
61.7
67.1
488.1
80.8
762 / 751
6
Table 6: The four identification methods on the X1 frame, from the same records, the same 2 % noise and the same distant initialization. Accuracy is the largest relative error over the three identifiable force densities, with the damping error beside it. Cost is the wall time of the run that produced the quoted accuracy, on the same hardware, taut value first; hyperparameter counting is defined in the text.
Figure 8: Validation on the hexagonal-prism test of Huang et al. [20] . (a) The rebuilt model of the structure, from the published parameters. (b) Elastic frequencies: the published theoretical values and those of the rebuilt model agree to about 2% across all eight modes, and the four measured peaks are best explained by a ring tension of 10.6 N (Section 4.10 ).
Figure 9: Identification on the biomimetic leg of Wang et al. [61] , from their published tables and records: (a) the leg drawn to scale from their published dimensions, with the antiparallelogram knee enlarged; (b) the knee spring identified from each case’s measured frequency pair against the rig-measured rate, with the ±10 % band shaded.
case
rig rate (N/m)
identified scale
error (%)
A
188.0
0.957
−4.3
B
188.0
1.030
+3.0
C
60.6
0.874
−12.6
D
60.6
0.934
−6.6
root-mean-square error
7.6
Table 7: The knee spring identified from the published measured frequencies of Wang et al. [61] , against the rate measured independently on their spring rig; a scale of 1.000 reproduces the rig value exactly. The identification target is selected in advance by the identifiability screening described in the text; Case E carries no springs and supplies that screening instead.
Figure 10: The tensegrity simplex of Małyszko and Rutkiewicz [15] . (a) The instrumented module, photographed by L. Małyszko and A. Rutkiewicz, with the nodes marked: the struts P1–P4, P2–P5 and P3–P6 on their threaded-rod ends, the horizontal cables of the two triangles, the cross cables with their turnbuckles, a force transducer in each of the nine cables, triaxial accelerometers at the three free nodes, and the hammer strike at node 4, azimuth 240∘ . (b) A single impact record, at level 4: the rotational projection, the envelope of the mechanism mode, and the time window used to measure the frequency.
Figure 11: Identification on the simplex of Figure 10 , from 13 impact records at 13 prestress levels: (a) the mechanism law: the measurement, the rebuilt model at the transducer reading, and the same model at the prestress obtained from static equilibrium in the 2020 publication, the shaded band between the two being that factor and the open marker the joint-supplied intercept; (b) frequency against force by mode family, log–log, the mechanism family in color with its 95 % confidence band and the dotted ideal slope of 1/2 anchored at the first point; the side strip shows each family’s exponent dlnf/dlnF , with its 95 % error bar, against the zero and one-half references; (c) prestress predicted against the transducer reading, the joint stiffness alone fitted leave-one-out ( ±10 % band); (d) the rotational spectrum at every level, each record normalized to its own peak, with the per-level prediction error as the color-graded column on the right, framed in black where it exceeds the transducer scatter.
level
measured (N)
±
predicted (N)
error (N)
error (%)
1
263
27
253
−10
−3.76
2
551
37
568
+17
+3.02
3
896
40
958
+62
+6.89
4
1167
38
1212
+45
+3.86
5
1434
35
1487
+53
+3.72
6
1695
31
1684
−11
−0.66
Table 8: Prestress identified from the vibration record at each of the 13 levels, against the force the cable transducers measured there. Bold marks the levels predicted inside their own transducer uncertainty.
Recovered model
max. σ error
margin error
Uc error
X1, taut
14 %
0.12 %
0.3 %
X1, slack-transition
87 %
0.04 %
0.2 %
X2, multi-record
90 %
6.0 %
22 %
Table 9: Propagation of identification error to the structural assessment.
force density (%)
damping (%)
bar-constraint weight, matched generator
wb
max
mean
1
29.50
23.11
16.59
10
7.42
5.99
0.62
100 †
3.25
2.69
1.45
1000
3.86
3.18
2.57
Table S1: Sensitivity of the identification to the loss weights, on the X1 taut record. Errors are over the three identifiable force densities; the inert base cable is excluded, being unidentifiable by construction. A dagger marks the value used throughout.
Parameter
Value
Source
Members, synthetic benchmarks
Cable radius, modulus
1 mm, 70 MPa ( EA=220 N)
benchmark definition
Cable damping ck
1.0 N s/m, taut cables only
benchmark definition
Bar radius, density
5 cm, 500 kg/m 3
benchmark definition
Bar treatment
rigid (length constraint)
§2.1
Hexagonal prism
EA=379 N, bar mass 42.3 g
Huang et al. [20]
Table S2: Parameters used throughout the main text, with the origin of each value. Section numbers refer to the main text; material values carry their published source. Per-run amplitudes and step counts are given in the main-text sections that use them.
Structure
Free
Mem.
Rank A
s
m
Resolvable
Class
λmin
X1 frame
4
6
4
2
0
3/4
stable
148.6
X2 frame
8
11
8
3
0
6/7
stable
211.6
X3 frame
12
16
12
4
0
9/10
stable
45.4
T3 prism
9
12
8
4
1
6/9
super-stable
28.2
T4 prism
12
16
11
5
1
8/12
super-stable
17.7
T6 prism
18
24
17
7
1
12/18
super-stable
8.4
Table S3: Free coordinates, members, rank of the equilibrium matrix, self-stress states s , mechanisms m , cables determined at first order, stability class, and margin λmin of the reduced tangent stiffness; rank tolerances are given in the text.
Figure S1: The classic tensegrity forms assessed alongside the identification benchmarks, struts heavy and cables light. None of these is identified; they extend the structural analysis only.
Figure S2: Normalized geometric-stiffness spectrum of each structure, green for super-stable spectra (all eigenvalues non-negative) and orange for merely stable ones.
Figure S3: Stability from the identified parameters: (a) the prism over a range of prestress; (b) the X2 frame over a range of the self-stress combination; (c) member utilizations of the steel X2 frame against prestress; (d) instantaneous stability class of the prism through a forced, damped motion (green super-stable, orange merely stable, red momentarily unstable), with (e) the cable-slack intervals.
Figure S4: Energy components of the X2 frame under resonant excitation (top) and the energy-balance residual (bottom); shaded bands mark slack intervals.
Condition
Seeds
max σ^ err (%)
c^ err (%)
2 % noise, 2 sensors
9
3.60±0.28
1.58±0.93
5 % noise, 2 sensors
4
4.81±0.76
4.10±2.10
10 % noise, 2 sensors
4
6.60±0.79
4.28±3.53
2 % noise, 1 sensor
4
4.26±0.67
3.57±2.95
Table S4: Robustness of the X1 inversion on the taut record. Each entry is the mean over seeds of the matched 12,000 -step protocol, with one standard deviation.
Taut record
Slack-transition record
Generator
σ
c
σ
c
matched
3.5±0.2
1.1
19.7±2.1
4.6
cable mass
7.0±0.2
2.1
21.0±2.8
5.7
flexible bars
5.3±0.4
1.9
17.0±1.9
6.0
tolerance 10−13
3.5±0.2
1.1
19.3±1.6
5.6
Table S5: Model-form mismatch on the X1 frame, as percentage errors. Each record is produced by a generator the residual cannot represent at any parameter value, and the inversion is left unchanged. The σ column is the maximum over the identifiable force densities, mean and standard deviation over three seeds; c is the damping coefficient.
Condition
Sensors
Noise
Samples
Forcing
σ error
baseline
2
2 %
500
harmonic
2.8 %
high noise
2
8 %
500
harmonic
4.7 %
single sensor
1
2 %
500
harmonic
4.9 %
sparse, dropout
2
2 %
106
harmonic
12.2 %
impulse
2
2 %
500
impulse
25.0 %
all four
1
8 %
120
impulse
24.9±4.2 %
Table S6: Adverse-sensing ablation on the X1 frame: each condition turns on one adverse factor against the baseline case, and the last turns on all four. Errors are the maximum over the identifiable force densities.
Tensegrity form-finding and physical property prediction are fundamental inverse problems in structural mechanics, which aim to determine equilibrium configurations and internal force distributions. These problems are challenging due to strong nonlinearity arising from the coupling between geometry and forces, the need to ensure structural stability, and the enforcement of constraints such as boundary conditions and symmetry. Moreover, traditional methods often lack robustness to noise and outliers. This paper proposes an energy-based learning framework for clustered tensegrity form finding and physical property prediction. The proposed approach incorporates total potential energy minimization and constitutive relations into the training objective, enabling the simultaneous prediction of equilibrium nodal configurations and associated physical quantities, including member forces and force densities. By incorporating energy-based physical losses directly into the learning process, the framework improves physical consistency, robustness, and data efficiency. Numerical experiments on tensegrity structures, including prism and lander systems, show the great potential of the proposed approach and demonstrate its capability for scalable form finding and accurate prediction of structural properties.
Jing Qin, Muhao Chen
Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA · Department of Mechanical and Aerospace Engineering, University of Houston, Houston, TX 77204, USA
Tensegrity structures possess intrinsic geometric symmetries that govern their dynamic behavior. However, most existing physics-informed neural network (PINN) approaches for tensegrity dynamics do not explicitly exploit these symmetries, leading to high computational complexity and unstable optimization. In this work, we propose a symmetry-reduced physics-informed neural network (SymPINN) framework that embeds group-theory-based symmetry directly into both the solution expression and the neural network architecture to predict tensegrity dynamics. By decomposing nodes into symmetry orbits and representing free nodal coordinates using a symmetry basis, the proposed method constructs a reduced coordinate representation that preserves geometric symmetry of the structure. The full coordinates are then recovered via symmetry transformations of the reduced solution learned by the network, ensuring that the predicted configurations automatically satisfy the symmetry constraints. In this framework, equivariance is enforced through orbit-based coordinate generation, symmetry-consistent message passing, and physics residual constraints. In addition, SymPINN improves training effectiveness by encoding initial conditions as hard constraints, incorporating Fourier feature encoding to enhance the representation of dynamic motions, and employing a two-stage optimization strategy. Extensive numerical experiments on symmetric T-bars and lander structures demonstrate significantly improved prediction accuracy and computational efficiency compared to standard physics-informed models, indicating the great potential of symmetry-aware learning for structure-preserving modeling of tensegrity dynamics.
Jing Qin, Muhao Chen
Department of Mathematics, University of Kentucky, Lexington, KY · Department of Mechanical and Aerospace Engineering, University of Houston, TX
Tensegrity robots offer compliance and adaptability, but their nonlinear, and underconstrained dynamics make state estimation challenging. Reliable continuous-time estimation of all rigid links is crucial for closed-loop control, system identification, and machine learning; however, conventional methods often fall short. This paper proposes a two-stage approach for robust state or trajectory estimation (i.e., filtering or smoothing) of a cable-driven tensegrity robot. For online state estimation, this work introduces a factor-graph-based method, which fuses measurements from an RGB-D camera with on-board cable length sensors. To the best of the authors' knowledge, this is the first application of factor graphs in this domain. Factor graphs are a natural choice, as they exploit the robot's structural properties and provide effective sensor fusion solutions capable of handling nonlinearities in practice. Both the Mahalanobis distance-based clustering algorithm, used to handle noise, and the Chebyshev polynomial method, used to estimate the most probable velocities and intermediate states, are shown to perform well on simulated and real-world data, compared to an ICP-based algorithm. Results show that the approach provides high fidelity, continuous-time state and trajectory estimates for complex tensegrity robot motions.
Edgar Granados, Patrick Meng, Charles Tang +4
Dept. of Computer Science, Rutgers University, NJ, USA. · Mech. Eng. & Material Sc. Yale University, New Haven, CT, USA.