Physics-Informed Identification of Tensegrity Prestress across Taut-Slack Transitions

Sandesh Lamsal; Sumit Kumar; Rubi Bhandari; Subhamoy Sen; Andrzej Rutkiewicz; Leszek Małyszko; Landolf Rhode-Barbarigos

cs.LGNov 2, 2026

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 ·

Abstract

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 ( $45^{\circ}$ , gravity)	6	3	9	6	uniquely fixed by equilibrium
Prism series ( $30^{\circ}$ , 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 $\beta=800$	$<1$ %
Simulator against classical-limit truss frequencies	$3\times 10^{-9}$
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,

\pm 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 $\hat{\sigma}$ err (%)	$\hat{c}$ 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 $\hat{\sigma}$ err (%)	$\hat{c}$ 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 $\hat{\sigma}$ err (%)	$\hat{c}$ 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

\pm 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.

Method	$\max\hat{\sigma}$	$\hat{c}$	$\max\hat{\sigma}$	$\hat{c}$	wall	hyper-
	Taut record		Slack-transition record		Cost and tuning
	err. (%)	err. (%)	err. (%)	err. (%)	(s)	parameters
Physics-informed identification	$1.0$	$0.3$	$1.3$	$6.2$	$2819$ / $4643$	$15$
Levenberg–Marquardt shooting	$97.4$	$6504$	$\mathbf{0.07}$	$\mathbf{0.2}$	$10\,700$ / $\mathbf{83}$	$\mathbf{2}$
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

\pm 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 $240^{\circ}$ . (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 $\mathrm{d}\ln f/\mathrm{d}\ln F$ , 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 ( $\pm 10$ % 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)	$\pm$	predicted (N)	error (N)	error (%)
1	263	27	253	$\mathbf{-10}$	$-3.76$
2	551	37	568	$\mathbf{+17}$	$+3.02$
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	$\mathbf{-11}$	$-0.66$

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. $\sigma$ error	margin error	$U_{c}$ 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.

bar-constraint weight, matched generator
	force density (%)		damping (%)
$w_{b}$	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 $c_{k}$	$1.0$ 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 $\bm{A}$	$s$	$m$	Resolvable	Class	$\lambda_{\min}$
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
$T_{3}$ prism	9	12	8	4	1	6/9	super-stable	28.2
$T_{4}$ prism	12	16	11	5	1	8/12	super-stable	17.7
$T_{6}$ 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

\lambda_{\min}

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 $\hat{\sigma}$ err (%)	$\hat{c}$ err (%)
$2$ % noise, 2 sensors	9	$3.60\pm 0.28$	$1.58\pm 0.93$
$5$ % noise, 2 sensors	4	$4.81\pm 0.76$	$4.10\pm 2.10$
$10$ % noise, 2 sensors	4	$6.60\pm 0.79$	$4.28\pm 3.53$
$2$ % noise, 1 sensor	4	$4.26\pm 0.67$	$3.57\pm 2.95$

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	$\sigma$	$c$	$\sigma$	$c$
matched	$3.5\pm 0.2$	$1.1$	$19.7\pm 2.1$	$4.6$
cable mass	$7.0\pm 0.2$	$2.1$	$21.0\pm 2.8$	$5.7$
flexible bars	$5.3\pm 0.4$	$1.9$	$17.0\pm 1.9$	$6.0$
tolerance $10^{-13}$	$3.5\pm 0.2$	$1.1$	$19.3\pm 1.6$	$5.6$

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

\sigma

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	$\sigma$ 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	$24.9\pm 4.2$ %

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.

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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.

Abstract

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