Adaptive piecewise Euler-Bernoulli beam model for tendon-actuated soft slender manipulator

doi:10.1007/s10483-026-3411-9

Abstract

Abstract:

Soft manipulators exhibit coupled geometric-material nonlinearities under large deflections, which complicate modeling, shape estimation, and closed-loop force control while incurring high computational costs. In this paper, a control-oriented reduced-order adaptive piecewise Euler-Bernoulli (APEB) beam model for tendon-driven slender soft arms is presented. A nonlocal end-force correction is explicitly embedded in the bending equilibrium to capture the projection and attenuation of distal tendon tension and external loads toward the proximal end. To curb the cost of high-dimensional discretization, we introduce curvature/strain-gradient-guided adaptive segmentation with local cubic-spline interpolation: nodes are refined only in regions of severe deformation, whereas fewer degrees of freedom (DoFs) are maintained elsewhere. Based on this discretization, we derive the complete weak-form residual, assemble the tangent stiffness (Jacobian) matrix, and formulate an iterative Newton update, which yields a solver that efficiently computes static equilibria with numerically stable convergence. The model enables fast skeleton reconstruction (shape estimation without external optical tracking) and direct end-effector Jacobian generation through local-to-global coordinate mapping, thereby mapping tendon inputs to the end-effector pose for closed-loop and force control. Simulations and benchtop experiments on a four-tendon continuum actuator show millimeter-level accuracy in the trunk shape and end-effector position across varying cable tensions, and the proposed approach outperforms conventional kinematic and classical beam-theory models in terms of both accuracy and computational efficiency. The model is intended for quasistatic, bending-dominated configurations in which torsional deformation and out-of-plane loading are negligible.

Key words: soft robot, kinetostatic modeling, three-dimensional (3D) shape reconstruction, statics, adaptive Euler-Bernoulli (EB) beam model

2010 MSC Number:

O342

Jin WANG, Yisong ZHAO, Fei GAO, Yongjian ZHAO. Adaptive piecewise Euler-Bernoulli beam model for tendon-actuated soft slender manipulator. Applied Mathematics and Mechanics (English Edition), 2026, 47(7): 1469-1486.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3411-9

https://www.amm.shu.edu.cn/EN/Y2026/V47/I7/1469

Figures/Tables 8

ξ,	curvature sampling value or strain sampling value of each segment, m^-1;
s,	arc-length coordinate along the beam ( $s ∈ [0, L]$ ), m;
L,	beam length, m;
x(s), y(s),	leftline coordinates in the global frame, x₀ and y₀ represent the base coordinates at s=0, m;
θ(s),	tangent angle of the leftline, rad;
$κ (s), ϕ (s),$	curvature, $κ (s) = ϕ (s) = θ ′ (s),$ m^-1;
EI or EI(s),	bending stiffness (constant or known function), N·m²;
$E I eff (s),$	effective stiffness: $E I eff = E I + α T + μ Q (s),$ N·m²;
α,	tension-stiffness coupling parameter ( $α T$ has units of N·m²), m²;
T,	axial tension (e.g., projection of the end force along the tangent), N;
μ,	curvature-energy coupling parameter (because Q has unit m^-1), N·m³;
$Q (s),$	curvature-energy integral: $Q (s) = ∫ s L κ (ξ) 2 d ξ,$ m^-1;
$F 0, P 0, θ 0,$	resultant end force, axial component, and global direction angle, respectively, N, N, rad;
A, D,	lumped constants: $A = F 0 cos θ 0 − P 0,$ $D = F 0 sin θ 0,$ N;
$w (s, ξ),$	nonnegative kernel (e.g., $w = e − η (ξ − s)$ ) that shapes nonlocal influence;
η,	decay rate used in $w = e − η (ξ − s),$ m^-1;
H(s),	nonlocal positional function $H (s) = ∫ s L w (s, ξ) cos (θ (ξ) − θ 0) d ξ,$ m;
γ,	weight for the nonlocal tip-load correction term $− γ D H (s);$
X_i,	node positions that split $[0, L],$ m;
$N j (s),$	shape function at node j;
$N (s),$	row of shape functions, $θ (s) = N (s) Θ;$
$B j (s),$	$B j (s) = d N j d s,$ m^-1;
B(s),	row of derivatives, $ϕ (s) = B (s) Θ,$ m^-1;
$Θ,$	vector of nodal angles $[Θ 0, Θ 1, Θ 2, ⋯, Θ M] T,$ rad;
$R (Θ),$	global residual vector from the weak form, m²;
$K (Θ),$	tangent stiffness: $∂ R ∂ Θ,$ m²;
$K (i), K (BC),$	i=1, 2, 3,4, decomposition terms of K (principal, coupling, geometric, load, and boundary, respectively);
$δ j M,$	Kronecker delta for selecting the tip node ( $δ j M = 1$ if $j = M$ and $δ j M = 0$ otherwise);
$R BC,$	boundary residual contribution $[N i E I eff θ ′] 0 L,$ N·m;
$s ext (s; Θ),$	external/nonlocal geometric load $(A sin θ − γ D cos (θ − θ 0)) N (s) T;$
$Δ Θ, ε,$	Newton update and convergence tolerance, respectively, rad, -;
T₁–T₄,	tendon tensions applied to the four actuation cables, N;
P,	endpoint position of the deformed backbone in Fig. 1, m;
$F x 0, F y 0, F z 0,$	components of the endpoint force in the global coordinate system, N;
M₀,	equivalent in-plane bending moment used in the APEB beam model, N·m;
$M x 0, M y 0, M z 0,$	components of the endpoint moment in the global coordinate system, N·m;
$φ,$	azimuth angle of the local bending plane in the global coordinate system, rad;
$x tip, y tip,$	tip position: $x tip = x (0) + ∫ 0 L cos θ d s,$ $y tip = y (0) + ∫ 0 L sin θ d s,$ m;
$J tip (Θ),$	tip-position Jacobian: $∫ 0 L [− sin θ cos θ] N (s) d s,$ m/rad;
$F tip, Q,$	tip force and generalized load, $Q = J tip T F tip,$ N, N·m;
$B, τ,$	actuation map and tendon tension with $Q = B τ,$ m, N;
$K tip,$	linearized Cartesian tip stiffness: $(J tip K − 1 J tip T) − 1,$ N/m.

Fig. 1

Fig. 2

Fig. 3

Table 1

Parameter estimation results with uncertainty quantification"

Parameter	Estimate	Unit	Standard error	95% CI (lower)	95% CI (upper)
$E I$	$6.399 993 × 10 − 3$	$N ⋅ m 2$	$8.252 × 10 − 9$	$6.399 977 × 10 − 3$	$6.400 010 × 10 − 3$
α	$7.126 164 × 10 − 9$	$m 2$	$2.906 × 10 − 9$	$1.430 909 × 10 − 9$	$1.282 142 × 10 − 8$
μ	$− 1.967 372 × 10 − 9$	$N ⋅ m 3$	$9.837 × 10 − 10$	$− 3.895 410 × 10 − 9$	$− 3.933 521 × 10 − 11$
Note: CI denotes the confidence interval

Table 1

Fig. 4

Table 2

Comparison of simulation accuracy and element efficiency"

Group	Tension inthe cables (unit: N)	Method	No. of Segments	End-effector position measured by the sensor (unit: mm)			End-effector position predicted by simulation (unit: mm)			Error component (unit: mm)			Absolute error (unit: mm) $∥ e ∥$
Group	Tension inthe cables (unit: N)	Method	No. of Segments	x	y	z	x	y	z	$Δ x$	$Δ y$	$Δ z$	Absolute error (unit: mm) $∥ e ∥$
1	$(0, 0, 0, 0) T$	APEB	1	0.00	0.00	$− 21.8$	0.00	0.00	$− 21.8$	0.00	0.00	0.00	0.00
		Traditional EB	–				0.00	0.00	$− 21.8$	0.00	0.00	0.00	0.00
		CC	–				0.00	0.00	$− 21.8$	0.00	0.00	0.00	0.00
2	$(0.98, 0, 0, 0) T$	APEB	5	$− 2.94$	$− 0.06$	$− 19.12$	$− 3.03$	0.04	$− 19.23$	$− 0.08$	0.10	$− 0.11$	0.17
		Traditional EB	–				$− 3.74$	0.06	$− 20.02$	0.80	$− 0.12$	$− 0.90$	1.21
		CC	–				$− 0.00$	$− 3.58$	$− 19.88$	2.94	$− 3.52$	$− 0.76$	4.65
3	$(1.96, 1.96, 0, 0) T$	APEB	6	$− 5.32$	$− 5.63$	$− 17.69$	$− 5.42$	$− 5.73$	$− 17.60$	$− 0.10$	$− 0.10$	0.09	0.17
		Traditional EB	–				$− 6.32$	$− 7.13$	$− 17.19$	1.00	1.50	0.50	1.87
		CC	–				6.16	$− 6.16$	$− 17.55$	$− 11.48$	0.53	$− 0.14$	11.49
4	$(2.94, 0, 0, 0) T$	APEB	6	$− 9.31$	$− 0.21$	$− 15.80$	$− 9.05$	0.05	$− 16.05$	0.25	0.26	$− 0.25$	0.44
		Traditional EB	–				$− 8.71$	0.14	$− 17.40$	$− 0.60$	$− 0.35$	$− 1.60$	1.74
		CC	–				$− 0.00$	$− 9.07$	$− 17.30$	9.31	$− 8.86$	$− 1.50$	12.93
5	$(3.92, 0, 0, 0) T$	APEB	8	$− 8.96$	$− 0.35$	$− 14.32$	$− 9.42$	0.11	$− 14.77$	$− 0.46$	0.45	$− 0.45$	0.79
		Traditional EB	–				$− 10.86$	0.25	$− 15.82$	1.90	$− 0.60$	$− 1.50$	2.49
		CC	–				$− 0.00$	$− 10.75$	$− 15.82$	8.96	$− 10.40$	$− 1.50$	13.81
6	$(2.94, 2.94, 0, 0) T$	APEB	11	8.12	$− 8.12$	$− 13.04$	7.63	$− 7.67$	$− 13.49$	$− 0.49$	0.45	$− 0.45$	0.80
		Traditional EB	–				5.82	$− 7.37$	$− 15.14$	2.30	$− 0.75$	$− 2.10$	3.20
		CC	–				8.73	8.73	$− 13.89$	$− 0.61$	$− 16.85$	0.85	16.89
7	$(2.45, 1.47, 0, 0) T$	APEB	14	8.72	$− 6.23$	$− 14.96$	8.45	$− 5.96$	$− 15.23$	$− 0.27$	0.27	$− 0.26$	0.46
		Traditional EB	–				6.02	$− 5.38$	$− 17.46$	2.70	$− 0.85$	$− 2.50$	3.78
		CC	–				6.47	9.06	$− 15.42$	2.25	$− 15.28$	0.46	15.46

Table 2

Fig. 5

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