Optimization of sensor placement and actuation strategy for a deployable origami structure through computational analysis and experimental investigation

Abstract

Origami-inspired deployable structures represent a promising field of research in structural engineering, offering innovative solutions for the design and development of versatile next-generation structural systems. Incorporating bio-inspired design principles from nature provides a unique pathway toward realizing such deployable systems. This paper presents an experimental approach informed by a computational study to investigate the actuation of the Origami Pill Bug structure, a plate-based modular deployable structure that leverages origami mechanics and is morphologically inspired by pill bugs. The research addresses three objectives: formulating an equivalent bar model for thick plate elements to enable computational simulations for form-finding using dynamic relaxation method; developing a multi-objective optimization algorithm to guide optimal sensor placement; and conducting experimental investigations to correlate actuation strategy with resulting strain development. The integrated framework enabled the investigation of multiple actuation rates, successfully identifying the rate of 2 cm/s as optimal for the Origami Pill Bug structure. By establishing this computational-experimental framework, this research provides an effective methodology for selecting actuation strategies in origami-enabled deployable structures.

Keywords

Deployable structures origami dynamic relaxation multi-objective optimization optimal sensor placement

Introduction

Deployable structures offer significant potential for sustainable designs by enabling efficient material usage while offering adaptability.^1
–3 These structures have garnered significant interest in various fields, including aerospace,⁴ architecture,⁵ and robotics.⁶ The deployability of these structures has been utilized in bridges,^7,8 roof systems,⁹ dynamic façades,¹⁰ and emergency shelters.¹¹ Recent advancements in materials science^12,13 and computational design¹⁴ have further extended their application. Despite these progresses, the actuation dynamics of deployable structures remain largely unexplored, presenting fertile ground for further investigation.

The principles of origami have been adapted to develop deployable solar panels,^15
–17 self-folding structures,¹⁸ and configurable metamaterials.¹⁹ Bar-and-hinge models have been developed to analyze the kinematics of these structures,^20
–22 while avoiding the computational complexity of full shell analysis. Kinematic modeling of origami often assumes zero-thickness panels; however, applications for load-bearing systems require finite thickness for structural stiffness.²³ Various methods have been investigated to incorporate thickness into origami design and analysis, such as synthesizing thick rigid panels as over-constrained spatial linkages²⁴ and integrating multiscale strategies by expanding the design space for thick origami.²⁵ Despite these advancements, accommodating thickness in computational origami models remains an area of active research, crucial for adapting origami designs for engineering applications.

Expanding upon the versatility of origami-inspired deployable systems, form-finding methods such as dynamic relaxation emerges as a powerful computational tool for analyzing structures with geometrical nonlinearity.^26,27 Originating from the seminal works of Otter²⁸ and Day,²⁹ it is a vector-based static analysis technique that circumvents the need for matrix inversion, thereby reducing the computational cost. Enhancements by Barnes³⁰ and Barnes et al.³¹ lead to improved convergence and the introduction of kinetic damping. The dynamic relaxation method has found utility in modeling the behavior of deployable structures.^32,33 While improvements have been made to the method for the analysis of deployable tensegrity structures,^34,35 potential of the dynamic relaxation method for analyzing the force distribution in origami structures during deployment has not yet been fully explored.

Actuation strategy and optimal actuation parameters are crucial for the effective deployment of these structures capable of controlled shape morphing.³⁶ Previous research has explored various approaches, including combining optimal control techniques with structural mechanics,^37,38 leveraging stochastic optimization, path-planning algorithms, and machine learning for shape adaptation.^39,40 Recent studies have formulated methods for designing minimum energy structures by optimizing actuation strategy to minimize both embodied energy and operational energy.^41,42 Accounting for large deformations, multi-objective optimization frameworks^43,44 and shape optimization techniques^45,46 have facilitated controlled morphing. However, there has not yet been a comprehensive experimental study correlating actuation rates and the resulting strain distributions for deployable origami structures.

Current research in structural health monitoring focuses on strategic sensor placement, emphasizing critical locations to maximize information gain while minimizing cost.⁴⁷ Various methodologies have been developed to identify these critical locations; such as strain concentration factor as a metric to identify critical locations in truss structures,⁴⁸ generalized frameworks to qualify alternate load path redundancy and identify critical members in steel truss bridges,⁴⁹ and strain energy distribution for identifying critical members in truss subjected to sudden member loss scenarios.⁵⁰ Numerical optimization techniques have been employed to determine optimal sensor placement for examining structural static responses.⁵¹ The primary focus of existing methods on static scenarios highlights a research gap that necessitates the development of novel methodologies to optimize sensor placement in dynamic deployable structures.

Effective utilization of deployable origami structures, such as the Origami Pill Bug,⁵² requires the development of tailored actuation strategies that ensure accurate shape control. The integration of active components^53,54 and sensitivity analysis⁵⁵ is crucial for advancing the design and development of such deployable systems. Developing a comprehensive framework to analyze these structures is essential for precise control over their deployment behavior. This research presents a combined approach of computational simulation and experimental investigation to achieve a deeper understanding of actuation dynamics. By examining the relationship between strain development and actuation rates, the study aims to determine optimal actuation strategies tailored to the unique characteristics of the structure. Deployable structures, like the Origami Pill Bug, offer significant potential for applications in adaptive architecture, including dynamic building façades¹⁰ and deployable shell structures¹¹ that can transition between compact and expanded forms.

The goal of this research is to develop a framework to optimize the actuation strategy of an origami-inspired deployable structures. This work focuses on the Origami Pill Bug (OPB), a novel deployable structure that integrates biomimetic and origami principles, providing a testbed for investigating the behavior of origami-inspired deployable systems. Three objectives have been identified: (1) formulate an equivalent bar model for the thick plate elements of the Origami Pill Bug structure, facilitating computational simulation using the dynamic relaxation method; (2) employ a multi-objective optimization approach based on computational simulations of both healthy and damaged structures to identify critical elements for optimal sensor placement; and (3) conduct experimental investigation to determine the relationship between the actuation strategy and the resulting strain distributions within the origami structure. The research aims to establish a computational-experimental framework, capable of guiding the development of tailored actuation strategies, illustrated through its effective application to the Origami Pill Bug structure. .

Structural description: Origami Pill Bug

The design of a novel origami-enabled structure is inspired by the conglobation behavior exhibited by pill bugs from the Armadillidiidae family as a defensive response to external stimuli. The functional morphology of pill bugs facilitates conglobation. Their sub-cylindrical body that restricts lateral bending and a segmented hard exoskeleton, enables this rolling⁵⁶ by contraction of the internal musculature, which pulls the body segments together to form a compact sphere (Figure 1(a)).

Figure 1.

(a) Pill bugs exhibiting conglobation as a result of muscle actuation⁵⁷ and (b) rolling of the OPB structure as a result of cable actuation.

The morphological characteristics of a pill bug are adopted by the Origami Pill Bug (OPB) structure through the utilization of origami principles. As depicted in Figure 1(b), the OPB structure is composed of segmented modules that can transform from an unrolled to a rolled configuration when actuated. The actuation of the OPB structure is carried out by increasing tension in the actuation cable connected at the base of the structure. This causes the segmented modules to fold and roll up into a compact shape, thereby mimicking the conglobation behavior of a pill bug. Investigation of the OPB structure and its actuation method holds promise for the advancement of origami-inspired structures and mechanisms that emulate the functional morphology observed in various organisms.

Meter-scale prototype

The research focuses on a meter-scale prototype of the OPB structure, as shown in Figure 2(a). The prototype is constructed from hardwood panels with a thickness of 0.635 cm. The material properties of the OPB prototype are tabulated in Table 1. The flat-folded prototype measures 100 cm in length and 40 cm in width, with a net mass of 6.4 kg.

Figure 2.

OPB prototype: (a) unrolled configuration showing the actuation system, data acquisition system (DAQ), and cables utilized for actuation, (b) flat-folded showing lockable brackets, hinges, actuation locations, actuation cables, and strain gauges. Typical panel specifications showing dimensions for: (c) panel type-A, (d) panel type-B, and (e) panel type-C.⁵⁷

Table 1.

Material properties for the OPB prototype.

Property	Values
Young’s modulus, $E$	16,428 MPa
Yield strength, $σ_{y}$	80.65 MPa
Density, $γ$	306 kg/m³
Poisson’s ratio, $ν$	0.3
Plate thickness, $T_{p}$	0.635 cm

The manufacturing process involves precise panel cutting and screw hole placement facilitated by a 60-W laser cutter (Universal Laser VLS4.60), ensuring dimensional accuracy, and minimizing fabrication errors.

For deployment, the design incorporates lockable brackets with 90° locking mechanisms at the side folds and hinges with angular stiffness of 0.13 Nm/rad along the fold lines. The hinges exhibit a linear response to the amount of rotation experienced by the structure. Steel fasteners secure both brackets and hinges, contributing to the high assembly precision.

Drawing inspiration from origami, the OPB design enables the structure to be flat folded (Figure 2(b)) from the operational state in the unrolled configuration (Figure 2(a)). The unrolled configuration of the OPB structure intentionally maintains partial rolling at its extremities to trigger rolling and facilitate efficient deployment. While a completely flat configuration would be theoretically possible, it would require either significantly higher actuation forces or multiple actuation points,^58
–60 compromising the simplicity and efficiency of the current design. Typical panel sections are detailed with their respective dimensions, including Panel Type-A (Figure 2(c)), Panel Type-B (Figure 2(d)), and Panel Type-C (Figure 2(e)). The panel specifications shown in Figure 2(c)–(e)) maintain dimensional consistency with a prior computational study by the authors⁵⁷ to enable direct comparison between simulated and experimental results.

Actuation mechanism

The meter-scale OPB prototype employs a static nylon cable actuation system connected to the structure. These cables function as tendons, translating the rotary motion of a central 90 V DC gearmotor into controlled structural deformation (Figure 3). A DC silicon controlled rectifier (SCR) board facilitates precise actuation speed control by regulating motor voltage and current.

Figure 3.

OPB prototype actuation mechanism alongside the data acquisition (DAQ) system.

The gearmotor drives a 1″ diameter axle via a bevel and pinion gear set in a miter gear arrangement. Two pillow block bearings support the axle, minimizing friction and ensuring smooth rotation essential for consistent actuation. Cable drums mounted on either side of the axle are responsible for winding and unwinding the actuation cables. A latching toggle switch with polarity reversal functionality enables control over motor direction. This bidirectional cable spooling capability on the drums facilitates actuation of the OPB prototype as necessitated by the research protocol. The apparatus is secured to a stabilized table with a low-friction surface covering to minimize vibration during testing while reducing frictional effects during deployment of the OPB.

Methodology

The research methodology adopts a synergistic approach, using computational simulations to inform the experimental investigations of the actuation strategy and strain development in a meter-scale Origami Pill Bug structure (Figure 4).

Figure 4.

Flowchart of research methodology.

To facilitate analysis using the dynamic relaxation method, an algorithm is devised to estimate the thick origami structure using a series of rigid bars connected by rotational hinges, effectively converting the structure into an equivalent bar model (i). This simplified representation enables the use of form-finding methods for the subsequent investigations.

The dynamic relaxation method is then employed to conduct deployment simulations of the Origami Pill Bug structure using the derived equivalent bar model (ii). These simulations allow for the assessment of strain development throughout the deployment process for both damaged and healthy versions of the origami structure. Based on the simulation results, three key metrics are used to determine the criticality of each member within the structure (iii). They are strain concentration factor (iii-a), load path redundancy (iii-b), and damage sensitivity (iii-c). This criticality assessment provides valuable information on the structural behavior and overall resilience of the origami design. Building upon the criticality assessment, a multi-objective optimization approach is undertaken to identify the most critical locations within the structure (iv). This guides the strategic placement of the sensors in the physical prototype for the experimental validation (v).

Experimental investigation is conducted to establish the relationship between the actuation strategy and the resulting strain distribution within the origami structure (vi). This empirical data is crucial for selecting the most suitable actuation approach for the Origami Pill Bug design.

The remainder of this paper is organized as follows: Each major topic—equivalent bar modeling, simulation via dynamic relaxation, sensor placement optimization, and experimental investigation—is presented as a combined section containing both methodology and results. This integrated approach allows for a focused discussion of each research objective in the context of its corresponding findings.

The application of computational simulations, criticality assessments, optimization techniques, and experimental validation enables a comprehensive understanding of the Origami Pill Bug’s structural behavior. The framework developed in this research—comprising the equivalent bar model formulation, dynamic relaxation analysis, and multi-objective sensor placement optimization—exhibits inherent scale invariance in its formulation. This property arises from the dimensionless nature of the governing equations and optimization criteria, which operate on normalized strain energy distributions, force equilibria, and geometric similarity principles. Together, these components provide an integrated methodology to determine effective actuation design strategies for achieving the desired controlled deployment.

Equivalent bar model for thick plates

With the methodological framework established, the next step involves developing and validating an equivalent bar model. To simulate the behavior of the OPB structure using the DR method, the continuous plate elements are discretized into an interconnected system of bar elements. While Schenk and Guest,²⁰ and Filipov et al.⁶¹ provide formulations for modeling thin-sheet origami as bar and hinge networks, these models do not accurately capture the behavior of thick plate structures. Thus, to approximate the behavior of thick plates using an equivalent system of pin-jointed bars for DR simulations, an optimization approach is developed.

The stiffness matrix of the thick plate elements (Kp) for plane stress is derived through finite element formulation, which accounts for the geometry of the plate and material properties. The stiffness matrix for the bar system (Kt) is also derived as a function of the unknown cross-sectional areas of the bar Ar(cm2). To find the optimal bar areas for the approximation, an optimization problem is formulated to minimize the difference between the stiffness matrices Kp and Kt.Ar, with a higher weight given to the diagonal terms which corresponds to axial stiffness:

\begin{array}{l} Minimize : \\ w (A r) = {‖ K_{t} A r - K_{p} ‖}_{F} + λ \sum | diag (K_{t} A r - K_{p}) | \\ Subject to : A r > 0 \end{array}

(1)

where:

${‖ ‖}_{F}$ : Frobenius norm, which represents the square root of the sum of the squares

$diag ()$ : extracts the diagonal elements of the matrix

$λ$ weighting factor emphasizing matching of the axial stiffness terms

This non-linear constrained optimization is implemented in MATLAB using quasi-Newton algorithm. For the optimization, the chosen value for $λ$ is 10 and the initial guess for $A r$ is also 10. The solution was found to be relatively insensitive to variations in these initial parameters. The optimization problem is solved numerically to obtain the optimal bar area such that the stiffness of the equivalent bar model, represented by $K_{t}$ , closely matches the stiffness of the thick plate element given by $K_{p}$ .

Equivalent bar model of the OPB

The findings for the three types of plate elements of the OPB structure are summarized in Table 2. Edge bars, shown in blue, represent the in-plane stretching and compression behavior along the edges of the plate, capturing its primary stiffness and resistance in these directions. Diagonal bars, shown in green, are designed to simulate shear and bending behavior within the plate, enabling the model to effectively represent coupled deformations and out-of-plane flexibility. By solving the optimization problem with a higher weight assigned to the axial stiffness terms, the equivalent bar model successfully captures the primary load transfer mechanism of the thick plate structure.

Table 2.

Equivalent bar model for thick plate elements of the OPB structure.

Plate element	Equivalent bar model	Optimal bar area
		A = 5.4 cm²
		B₁ = 8.4 cm²
		B₂ = 7.3 cm²
		B₃ = 4.2 cm²
		C₁ = 5.7 cm²
		C₂ = 11.4 cm²
		C₃ = 12.7 cm²
Edge elements	Diagonal elements	Nodes

Preliminary investigations using finite element analysis revealed good agreement between the in-plane and out-of-plane deflections of the plate elements and their equivalent bar models, with discrepancies limited to under 4% in both cases. This alignment demonstrates that the equivalent bar model successfully represents the essential stiffness behavior of the plate elements, thereby enabling efficient simulation using form-finding methods like DR.

Simulation using dynamic relaxation

Having verified the equivalent bar model, the dynamic relaxation (DR) method is applied for quasi-static form-finding of the meter-scale OPB. A novel module integrated with the DR method⁵³ which incorporates angle stiffness calculations within the residual forces, effectively characterizing bending and folding stiffness. This enables the development of iterative actuation algorithms for such structures. Convergence is determined through criteria involving residual forces, kinetic energy, and iteration limits. The algorithm also includes an element collision condition to prevent nodes from approaching closer than a predetermined threshold and a hard deck condition to ensure nodes do not penetrate a specified lower boundary. Previous research has documented the fundamental formulation as well as the efficacy of the DR method for quasi-static form-finding of the OPB.⁵²

This investigation employs the DR method to perform non-linear static analysis on the OPB structure under various damage scenarios. In damaged instances, the model assumes a complete loss of stiffness in a designated bar element, rendering it incapable of resisting any stress. Within the DR method, this is implemented by assigning a value of zero to the affected bar’s area, which effectively removes the element from the computational model. This study focuses solely on the failure of a single element at a time and does not consider the simultaneous failure of multiple elements within the same model.

Deployment analysis of the OPB

The modified DR method is used to simulate the configurations of the OPB throughout its deployment. Figure 5(a) shows the initial unrolled configuration and Figure 5(b) shows the final rolled configuration of the OPB, both obtained from the DR simulation. The dashed blue line represents the actuation cable. To achieve the desired rolled configuration in the computational study, specific boundary conditions are applied, as depicted in Figure 5. These boundary conditions are designed to represent the physical model. Nodes shown by are restricted in Y direction to represent the constraint due to the lockable brackets in the prototype. Nodes shown by represent pinned ends of the prototype, while indicate nodes restricted in both Y and Z directions, corresponding to the end of the prototype that translates along the X-axis during actuation. The DR simulation provides internal force data for each element of the OPB structure, encompassing both healthy and damaged cases, as well as each step of the deployment process.

Figure 5.

(a) Unrolled state and (b) rolled state of the OPB structure (solid black) from DR method, showing actuation cables (dashed blue) and imposed boundary conditions.

Sensor placement optimization

The simulation results provide the basis for identifying structurally critical regions, which informs the subsequent optimization of sensor placement for the meter-scale OPB. Optimization of sensor placement is a crucial step in experimental investigations to ensure efficient and comprehensive data collection while minimizing the sensor count. In this work, a multiobjective optimization (MOO) approach is presented to identify the critical elements within the structure for optimal sensor placement. Since critical elements represent the areas most vulnerable to failure and exhibit a greater sensitivity to damage, they serve as reliable indicators of structural health.

Three key metrics—strain concentration factor,⁴⁸ load path redundancy,⁴⁹ and damage sensitivity⁵⁰—are utilized to quantify the criticality of each element. Each of these metrics provides a unique understanding of the structural integrity and a comprehensive assessment of the criticality of structural elements.

Strain concentration factor

The strain concentration factor (SCF) is a metric used to quantify the deviation of the strain in a particular element from the average strain in the healthy structure during deployment. This metric highlights the susceptibility of elements to greater localized strains, which could lead to potential failure.

The strain in member i of the equivalent bar model of the OPB at deployment stage j, $ε_{i j}$ is calculated based on the internal force ( $f_{i j}$ ), Young’s modulus ( $E_{i}$ ), and cross-sectional area ( ${Ar}_{i}$ ) of the member.

The SCF for each member is then computed as the ratio of the member’s absolute maximum strain across all deployment stages ( $ε_{i, \max}$ ) to the average maximum strain across all members throughout deployment ( $ε_{a v g}$ ) as:

S C F_{i} = \frac{ε_{i, \max}}{ε_{a v g}}

(2)

Elements with high SCF values are considered more critical, as they are susceptible to higher localized strains that could potentially lead to failure and therefore require closer examination.

A normalized criticality score, denoted as $\bar{S C F}$ , is assigned to each element of the OPB structure. This score is derived from the individual SCF values and is scaled to range from 0 to 1, where 0 represents the least critical elements and 1 indicates the most critical elements.

Load path redundancy

The load path redundancy (LPR) is a metric that quantifies the ability of the structure to redistribute internal forces when an element is removed. This metric highlights the criticality of elements, as lower LPR values indicate a reduced capacity to redistribute forces.

Sum of absolute forces in the healthy structure ( $F_{i, h}$ ) is calculated for each member as:

F_{i, h} = \sum_{j = 1}^{N} | f_{i j, h} |

(3)

where:

$f_{i j, h}$ : internal force in member i at deployment stage j of the healthy structure

N: Number of deployment stages

For damage cases, only one element is removed for each corresponding damage case. For damage cases, the internal forces are determined in the damaged structure. The load redistribution ( $δ f_{i j, k}$ ) for each damage case is calculated as the absolute difference between the healthy and damaged forces as:

δ f_{i j, k} = | f_{i j, h} - f_{i j, k} |

(4)

where:

$f_{i j, h}$ : internal force in member i at deployment stage j of the healthy structure

$f_{i j, k}$ : internal force in member i at deployment stage j of damage case k

The total load redistribution factor ( $Δ f_{i k}$ ) for each member is computed for each damage case as:

Δ f_{i k} = \sum_{j = 1}^{N} δ f_{i j, k}

(5)

The load path redundancy value ( $L P R_{i k}$ ) for each member in each damage case is calculated using the equation:

L P R_{i k} = 1 - (\frac{Δ f_{i k}}{\sum_{i = 1}^{M} F_{i, h}})

(6)

where k represents the damage case, i represents the member, and M is the total number of elements in the structure. For the load path redundancy value ( $L P R_{k}$ ), the minimum value across all members is selected:

L P R_{k} = \min_{i} (_{i k})

(7)

Low LPR values indicate higher criticality, as the structure exhibits a lower ability to redistribute internal forces in these damage cases. Thus, these elements should be the focus of targeted investigation to improve the overall structural resilience.

A normalized criticality score, denoted as $\bar{L P R}$ , is assigned to each element of the OPB structure. This score is derived from the individual LPR values and is scaled to range from 0 to 1, where 0 represents the least critical elements and 1 indicates the most critical elements.

It is important to note that while lower raw LPR values indicate higher criticality, the normalization process inverts this relationship, making higher $\bar{L P R}$ scores (close to 1) indicative of more critical elements. Therefore, maximizing this normalized score in the optimization framework correctly identifies the most critical structural components.

Damage sensitivity

The damage sensitivity (DS) provides a measure to quantify the change in strain energy density of the structure when an element is removed or damaged. This metric provides information into the impact of element removal or damage on the overall structural behavior.

Strain energy density is defined as the area under the stress-strain curve as follows:

U = \int σ d ε = \int E ε d ε = \frac{1}{2} E ε^{2} = \frac{1}{2} \frac{f^{2}}{E A r^{2}}

(8)

The strain energy density for the healthy structure ( $U_{i j, h}$ ) is calculated for each member and deployment stage using the equation:

U_{i j, h} = \frac{1}{2} \times (\frac{{(f_{i j, h})}^{2}}{E_{i} \times A r_{i}^{2}})

(9)

where:

$f_{i j, h}$ : internal force in member i at deployment stage j of the healthy structure

$E_{i}$ : Young’s modulus of the material for member i

$A r_{i}$ : Cross-sectional area of member i

The strain energy density represents the energy stored per unit volume of the material due to deformation. Strain energy for each member of the healthy structure ( $W_{i, h}$ ) is obtained by summing the strain energy density of the member over all deployment stages as:

W_{i, h} = \sum_{j = 1}^{N} U_{i j, h}

(10)

The process is repeated for each damage case, where only one element is removed for each corresponding damage case. For each damaged case (k), the strain energy density in the damaged structure ( $U_{i j, k}$ ) is calculated in a similar manner, and the strain energy for each member in the damaged case ( $W_{i, k}$ ) is also determined. The net change in strain energy for each member for each damage case as compared to the healthy structure is then calculated as:

Δ W_{i, k} = | W_{i, k} - W_{i, h} |

(11)

The damage sensitivity scores for each damage case ( $D S_{k}$ ) is therefore computed as:

D S_{k} = \sum_{i = 1}^{M} Δ W_{i, k}

(12)

Elements whose damage leads to higher damage sensitivity values ( $D S_{k}$ ) are considered more critical. This metric provides valuable information on the relative impact of different damage scenarios on overall structural stability.

Individual DS values are normalized to a scale of 0 (least critical) to 1 (most critical), resulting in a normalized criticality score ( $\bar{D S}$ ) for each OPB element.

Multi-objective optimization

A multi-objective optimization (MOO) approach is employed to find a set of optimal solutions that balance the criticality value from the three metrics. The goal is to find a set of weights, $w$ , that simultaneously maximizes the objective functions corresponding to the three metrics.

The objective functions are defined as:

Objective 1 : Maximize \bar{L P R} = \sum (\bar{L P R} \cdot w)

(13)

Objective 2 : Maximize \bar{D S} = \sum (\bar{D S} \cdot w)

(14)

Objective 3 : Maximize \bar{S C F} = \sum (\bar{S C F} \cdot w)

(15)

To solve this MOO problem, the gamultiobj function from MATLAB’s Global Optimization Toolbox is utilized. This function employs a controlled, elitist genetic algorithm (a variant of NSGA-II) with a maximum number of generations of 250 and a population of 250 to find the Pareto-optimal set of solutions.

Pareto fronts are obtained from the multi-objective optimization (MOO) algorithm aimed at maximizing three objectives: maximize $\bar{S C F}$ , maximize $\bar{L P R}$ , and maximize $\bar{D S}$ . The $\bar{S C F}$ , $\bar{L P R}$ , and $\bar{D S}$ are the normalized metrics used to evaluate and rank the OPB elements based on their criticality throughout deployment. Each point on the Pareto front represents a unique set of weights ( $w$ ) that provide the best trade-off between the three objective functions. The MOO approach generates multiple optimization scores for each element across the objectives as the algorithm explores various combinations of weights. Each combination prioritizes the objectives differently, leading to unique optimal solutions.

The 3D contour plot of the Pareto fronts in Figure 6 visualizes the set of Pareto-optimal solutions in the space of the three objective functions for three representative elements: (A) Element Number 7 (low impact), (B) Element Number 20 (medium impact), and (C) Element Number 21 (high impact). These specific elements were selected to illustrate a range of impacts on the system, based on a categorization that is detailed in the subsequent section of this study.

Figure 6.

3D Pareto fronts from multi-objective optimization maximizing $\bar{S C F}$ , $\bar{L P R}$ , and $\bar{D S}$ for: (a) element numbers 7 (low impact), (b) element numbers 20 (medium impact), and (c) element numbers 21 (high impact).

Critical element identification

After obtaining the Pareto-optimal set, the composite scores for each element are calculated by aggregating the three objective function values using a weighted sum method. For this, the normalized scores for the three metrics ( $\bar{S C F}, \bar{L P R,}$ and $\bar{D S}$ ) are multiplied by the corresponding elements in the Pareto-optimal set, and then summed together to obtain the composite score for each element. The composite score is then used to rank all the elements in the structure based on their criticality.

Figure 7(a) presents a stacked bar plot showing the composite score of each element of the OPB structure. Each section of the stacked bar plot represents the contribution of each of the three metrics— $\bar{S C F}, \bar{L P R},$ and $\bar{D S}$ —toward the composite score, as determined by the multi-objective optimization.

Figure 7.

(a) Composite scores for elements of the OPB obtained from multi-objective optimization. The dashed lines represent the mean (green), mean+ $σ$ (blue), and mean+2 $σ$ (red) and (b) critical elements highlighted in the OPB structure.

Three thresholds are considered for the criticality assessment using composite score: mean (dashed green line), one standard deviation above mean (dashed blue line), and two standard deviation above mean (dashed red line). The elements with composite score higher than the thresholds under consideration are highlighted in the plot. This approach is statistically grounded in identifying outliers that deviate from the central tendency, indicating areas of heightened importance or potential concern within the OPB structure. .

Elements above the mean composite score are considered outliers in the positive direction, suggesting their potential criticality. For the OPB structure, 20 elements exhibited composite scores above the mean value and are designated as critical elements. Figure 7(b) depicts the OPB structure with the critical elements (7, 18, 20–27, 40, 51, and 53–60) highlighted, indicating their location within the OPB structure. The regions in the structure corresponding to the critical elements are designated as critical regions in the OPB structure.⁶¹

The placement of sensors is optimized based on the results of the identified critical regions (Figure 7(b)). Leveraging the structural symmetry, five critical locations are selected for sensor placement: Element Numbers 7, 24, 25, 26, and 27. At these strategically chosen locations, five strain gauges are installed, each with a resistance of 120 $Ω$ and a gauge factor of 2.11. This optimized configuration is used for the experimental investigation as depicted in Figure 8, and is designed to maximize data collection efficiency while minimizing the number of strain gauges required.

Figure 8.

Optimized placement of five numbered strain gauges (1–5) on the OPB structure.

Experimental investigation

Based on the optimized sensor locations, the experimental phase is conducted to investigate the relationship between actuation rates and the resulting deployment force and strain development in the OPB structure. As depicted in Figure 9, a compression load cell (model 266AA-50 kg from Anyload) is utilized to directly measure the actuation force applied during the deployment process. The placement of strain gauges on the structure is optimized based on a multi-objective optimization algorithm that identifies critical elements with the OPB structure. Figure 9 illustrates the experimental setup for the OPB prototype and its components.

Figure 9.

Schematic diagram OPB prototype experimental setup, illustrating all components.

National Instruments compact data acquisition (DAQ) system (NI cDAQ-9178) is employed for data acquisition. It uses the NI 9235 module to acquire the strain data from the five gauges and the NI 9237 module to measure the force from the load cell throughout the deployment cycle. The DAQ system is configured with a sampling rate of 200 Hz to capture any strain oscillations. The measured force and strain data are processed and analyzed to evaluate the structural behavior during actuation.

To effectively capture the actuation performance and to identify key trends, three representative actuation rates are selected for evaluation: 1.5, 2.0, and 4.0 cm/s. The actuation rate is defined as the rate at which the actuation cables are shortened to deploy the structure from its unrolled state. To ensure experimental repeatability and minimize random error, five independent trials are conducted for each actuation rate. Subsequently, the data are averaged to obtain a robust measure of the effect. Two key metrics are captured during deployment: actuation force, measured as the total tension in the cables, and the strain experienced by the structure’s elements.

Results: Actuation force and strain response

Figure 10 presents the time history plots of the actuation force (N) and strain measurements from five strain gauges for actuation rates of 4.0, 2.0, and 1.5 cm/s. The force exhibits a generally increasing trend as the actuation cables are shortened. The strain measurements show more complex behavior, with both tensile and compressive strains developing simultaneously at different locations within the structure during deployment.

Figure 10.

Time history plots for actuation force and strain responses from five different strain gauges across varying actuation rates: (a), (c), and (e) show the actuation force profiles at actuation rates of 4.0, 2.0, and 1.5 cm/s, respectively. (b), (d), and (f) present the strain profiles at actuation rates of 4.0, 2.0, and 1.5 cm/s, respectively.

The structural response under different actuation rates is summarized in Table 3. For an actuation rate of 4.0 cm/s, the peak actuation force reaches 286.5 N, with maximum tensile and compressive strains of 0.242‰ and −0.254‰, respectively. When the actuation rate is set to 2.0 cm/s, the peak force reduces to 139.3 N, while the maximum tensile strain increases to 0.331‰ and the maximum compressive strain decreases to −0.166‰. At the lowest actuation rate of 1.5 cm/s, the peak actuation force is 133.4 N, and the tensile and compressive strain extremes are 0.365‰ and −0.120‰, respectively.

Table 3.

Summary of peak actuation force, maximum tensile strain, and maximum compressive strain for different actuation rates during deployment of the OPB prototype.

Actuation rate (cm/s)	Peak actuation force (N)	Highest tensile strain (+)	Highest compressive strain (−)
4	287	0.242‰	−0.254‰
2	139	0.331‰	−0.166‰
1.5	133	0.365‰	−0.120‰

Actuation force

The actuation force plots (Figure 10(a), (c), and (e)), corresponding to actuation rates of 4, 2 and 1.5 cm/s respectively, show a non-linear increase as the cable-actuated structure deploys. The undulations in the force plots stems from the complex geometrical coupling between the panels and the intricate folding kinematics facilitated by the cable routing mechanism. The initial steep rise in actuation force can be attributed to the high forces required to overcome the inherent stiffness of the structure in the unrolled configuration and initiate the folding process. As deployment progresses, the force increase becomes more gradual. This suggests a reduction in the resistance to actuation as the panels near their final deployed configurations.

Strain profiles

The strain profiles (Figure 10(b), (d), and (f)), corresponding to actuation rates of 4, 2, and 1.5 cm/s respectively, offer information into the localized deformation mechanisms and load transfer pathways within the structure during deployment. The relatively low strain magnitudes observed in Gauges 2 and 3, located on the partially folded panels in the unrolled state, indicate that these panels experience limited deformation during deployment. Conversely, Gauge 1 is vertically positioned and undergoes predominantly compressive strains as the structure rolls. Gauges 4 and 5 show relatively high strain magnitudes reflecting the higher degree of folding experienced by the respective panels during deployment. The strain distributions across Gauges 4 and 5 also highlight the contrasting deformation modes of the panels. Gauge 5 undergoes tensile strains, while Gauge 4 experiences primarily compressive strains, reflecting the inward folding motion of the panels. This behavior can be attributed to the specific cable routing and panel connectivity employed in the structural design.

The cyclic nature of the strain profiles arises from the specific folding sequence and coupled deformation of the panels. The coupled nature of deformations in adjacent panels is evident from the simultaneous occurrence of strain peaks and valleys in gauge 4 and 5. This coupling arises from the hinges between adjacent panels and the physical connections through the cable actuation mechanism. The residual strain values in the stabilized phase indicate that different panels experience varying degrees of residual deformation and stress states in the final configuration. For Gauges 1, 2, and 3, the residual strains remain largely consistent across all tested actuation rates. In contrast, for Gauges 4 and 5, the residual strain shifts progressively toward the tensile side as the actuation rate decreases.

The variations in strain magnitude across the gauges reflect the intricate folding kinematics and the evolving deformation modes experienced by different regions of the structure. The strain distributions are governed by the interplay between the geometry, hinge locations, cable routing design, and the specific folding sequence dictated by the actuation mechanism. These observations enhance the understanding of structural behavior, load transfer pathways, and deformation mechanisms, thereby enabling informed design iterations and optimization efforts for deployable structures of this nature.

Discussion

Comparison of actuation rates

Comparison of the actuation force plots and strain profiles across the three actuation rates—4.0, 2.0, and 1.5 cm/s—reveals some notable differences. As the actuation rate increases, the overall magnitude of the actuation force required for deployment increases. At lower actuation rates, the initial rise in actuation force becomes more gradual, suggesting a slower initiation of the folding process. The overall shape of the force profile also becomes more linear, indicating a more uniform distribution of resistance throughout deployment.

In terms of the strain profiles—the highest tensile strain magnitudes decrease with increasing actuation rate, while the highest compressive strain magnitudes increase with increasing actuation rate. Lower actuation rates appear to allow for more redistribution of strains in certain regions, potentially due to reduced dynamic effects and inertial forces during deployment. Higher actuation rates tend to amplify compressive residual strains, whereas lower actuation rates facilitate a shift toward tensile residual strains. This indicates that deployment velocity can influence the final stress state in specific regions of the structure, potentially leading to localized variations in residual deformation.

Optimal actuation rate

Among the three actuation rates investigated, an actuation rate of 2.0 cm/s is selected as the suitable choice for deploying the cable-actuated OPB structure based on the following considerations:

Actuation force profile: At 2.0 cm/s, the actuation force profile exhibits a gradual and linear increase. The peak actuation force decreases to less than 50% between actuation rates of 4.0 and 2.0 cm/s, whereas it shows minimal variation between 2.0 and 1.5 cm/s.

Strain magnitudes: The strain profiles at 2.0 cm/s show moderate peaks for both tensile and compressive strain magnitudes, avoiding the extremes observed at higher (4.0 cm/s) and lower (1.5 cm/s) actuation rates.

Residual strains: The 2.0 cm/s actuation rate offers a balanced compromise, by avoiding the high compressive residual strains of 4.0 cm/s and the pronounced tensile shift of residual strains at 1.5 cm/s.

Deployment time: While the deployment time is longer than the highest actuation rate of 4.0 cm/s, the deployment process at 2.0 cm/s is reasonably fast as compared to 1.5 cm/s.

Conclusion

The research effectively established a comprehensive framework for optimizing the actuation strategy of the deployable origami structure through a synergistic approach that combined computational analysis with experimental validation. The equivalent bar model effectively captured the core load transfer mechanisms of the thick origami structure and enabled simulation using the dynamic relaxation method. Multi-objective optimization approach successfully identified key elements for optimal sensor placement, providing valuable data on deformation and strain distribution. The experimental framework facilitated the investigation of various actuation rates, leading to the identification of 2 cm/s as the optimal rate for effective cable actuation of the meter-scale structure. The findings contribute to the development of tailored actuation strategies in origami-inspired deployable structures, paving the way for their broader application in engineering systems.

Footnotes

Acknowledgements

The authors would like to thank Prof. Jacob Henschen for his guidance and support, and Mr. Shizhao Xu for his contributions in the laboratory toward experimental testing of the Origami Pill Bug.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by funding provided by the Grainger College of Engineering at the University of Illinois Urbana-Champaign.

ORCID iD

Angshuman C Baruah

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