Mechanical Solutions to Variable Stiffness Robotic Arms

Xianpai Zeng1, Tyler Morrison2 and Hai-Jun Su3

Abstract— Collaborative robots or Co-robots are widely usedin automotive industries and material handling. Co-robots aredesigned to work with human workers side by side. However,safety is a challenging major concern. Rigidity is requiredto carry a payload and achieve accuracy in motion, whileflexibility is often required for safe human interaction. Robotswith control over their stiffness could combine some of theadvantages seen in compliant robots with the performanceof traditional rigid robots. This paper aims at presentingseveral variable stiffness solutions developed in the Design,Innovation, and Simulation Laboratory (DISL) at The OhioState University. These methods are classified based on theworking principles and compared quantitatively based on thecriteria such as stiffness range, stiffness ratio, and responsetime. Pros and cons are also described for each method.

I. INTRODUCTION

Close cooperation between intelligent assistive robots andhuman workers could improve the efficiency of complexproduction processes. This cooperation involves physicalhuman-robot interaction (pHRI) which requires deliberatedesign of production equipment around human operators [1].Hybrid assembly – the process of robots and human workersjointly performing a handling and/or assembling task – isproven to have higher efficiency and lower cost comparedto pure robotic or human assembly [2]. In order for co-robots to work safely and effectively alongside humans,flexibility and rigidity are both required. Flexible structuresensure safe human-robot interaction while rigidity is a criticaldesign consideration for load carrying capability and highmotion accuracy. Existing research shows that robotic armsdesigned with greater compliance are less likely to causeinjury to humans [3], [4]. Some authors also argue that,through an effective control policy on stiffness variation,total operating time for a typical rest-to-rest task can beminimized by appropriately blending slow-and-stiff and fast-and-soft phases, [3], [5] but the variability and controllabilityof such compliance remains challenging [6].

Compliant robotics is an important frontier of co-roboticresearch. Compliant robots are typically constructed frommaterials such as textiles and elastomers, but can be made

This material is based upon work supported by the National ScienceFoundation under Grant No: CMMI-1637656. Any opinions, findings, andconclusions or recommendations expressed in this material are those of theauthor(s) and do not necessarily reflect the views of the funding agencies.

1Xianpai Zeng is with the Department of Mechanical & AerospaceEngineering, The Ohio State University,Columbus, OH 43210, USA.zeng.108@osu.edu

2Tyler Morrison is with the Department of Mechanical & AerospaceEngineering, The Ohio State University,Columbus, OH 43210, USA.morrison.730@osu.edu

3Hai-jun Su is with Faculty of the Department of Mechanical &Aerospace Engineering, The Ohio State University,Columbus, OH 43210,USA. su.298@osu.edu

from any highly compliant material. Due to their compliantnature and theoretically infinite degrees of freedom, compli-ant robots can be extremely durable and more dexterous, anddo not require careful control and environmental awarenessto avoid damaging payloads or harming humans [7], [8].

Compliant robots can also be less expensive, lighter, andeasier to customize for different applications when comparedto traditional robots [9]. However, there are some limitationsassociated with the high compliance. One challenge withcompliant robotics is maintaining high positional accuracy.The inherit flexibility makes it difficult to control the exactposition of the soft robot’s appendages. The theoreticallyinfinite degrees of freedom makes applying traditional sensorsetups such as encoders a challenge. Additionally, compliantrobot tends to have large deformation under external load.This prevents compliant robots from matching the load-carrying capabilities and acceleration speeds of traditionalrigid robots, which limits their potential uses.

Due to the performance limitations of compliant robots,there is interest in variable stiffness technologies. Robotswith control over their stiffness could combine some of theadvantages seen in compliant robots with the performanceof traditional rigid robots. For example, a robot capable ofvarying its stiffness can become more rigid when higherpositional accuracy or load capabilities are needed, but canalso become more complaint to prevent injuring surroundinghumans or damaging the environment. The additional safetythat variable stiffness allows for is of increasing interest asrobots which interact with humans become more common.For robots in assistive, rehabilitation, and domestic roles,safety is a critical design concern. This safety risk must beaddressed before robots which work alongside humans canbecome more prevalent, and variable stiffness capabilitiesmay be an important solution.

This work proposes a review of four variable stiffnessconcepts and six prototypes from the Design Innovationand Simulation Laboratory (DISL). Blanc et al. classifiedcontrollable stiffness mechanisms on the basis of intrinsicproperties change [5]. The solution of the first main familyis to change the second moment of the area, and the secondmain family’s solution is to change the structure’s elasticproperties. This approach to classification and description isa major inspiration to our approach in this paper. Section IIintroduces a classification of the four concepts based on theunderlying working principles including a summary of thedesign and experimental results for each concept. Section IIIgives a quantitative comparison.

II. DESIGNS

A. Shape Morphing Cross-Section

1) Concept: The shape morphing concept varies stiffnessby changing the cross-sectional shape of the beam. Theprototype (SM1) proposed in [10] is comprised of a servomotor, four pairs of bearing frames, two universal trans-mission shafts, two flexible beams, and three cable-drivenmechanisms as shown in Fig. 1. The flexible beams arethe key concept for achieving variable lateral stiffness. Incompliant mode, these beams relaxed and flat. In stiff mode,they are morphed into a curved shape which has a higherstiffness for lateral loads just as a sheet of paper can befolded to increase its stiffness.

The lateral stiffness of a single fixed-guided shape-morphing beam is

k =12EI

L3(1)

where E is the elastic modulus, L is the beam length,and I is the appropriate second moment of area of thecross-section. According to (1), changing I changes stiffnessproportionally. To determine the required actuator position toachieve a given stiffness, the shape of the beam cross-sectiondue to a deflection of its tip via the cable-driven mechanismcan be determined by applying classical beam theory, and Ican be determined by integrating the cross-section.

However, achieving variable stiffness is not useful if thedevice cannot support loads in other directions. Therefore,four prismatic bearing frames are added to connect the cable-driven mechanisms. The proximal cable-driven mechanism isfixed to the base housing and the distal cable-driven mech-anism is attached to the end housing. The bearing framesare designed as a square rail with linear guides permittingprismatic extension and contraction, and the connectionsbetween the bearing frames and cable-driven mechanisms arerotational joints. As a result, the shape morphing arm behavesas a compliant parallel-guided mechanism in the lateraldirection with variable stiffness while it remains rigid in thevertical direction and the perpendicular lateral direction.

In order to morph the flexible beams uniformly, threecable-driven mechanisms are distributed along the length ofthe morphing beams. To reduce the number of actuators, twouniversal transmission shafts are designed to couple the threecable-driven mechanisms such that a single servo motor inthe base housing connected to the shaft can actuate all threecable-driven mechanisms simultaneously.

The center-line of the flexible beams is fixed to the cable-driven mechanisms, while the upper and lower edges ofthe flexible beams are connected to the transmission systemwith cables so that they are pulled inward and flexed intoa concave shape (see Fig. 1(b)). The base housing, endhousing, cable-driven mechanisms and bearing frames canbe considered the rigid “skeleton,” while the flexible armsare like “muscles.”

2) Results: The prototype of this concept (SM1) is con-structed from plastic with an elastic modulus of 1.2GPa.In its most compliant mode, the stiffness of the prototype

a)

b)

Fig. 1. (a) Prototype of the shape-morphing cross-section concept showingall components [10]. (b) Cross-section of one of the three sections ofcompliant mechanism used in the beam. A center pulley connected to theservo via the transmission shaft pulls a set of four cables that each flex thebeam into a concave shape when activated.

is 0.207N/mm while in its stiff mode, the stiffness is0.458N/mm. This results in a stiffness ratio of 2.21. Apseudo-rigid-body model is proposed in [10] which accountsfor the morphing angle of the cross-section and closelymatches experimental results.

Another version of the same concept (SM2), shown inFig. 2, is presented in [11]. In this work, the cable drivenapproach is replaced with a set of four-bar mechanisms. Thisbeam, made from the same plastic, can vary lateral stiffnessfrom 0.540N/mm to 1.936N/mm and therefore achieves astiffness ratio of 3.6

3) Pros and Cons:Pros• Control — Control of the stiffness is easily obtained

by modeling of the relationship between stiffness andmorphing actuator angle, and closed loop control ofmorphing actuator.

• Actuation Time — Actuation is quick and is directlyrelated to the power of the servo motor.

• Scalability — Scaling of the stiffness range of thisdesign can be accomplished by changing the materialselection, beam cross-section dimensions or length.

Cons• Mechanical Complexity — The mechanical complex-

ity of the linkage and transmission system needed touniformly morph a beam introduces design and massoverhead and additional failure points.

a)

b)

Fig. 2. (a) Prototype of the shape-morphing cross-section concept showingall components [11]. (b) Cross-section of one of the three sections ofcompliant mechanism used in the beam. A central mechanism pulls theedges of the beam inward into a concave shape.

• Actuator Effort — Even without the presence of anexternal load, the actuators need to exert a force/torqueto hold the beam in its morphed shape.

B. The Rotating Beam Link (RBL)

1) Concept: Fig. 3 shows an overview of the prototypedesign of the rotating beam link (RBL). Four parallel thinaluminum beams are joined at each end to a hub whichcontains a gearbox for rotating the beams with the servoin each hub. This design is motivated by considering thetransformation of area moments of inertia under rotationby θ from the beam-fixed x-y frame to the link-fixed x′-y′ frame centered on the beam. For a symmetric beam witha rectangular cross-section and the x-y coordinate system atits centroid, Ixy = 0 and

Iy′ =Ix + Iy

2− Ix − Iy

2cos(2θ) . (2)

where Ix and Iy are the area moments of inertia in therotating frame and Iy′ is an area moment of inertia in thefixed frame.

If we assume that Ix > Iy as indicated by Fig. 3,intuitively, (2) has minima at the angles θ = jπ, and maximaat the angles θ = (π/2 + jπ) , ∀j ∈ Z. As a result, wecan vary the lateral stiffness as a sinusoid by symmetricallyrotating the beams.

0 30 60 90 120 150 180

(deg.)

0

0.5

1

1.5

2

2.5

kX(

) (N

/mm

)

Analytical Model

Experiment

e)

Fig. 3. (a) Illustration of the RBL concept [12] with sub-figures (b-d)showing the cross-section of the link from the top-down view. Four parallelbeams form the structure of the link. When rotating the beams by θ in syncas in from (b) to (c) and (c) to (d), their alignment with the lateral loadFX changes the effective area moment of inertia and therefore the resultinglateral stiffness. (e) The link lateral stiffness as a function of beam angle.

2) Results: In [12], we demonstrate a prototype of ournew variable stiffness link concept achieves a stiffness ratioof 13.9 between its extreme positions, and a ratio of 8.6between its neutral configuration and its compliant config-uration. However, this is a significant difference from theideal theoretical ratio of the prototype which is shown tobe 122. The difference in these ratios is due to parasiticcompliance introduced by the components in the drive-trainthat are necessary to actuate the rotating beams.

Finite element analysis suggests the maximum torsionalstiffness of the link is about 19N-m/rad, and experimentaltests show the link can support 20N of axial compressionwithout buckling, and substantially more at other beamangles. If necessary, torsional load capacity may be improvedby adding an extra mechanism between the two hubs of therobotic arm.

The model we developed for the lateral stiffness of thelink as a function of the beam angles includes a completeaccounting of the compliance in each beam as well as buck-ling phenomena and it accurately predicts stiffness behaviorof the prototype. We also show that the model can beused to help characterize the effect of parasitic compliance:an essential challenge of implementing this concept into aworking design.

3) Pros and Cons:Pros• Control — Control of the stiffness ratio is easily ob-

tained by modeling of the relationship between stiffnessand beam angle, and closed loop control of beam angle.

• Actuator Effort — Actuator effort to maintain a stiffnessis only required under external loads.

• Actuation Time — Actuation is quick. Beams only needto rotate 90 degrees to obtain largest stiffness change.

• Scalability — Scaling the stiffness ratio and range ofthe design can be accomplished by changing materialselection, beam cross-section dimensions, beam length,or number of beams.

Cons• Mechanical Complexity — The mechanical complexity

of the transmission system adds mass overhead and in-troduces parasitic compliance which decreases stiffnessthrottling performance.

• Buckling — The reliance of the design on several long,slender beams introduces an opportunity for elasticbuckling in certain loading modes which decreasesstiffness performance and induces non-linearities whichimpede modeling.

C. Slider on Parallel-Guided Arm

1) Concept: The slider variable stiffness robotic arm isbased on a parallel guided beam, and consists of two end-guided flexures with a rigid connection in between. Thestiffness of the arm is calculated as:

k =24EI

L3(3)

where, E is the elastic modulus of the beam material, I isthe cross-section area moment of inertia of each sheet, andL is the effective beam length. The concept of the design isto change the effective length of the parallel guided beam toimplement stiffness variation.

The first prototype based on this concept is shown inFig. 4 [13]. The stiffness change is enabled by changing theeffective length of beams through a roller carriage which isactuated by a screw drive and an electric motor. The arm hastwo parallel Al 7075 sheet flexures. The fixed end is designedto be the motor and transmission housing. A power screw isassembled through the slider. To allow the bending of the twoflexures, a small nut featuring with grooves is added to thefree end containing spherical balls as rolling supports. As thecarriage is moved towards the free end, the free length of theflexures decreases, leading to an increase in stiffness sincethe support bar has high stiffness. The overall dimensions ofthe prototype are 406mm × 95mm × 104mm and the totalweight is 952 g.

Fig. 5 [14] shows the second prototype which is basedon the same concept. A low-weight, high-speed pneumaticcylinder is used to decrease the time of stiffness variation.The carriage runs along the support bar, with linear bearingsto reduce friction. It makes contact with the actuator onlyin the axial direction, and the bar carries all lateral loads.The boundary conditions are enforced using roller bearingsthat make contact with the sheets. The free ends of thesheet flexures are connected by a single part, which is thenallowed to move with respect to the cylinder by rollingcontact through metal balls. The overall dimensions of the

(a)

(b)

Fig. 4. (a) Prototype of the parallel-guided arm with screw driven slider[13]. (b) Link stiffness as a function of effective length for the design withscrew and motor driven slider.

arm are 450mm × 100mm × 100mm. The total mass is450 g.

2) Results: The link with the slider actuated by motorand screw drive is capable of stiffness change by 20 fold,with the maximum static stiffness at 10.049N/mm and theminimum value at 0.500N/mm. The link with the slideractuated by pneumatic cylinder could achieve a relatively lowstiffness change of 10 fold, but it can be achieved in 0.6 s.The maximum stiffness is 3.407N/mm and the minimumstiffness is 0.358N/mm.

3) Pros and Cons:Pros• Payload — High load capacity is obtained due to the

parallel-guided configuration.• Stiffness Range (Screw Driven) — For the design that

uses the screw driven slider, very large stiffness range isfeasible. The power screw shaft has very large stiffness

(a)

(b)

Fig. 5. (a) Prototype of the parallel-guided arm with pneumatic cylinderdriven slider [14]. (b) Link stiffness as a function of effective length for thedesign with pneumatic cylinder driven slider.

which adds to the overall stiffness significantly whenfully coupled with the parallel-guided beam.

• Response (Pneumatic Driven) — The effective lengthcan be changed quickly since the slider can move veryfast when driven with the pneumatic cylinder.

• Control — Control of the stiffness is directly relatedwith the effective beam length.

Cons• Response (Screw Driven) — The moving speed of the

slider driven with screw and motor is related to thescrew pitch, motor power. Motors with higher powertend to be bulky and expensive which limits the overallresponse time, and adds mass to the link.

• Actuator Effort — Extra actuator effort is needed tomove the slider while the link is deflected.

D. Layer Jamming on Parallel-Guided Arm

1) Concept: This section introduces a parallel-guidedrobotic arm that can achieve a large stiffness variation

ratio – up to 75 times – through pneumatic actuated layerjamming [15]. Fig. 6 (a) illustrates the basic configurationof the parallel-guided arm. The robotic arm is composed oftwo 3D printed flexible beams securely clamped on bothends. Juxtaposing two beams in parallel makes the roboticarm more resilient to bending moments due to the endconstraints and results in a higher vertical load capabilitycompared to a single beam. A novel cross-section thatfeatures interconnected stiff and soft sections is proposed. Asillustrated in Fig. 6 (b), the hourglass shaped stiff sectionsare connected through the relatively longer and thinner softsections. With this novel design, the beam has large thicknessbut retains high flexibility. Fig. 6 (c) illustrates the interleavedarrangement of jamming layers. On both sides of the beam,jamming layers are laid on top of the hourglass shapedsections. One group of layers, shown in green, is bondedto the left end of the beam, the other group of layers, shownin yellow, is attached to the right end. The two groups areshuffled evenly so that the layers are on top of each other.The effect of layer jamming is augmented by the enhancedbeam thickness due to the increased leverage of friction forceexerted by the jammed layers.

2) Results: Force and deflection data have been collectedwhile the arm is deflecting from its initial position to atip deflection of 20mm. Multiple data collections are madefrom 0 psig to 12.5 psig with an increment of 2.5 psig.Experimental result is presented in Fig. 7. For a givenvacuum pressure, the stiffness of the parallel-guided arminitially increases linearly followed by a non-linear increasephase, in which the rate of increase starts to slow down. Andfinally the stiffness comes to saturate as the tip deflectionreaches 20mm. During the unloading process, the arm tendsto restore its initial position due the strain energy stored inthe flexible backbone.

At 0 psig vacuum pressure, the arm stiffness is defined asthe base stiffness which is used as the denominator to calcu-late stiffness variation ratio. The base stiffness characterizesthe flexibility of the center backbone. Stiffness increasesas the applied vacuum pressure is increased. The stiffnessvariation ratio is calculated as the stiffness corresponding to12.5 psig divided by the base stiffness. Experiment on theprototype shows that a stiffness ratio of 75 is achieved. Theminimum stiffness and maximum stiffness are 0.0803N/mmand 6.05N/mm respectively.

Quick actuation of the jamming layers is also an importantdesign consideration. To remove the air from the vacuumbag in a quick manner, channels as an internal feature ofbeams have been designed to facilitate fast air removal.The internal air channels are made during the 3D printing,with no extra fabrication process needed. The other measurefor generating vacuum quickly is to use vacuum generatorrather than vacuum pumps. The flow rate of regular motor-based vacuum pumps is usually around 2 cfm (cubic feetper minute), with a few up to 10 cfm. However, air-poweredvacuum generators can go up to 30 cfm. This significantlyenhance the performance on adjusting the vacuum level. Thejamming layers can be fully actuated in 0.25 second.

(b)

(c)

Fixed End

External Load

PressureJamming Layers

Vacuum Bag Pressure

(a)

Fixed End

External Load

Vacuum Port

Fig. 6. Design concept of the parallel-guided arm with layer jamming [15].(a) Basic configuration. (b) Deformed parallel-guided arm with one end fixedwhile the other moving laterally under external load. (c) Configuration ofthe vacuum membrane and jamming layers.

0 5 10 15 20

Tip Deflection (mm)

0

10

20

30

40

50

Ex

tern

al L

oad

(N

)

0 psi

2.5 psi

5.0 psi

7.5 psi

10.0 psi

12.5 psi

Fig. 7. Load-deflection measurement of the parallel-guided arm with layerjamming [15].

(a)

(b) (c)

Fig. 8. Applications of layer jamming on (a) morphing structure withtunable stiffness [16] and (b) robotic gripper with variable stiffness [17].

Layer jamming for variable stiffness has also been appliedto morphing structures and robotic grippers. Fig. 8 (a)shows a prototype [16] that is capable of both morphingits curvature and varying its stiffness. The structure is ac-tuated via pneumatic air muscles to achieve high levels ofcurvature. Integrating the McKibben actuators inside the 3Dprinted structure allows this design to achieve much higherloading forces compared to previous morphing structures insoft robotics. In addition, this design achieved a maximumstiffness variation of 75 times. Fig. 8 (b) and (c) presenta robotic grasper [17] with tunable stiffness by combininglayer jamming with a cable-driven compliant mechanism.The stiffness and load capacity of the gripper are greatlyincreased. A 24-fold and 30-fold increase are observedrespectively.

3) Pros and Cons:Pros• Payload — High load capacity is obtained due to the

parallel-guided configuration.• Mechanical Complexity — The design is simple and

compact with few moving parts.• Response — Specific design and part selection is used

to facilitate fast air removal and vacuum pressure reg-ulation.

• Scalability — Stiffness range and ratio could be variedby changing the beam dimensions, layers materials andnumber of jamming layers.

Cons• Modeling — Analytical model for unloading is difficult

due to the presence of hysteresis.

Fig. 9. Comparison of the stiffness ratio, stiffness range and response timefor tunable stiffness solutions.

• Control — Due to the use of high flow rate vacuumgenerator, sophisticated pressure control algorithm andhardware are required.

III. DISCUSSIONS

In Table I, each method is classified, and various propertiesare quantified. Fig. 9 gives a visual overview of the relativestrengths and weaknesses of each approach in terms ofstiffness and response time. It should be noted that not alldesign parameters are held constant between the variousprototypes. Mass, length, material properties, etc., all varysomewhat between the devices which affects their variablestiffness performance and makes them difficult to compare.However, we can still draw some broad conclusions.

Consider the response time of the various methods. Theshape morphing, rotating beam, pneumatic cylinder drivenslider, and layer jamming designs have the advantage of arelatively quick response. The motors of the shape morphingand rotating beam links drive the mechanical actuation mech-anism through cables, linkages or gears. With the selectionof sufficient motors and driving mechanisms, response timecould be further optimized. However, increasing the com-plexity of the driving mechanisms can have a negative impacton the stiffness ratio and mass. The pneumatic cylinderdriven slider and layer jamming concepts have a relativelysimpler configuration in terms of the number of movingparts and compactness. The pneumatic cylinder driven slidermoves much faster than the screw driven slider on the linearguide thus achieving a significant performance gain at thecost of more extensive work on control and calibration. Theresponse time of the layer jamming method is related to theamount of air that needs to be removed, the resistance in thefluid system, and the pump performance. The proposed layerjamming link optimizes the response time by using a tightvacuum bag, embedding internal air channels, and using ahigh flow rate vacuum generator. Achieving fast and precise

pressure regulation also requires effort designing the controlscheme.

Among the presented methods, the best stiffness ratiois attained by the layer jamming link. This is a result ofboth the relatively higher maximum stiffness – with nolosses to parasitic compliance in any mechanical linkages– and the low minimum stiffness. The stiffness ratio ofthe layer jamming link depends on the beam dimensions,material selection of both the beam and jamming layer,number of layers, and available vacuum level. To design formaximum stiffness ratio, a comprehensive analytical modeland extensive experiment are necessary. The links based onmorphing shape require high wire tension or linkage bendingstress to achieve high stiffness ratio. The rotating beam linkhas potential for much higher stiffness change, but design fora more secure beam boundary condition remains challenging.The links based on effective length change using sliders alsoneed to improve the slider mechanism to provide large andsecure clamping force on the beams.

Finally, the layer jamming approach has a unique benefit.It can change the stiffness while the link is deflected underexternal load by simply increasing or decreasing the vacuumpressure. Other links have more difficulty changing stiffnessunder external load. The wire and linkage driven mechanismsand the servo motor on the rotating beam link need largermotor torque to deform or actuate their pre-loaded beams.Similarly, a slider that moves on a deflected parallel-guidedbeam will have to overcome larger resistance.

IV. CONCLUSIONS

As in all engineering problems, the best variable stiffnessdesign depends on the problem objectives and technicalconstraints. This paper gives an overview of several candidatedesigns based on the concept of varying the stiffness of thestructure of what would otherwise be a rigid link. Thesedesigns are prototypes. Additional work is necessary to turneach of them into a mature technology, but many showpromise. The layer jamming approaches have a high stiffnessratio and range and a fast response at the cost of implement-ing pneumatic pressure control. The slider approache withscrew drive is simple but have slower response time. Theother slider method with pneumatic driven slider is quick butneed complex control algorithm. The variable cross-sectionbased approaches have good stiffness ratios and responsetimes, but suffer from mechanical complexity.

Promising future work involves implementing one or moreof these concepts to solve a practical engineering problemwith variable stiffness, such as multiple-section robotic arms,and iterating to improve the prototype designs and perfor-mance.

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TABLE ICOMPARISON OF PRESENTED VARIABLE STIFFNESS METHODS

Prototype Properties

Classification AcronymStiffness

Ratio1

Actuation

Time2

Stiffness

Range (N/mm)

Modeling

DifficultyReferences

Change ingeometry

Cross-sectionShapemorphing

SM1 2.2 <1 s 0.207 to 0.458 medium Sec. II-A, [10]

SM2 3.6 <1 s 0.540 to 1.936 medium Sec. II-A, [11]

Rotatingbeam link

RBL122 (ideal)

13.9 (prototype)<0.25 s 0.1613 to 2.245 medium Sec. II-B, [12]

Beam lengthSlidingbeam

SL1 20 10 s 0.500 to 10.049 easy Sec. II-C, [13]

SL2 10 0.6 s 0.358 to 3.407 easy Sec. II-C, [14]

Changein elasticproperties

StructuralInteractions

Layerjamming

LJ 75 0.25 s 0.0803 to 6.05 difficult Sec. II-D, [15]

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