Design and Modelling of a Cable-Driven Parallel-Series ...€¦ · ing the compliance. The “Jack...

Mech. Sci., 8, 65–77, 2017www.mech-sci.net/8/65/2017/doi:10.5194/ms-8-65-2017© Author(s) 2017. CC Attribution 3.0 License.

Design and Modelling of a Cable-Driven Parallel-SeriesHybrid Variable Stiffness Joint Mechanism for Robotics

Cihat Bora Yigit and Pinar BoyrazDepartment of Mechanical Engineering, Istanbul Technical University, Inonu Cd. No:65, 34437, Beyoglu,

Istanbul, Turkey

Correspondence to: Cihat Bora Yigit ([email protected]) and Pinar Boyraz ([email protected])

Received: 25 November 2016 – Revised: 12 February 2017 – Accepted: 24 February 2017 – Published: 22 March 2017

Abstract. The robotics, particularly the humanoid research field, needs new mechanisms to meet the crite-ria enforced by compliance, workspace requirements, motion profile characteristics and variable stiffness usinglightweight but robust designs. The mechanism proposed herein is a solution to this problem by a parallel-serieshybrid mechanism. The parallel term comes from two cable-driven plates supported by a compression spring inbetween. Furthermore, there is a two-part concentric shaft, passing through both plates connected by a universaljoint. Because of the kinematic constraints of the universal joint, the mechanism can be considered as a serialchain. The mechanism has 4 degrees of freedom (DOF) which are pitch, roll, yaw motions and translationalmovement in z axis for stiffness adjustment. The kinematic model is obtained to define the workspace. Thehelical spring is analysed by using Castigliano’s Theorem and the behaviour of bending and compression char-acteristics are presented which are validated by using finite element analysis (FEA). Hence, the dynamic modelof the mechanism is derived depending on the spring reaction forces and moments. The motion experiments areperformed to validate both kinematic and dynamic models. As a result, the proposed mechanism has a potentialuse in robotics especially in humanoid robot joints, considering the requirements of this robotic field.

1 Introduction

In recent years, there has been an emergent need in roboticsto develop new mechanisms that go beyond the conven-tional structures, focusing on compliant, lightweight and en-ergy efficient designs. Following this need, increasing num-ber of studies related with non-conventional robot mecha-nism design are reported (Grioli et al., 2015). In addition,large workspace, smooth motion profiles, and new mechan-ical structures with certain redundancies to ease the controlapplications can be considered as desired properties of suchmechanisms. Mizuuchi et al. (2002), Yang et al. (2005), Hamet al. (2009) and Vanderborght et al. (2013) also emphasizesthat such joint designs are needed in robotics instead of con-ventional structures.

Most of non-conventional mechanisms are studied in con-tinuum and hyper-redundant robotics and used especially inminimally invasive surgery (MIS) robotics. These structurescombine the compliancy and lightweight compact design re-quirements, although they may fall short of controllability

and introduce more complexity in modelling. For instance,in Gravagne et al. (2003) a planar continuum robot is in-troduced accounting the large deflection dynamics, examin-ing the dynamics of a planar backbone section. Another ap-proach for designing compliant and lightweight design canbe seen in Wendlandt and Sastry (1994), employing a paral-lel kinematic mechanism (PKM) together with a spring anda spherical joint in the middle. Instead of including PKMor spring, a typical approach is to propose multi-sections asgiven in Jones and Walker (2006) in order to increase thecontrollability of the mechanical structure. Additionally, aPKM mechanism, called cable-driven universal joint (CPUJ),is presented as a module (Lim et al., 2009) and multisec-tion properties are also examined in Lim et al. (2012). Thesemechanisms are used widely in continuum robotics, althoughsatisfying most of the requirements that come with the dis-advantage of complex dynamics. One way to overcome theproblem of complex and highly-nonlinear dynamics, spatialmodels considering the large deformation are used (Tunay,

Published by Copernicus Publications.

66 C. B. Yigit and P. Boyraz: Design and Modelling of CDPS Mechanism

2013). On the other hand, an extra difficulty comes fromthe compactness requirements. Since the continuum back-bone structures have often small diameter in nature, there isvery limited space to fit the actuation units within the mod-ule. Therefore, a cable-driven remote manipulation is pre-ferred with often an antagonist arrangement (Potkonjak et al.,2011). Although, it can be used for a different purpose, a verysimilar humanoid neck mechanism is proposed in Gao et al.(2012b) and is based on a compression spring and two par-allel plates, driven by four cables. The same research groupalso studied inverse kinematics of the structure in Gao et al.(2012a); however, it differs from the mechanism proposedin this work since in their mechanism, there is no serial linkto restrict the highly complex spring movement. A similarmechanism is introduced in Nori et al. (2007) and a partialkinematic model with control strategy is presented. In orderto handle the nonlinear dynamics of continuum robots, theelastic rod dynamic behavior can be taken as a model us-ing Cosserat Theory in Cao and Tucker (2008). The contin-uum robotics literature is diverse in terms of pointing the newdirections in mechanism design and a detailed review canbe found in Walker (2013). However, there are still alterna-tives to elastic back-bone and continuum structures that maylead to more feasible structures. For example novel 3 DOFfully parallel manipulators with rotational capabilities whichis given in Liu et al. (2005) can also be considered as goodcandidates for especially humanoid neck and joint design.The alternative multi-section designs can be structures suchas given in Woehrmann et al. (2013) with effective magneticactuation and interleaved continuum-rigid manipulators aspresented in Conrad et al. (2013). Moreover, a non-compliantbut similar structure to the mechanism proposed in this paper,known as 3-SPS mechanism is studied in Alici and Shirin-zadeh (2004) and Kim et al. (2015).

In addition to given studies in continuum robotics andcable-driven robotic mechanisms, robotic joint with vary-ing stiffness/compliance is also required. In the last decade,many inspiring developments occured in this area. In Hamet al. (2009) and Vanderborght et al. (2013), the most impor-tant ones are summarized and classified. Among this classifi-cation, the structure-controlled stiffness in Ham et al. (2009)uses the natural characteristics of the elastic element provid-ing the compliance. The “Jack Spring” mechanism in Hol-lander et al. (2005) controls the number of active coils byusing a screw mechanism to adjust the stiffness without us-ing any additional elements in the mechanism ,which simpli-fies the design. The proposed mechanism in this paper canbe included in this class since it uses natural mechanical be-haviour of the helical spring under bending and compressioneffects.

In this study, a cable-driven, compression spring-supported hybrid mechanism is proposed. The advantagesof the proposed mechanism are the unique combination ofmultiple traits of variable compliancy, hybrid parallel-serialstructure for better controllability and lightweight design.

The mechanism can be used in robotic joints especially inhumanoid design due to its smooth motion profile, its poten-tial in design of a multi-section robot as a module or sec-tion which is individually controllable. The main contribu-tion of this paper is to present a new joint mechanism de-sign which combines advantages of two different joint de-sign approaches. The first approach, which is given in Noriet al. (2007) and Gao et al. (2012b), is to use compressionspring and cable-driven actuation. The most important disad-vantage of these structures is modelling and control difficul-ties. These mechanisms do not have accurate mathematicalmodels as serial manipulators have. In the proposed mecha-nism, the additional shaft and the universal joint constrainsthe motion, which in turn allows treating it as a serial mech-anism and facilitates calculations of the kinematic and dy-namic modelling which are also presented. Furthermore, theshaft allows transferring yaw motion directly. In the secondapproach, Yang et al. (2005), Lim et al. (2009, 2012), Aliciand Shirinzadeh (2004) and Kim et al. (2015) use the two partshaft and the universal joint however the absence of compres-sion spring results in a stiff structure. On the other hand, pro-posed mechanism shows a compliant behaviour as a resultof the additional translation motion and the helical compres-sion spring. Linear helical compression spring provides non-linear stiffness characteristics under combined bending andcompression effects. Besides, nonlinear stiffness character-istics is essential for a variable stiffness actuator design andmost of the designs make use of complex nonlinear springmechanisms in Ham et al. (2009) and Vanderborght et al.(2013). Another contribution of this study to design literatureon variable stiffness actuators is the simplification of nonlin-ear stiffness mechanism. Therefore, the helical spring can beassumed as nonlinear stiffness mechanism under compres-sion and bending effects. Although it is a commonly usedmachine element, there exist few studies analysing combinedeffect of bending and compression on a helical spring. InLeech (1994), shape memory alloy wires are used as actua-tors and two different spring loading scenarios are analysed.First a single-sided load is applied, second a pure bendingis analysed. Both solutions are based on Castigliano’s The-orem. In this study, the same theorem is used and the solu-tion is improved numerically with the combination of bend-ing and compression scenario.

This paper first presents the main ideas in the conceptof cable-driven parallel-series hybrid mechanism (CDPS) inSect. 2. Then, the kinematic modelling is presented in Sect. 3.Next, in Sect. 4, the full dynamic modelling of this hybridmechanism is reported. In Sect. 5, both experimental struc-ture and results are detailed. Finally, in Sect. 6, the conclu-sions are drawn and further work on the mechanism designare proposed.

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C. B. Yigit and P. Boyraz: Design and Modelling of CDPS Mechanism 67

2 Description of Cable-Driven Parallel-Series(CDPS) Hybrid Mechanism

The proposed mechanism includes a lower plate and an upperplate which are used to hold the compression spring in con-centric position as shown in Fig. 1. The structure is driven bythree cables pulled or set free by three servo motors locatedunderneath the CDPS mechanism. The upper plate is able toperform rotation in two axes providing roll and pitch angles.This configuration is further supported by a concentric shaftwith a universal joint in the middle, which passes throughthe mechanism restricting and defining the bending motionof the compression spring. The concentric shaft is actuatedby a fourth motor to provide the movement about z axis –yaw angle. These four motors are encapsulated in a separatecase and can be installed at a far location from the mecha-nism which makes the mechanism lightweight and remotely-actuated. Despite the fact that the mechanism can be classi-fied as a parallel mechanism because of its main construc-tion properties, it is modelled as a serial mechanism due toserial kinematics imposed by the middle shaft and the univer-sal joint. Therefore, the mechanism is called as CDPS hybridmechanism.

The mechanism has 4 DOF in total, 3 of them are relatedwith motion and the other one is a translational motion alongz axis for adjusting the stiffness. Two rotations of upper plate(roll and pitch) are actuated in a cable-driven way, howeverthe motion is constrained by the shaft inside the spring andthe universal joint. The restriction in the motion simplifiesthe kinematic and dynamic calculations. The last DOF ofthe mechanism is the translational motion of the upper platealong z axis and is designed to adjust the stiffness value of thespring which determines the combined stiffness of the mech-anism in pitch and roll axes. Within this study, the control isachieved over stiffness via a structure-dependent way with-out using additional screw mechanism as in Hollander et al.(2005). The yaw motion herein is not inherently compliant,therefore modelling and experiments sections do not includeyaw motion in this work and examine other 3 DOF (pitch,roll and translation) . However, when required, the compliantbehavior can be induced using a torsional spring at the lowerplate.

3 Kinematic Modelling

The upper and lower plates of the CDPS mechanism are con-nected by a shaft with a universal joint. Therefore, excludingthe yaw motion, the neck mechanism is essentially consid-ered as a serial manipulator with rotation around two differ-ent axes at the center of the shaft such as pitch and roll and atranslation at the end of the shaft, summing up to 3 DOF. Theshaft having the universal joint in the middle of the structuretransmits the torque to change the yaw angle while providinga geometric constraint for resolving the spring forces becauseit determines the bending point. Therefore, in this mechanism

Figure 1. CDPS mechanism and its components, with motors, bear-ings and capstans.

the amount of the compression can be taken into account andbe described by distance between the center of the universaljoint and the upper plate. The yaw motion of shaft does notaffect the upper plate, because it is directly transmitted to therobotic head platform. The pitch and roll (θ , φ) angles of theupper plate are determined by the cable lengths. Therefore,the relative position of the upper plate with reference to thelower plate can be defined by using three generalised coordi-nates (θ , φ, d2). They are included in a vector which is de-noted as q, noting that the vectors are shown with bold italiccharacters. The variable d2 represents the distance betweenthe center of the universal joint and the upper plate. The ge-ometric variables used in deriving the kinematic model aregiven schematically in Fig. 2.

It is clearly seen in Fig. 3 that the neck mechanism is anRRP structure with 3 DOF, having the variables roll, pitchand translation (θ , φ, d2). The local frames (X1, Y 1, Z1) and(X2, Y 2, Z2) intersect and one of them is rotated by angleπ/2 with respect to the other one. The distance d1 is constantsince it is structurally fixed.

Having defined the variables of the kinematic model, nowone can formulate and solve the forward and inverse kine-matics. In forward kinematics, the cable lengths (l1, l2, l3) orthe motor shaft angles are inputs and the roll, pitch anglesand translation of the upper plate (θ , φ, d2) are the outputs.On the other hand, in inverse kinematics, (θ , φ, d2) are in-puts and the cable lengths or necessary motor shaft anglesare outputs. In order to derive the kinematics, we can usetransformation between the upper plate and the lower plate.Since the points where the cables are attached on the upperplate (P 1, P 2, P 3) can define the upper plane and the points

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Figure 2. Neck mechanism with variables, local and global axes,cable lengths, upper and lower plate cable assembly points.

where the cables pass through at the lower plate can definethe lower plane (O1, O2, O3), the transformation matrix isused to define P i according to Oi . The translation (Tr1) ofconstant length of the lower part of the shaft (d1), the rota-tions (Rot1) of pitch and roll (θ , φ) and the translation (Tr2)of the upper plate (d2) are taken into account in order to ob-tain this transformation. These transformations are given inEq. (1), all together forming the transformation matrix T03 inEq. (2).

Tr1 =

1 0 0 00 1 0 00 0 1 d10 0 0 1

,

Rot1 =

cθ sθsφ sθcφ 00 cφ −sφ 0−sφ cθsφ cθcφ d1

0 0 0 1

,

Tr2 =

1 0 0 00 1 0 00 0 1 d20 0 0 1

(1)

T03 =

cθ sθsφ sθcφ sθcφd20 cφ −sφ −sφd2−sφ cθsφ cθcφ cθcφd2+ d1

0 0 0 1

(2)

Then, the transformation matrix is used to calculate P i =[Pi,x,Pi,y,Pi,z]

T using Oi = [Oi,x,Oi,y,Oi,z]T via Eq. (3).

[P i1

]= T03

[Oi

1

](3)

Figure 3. Reduced serial kinematic model of neck mechanism withRRP structure (pitch, roll and translation).

In order to relate the cable lengths (l1, l2, l3) with upperplate position (θ , φ, d2), the definition of the Euclidean dis-tance between Oi and P i is used. According to this defini-tion, any cable length can be defined by the expression givenin Eq. (4).

li =

√(Pi,x −Oi,x)2+ (Pi,y −Oi,y)2+ (Pi,z−Oi,z)2,

where i = 1,2,3 (4)

If we could obtain the correct P i coordinates using Oi

from the kinematic solution, the nonlinear error function fidefined by Eq. (5) must be zero.

fi = li −

√(Pi,x −Oi,x)2+ (Pi,y −Oi,y)2+ (Pi,z−Oi,z)2

= 0 (5)

Using Eqs. (3) and (5), any variable can be obtained nu-merically using recursive Newton-Raphson algorithm givenby Eq. (6).

qk = qk−1− J−1F (6)

where J is the Jacobian matrix defined by Eq. (7) and qkis the kth iteration of the solution for the vector of the gen-eralised coordinates, while vector F is formed by equationsfi .



J=

∂f1θ

∂f1φ

∂f1d2

∂f2θ

∂f2φ

∂f2d2

∂f3θ

∂f3φ

∂f3d2

(7)

The solution of inverse kinematics is straightforward byusing Eq. (5) when desired state variables are known. Addi-tionally, a simple kinematic model is included, which mini-mizes the compression of the spring. The model assumes onecable is fixed at the initial position and takes only an orien-tation input. It is indeed possible to realize all the pitch androll angles by manipulating only two cables. For each cable,a separate solution loop is defined according to the selectednon-moving (i.e. idle) cable. The solution proceeds in the ap-propriate direction (i.e. towards minimizing approximationerror) for finding the lengths of the remaining cables to re-alize the roll and pitch angles given in the inverse kinematicproblem. According to this method, only two selected cablesare manipulated at any motion command. This approach isalso reflected at the simplified Jacobian matrix used in theinverse kinematic problem as given in Eq. (8).

J=

∂f1l2

∂f1l3

∂f1d2

∂f2l2

∂f2l3

∂f2d2

∂f3l2

∂f3l3

∂f3d2

=0 0 ∂f1

d2

1 0 ∂f2d2

0 1 ∂f3d2

(8)

The simplified kinematic model can be better understoodby looking at the Fig. 4 where top view of the lower plateis given. The virtual lines between the points of Oi and thecenter divides the plate into three areas. Desired pitch androll motions can be shown as a vector to determine the mov-ing and fixed cables. Two cables neighbouring the area whichincludes the motion vector, are pulled to perform the desiredmotion. Besides, opposite cable length is set to default ini-tial value. For example, to complete a 30◦ rotation in bothpitch and roll axes, the distance between upper plate and thelower plate has to be decreased in the direction of vector E.Thus, the neighbouring cables of this area which includes thevector are cable 1 and 3. The amount of pull or motor shaftrotation for these cables are calculated by using Eqs. (7) and(8). The length of cable 2 is set to default value.

The mechanism is implemented as a humanoid neck plat-form and it is used throughout the study. The same param-eters with the implemented mechanism, which are given inTable 1, are used for workspace simulation. The positions ofthe midpoint and the orientations of the upper plate are calcu-lated, using simplified forward kinematics, i.e., keeping onecable length constant at default value (95 mm) while chang-ing the other two within 50–95 mm range, as seen in Fig. 5.

The workspace for the simplified kinematic algorithm isgiven in Fig. 5a. The workspace is obtained as a sum ofthree distinct cases. In each case, one of the cable is heldat 95 mm. The midpoint of the upper plate is shown with dif-ferent colours and symbols for each case. Figure 5b and c

O1

O2O3

Y0

X0

E

Figure 4. Top view of the lower plate and an example motion com-mand vector.

Table 1. Simulation parameters in workspace analysis.

Variable Value [unit]

Lower shaft length d1 47 [mm]Length of constant cable 95 [mm]Distance between lower plate center and Oi points 40 [mm]Distance between upper plate center and Pi points 40 [mm]

show Y−Z andX−Z views of the same figure, respectively.The cable numbers 2 and 3 are placed symmetrically withrespect to x axis which results a symmetrical distribution inY −Z plane. Since the cable number 1 is on the x axis andthere is no cable on the opposite side, the midpoints are dis-tributed asymmetrically inX−Z plane. Thus, single roll mo-tions require at least two cables (2 and 3) have to be pulled.Even though, single positive pitch motion requires action oftwo cables similar to roll motion, only cable 1 is respon-sible for single negative roll motions. Figure 5d shows therange of motion in pitch and roll angles with respect to givenvariations of cable lengths. Obviously, each orientation valuegiven in the range of motion (between −30 and 30◦) is ob-tained by holding one cable length constant at default value.

Although, the range of motion (ROM) of the human neckdiffers from person to person regarding to their ages, gen-ders and physical attributes, average values for pitch (flexion-extension), roll (lateral bending) and yaw (axial rotation)axes are reported around 60, 40 and 80◦, respectively in thestudy of Ferrario et al. (2002). On the other hand, Bennettet al. (2002) shows that only a limited amount of the range(between 30 and 50 %) are utilised to complete daily tasks.The workspace of the humanoid neck platform is calculatedbetween −30 and 30◦ yielding 60◦ in total for pitch and rollmotions because of practical reasons such as strength of ca-bles and stall torque of motors. Additionally, the lower plateblocks the upper plate when roll and pitch angles are in-creased to a certain value which is related with the plate di-mensions. Thus, the main drawback of the mechanism can be



4020

X [mm]

0–20

–40

(a)

–40

–20

0

Y [mm]

20

60

80

90

100

110

70

40

Z [m

m]

l1 cnstl2 cnstl3 cnst

Y [mm]–30 –20 –10 0 10 20 30

Z [m

m]

60

70

80

90

100

110(b)


X [mm]–40 –30 –20 –10 0 10 20 30

Z [m

m]

60

70

80

90

100

110(c)


Pitch angle [deg]–50 0 50

Rol

l Ang

le [d

eg]

–40

–20

0

20

40(d)


Figure 5. Workspace of the proposed mechanism in (a) 3-D, (b) Y −Z, (c) X−Z view, (d) Pitch and Roll Angles.

considered as the motion boundaries when compared to thefull ROM of humans. However, it is enough for performingdaily activities when the proposed mechanism is used as ahumanoid neck joint.

4 Dynamic Modelling

In dynamic modelling, the result of kinematic model (posi-tion, velocity and acceleration) obtained in Sect. 3 is usedto define the dynamic motion of the mechanism under cablebased forces. The dynamic modelling is composed of threesequential steps: (i) reduction of 3-D model into a 2-D modelwithout information loss, (ii) force analysis on a bending he-lical spring using Castigliano’s Theorem, and (iii) completedynamic model.

4.1 Dimension Reduction of the Model

To decrease the complexity of computations and its cost dur-ing the analysis of helical spring under bending and com-pression effects, it is possible to reduce the dimension with-out information loss. Deformations of the helical spring canbe considered and calculated in a 2-D model because ofits cylindrical shape. In order to simplify the computationof bending of helical spring in 3-D coordinates, a differentEuler angles convention is used as follows: Two rotationsof mechanism around X0 and Y 0 axes are defined in Euler(X−Y −Z) convention and called roll and pitch angles, re-

spectively. However, these two angles can be defined in Eu-ler (Z−X−Y ) convention. Thus, the X−Y plane is rotatedaround Z0 axis so that bending of the helical spring appearsonly in this plane. Then the bending or rotation around newx axis is called as deflection angle and denoted with ρ.

4.2 Helical Spring Analysis

The helical spring is subjected to bending effects rather thanbuckling. The cables are assumed to have constant lengths,hence no plastic/elastic deformation are allowed for the ca-bles. The universal joint in the middle determines the bend-ing point of the helical spring. Figure 6 shows geometricalrelations and frames which are used to calculate the dynamicmodel of the system.

The frame xkykzk is attached to the bottom center all of thecoils. The variable k is the index number of the coils so thatx0y0z0 and X0Y0Z0 are coincident on the lower plate. Suc-cessive coil frames are rotated around their x axes in equallyso that zn−1 and Z3 are tangent. The vectorNk stands for theorigins of all coils.

The same idea is used for helical spring analysis as inLeech (1994) and each coil of the spring is separately anal-ysed. In order to use Castigliano’s Theorem, infinitesimal el-ements are defined in the helical spring. Figure 6 shows thehelical spring and the frame definitions. Angular position ofthe infinitesimal elements on kth coil is defined as α on xkykplane.



Figure 6. (a) Local frames on spring and infinitesimal element.(b) Top view of a spring coil.

In order to define strength properties of the infinitesimalelement on the spring coils, a 2-D frame is attached to all el-ements. The vector εx is placed on tangential direction andthe vector εy is placed on normal direction. This frame andthe other variables which are used to calculate the deforma-tion of the spring are shown in Fig. 6.

The cable forces are shown in Fig. 7 which are acting onthe upper plate are defined as T 1, T 2 and T 3. These forcesare unidirectional and are reduced to generalized forces def-inition which includes a force on −zn−1 direction which isF eq and a moment around xn−1 axis which is Meq. To cal-culate the relation between the deformation of the springand the generalized forces, the distances between upper platecenter point and central points of the infinitesimal elementshave to be defined. The position vector of the central pointof the upper plate P0 with respect to frame k is representedas krP0 . The position of an infinitesimal element “µ’ on kthcoil of the spring, krµ, is defined as in Eq. (9),

krµ=krα+

krP0 (9)

where krα is described as the position of the infinitesimalelement in the particular coil and is given in Eq. (10).

krα =

Rcos(α)Rsin(α)

(Nk,z+Nk+1,z−(Nk,z)α

25

(10)

The moment vector, kMµ, which is resulting from the equiv-alent force vector and is acting on the element µ is calculatedby using cross product given in Eq. (11).

kMµ=krµ×

kF eq (11)

Next, the total moment acting on the element (kMµt) is cal-culated as given in Eq. (12).

kMµt=kMµ+

kMeq (12)

In order to use the moment in Castigliano’s Formula, it hasto be defined in the element specific frame εxεy . Since the

Figure 7. Spring and cable forces acting on the mechanism.

given moments are defined in coil frame, a rotation is neededas α about zk axis, using the rotation matrix (Rotz(α)). Therotated moment vector (kMc) can be obtained as

kMc = Rotz(α)kMµt. (13)

Using the Castigliano’s Theorem, bending ((Ub1)k , (Ub2)k)and torsional ((Ut)k) strain energies are given in Eqs. (14),(15) and (16), respectively. The material properties which aremodulus of elasticity and shear modulus are represented asE andG. The geometric quantities I , J and R are defined asarea moment inertia, polar moment inertia and radius of thehelical spring.

(Ub1)k =

25∫0

(kMc,y)2

2EIRdα (14)

(Ub2)k =

25∫0

(kMc,z)2

2EIRdα (15)

(Ut)k =

25∫0

(kMc,x)2

2GJRdα (16)

Summation of three strain energies give full strain energy ofa single coil ((U )k) and it is added for every coil to calculatethe total strain energy of the spring (U ), which are given inEqs. (17) and (18), respectively.

(U )k = (Ub1)+ (Ub2)+ (Ut)k (17)

(U )=n−1∑k=0

(U )k (18)



The Castigliano’s Theorem states that derivatives of thestrain energies equal to the deformations which are the com-pression distance 1d2 and the 2-D bending angle ρ which isprojection of pitch and roll angles on 2-D plane.

1d2 =∂(U )∂Feq

(19)

ρ =∂(U )∂Meq

(20)

As a result, the nonlinear force-deformation relation or stiff-ness equation of the spring can be written in given form inEq. (20).[FeqMeq

]= [K]

[1d2ρ

](21)

The analysis is performed by using the Matlab SymbolicToolbox for each1d2 (between 0◦ and 32 mm with 1 mm in-crements) and ρ (between 0 and 40◦ with 1◦ increments). Theresult of 2×2 K matrices are collected within a look-up table.Wire and spring radii are 1.5 and 16.5 mm, respectively. Thenumber of active coils are 8 and the material of the springis ASTM A227. In the dynamic model, the algorithm usesthe appropriate K matrix values according to the generalisedcoordinates. With the definition of K matrix as in Eq. (22),the K12 and K21 elements are same due to the symmetricalstructure of the stiffness matrix. The simulation results forK11, K12 and K22 elements are given in Fig. 8.

[K]=[K11K12K21K22

](22)

K11 can be interpreted as the relation between the transla-tional motion of the upper plate 1d2 and the reaction forceof the spring in the same direction. Since the stiffness matrixis symmetric, K12 and K21 are equal. K12 is the relation be-tween combined rotation ρ and reaction force of the spring.Similarly, K21 connects 1d2 and the reaction moment. Thecombined rotation and the reaction moment are connected byK22. According to the Fig. 8, when both roll and pitch anglesare zero, the translational motion has no effect on the changeof the elements of stiffness matrix. Thus, this configurationof the mechanism can be called as a singular position forstiffness. On the other hand, all of the elements of the ma-trix decreases with the increase in the value of 1d2, whichmeans the spring gets softer with the compression. Comparedto other parameters, K11 responds to the change of ρ differ-ently. The decrease ofK11 is obvious with the increasing val-ues of ρ while the other parameters increase. So, the bendingmotion softens the spring for compression effect and it getsstiffer for rotation. Although the stiffness of the mechanism isadjusted with the translational motion of the upper plate, thebending angle has more effect on the stiffness of the mech-anism. As a result of this, the stiffness value depends on theconfiguration of the mechanism. But, it also changes with the

4030

rho [deg]

2010020

30

40

50

d2 [mm]

3000

3200

3400

3600

60

K11

[Nm

–1]

4030

rho [deg]

2010

020

40

d2 [mm]

0

20

40

60

K12

| K

21 [N

]

4030

rho [deg]

2010

020

40

d2 [mm]

0.8

1

1.2

1.4

60

K22

[Nm

]

Figure 8. Stiffness matrix change with respect to bending angle andcompression length.

variation of 1d2 which gives variable stiffness ability to themechanism.

Since finite element analysis (FEA) is commonly used toolfor structural mechanics, this technique is used to validatethe results of the stiffness analysis proposed in this study. InFig. 9a and b, initial shape and the deformed shape of themechanism are shown, respectively. The upper and the lowerplates and the shaft are assumed to be rigid parts to isolatespring in analysis particularly and to simplify the model. Ad-ditionally, the spring is fixed to lower and upper plates. A rev-olute joint relation is defined between two shaft parts. Simi-larly, a translational joint constraint is described between theupper plate and the lower part of the shaft. A prescribed mo-tion profile is applied to these relations as 10 mm translationand 40◦ rotation. As a result, reaction forces and moments ofthe spring are obtained during the motion.

FEA and proposed stiffness analysis technique are com-pared in Fig. 10. Both translation and rotation motions areapplied as a linearly increasing function, i.e. ramp function.Thus, x axes of Fig. 10a and b are given as compressionin millimetres and bending in degrees. The reaction forces



Figure 9. The models of the mechanism in finite element analysis.(a) Initial, (b) deformed.

and moments are calculated using FEA and stiffness valueswhich are given in Fig. 8. The mean errors between FEAand the proposed method are 2.2 % for reaction force and1.5 % for reaction moment. A convergence problem can beseen in the initial steps of FEA method. Moreover, quadraticfunction behaviour can be seen from both force and momentgraphs which is a result of linearly changing stiffness val-ues. Subsequently, the results of FEA demonstrates that lin-ear helical spring shows a nonlinear behaviour subjected tobending and compression effects.

4.3 Complete Dynamic Model

The equation of motion of a serial manipulator is defined asin Eq. (23).

M(q)q̈ +C(q, q̇)q̇ +G(q)= B(q)T +F s(q) (23)

The vector q is the generalized coordinates vector [θ,φ,d2]T,

q̇ and q̈ are the velocity and acceleration vectors, respec-tively. M is the mass matrix, C includes Coriolis and cen-trifugal forces and G is the gravity matrix. There are no mo-tors connected joints directly. Cable tensions are taken intoaccount with a mapping matrix B which is given in Eq. (24).F s is the generalized force vector of helical spring and si isthe unit vector on the ith cable direction. The variables Pi,x ,Pi,y , Pi,z are the positions of connection points of upper plateand each cable.

B(q)=[s1,z s2,z s3,z

−P1,zs1,y +P1y s1,z −P2,zs2,y +P2y s1,z −P2,zs2,y +P2y s2,zP1,zs1,x −P1x s1,z P2,zs2,x −P2x s2,z P2,zs2,x −P2x s2,z

](24)

5 Experiments and Results

In this section, the implementation of the proposed mech-anism is explained. First, the mechanical construction and

Bending [deg]0 5 10 15 20 25 30 35 40

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Figure 10. Behaviour of the spring under compression and bendingeffects. (a) Reaction force. (b) Reaction moment.

components are introduced. Then control structure, sensorsand the software are given. In the experimental results sec-tion, the mechanism performs a controlled motion where theparameters are measured to validate the kinematic and dy-namic models. The implemented mechanism is used as hu-manoid neck in UMAY project (Boyraz et al., 2013).

5.1 Experimental Setup

The mechanism is constructed in order to perform validationof both kinematic and dynamic models. Aluminum (7075)is used in upper and lower plates. The shafts are made of



Figure 11. UMAY neck mechanism.

chromed steel (AISI E 52100). Four Robotis Dynamixel MX-28 motors, which has 2.5 Nm stall torque at 12 V, are used.One of the motors are directly connected to shaft in orderto perform yaw motion. Other three motors are connectedto the cables with 14 mm radius capstans. Fishing lines (with0.5 mm radius) are used as cable material which have a 100 Npayload capacity. The friction force between the lower plateand the cables are reduced using PTFE tubes (with 2 mm in-ner radius) in O1, O2 and O3 points. A picture of the imple-mented mechanism is given in Fig. 11 and a video is pre-sented in the Supplement. The video is composed of ran-domly chosen pitch and roll motion combinations. Althoughthe translational motion of the upper plate in zero positionsof pitch and roll angles neither affects the stiffness nor theorientation, it is illustrated in the video to emphasize the mo-tion. Three cables are pulled equal amount in length to obtainpure translational motion in the video.

5.2 Results

The experimental setup is equipped with Razor 9 DOF iner-tial measurement unit (IMU) which is attached on the upperplate along the x axis. The IMU is used to collect the ori-

GUI nodeCubic trajectory

nodeDynamixel

trajectory node

Dynamixel motors

CDPS mechanismIMU

Inversekinematic server

User input(pitch and roll)

Trajectory foreach motor

Motorangles

Pitchroll

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Pitch and rollangles

Motorcurrents

Figure 12. Control structure used in experiments.

entation data of output shaft in pitch and roll axes. IMU isused to create a truth table to validate the orientation outputs.The only feedback devices are the encoders of Dynamixelmotors.

The whole position control structure used in experimentsis programmed within Robot Operating System (ROS) whichcan be seen in Fig. 12. It is worth noting that the con-trol scheme considers only kinematics of the mechanism. Agraphical user interface (GUI) node is designed to collectdata from the human user. The commands are sent to cu-bic trajectory node which enquires necessary motor anglesfrom inverse kinematic server. Since the aim is to validatethe models, the server calculates the necessary motor anglechanges with the simplified solution of inverse kinematics asexplained in Sect. 3. In this solution, one of the cables areheld at a constant position. Therefore, the inverse kinematicserver returns with the other cable lengths and d2 parameterwhile ignoring the stiffness value. The cubic trajectory nodetakes these arguments as input and calculates a cubic poly-nomial for motor angles. The total trajectory time is given as2 s for each trajectory which is a feasible amount of time forgiven workspace and motors. The cubic trajectory of motorshaft angles are sent to Dynamixel trajectory node.

In this section, all given data are obtained from a singleone of several experiments. Thus, all figures are connected toeach other. The experimental results are given for pitch/rollangles and related motors. The yaw axis motion and dynam-ics are excluded because they do not affect the motion of theupper plate and stiffness. The experiment is performed for80 s which is long enough for an arbitrary duration to validaterandom pitch and roll input commands. A user gives randompitch and roll angle commands using GUI which is given indetail in this section. The experimental results are presentedin two categories. First, in order to validate kinematic model,orientation of the output shaft and forward kinematics arecompared. Second, motor load data and the inverse dynam-ics solution are examined.

Kinematic results of the experiment are given in Fig. 13for pitch and roll axes. The user command is shown as stepfunction in figures which starts a 2 s cubic trajectory. Forward



Time [s]0 10 20 30 40 50 60 70 80

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Figure 13. Angular positions of the output shaft in pitch and rollaxes.

kinematic solutions are calculated using motor shaft angleswhich are obtained from experiment.

Inverse dynamic algorithm is performed using user com-mands collected from GUI in order to estimate cable ten-sions for each motor. Additionally, the load data is used toestimate cable tensions which is provided by Dynamixel mo-tors and obtained from motor current values. Two cable ten-sion estimations are compared for front cable, left cable andright cable as can be seen in Fig. 14. The general behaviorof these data for each motor are similar, but there are errorsin some parts of these results because of three reasons. First,the motor current measurements are very noisy due to theuse of pulse width modulation of the motor low level con-troller. Second, neglected effects such as friction in gearboxof motor, friction between cable and PTFE parts appears dif-ference between torque estimation from motor current andinverse dynamics model result. Finally, DC motors have noload current which is not taken into account in this estima-tion.

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Figure 14. Cable tension estimations from inverse dynamics andmotor currents.

6 Conclusions

A cable-driven parallel-series hybrid variable stiffness jointmechanism is presented in this paper. First, the mechanismis explained reporting advantages and disadvantages. Then,kinematic analysis of the mechanism is given considering itas a serial manipulator, simplifying the solution. By usingthe forward kinematics the workspace is computed. Then,dynamic model is presented. The stiffness analysis of heli-cal spring between upper and lower plates is explained andfor the spring which is used in the experimental setup, thelook-up table of stiffness matrix according to compressionand bending angle is calculated. To validate the results of theproposed helical spring analysis method, FEA is applied fora motion scenario and the results of both analysis are com-pared.



The proposed mechanism is implemented and operated tovalidate the models given in the paper. Experiments are stud-ied for kinematics and dynamics. In kinematics part, the ori-entation of the output shaft is compared to forward kinemat-ics computations. It is shown that forward kinematics canpredict motion profile of the output shaft. In dynamics sec-tion, two estimations of cable tensions, one from motor cur-rent and the other one from inverse dynamics calculations,are examined. It can be seen that, the results from the inversedynamic calculations are similar to the motor current values.

In future work, the closed-loop stiffness control and hy-brid position-force control techniques are going to be ap-plied to the mechanism. With its large workspace, smoothmotion profiles, lightweight mechanical structure and vari-able stiffness properties, the possible application area to thismechanism could be humanoid robots. It is indeed used asa humanoid neck mechanism in UMAY project. Humanoidrobots interact with their environments and humans. Forceinteraction with humans requires safe operation conditions.After applying force control techniques, it can be used in hu-manoid joints where the force interaction occurs such as armsand hands in a safe manner as a result of compliant nature ofthe mechanism.

Data availability. All the data used in this manuscript can be ob-tained by requesting from the corresponding author.

The Supplement related to this article is available onlineat doi:10.5194/ms-8-65-2017-supplement.

Competing interests. The authors declare that they have no con-flict of interest.

Acknowledgements. The project is supported by ScientificResearch Projects Unit of Istanbul Technical University YoungResearcher Award.

Edited by: M. WojtyraReviewed by: two anonymous referees

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