International Journal of Advanced Robotic Systems ARTICLE

Conceptual Design and Analysis of Four Types of Variable Stiffness Actuators Based on Spring Pretension

Regular Paper

Jishu Guo1 and Guohui Tian1*

1 School of Control Science and Engineering, Shandong University, Jinan, Shandong, China *Corresponding author(s) E-mail: [email protected]

Received 12 November 2014; Accepted 17 March 2015

DOI: 10.5772/60580

© 2015 The Author(s). Licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract between the output stiffness and the angular deflection is fairly linear. The conceptual layouts and working princi‐ The variable stiffness actuator (VSA) is an open research ples are elaborated for the four actuators. The characteris‐ field. This paper introduces the conceptual design and tics of torque and stiffness of the VSAs are presented, and analysis of four types of novel variable stiffness actuators the mechanical solutions are illustrated. (VSAs). The main novelty of this paper is focused on the convenience control of the torque and stiffness of the Keywords Variable stiffness actuator, Approximate actuator. We do not need to design complicated control quadratic spring, Characteristics of torque and stiffness, strategies to achieve the desired actuating torque and Convenient to control, Mechanical solution output stiffness. This feature is beneficial for real-time control. In order to achieve this advantage, three types of approximate quadratic springs and four types of stiffness regulation mechanisms are presented. For the first VSA, the 1. Introduction relationship between the output stiffness and angular deflection is close to linear, and the fitted quadratic spring In recent years, interest in compliant actuation systems has is easy to implement and compact. For the second VSA, the grown due to a change of trends in robotic applications. output stiffness is only related to the working radius, and Since classical stiff actuators lack adaptability, energy the relationship between the exerted torque and angular efficiency and interaction safety, compliant actuators are deflection is linear. For the third VSA, two equivalent strongly recommended when robots coexist with human quadratic torsion springs are used. Compared with the beings or physically interact with them [1, 2]. Among the existing quadratic torsion springs, the design method is compliant actuators, the requirement of mechanically simple and the structure size is compact. The stiffness and variable stiffness joints is fulfilled by a new generation of torque is a linear function of the kinematic parameters of actuators called VSAs. The VSA is characterized by the the VSA. The novelty of the fourth VSA is the use of the ability for the output stiffness to be varied independently tension spring set. The approximate quadratic tension from its position. The intrinsic compliance of the VSA can spring is compact and easy to implement. The relationship ensure the safety of human-robot or robot-environment

Int J Adv Robot Syst, 2015, 12:62 | doi: 10.5772/60580 1 interaction, even for unexpected collisions. The optimal of the actuators, and the nonlinear characteristics of the criteria of the VSA are elaborated as follows: (1) Stiffness actuating torque and output stiffness require an advanced should be changed without energy consumption. (2) and complicated control strategy, these actuators are still Energy efficiency of stiffness adjustment should be 100%. widely used for different application requirements. (3) No energy should be required to maintain the stiffness. In this paper, four types of variable stiffness actuators (4) Stiffness should be independent from the external load. based on spring pretension are presented. Different from (5) Range of stiffness should be unlimited. (6) No inertia the most existing VSAs, the novelty of the proposed VSAs should be added to the output link due to the stiffness is that the proposed internal elastic elements and stiffness adjustment mechanism. (7) Maximum capacity of the adjustment mechanisms are convenient for controlling the energy storage should be accessible [3, 4]. actuating torque and output stiffness. Therefore, the In order to achieve the features described above, many control algorithms for torque and stiffness are relatively research efforts have been put into the development of the simple. The real time and convenience of the stiffness VSA. Some designs rely on changing the transmission ratio control is beneficial for improving the efficiency of task between the output and the internal elastic elements. The execution. In order to achieve this feature, four different AwAS-II [5], CompAct-VSA [6], CompAct-ARS [7], HDAU stiffness adjustment mechanisms and three different [8], the vsaUT [9], the vsaUT-II [10], and the mVSA-UT [11] approximate quadratic elastic elements were designed for use a lever arm of variable effective length to regulate the satisfying the kinematic structure requirements. transmission ratio. As is elaborated in [12], the output The common characteristics of the four types of VSAs are stiffness of these variable stiffness actuators can be changed described as follows. 1) The output stiffness can be changed without any energy injection into or extraction from the independently of output position. 2) By using the screw- internal elastic elements. This implies that all the energy nut mechanism or worm and worm wheel mechanism, supplied by the internal actuated degrees of freedom can energy is not required to maintain the stiffness. 3) By be used to do work on the output without being stored in limiting the deformation of the spring, the output stiffness the internal springs. This is an energy efficient way to of the VSA can be regulated to infinite. This situation is regulate the output stiffness, especially in mobile applica‐ necessary in cases where compliance is not desired. 4) The tions. However, from a control perspective, the nonlinear stiffness regulation is based on the principle of the spring relationship between torque and angular deflection or pretension. 5) The error of torque and stiffness between the stiffness and angular deflection leads to difficulties in prescribed and actual achieved is small. Therefore, the applications. Besides, the output stiffness is often related to proposed approximate quadratic springs are used to the exerted torque or angular deflection. To address this replace the desired purely quadratic spring. problem, significant research has been carried out on the control of these actuators. The complicated control strat‐ This paper is organized as follows. Section 2 introduces the egies may be used for achieving the desired output working principles of the proposed approximate quadratic stiffness, since the stiffness characteristic is intrinsically springs. Section 3 describes the conceptual design and nonlinear. The online stiffness estimation method is analysis of the first VSA. In this section, the conceptual proposed, since there are no sensors available for a direct layout and working principles of the VSA are presented measurement of the varying stiffness online [13]. The time and the characteristics of the actuating torque and output delay caused by the complicated control strategy is not stiffness are illustrated. The mechanical solution of the VSA beneficial for real-time control. Real time, and the conven‐ is described. Section 4 illustrates the conceptual design and ience of the control method, is also important for the VSA. analysis of the second VSA. Section 5 presents the concep‐ tual design and analysis of the third VSA. Subsequently, Compared to the actuators mentioned above, other designs the conceptual design and analysis of the fourth VSA is are based on changing the pretension of the internal elastic given in section 6. Finally, conclusions and future works elements, e.g., MACCEPA 2.0 [14], VSA-II [15], QA-Joint are discussed in section 7. [16], DLR-FSJ [17], and the VSA-HD [18]. A characteristic of these systems is that the energy is put into or extracted 2. Working principles of the approximate quadratic from the elastic elements during stiffness changes. Accord‐ springs ing to the functional concept and layout, they can be classified into antagonistic configurations or serial config‐ Based on the mechanical solutions, for the first VSA, the urations. In antagonistic configurations, two elastic relationship between the output stiffness and the angular elements act in opposite directions at the joint, and the deflection is linear for quadratic spring configuration. For equilibrium position and output stiffness of the joint are the third VSA, the linear relationship between the exerted controlled by two motors. In series setups, one motor is torque and the angular deflection can be achieved for used to change the position and the second motor is used quadratic torsion spring configuration. For the fourth VSA, to change the stiffness by setting the pretension of an elastic the relationship between the output stiffness and the element. Although the pretension of the internal elastic angular deflection is linear for the quadratic tension spring elements has a negative influence on the power-flow ratio configuration. Therefore, for the convenience of control, the

2 Int J Adv Robot Syst, 2015, 12:62 | doi: 10.5772/60580 relationship between the output stiffness and the angular criteria of the compression spring can be seen in [22]. The deflection is linear for quadratic spring configuration. For parameters for the compression spring set were designed the third VSA, the linear relationship between the exerted and presented in Table 1. torque and quadraticthe angular springs deflection in these VSAs can are be essential. achieved However, for the quadratic torsionreliable manufacturingspring configuration. of the quadratic For springthe fourth is difficult, VSA, the relationshipand the commercial between availability the output of stiffness a purely and quadratic the angular springdeflection is extremely is lin earlimited. for Thethe quadratic quadratic spring tension construct‐ spring configuration.ed by the linearTherefore, springs for and the a nonlinearconvenience transmission of control, the mechanismquadratic springs(i.e., cable, in camthese or VSAscurved are surface, essential. etc.) often has a relatively large size and weight, which ultimately However, thelimits reliable their implementation manufacturing in multi-DOF of the quadraticrobotic systems spring is difficult, and the commercial availability of a [19, 20, 21]. Moreover, the special frame or cams may purely quadraticrequire relativelyspring complicatedis extremely design limited. and manufacturing. The FigureFigure 1. Schematic 1. Schematic ofof the the fitted fitted quadratic quadratic compression compression spring spring. quadratic springIn order constructed to realize the by quadratic the linear spring springs from a linearand aspring As shown in Figure 1, the number of the compression nonlinear transmissionand reduce the mechanismdifficulty in the (i.e., design, cable, three camnovel typesor of As shown in Figure 1, the number of the compression spring is 12. A different initial height for each spring inside curved surface,approximate etc.) often quadratic has a springs relatively are proposedlarge size and and demon‐ spring is 12. A different initial height for each spring strated. the spring set pedestal (2) is pre-set. The working distance weight, which ultimately limits their implementation in insidebetween the spring the two set adjacent pedestal springs (2) is Δxpre-set.=2mm. TheThe workingmaxi‐ multi-DOF robotic systems [19, 20, 21]. Moreover, the 2.1 Conceptual design of the fitted quadratic spring distancemum amountbetween of deformationthe two adjacent of the spring springs set is x maxis =24mm.Δx=2mm. special frame or cams may require relatively complicated The Whenmaximum both sides amount of the of spring deformation pedestal ofguide the sleevesspring (1) set is design andThe manufacturing. first type of approximate In order quadraticto realize spring the is the xmax=24mm.contact with When each both other, sides the of springs the spring could pedestal not be com‐ guide quadratic springcompression from springa lin setear in spring parallel andtype. Sincereduce springs the with sleevespressed. (1) contactIn order withto reduce each the other, friction the between springs the could spring not different parameters can be used to constitute a spring set set pedestal (2) and the guide groove (3), and to ensure the difficulty in the design, three novel types of approximate be compressed. In order to reduce the friction between with arbitrary stiffness, the linear compression spring is spring set pedestal can withstand a relatively uniform quadratic springsused with are differentproposed parameters and demonstrated. to constitute an approxi‐ the pressure,spring set the pedestal arrangement (2) andof the the springs guide in groovethe spring (3), set and mate quadratic compression spring. to ensurepedestal shouldthe spring be optimized, set pedestal as shown canin Figure withstand 1. The a 2.1 Conceptual design of the fitted quadratic spring relativelyrelationship uniform between pressure, force F(x) the (N) arrangement and displacement of x the

springs(mm) in of the the spring spring set set pedestal is formulated should in equation be optimized, (1). By as The first typeID of approximat d D e nquadratic k springt is theH 0 shownusing in theFigure least 1.squares The relationship method, the equation between for force the fitted F(x) (N) compression spring(mm) set (mm)in parallel type.(N/mm) Since(mm) springs(mm) quadratic spring F (x) and the simplified equation F (x) are and displacement x1 (mm) of the spring set is formulated2 with different1 parameters 0.5 4 can 30be used 0.3215 to constitute 1.975 a 60 formulated in equation (2). in equation (1). By using the least squares method, the spring set with2 arbitrary 0.6 stiffness, 4 30 the linear 0.6666 compression 1.9 58 equation for the fitted quadratic spring F1(x) and the spring is used with different parameters to constitute an ìk1 x 0

definitions in Table 1 are described as follows: D represents F [N] Table 1. Specifications of the compression spring set. compression spring can be used in6 the VSA. the middle diameter of the spring, d is the diameter of 0.15

0.1 4 spring wire, k represents the stiffness of the compression Piecewise linear spring set Piecewise linearReal spring stiffness set In the design, the end portion of the spring is flat. The 0.25 0.05 8 spring, t is the pitch between two adjacent active coils, n is 2 Fitted stiffness 0.2 0 K:RK-FK number of the supporting coil is one. The compression F [N]

the number of active coils of the spring, and H0 is the free  60 0.15 -0.05

spring materiallength is of thepian spring.o wire, The and main D design grade. criteria The of the force output Difference

-0.1 K [N/mm] of stiffness Comparison 4-2 parameter definitionscompression in spring Table can 1 beare seen described in [22]. The as parametersfollows: for 0.1 0 5 10 15 20 25 0 5 10 15 20 25 Deformation amount of the spring set x[mm] Deformation amount of the springReal setstiffness x[mm] 0.05 D representsthe the compression middle diameter spring set of were the designed spring, andd is presented the 2 Fitted stiffness Figure 2. Characteristic of the fitted quadratic spring in Table 1. 0Figure 2. Characteristic of the fitted quadratic spring K:RK-FK diameter of spring wire, k represents the stiffness of the 0 -0.05 compression spring, t is the pitch between two adjacent force output Difference

-0.1 KComparison of stiffness [N/mm] -2 Jishu Guo and Guohui Tian: 3 Conceptual Design and Analysis of Four2.2.0 Types Conceptual5 of 10Variable15 Stiffness design20 25Actuators of 0the Based approximate5 on 10Spring15 Pretension20 quadratic25 torsion spring active coils, n is the number of active coils of the spring, Deformation amount of the spring set x[mm] Deformation amount of the spring set x[mm] and H0 is the free length of the spring. The main design FigureThe 2. calculationCharacteristic scheme of the of fitted the quadratic quadratic torsion spring spring is shown in Figure 3, and the parameter definitions are presented in Table 2. The cam is being deflected by an angle θ due to the applied actuating torque τ(θ). The follower is being pushed up by the force Fnn, and the spring is compressed in the y direction. The amount of deformation of the spring depends on the pitch radius of the pitch profile. The force at the equilibrium state of the follower is obtained such as:

 F=KSSpp0S r  r =Ks  . (3)      sin   0  FNNykkrFFcosSn12 F r

Figure 3. Mechanism schematic of the quadratic torsion spring

Symbol Description

KS Stiffness of the compression spring Fs Spring force acting on the follower in y direction

N1 Reaction force from linear bearing to the follower

N2 Reaction force from linear bearing to the follower

μk Kinetic friction coefficient of the linear bearing

μr Rolling friction coefficient between roller and cam s Amount of deformation of the spring

smax Allowable maximum deformation of the spring Fn Normal force on the cam surface Fr Friction force between the roller and cam

rpθ Pitch radius at angular deflection θ

rp0 Radius of base circle of the cam α Pressure angle of the cam-roller mechanism θ Angular deflection of the cam-roller mechanism

rt Radius of the roller Table 2. Symbols used for schematic description

In the mechanical solution, the rolling bearing and the linear bearing are used in the follower. The friction force can be neglected in the calculation due to the relative small friction coefficients. To simplify the calculation, the inertia of the 2.2 Conceptual design of the approximate quadratic torsion Symbol Description spring KS Stiffness of the compression spring

The calculation scheme of the quadratic torsion spring is Fs Spring force acting on the follower in y direction shown in Figure 3, and the parameter definitions are N Reaction force from linear bearing to the follower presented in Table 2. The cam is being deflected by an angle 1 θ due to the applied actuating torque τ(θ). The follower is N2 Reaction force from linear bearing to the follower

being pushed up by the force Fn, and the spring is com‐ μk Kinetic friction coefficient of the linear bearing pressed in the y direction. The amount of deformation of μr Rolling friction coefficient between roller and cam the spring depends on the pitch radius of the pitch profile. s Amount of deformation of the spring The force at the equilibrium state of the follower is obtained such as: smax Allowable maximum deformation of the spring

Fn Normal force on the cam surface

ì Fr Friction force between the roller and cam ïFS =K S( r pq - r p0) =s K S í . (3) å = - +a - m - m + msin a = 0 rpθ Pitch radius at angular deflection θ îï Fy FS F n cos kNN1 k 2 rFr θ Angular deflection of the cam-roller r Radius of base circle of the cam FS =K S (r p  r p0 )=s K S p0 mechanism  . (3) rαt Pressure Radiusangle of theof cam-rollerthe roller mechanism FNNy FS F n cos  k 1  k 2  r F r sin  0

θ Angular deflection of the cam-roller mechanism Table 2. Symbols used for schematic description.  rt Radius of the roller N2 k Fs smax N2 TableIn the 2. Symbols mechanical used for solution, schematic thedescription rolling bearing and the Fn s linear bearing are used in the follower. The friction force Fr N1

N1 k can be neglected in the calculation due to the relative r ìt( q) =FStan a r pq = F n sin a r pq small frictionï coefficients. To simplify the calculation, the y r p í ¶t( q ) (4) θ inertia of theïK =moving bodies is not considered, and the x α î ¶q O mass of the spring is considered null. The characteristics () O of actuating torque and output stiffness of the equivalent rp0 Thequadratic characteristic torsion of spring a purely can bequadratic expressed torsion as follows spring: is 2 defined by τ(θ)=aθ . From the principle of virtual work: (  ) F tan  r  F sin  r (a) Principle analysis of the equivalent quadratic torsion  S p n p spring  () t( q)d q= l s K ds (4) K  S (5)  

where s is the amount of deformation of the spring with () The characteristic of a purely quadratic torsion spring is respect to its natural length. The number of the springs defined by τ(θ)=aθ2. From the principle of virtual work: compressed during the rotation of the cam is defined as λ.

s( ) It is noted that the profile of the cam is not affected by the 0 (  )d  =  s K ds (5) A1 A S y number of compression springs. Suppose that τ(θ)>0 and rt B1 Y B0 rp0 s>0. The expression of s and α is formulated such as: θ where s is the amount of deformation of the spring with respect to its natural length. The number of the springs x X O y compressedì during the rotation of the cam is defined as λ. ï 2 l 2 x t q q= q q = O It is notedï ( that)d the a profile d of K theS d( scam) is not affected by the 2 numberï of compression springs. Suppose that τ(θ)>0 and q q ï 2 2 2 2 2 s>0. The expressiont ()u du= of s and au α is du formul= s - sated0 such as: (b) Cam profile calculation scheme ïlK ò0 lK ò0 ï S S ï 3 ï 2aq 2 Figure 3. Mechanism schematic of the quadratic torsion ís=22 + s0  (6) ()() d 3 lK a  d   KS d s Figure 3. Mechanism schematic ofspring the quadratic torsion spring ï S 2 ï 2   223lK s 2 2 2 In the mechanical solution, the rolling bearing and the ïa = ()uS max du  au du  s  s ï 003 0 Symbol Description KKSS2qmax linear bearing are used in the follower. The friction force ï KS Stiffness of the compression spring 3 æ ö can be neglected in the calculation due to the relative small ï 2a 21 ds ïssa= arctanç0 ÷ (6) Fs Spring force acting on the follower in y 3K ç+ q ÷ friction coefficients. To simplify the calculation, the inertia îï S ès rp0 d ø direction 2 of the moving bodies is not considered, and the mass of the  3Ks a  S max springN1 is consideredReaction null. force The from characteristics linear bearing of toactuating the 3  2max torque and output stiffness offollower the equivalent quadratic where  s0 is the displacement of the spring when θ=0, and 1 ds torsionN2 springReaction can be expressed force from as linear follows: bearing to the s0=0. In the arctan(calculation of τ(θ) ) , the units of KS, smax, θmax, and  s r d follower  p0 4 Int J Advμk Robot Syst,Kinetic 2015, friction 12:62 | doi:coefficient 10.5772/60580 of the linear bearing where s0 is the displacement of the spring when θ=0, and μr Rolling friction coefficient between roller s0=0. In the calculation of τ(θ), the units of KS, smax, θmax, and cam and θ are N/m, m, rad, and rad, respectively. In order to s Amount of deformation of the spring more clearly express the profile of the cam, in the smax Allowable maximum deformation of the calculation of the s, the units of KS, smax, θmax, and θ are spring N/mm, mm, rad, and rad, respectively. Therefore, the Fn Normal force on the cam surface unit of s is mm. As is described in Figure 3 (b), the profile Fr Friction force between the roller and cam of the cam can be determined by considering the motion of the follower relative to the cam profile's frame of rpθ Pitch radius at angular deflection θ reference (i.e., by assuming the cam as fixed). At the rp0 Radius of base circle of the cam initial position, the tangency point of the roller on the α Pressure angle of the cam-roller mechanism cam profile is denoted with B0. When the follower is The cam parameters are defined as: rt=6mm, rp0p0=25mm, αmaxmax=33.1°, λ=4, and a=12.35. The maximum allowable angular deflection is ±0.7454rad. The profiles of the cams and the curves of pressure angles are shown in Figure 4.

θ are N/m, m, rad, and rad, respectively. In order to more Profile of cam for two springs configuration Profile of cam for four springs configuration 90 50 90 50 clearly express the profile of the cam, in the calculation of 120 60 120 60 Theoretical contour 25 150 25 the s, the units of KS, smax, θmax, and θ are N/mm, mm, rad, 150 30Inner enveloping line 30 and rad, respectively. Therefore, the unit of s is mm. As is Radius of base circle described in Figure 3 (b), the profile of the cam can be 180 0 180 0

determined by considering the motion of the follower 210 330 210 330

relative to the cam profile's frame of reference (i.e., by 240 300 240 300 270 270 assuming the cam as fixed). At the initial position, the Pressure angle for two springs configuration Pressure angle for four springs configuration tangency point of the roller on the cam profile is denoted 20 35 30 with B0. When the follower is rotated by an angle θ in the 15 rotated by an angle θ in the CCW direction around the spring is equipped with two linear25 bearings (4) and two CCW direction around the cam, the tangent point is shifted [degree] [degree] α α 20 cam, the tangent point is shifted from point B0 to point B1. guide10 shafts (1). The spring is placed in the pedestal (2), from point B0 to point B1. Let the coordinates of the tangen‐ 15 1 Let the coordinates of the tangency point B be X and Y, and the cam (3) is used to drive10 the roller. The pressure cy point B1 be X and Y, and the coordinates of the A1 is 5 and denotedthe coordinates by x and of y. theFinally, A1 is the denoted profile byof thex and cam y . can be angle is an important parameter5 for evaluating the Pressure angle Pressure angle 0 0 Finally,expressed the profile as offollows: the cam can be expressed as follows: friction0 dissipation,0.5 1 and1.5 the2 range0 of 0.2angular0.4 deflection0.6 0.8 is an importantAngular deflection measure of the torsion springof compliance. [rad] Angular deflection In of the the torsion design spring  [rad]of the x( s  r )sin  thirdFigure VSA, 4. Cam theprofiles first and type pressure of equivalentangle curves quadratic torsion  p ìx= s + r sinq Figure 4. Cam profiles and pressure angle curves  ( pq ) y( s  r )cosï  spring will preferably be used as the elastic unit.  p ï = + cosq y( s rpq ) The mechanical solutions for the two types of equivalent torsion springs are shown in Figure 5. Each compression spring  ï dy d X x rt ï dy dq is equipped with two linear bearings (4) and two guide shafts (1). The spring is placed in the pedestal (2), and the cam (3)  ïXdx= x + rt dy is used to drive the roller. The pressure angle is an important parameter for evaluating the friction dissipation, and the  ï()()22 2 2 (7) í ædx ö ædy ö (7) range of angular deflection is an important measure of compliance. In the design of the third VSA, the first type of  ddç ÷ + ç ÷ ï èdq ø è d q ø equivalent quadratic torsion spring will preferably be used as the elastic unit.  ï dx d Y y rt ï dx dq  Ydx= y - rt dy ï()()22 2 2  ï ædx ö ædy ö  ddç ÷ + ç ÷ îï èdq ø è d q ø

Two types of mechanical solutions for the quadratic torsionTwo spring types are of mechanicaldesigned. Thesolutions first typefor the of quadratic equivalent torsion spring are designed. The first type of equivalent torsion torsion spring consists of two compression springs. The Figure 5. CAD assembly of the quadratic torsion spring spring consists of two compression springs. The spring spring parameters are defined as: H0=32mm, d=2.5mm, Figure 5. CAD assembly of the quadratic torsion spring. parameters are defined as: H0=32mm, d=2.5mm, D=14mm, D=14mm, n=5, KS=28.13N/mm. Suppose that θ=0.5πrad n=5, K =28.13N/mm. Suppose that θ=0.5πrad corresponds 2.3 Conceptual design of the approximate quadratic tension correspondsS to s=12.5mm. The maximum amount of 2.3 Conceptual design of the approximate quadratic tension spring to s=12.5mm. The maximum amount of deformation of the spring Figure 5. CAD assembly of the quadratic torsion spring deformationspring is of12mm the springby the mechanical is 12mm by limit, the andmechanical the maximum The third type of approximate quadratic spring, which is limit, andallowable the maximum angular deflection allowable is ±1.527rad.angular deflection The cam param‐ is The third type of approximate quadratic spring, which is constituted2.3. Conceptual by a tension design spring of the set approximate with a triangular quadratic tension spring ±1.527etersrad. areThe defined cam parameters as: r =6mm, arer =22mm, defined α as:=19.1°, rt=6mm, λ=2, and constituted by a tension spring set with a triangular t p0 max structure, is illustrated in Figure 6. Compared with the rp0=22mm,a=3.402. α Themax= 19.1° second, λ =2, quadratic and a= torsion3.402. The spring second consists of structure, is illustrated in Figure 6. Compared with the existingThe third quadratic type of springapproximate based quadraticon a curved spring, surface, which cam is constituted by a tension spring set with a triangular structure, quadraticfour springs.torsion spring The spring consists parameters of four springs are defined. The as: existing quadratic spring based on a curved surface, cam or isa varying-radiusillustrated in Figure shaft, 6. this Compared method iswith easily the implement‐ existing quadratic spring based on a curved surface, cam or a varying- H =26mm, d=1.6mm, D=14mm, n=4, and K =5.9N/mm. The or a varying-radius shaft, this method is easily spring parameters0 are defined as: H0=26mm,S d=1.6mm, edradius and the shaft, structure this method is compact. is easily The implementeddesign of the quadraticand the structure is compact. The design of the quadratic spring can be maximum amount of deformation of the spring is 12mm by implemented and the structure is compact. The design of D=14mm, n=4, and KS=5.9N/mm. The maximum amount springachieved can beby achieved optimizing by optimizingthe design theparameters design parame‐to minimize the error between the prescribed and achieved load- the mechanical limit. Suppose that θ=0.25πrad corresponds of deformation of the spring is 12mm by the mechanical thetersdisplacement quadratic to minimize spring function. the can error be betweenachieved the by prescribedoptimizing and the to s=13mm. The cam parameters are defined as: r =6mm, limit. Suppose that θ=0.25πrad corresponds to s=13mm.t achieveddesign parameters load-displacement to minimize function. the error between the rp0=25mm, αmax=33.1°, λ=4, and a=12.35. The maximum The cam parameters are defined as: rt=6mm, rp0=25mm, prescribed and achieved load-displacement function. allowable angular deflection is ±0.7454rad. The profiles of

αmax=33the.1° cams, λ=4, and and the a =curves12.35. Tofhe pressure maximum angles allowable are shown in angularFigure deflection 4. is ±0.7454rad. The profiles of the cams KE and the curves of pressure angles are shown in Figure 4. F The mechanical solutions for the two types of equivalent Figure 6. Schematic description of the second quadratic spring torsion springs are shown in Figure 5. Each compression Profile of cam for two springs configuration Profile of cam for four springs configuration L 90 50 90 50 Suppose that the approximate quadratic spring element constitutes eight of the same tension springs. The bulge (1) is spring120 is equipped60 with two linear120 bearings60 (4) and two guide shafts (1). TheTheoretical spring contour is placed in the pedestal (2), and used for spring initial pretension. The guide rod (2) and the pedestal (3) are connected by the linear bearing, so the 25 150 25 150 30Inner enveloping line 30 friction is neglectedy in the calculation. The stiffness of the cable (4) is considered infinite (no stretching). The x and y the cam (3) is usedRadius to ofdrive base circle the roller. The pressure angle is x 180 an important parameter0 for evaluating180 the friction 0dissipa‐ tion, and the range of angular deflection is an important 210 measure of compliance.330 In the 210design of the third330 VSA, the

first240 type of300 equivalent quadratic240 torsion300 spring will Figure 6. Schematic description of the second quadratic 270 270 Figure 6. Schematic description of the second quadratic spring Pressurepreferably angle for two springs be used configuration as the Pressureelastic angle unit. for four springs configuration spring. 20 35

30 Jishu Guo and Guohui Tian: 5 15 Suppose that the approximate quadratic spring element 25 Conceptual Design and Analysis of Four Types of Variable Stiffness Actuators Based on Spring Pretension constitutes eight of the same tension springs. The bulge (1) 20 10 15 is used for spring initial pretension. The guide rod (2) and the pedestal (3) are connected by the linear bearing, so the 5 10

5 friction is neglected in the calculation. The stiffness of the

Pressure angle α angle Pressure [degree] α angle Pressure [degree] 0 0 cable (4) is considered infinite (no stretching). The x and y 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 Angular deflection of the torsion spring [rad] Angular deflection of the torsion spring [rad] represent the initial position of the spring on the frame. The spring constant is Ks, the amount of deformation of Figure 4. Cam profiles and pressure angle curves the spring along the direction of the force F is L, and the equivalent stiffness of the spring set along the direction of The mechanical solutions for the two types of equivalent the force F is KE. The KE can be calculated as follows: torsion springs are shown in Figure 5. Each compression represent the initial position of the spring on the frame. The spring constant is Ks, the amount of deformation of the spring along the direction of the force F is L, and the equivalent stiffness of the spring set along the direction of the force F is KEE. The KEE can be calculated as follows:

 2 222 yL FKsyLx8  xy  2  y Lx2  (8)  F KE   L

To achieve a prescribed load-displacement function (i.e., the purely quadratic spring F(L)= aL22), the error function ΔF and ΔK are defined as:

 2 222yL 2 FKsyLx8     xy   aL  22  ()yL x  . (9)  22 2  xx y  21    2  KKE aLKs3 aL 2  yL x2 2  

By optimizing the design parameters (i.e., x, y, Ks and allowable amount of deformation L) of the spring element, the error is small. Finally, the parameters are obtained as: x=33mm, y=5mm, Ks=1.5N/mm, and L is defined from 0 to 24mm. The prescribed load-displacement function is F(L)=0.1452L2. The differences between the desired purely quadratic spring and actual approximate quadratic spring are shown in Figure 7. The differences are relatively small. Therefore, the characteristics of the torque and stiffness of the proposed fourth VSA can be calculated by using the tension spring set, and the high level of linearity of the output stiffness-angular deflection curve can be achieved.

Suppose that the approximate quadratic spring element Tension spring set with triangular structure Tension spring set with triangular structure 100 7 constitutes eight of the same tension springs. The bulge (1) 6 80 is used for spring initial pretension. The guide rod (2) and Real force-displacement curve 5 Fitted force-displacement curve the pedestal (3) are connected by the linear bearing, so the 60 4

friction is neglected in the calculation. The stiffness of the 40 3 2

cable (4) is considered infinite (no stretching). The x and y Output force F[N] Real stiffness-displacement curve 20

represent the initial position of the spring on the frame. The Output stiffness K [N/mm] 1 Fitted stiffness-displacement curve 0 0 spring constant is Ks, the amount of deformation of the 0 5 10 15 20 25 0 5 10 15 20 25 Deformation of the tension spring set x[mm] Deformation of the tension spring set x[mm] spring along the direction of the force F is L, and the Tension spring set with triangular structure Tension spring set with triangular structure equivalent stiffness of the spring set along the direction of 0.4 0.3 F [N] 

the force F is K . The K can be calculated as follows: [N/mm] K E E 0.2 0.2 

0.1 ì é 2 2 2 2 ù y+ L 0 F=8 Ks( y + L) + x - x + y ï ê ú 2 0 ï ë û + + 2 (y L) x -0.2 í (8) -0.1 ï ¶F ïKE = -0.4 -0.2

Difference of the output force force output the of Difference 0 5 10 15 20 25 0 5 10 15 20 25 î ¶L stiffness the of Difference Deformation of the tension spring set x[mm] Deformation of the tension spring set x[mm]

To achieve a prescribed load-displacement function (i.e., FigureFigure 7. Comparison7. Comparison of the of desiredthe desired quadratic quadratic spring spring and the an approximated the approximate quadratic tension spring set quadratic tension spring set the purely quadratic spring F(L)= aL2), the error function ΔF and ΔK are defined as: R 3.J. The Conceptual initial distance design of the and two analysisspring guide of sleevesthe first is S.VSA When the two-spring guide sleeves contact with each other, ì é 2 2 2 2 ù y+ L 2 DF =8 Ks( y + L) + x - x + y - aL the springs cannot be compressed any further, and the ï ëê ûú 2 2 3.1. Conceptual layout of the first VSA ï ()y+ L + x actuator will switch into the stiff operation mode from the ï ï æ ö compliance mode. The movable pulley rod (10) and the í 2 2 2 (9) The actuator discussed in this section is based on the use of the tendon tensioning mechanism and spring preload ç x x+ y ÷ ï ç ÷ movable baffle (7) are used for spring pretension, and to DK = KE -2 aL = Ks 1 - - 2aL mechanism. A simple schematic is shown in Figure 8. The output disc (1) and the driving pulley (2) are coaxially ï ç 3 ÷ 2 2 2 ensure that the tendons will not become slack when the ï ç é(y+ L) + x ù ÷ connected by bearing. The stiffness regulation mechanism components are located in (1). The two ends of the tendon are ïî è ë û ø springsconnected are with compressed. the spring Aspedestal is shown (8) and in the Figure pulley 8(b) (2), respectively. The tendons (3) and (4) are used for counter andFigure 8(c), the (2) is rotated at an angle β with respect

By optimizing the design parameters (i.e., x, y, Ks and to the (1) under the action of the applied external torque τext. allowable amount of deformation L) of the spring element, The amount of deformation of the spring is Δh. Because the the error is small. Finally, the parameters are obtained as: relationship between the displacement amount x of the (10) x=33mm, y=5mm, Ks=1.5N/mm, and L is defined from 0 to and the displacement amount ΔL of the (7) is nonlinear, the 24mm. The prescribed load-displacement function is screw transmission mechanism and roller-cam mechanism F(L)=0.1452L2. The differences between the desired purely with self-locking function are proposed. quadratic spring and actual approximate quadratic spring are shown in Figure 7. The differences are relatively small. Therefore, the characteristics of the torque and stiffness of the proposed fourth VSA can be calculated by using the tension spring set, and the high level of linearity of the output stiffness-angular deflection curve can be achieved.

3. Conceptual design and analysis of the first VSA

3.1 Conceptual layout of the first VSA

The actuator discussed in this section is based on the use of the tendon tensioning mechanism and spring preload mechanism. A simple schematic is shown in Figure 8. The output disc (1) and the driving pulley (2) are coaxially connected by bearing. The stiffness regulation mechanism components are located in (1). The two ends of the tendon are connected with the spring pedestal (8) and the pulley (2), respectively. The tendons (3) and (4) are used for

counter clockwise rotations while the (5) and (6) are used Figure 8. Conceptual principle of the first VSA: (a) Initial state. (b) Rotate an for clockwise rotations. The two pairs of tendons are in angle β under the applied external torque. (c) Stiffness control. (d) Rotate an different planes to avoid crossing. The radius of the (2) is angle θ after stiffness adjustment.

6 Int J Adv Robot Syst, 2015, 12:62 | doi: 10.5772/60580 3.2 Mathematical modelling of the first VSA The pedestal (1) of the movable pulley rod is driven by a threaded shaft (2). The number of teeth of the bevel gear As is illustrated in Figure 8(d), under the applied external fixed with the (2) is Z2 and the pitch of the threaded shaft torque τ’ , the angular deflection between the (2) and the ext (2) is P1. The cam (3), the spur gear (4), the rack (5), the (1) is θ. The actuating torque TE and output stiffness K at threaded shaft (6), the movable baffle, the roller mounted the elastic transmission for the linear spring configuration on the baffle, and the bevel gear pair constitute the spring are formulated as follows. The linear spring constant is KS. preload mechanism. The (3) is fixed with the cam (4). The

number of teeth of the bevel gear fixed with the (6) is Z3 and ì TKRRLESJJ= 2 ( q +2 D ) the pitch of the threaded shaft (2) is P2. RC is the radius of ï í ¶T (10) the pitch circle of (4). A Cartesian coordinate system is TKRRL2E (   22  ) number of teeth of the bevel gear fixed with the (6) is Z3  ESJJïKKR= =2 SJ established and the centre of rotation of the cam is the î ¶q and the pitch of the threaded shaft (2) is P2. RC is the  TE 2 (10) KKR= =2 origin. The positive axis of xc is defined as the direction of  SJ radius of the pitch circle of (4). A Cartesian coordinate   motion of the movable baffle for rising travel. The angle of In addition, the calculation formulas of the torque TE’ and system is established and the centre of rotation of the cam rotation of the (3) is βC, which corresponds to the amount stiffness K’ for quadratic spring configuration (i.e., F(x) = is the origin. The positive axis of xc is defined as the In addition, the calculation formulas of the torque TE’ and of displacement x of the (1). In the cam-roller mechanism, ax2) are shown as follows: direction of motion of the movable baffle for rising travel. stiffness K’ for quadratic spring configuration (i.e., F(x) = suppose that the eccentricity is zero, the radius of the base ax2) are shown as follows: The angle of rotation of the (3) is βC, which corresponds to 2 circle is rb, and the radius of the roller is rt. Finally, the ìT'= 2 aR Rq + 2 D L the amount of displacement x of the (1). In the cam-roller ï E JJ( ) equation describing the profile of the cam for spring í 2 (11) mechanism, suppose that the eccentricity is zero, the TEJJ' 2 aR¶ (T R '  2  L ) pretension can be expressed as:  ïK'=E =4 aR3q +8 aR2 D L radius of the base circle is rb, and the radius of the roller  T ' J J (11) î E ¶q 32 is rt. Finally, the equation describing the profile of the K'= =4 aRJJ  8 aR  L   cam forì springPZR pretension can be expressed as: = 1 3 C b As is illustrated in Figure 8(c), the movable pulley rod (10) ïx C PZ2 2 andAs is the illustrated movable in baffleFigure (7) 8(c), are the used movable to ensure pulley that rod (10) the ï  ïPZRD13C = D b x L( x) L( C ) tendonsand the always movable maintain baffle contact(7) are used with t othe ensure driving that pulley. the  ï PZ C  22 Thetendons related always design maintain and kinematic contact with relationship the driving are pulley. shown ïxc=( D L(b C) + rb)cos b C L()()ï x   L C inThe Figure related 9. design and kinematic relationship are shown  ïy= - D Lb + rb sin b x( c L ( ( )  rb( )cos C)  ) C in Figure 9.  cï C C  í dyc db C (13) ycX ( = L x(  C - )rt  rb )sin  C  ï c c 2 2  ï dyæcCdx dc ö ædy c ö (13) Xcc x rt ç ÷+ ç ÷  ï dxdb dy d b  ï ()()ccè22C ø èC ø dd  ï CCdx db  Y= y + rt c C ï c c 2 2  dxcC d Yccï y rt ædx ö ædy ö  ï dxçc dy÷+ çc ÷ ()()cc22b b  îï èdC ø è d C ø  ddCC

wherewhere the ((xxcc,, yyc c)) representrepresentss the the theoretical theoretical profile profile of ofthe the camcam,, andand thethe actual actual profile profile is is represented represented by by (X (CX, CY, CY).C The). The radiusradius ofof thethe base circle circle and and the the radius radius of ofthe the roller roller are are Figure 9. Schematic description of the stiffness adjustment mechanism Figure 9. Schematic description of the stiffness defineddefined as rb=8mm as rb= 8andmm rt and=3mm, rt=3 mmrespectively., respectively. adjustment mechanism Figure 9 shows the relationship between the x and the ΔL.

Suppose that the movable pulley rod is moved from initial 3.3 Characteristics3.3 Characteristic ofs torqueof torque and and stiffness stiffness of of thethe first VSAVSA Figure 9 shows the relationship between the x and the ΔL. position point O to point O'. Since the radius of the fixed Suppose that the movable pulley rod is moved from In the design of the mechanical solution, the radius RJ of the pulley is small, the contact point A is considered to remain In the design of the mechanical solution, the radius RJ of initial position point O to point O'. Since the radius of the driving pulley is defined as 30mm. For the linear spring unchanged. The radius of the movable pulley rod is the driving pulley is defined as 30mm. For the linear fixed pulley is small, the contact point A is considered to configuration, the compression spring stiffness is 4N/mm. considered to be null. The angle α, |AO|, |CO|, |DO|, and spring configuration, the compression spring stiffness is remain unchanged. The radius of the movable pulley rod The maximum amount of deformation of the spring is the tendon length from point A to point B are known. The 4N/mm. The maximum amount of deformation of the is considered to be null. The angle α, |AO|, |CO|, |DO|, spring24mm isand 24mm the andmaximum the maximum angular angular deflection deflection θ is ±0.8rad. θ is related calculation can be expressed as follows: and the tendon length from point A to point B are known. The±0.8rad maximum. The maximum elastic output elastic torque output is torque 5.76Nm. is 5.76Nm. The linear The related calculation can be expressed as follows: relationshipThe linear between relationship the stiffness between K the and stiffness angular K deflectionand ì 2 2 θ can be achieved for the purely quadratic spring (i.e., AO' = x + AO - 2 x AO cosa angular deflection θ can be achieved for the purely ï 2  AO' x2  AO2  2 x AO cos  quadraticF(x)=0.1661 springx ) configuration. (i.e., F(x)=0.1661 Figurex2) configuration. 10 shows the Figure charac‐ ï  2 2 ï EO' =( x + OD) - RJ teristics10 show ofs actuatingthe characteristics torque andof actuating output stiffnesstorque and for the ï  22  EO'() x  OD  RJ different types of spring configurations (linear, quadratic í  æ ö (12) output stiffness for the different types of spring ï  p RJ (12) EB» = R (ç  - arccos RJ ÷ andconfiguration piecewise linears (linear, compression quadratic springand piecewise set). As illustratedlinear EBJ Rç ( arccos÷ ) ï èJ2 x+ OD ø ï  2 x OD compressionin Figure 10(c), spring the set ). output As illustrated stiffness-angular in Figure 10 deflection(c), the ïDL = AO'' + EO + EB» - AO - OC - CB» = D Lx( ) curveoutput corresponding stiffness-angular to the deflection piecewise curve linear corresponding spring set is in î L  AO''()  EO  EB  AO  OC  CB   L x discontinuousto the piecewise straight-line linear spring segments. set is in discontinuous The differences straight-line segments. The differences between the The pedestal (1) of the movable pulley rod is driven by a desired and achieved stiffness areJishu relatively Guo and small.Guohui Tian: 7 Conceptual Design and Analysis of Four Types of Variable Stiffness Actuators Based on Spring Pretension threaded shaft (2). The number of teeth of the bevel gear fixed with the (2) is Z2 and the pitch of the threaded shaft (2) is P1. The cam (3), the spur gear (4), the rack (5), the threaded shaft (6), the movable baffle, the roller mounted on the baffle, and the bevel gear pair constitute the spring preload mechanism. The (3) is fixed with the cam (4). The Linear compression spring configuration L[mm] Linear compression spring configuration 6 between the desired and achieved7.2 stiffness are relatively isand small. the outer The allowable race of the range spring of is rotation small. ofThe the allowable stiffness

5 small. 20 7.2 rangemotor of with rotation respect of theto the stiffness fixed housingmotor with is limited respect by to the the 4 fixedcables housing attached is to limited it. The allowableby the cable angles attached range of to rotation it. The 15 7.2 Linear3 compression spring configuration L[mm] Linear compression spring configuration ofallowable the first VSA angle is defined range ofas rotation±150°. of the first VSA is 6 7.2 Linear compression spring configuration10 L[mm]7.2 Linear compression spring configuration 2 6 7.2 and the outer race of thedefined spring asis small. ±150°. The allowable 5 20 7.2 range of rotation of the stiffness motor with respect to the 1 5 5 207.2 7.2 4[Nm] Te torque Output 4 15 [Nm/rad] K 7.2stiffness Output fixed housing is limited by the cables attached to it. The 0 157.2 7.2 3 0 0 Linear0.23 compression0.4 spring0.6 configuration0.8 L[mm] 0 Linear0.2 compression0.4 spring configuration0.6 0.8 allowable angle range of rotation of the first VSA is Angular6 deflection θ[rad] 10 7.2 7.2 Angular deflection θ[rad] 10 7.2 2 2 defined as ±150°. 5 20 7.2 Purely quadratic spring configuration L[mm]5 7.2Purely quadratic spring configuration L[mm]

1 1 5 7.2 Output torque Te [Nm] Te torque Output

Output torque Te [Nm] Te torque Output 6 4 15 Output stiffness K [Nm/rad] K stiffness Output (a) Characteristics for linear[Nm/rad] K stiffness Output 15 compression7.2 spring 0 7.2 05 3 020 7.20 20 0 0.2 0 0.40.2 0.60.4 0.60.8 0.8 0 0 0.2 0.2 0.4 0.4 0.60.6 0.80.8 Angular deflectionconfiguration θ[rad] 10 Angular deflection θ[rad] Angular deflection θ[rad] 7.2 Angular deflection θ[rad] 4 2 10 15 15 Purely quadratic spring configuration L[mm] Purely quadratic spring configuration L[mm] 1 5 7.2 Purely3 quadratic[Nm] Te torque Output 6 spring configuration L[mm] Purely15 quadratic spring configuration L[mm]

(a) Characteristics for linear[Nm/rad] K stiffness Output compression spring 6 10 15 10 05 020 7.2 20 2 0 0.2 0.4 0.6 configuration0.8 5 0 0.2 0.4 0.6 0.8 5 Angular deflection θ[rad] 20 Angular deflection θ[rad] 20 4 5 10 5 1 15 15 Output torque Te' [Nm] Te' torque Output 4 Purely quadratic spring configuration L[mm]10 Purely quadratic spring configuration L[mm] 3 15 [Nm/rad] K' stiffness Output 15 6 15 0 0 100 010 3 0 0.22 0.4 0.6 0.8 0 5 0.2 0.4 0.6 0.8 Angular5 deflection θ[rad] 10 20 Angular deflection θ[rad] 1020 5 5 (a) Front and back views of the stiffness adjustment 1 2 [Nm] Te' torque Output 5 4 10 15 [Nm/rad] K' stiffness Output 15 mechanism 5 5 1 0 0 0 0 Output torque Te' [Nm] Te' torque Output 3 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8

Angular deflection θ[rad] [Nm/rad] K' stiffness Output 10 Angular deflection θ[rad] 10 (a) Front and back views of the stiffness adjustment 0 2 0 0 5 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 mechanism 5 5 Angular1 deflection θ[rad] Angular deflection θ[rad]

Output torque Te' [Nm] Te' torque Output Output stiffness K' [Nm/rad] K' stiffness Output 0 0 0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 (b) CharacteristicsAngular deflection forθ[rad] the purely quadraticAngular deflection spring θ[rad]

configuration (b) Characteristics for the purely quadratic spring

Comparison of the output stiffness L=0mmconfigurationComparison of the output stiffness L=5mm 15 15 Comparison of the output stiffness L=0mm Comparison of the output stiffness L=5mm 15 15 10 10 10 10 5 5 Real stiffness curve:K-θ Fitted stiffness curve:K-θ 5 5 Real stiffness curve:K-θ (b) Section view of the first VSA Difference:RK-FKFitted stiffness curve:K-θ (b) Section view of the first VSA 0 0 Difference:RK-FK 0 0 Figure 11. CAD assembly of the first actuator

Output stiffness K [Nm/rad] K stiffness Output Figure 11. CAD assembly of the first actuator. Output stiffness K [Nm/rad] K stiffness Output

Output stiffness K [Nm/rad] K stiffness Output Figure 11. CAD assembly of the first actuator. Output stiffness K [Nm/rad] K stiffness Output -5 -5 0 0.2-5 0.4 0.6 0.8 0 -5 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 Angular deflection θ[rad] Angular deflection θ[rad] 4. Conceptual design and analysis of the second VSA Angular deflection θ[rad] Angular deflection θ[rad] 44. Conceptual. Conceptual design design and andanalysis analysis of the ofsecond the second VSA VSA

(c) Stiffness(c) Stiffness characteristic characteristic for forthe the piecewise piecewise linear linear 4.1 Conceptual4.14 Conceptual.1 Conceptuallayout oflayout the layout secondof the ofsecond VSA the second VSA VSA compressioncompression spring spring set setconfiguration configuration In this section, a VSA based on an adjustable moment arm In this section, a VSA based on an adjustable moment FigureFigure 10. Characteristics 10. Characteristics of torque andof torque stiffness and of the stiffness first VSA forof differentthe mechanismIn this section, is presented. a VSA based This is on an an equivalent adjustable model moment of the Figuretypes 10 of. spring Characteristics configurations of torque and stiffness of the arm mechanism is presented. This is an equivalent model Real stiffness curve:K-θ armlever mechanism arm mechanism. is presented. This VSA This is is characterized an equivalent by model the first VSA for different types of spring configurations of the lever arm mechanism. This VSA is characterized by first VSA for different types Realof springstiffnessFitted stiffness curve:K-θ configuration curve:K-θ s property that the proportion of the two moment arms can Fitted Difference:RK-FK stiffness curve:K-θ theof theproperty lever that arm the mechanism. proportion of Th theis twoVSA moment is characterized arms by 3.4 Mechanical structure solution Difference:RK-FK of the first VSA be changed, and this feature is implemented by the 3.4 Mechanical structure solution of the first VSA thecan propertybe changed, that and the this proportion feature is implemented of the two bymoment the arms 3.4 Mechanical structure solution of the first VSA connecting rod and slider mechanism with self-locking The mechanical structure solution is shown in Figure 11. connectingcan be changed, rod and andslider this mechanism feature withis implemented self-locking by the The mechanical structure solution is shown in Figure 11. function. In this design, the output stiffness of the VSA is This model is essentially composed of an output disc (2) function.connecting In this rod design, and tsliderhe output mechanism stiffness of with the VSA self -islocking The mechanicalThis model structure is essentially solution composed is shown of an outputin Figure disc 1(2)1. only related to the effective working radius for linear functioning as the frame, where the stiffness adjustment function.only related In tothis the design, effective the working output radius stiffness for linear of the VSA is functioning as the frame, where the stiffness adjustment compression spring configuration, but not to the exerted This mechanismmodel is essentially components composed are located. of a Then output output disc disc (2) (2) is compresonly relatedsion spring to theconfiguration effective , workingbut not to radiusthe exerted for linear mechanism components are located. The output disc (2) is torquetorque or orangular angular deflection deflection.. Moreover Moreover,, the relationsh the relationshipip functioningfixedly connectedas the frame, with where the rotatable the stiffness housing adjustment (10) through compression spring configuration, but not to the exerted fixedly connected with the rotatable housing (10) through betweenbetween the the actuating actuating torque torque and angular and angular deflection deflection is is mechanismthe screws. components The (10) is are connected located. with The theoutput fixed disc housing (2) is (9) the screws. The (10) is connected with the fixed housing lineartorquelinear.. This Thisor feature angular feature makes deflection makes the torquethe. torqueMoreover control control and, the stiffness and relationsh stiffness ip fixedlyby connected the bearing with (8). Thethe motorrotatable (4) ishousing fixedly (10)connected through with (9) by the bearing (8). The motor (4) is fixedly connected regulationregulationbetween more the more convenient.actuating convenient. torque The woThe rkingand working angular principle principle deflection of the of the is the screws.the (10) Thethrough (10) theis connected bracket (5) with and screws.the fixed The housing bevel gear with the (10) through the bracket (5) and screws. The linearsecond.second This VSA feature VSAis demonstrated is demonstratedmakes the in torque Figure in Figure control 12. 12. and stiffness (3) is driven by the motor (4). The driving pulley (1) and the (9) by thebevel bearing gear (3)(8). i sThe driven motor by the (4) motor is fixedly (4). The connected driving regulation more convenient. The working principle of the (2) are coaxially connected by the bearing. The motor (6) is The output disc (1) and the driving disc (3) are coaxially with thepulley (10) through(1) and the the (2) bracket are coaxially (5) and connected screws. by The the second VSA is demonstrated in Figure 12. fixedlybearing. connected The motor with (6) theis fixedly (9) by connectedthe bracket with (7) and the (9)screws. by connected by the bearing (2). The motor used for regulating bevel gear (3) is driven by the motor (4). The driving pulleyThethe motor (1) bracket and (6) (7)the is usedand (2) screws.are to drive coaxially The the motor (1) connected and (6) control is used by theto the drive output the compliance is mounted on the (3) and drives the screw the (1) and control the output position. To reduce the bearing.position. The motor To reduce (6) is thefixedly vibration connected of the with springs, the (9)the by gap shaft (4) and the nut (5). The (5) is connected with the (3) by betweenvibration the of guidethe springs, sleeve the and gap the between outer racethe guide of the sleeve spring the vertical guide rod. One end of the connecting rod is the bracket (7) and screws. The motor (6) is used to drive 8the Int(1) J Advand Robot control Syst, the 2015, output 12:62 position| doi: 10.5772/60580. To reduce the vibration of the springs, the gap between the guide sleeve where F1 and F2, respectively, represent the force generated

by the elastic deformation of the springs, and KS is the spring constant. The output stiffness K depends on the working radius R. The relationship between the ΔR and the Δx is demonstrated in Figure 13.

x

R Figure 12. Conceptual diagram of the second VSA: (a) Initial state. (b) Rotate an angle θ under the action of actuating torque. (c) Control of working radius of the driving disc. (d) Rotate the same angle θ after changing the radius. connected to the (5) by a rotatable pin and another end is connected with the slider. The sliders can be moved along Figure 13. Relationship description between the ΔR and the Δx the horizontal guide rod (6). The outer contours of the sliders form an approximate circular shape. There are four GivenFigure that there 13. areRelationship 20 sliders in the description mechanical solution, between the ΔR and the Δx linear compression springs used in the elastic elements. an ideal circular is used to replace the actual approximate Two ends of the cable are connected to each of the spring circularGiven contour that in the therecalculation. are Suppose 20 sliders that the radius in the mechanical solution, an ideal circular is used to replace the actual approximate and the slider. H0 is the resting (no-load) length of the of the diverting pulley is Rs, the centre of rotation of the spring, ΔH is the initial amount of deformation of the divertingcircular pulley iscontour point O', andin thethe centre calculation. of rotation of Suppose that the radius of the diverting pulley is Rs, the centre of rotation of the spring, and the maximum amount of deformation of the the VSAdiverting is point O. Whenpulley the radiusis point is changed O', fromand R 0the to centre of rotation of the VSA is point O. When the radius is changed from R00 to R, spring corresponding to the initial working radius R0 is R, the tangent point between the cable and the diverting

ΔLmax. To avoid cable slack, we define this relationship, i.e., pulleythe is changed tangent from point I to between point C. The pointthe Dcable and and the diverting pulley is changed from point I to point C. The point D and the the point B are the tangent points corresponding to working ΔLmax<ΔH. When the working radius of the (3) is changed point B are the tangent points corresponding to working radius R00 and R, respectively. In addition, the radius of the from R0 to R, the additional amount of deformation of the radius R0 and R, respectively. In addition, the radius of the spring is Δx, and the allowable maximum amount of cablecable is considered is consider null, and ed the null,|OW| andand |WO'| the are|OW| and |WO'| are known in the mechanical solution. The related calculations are known in the mechanical solution. The related calculations deformation of the spring is changed from ΔLmax to ΔL’max. formulated as: The relationship between the incremental working radius are formulated as:

ΔR (i.e., |R-R0|) and the incremental amount of deforma‐ tion of the spring (i.e., Δx) is nonlinear. When the springs ì OW could not be compressed because of the mechanical stop or ïÐO' OW = arccos OW OOW' arccos2 2 the effective working radius reached the maximum ï OW+ WO' 22 ï allowable value, the VSA would move into the stiff mode RR0 - OW WO' ïÐO' OE = S from the compliance mode. ï 2 2 ï OW+ WORR'0  S ïOOE' p íÐDOH = - Ð O'' OE+ Ð22 O OW (15) 4.2 Mathematical modelling of the second VSA ï 2 OW WO' ï NI» = R Ð NO' I= R(p -Ð DOH) As is shown in Figure 12(d), when the external torque τ’ext ï S  S ï (15) is applied to the output disc (1), the elastic torque T and DOH2 2 O'' OE2 + O OW E ïID= OW + WO' +( R - R ) the output stiffness K can be described by: ï 2 0 S ï =p - Ð + + Ð îLNH R S ( DOH) IDRDOH0 NI RSS NO' I= R  DOH ì  ïT =2(F1 -F 2 ) R 22 2 ï E where LNH is the length of the cable from point N to point ï 2 ID OW WO'  R0 RS í =2ëé( D H+ Dx + Rq ) -( D H+ Dx -Rq ) ûù KSS R = 4 K R q (14) H. When the working radius is changed from R0 to R, the ï  ¶ length of the cable from point N to point A can be obtained ï TE 2  KKR= = 4 LR DOHIDRDOH0 îï ¶q S by:  NH S

Jishu Guo and Guohui Tian: 9 Conceptual Design and Analysis of Four Typeswhere of Variable LNHNH Stiffness is Actuators the lengthBased on Spring of thePretension cable from point N to point H. When the working radius is changed from R00 to R, the length of the cable from point N to point A can be obtained by:

 22 2 OG'' BC OW  WO  R RS  RR DOB O'arccos OE S 22   '  OW WO MOB DOB DOH . (16)  AB R DOB  DOH  NCRNOCRMOBSS'     LABBCNCNA 

Finally, the relationship between Δx and R is given by:

xL= NA  L NH  f R f  R R0  f  R. (17)

As mentioned above, one end of the cable is connected with the slider, and the connecting point withstands the tension force from the external torque, so a ball screw mechanism with a self-locking function is designed for changing the radius. The related design is shown in Figure 14. In the mechanical solution, the stiffness motor (the one which tunes the output stiffness) is attached to the driving pulley and can change the position of the nut (1) by driving the threaded shaft (2), and the working radius is changed under the action of the connecting rod (3). Suppose that the output angle of the stiffness motor is β and the ratio of the reducer is η. The pitch of the nut (1) is PVRVR and the displacement is LVRVR. The length of the connecting rod (3) is |ab|=|a'b'|. The a, b, c, a', b', and c' are defined in Figure 14.

in (14), the output stiffness K for linear spring configuration ì 2 2 2 ' ' ïO G= BC = OW + WO -( R - RS ) is obtained by: ï RR- ïÐDOB = Ð O' OE - arccos S 2 2 ï OW+ WO' ¶T 2 ï K= =4 KS éë f (b ) ùû (19) Ð =p - Ð - Ð ¶q í MOB DOB DOH (16) ï ïAB» = R( Ð DOB + Ð DOH) ï 4.3Figure Characteristics 14. Kinematic of diagram torque and of the stiffness variable of radiusthe second mechanism VSA ïNC¼ = RS Ð NO' C = RS Ð MOB ï InThe this relationship section, the between characteristics the angle of β actuatingand the effective torque workingand radius R is given by: îLNA = AB» + BC + NC¼ the output stiffness of the actuation system are analysed.  The variationPPVR range VR VR of the effective working radius is Finally, the relationship between Δx and R is given by: LVR  =  22  defined from 30mm to 40.5mm. The number of the(18) sliders is 20. The spring222 constant is2 3.24N/mm, and the maximum Dx= LNA - L NH = f( D R) = f( D R + R0 ) = f( R) (17) RR0 ab''  cdL VR  abcdf   amount of deformation of the spring is defined as 32mm by mechanical limit. The initial amount of deformation of the As mentioned above, one end of the cable is connected springwhere isthe 17mm. radius The R is allowable a function maximum of angular angular displacement deflec‐ of the stiffness motor. By substituting equation (16) and (17) in with the slider, and the connecting point withstands the tion(14), will the decreaseoutput stiffness with the K increasefor linear in spring working configuration radius, and is obtained by: tension force from the external torque, so a ball screw the maximum allowable angular deflection between the mechanism with a self-locking function is designed for T 2 driving disc4 and  output  link is ±0.5rad. When the working changing the radius. The related design is shown in KKfS  . (19) radius R has reached 40.5 mm, the spring cannot be Figure 14. In the mechanical solution, the stiffness motor compressed, no angular deviation is mechanically allowed, (the one which tunes the output stiffness) is attached to and the actuator will go into the stiff operation mode. the driving pulley and can change the position of the nut 4.3. Characteristics of torque and stiffness of the second VSA

(1) by driving the threaded shaft (2), and the working When R0 =30mm, |WO'|=46mm, |OW|=39mm, RS=3mm, radius is changed under the action of the connecting rod In this section, the characteristics of actuating torque and the output stiffness of the actuation system are analysed. The ab=a'b'=20mm, cd=19.7mm, the characteristics of torque (3). Suppose that the output angle of the stiffness motor variation range of the effective working radius is defined from 30mm to 40.5mm. The number of the sliders is 20. The and output stiffness of the VSA are shown in Figure 15. The is β and the ratio of the reducer is η. The pitch of the nut spring constant is 3.24N/mm, and the maximum amount of deformation of the spring is defined as 32mm by mechanical mechanicallylimit. The initial allowable amount torqueof deformation range at of the the elastic spring trans‐ is 17mm. The allowable maximum angular deflection will decrease (1) is PVR and the displacement is LVR. The length of the missionwith the mode increase is from in working0Nm to 5.83Nm,radius, andand the maximum allowable angular deflection between the driving disc and connecting rod (3) is |ab|=|a'b'|. The a, b, c, a', b', and c' elasticoutput torque link is will ±0.5rad. change When with the the working changes radius of the workingR has reached 40.5 mm, the spring cannot be compressed, no angular are defined in Figure 14. radiusdeviation R. The is mechanically mechanically allowed, allowable and range the actu of atorstiffness will goat into the stiff operation mode. the elastic transmission of the VSA is from 11.66Nm/rad to 20.73Nm/rad.When R0 0 =30mm, The output|WO'| =46mm,stiffness |OW|K is kept=39mm, constant RSS=3mm, in the allowable angular deflection range if the working radius R isab = unchanged.a'b'=20mm, Comparedcd=19.7mm, withthe characteristics the other actuators, of torque the and output stiffness of the VSA are shown in Figure 15. The controlmechanically method allowable of stiffness torque is very range convenient at the elastic even transmis in the sion mode is from 0Nm to 5.83Nm, and the maximum elastic presencetorque will of thechange disturbances. with the changes This feature of the isworking beneficial radius for R. The mechanically allowable range of stiffness at the elastic real-timetransmission regulation of the VSAof the is output from 11.66Nm/ stiffness.rad Therefore, to 20.73Nm/rad. we The output stiffness K is kept constant in the allowable doangular not need deflection a complicated range if controlthe working strategy radius to achieveR is unchanged. the Compared with the other actuators, the control method desiredof stiffness stiffness, is very and convenient only need even to in change the presence the effective of the disturbances. This feature is beneficial for real-time regulation workingof the output radius stiffness. R. Therefore, we do not need a complicated control strategy to achieve the desired stiffness, and only need to change the effective working radius R.

Torque characteristic curve： Te-θ Stiffness characteristic curve： K-θ 6 22

5 R=30mm 20 R=31mm 4 R=32mm 18 Figure 14. Kinematic diagram of the variable radius mechanism R=33mm 3 R=34mm 16 R=35mm The relationship between the angle β and the effective 2 R=36mm 14 R=37mm working radius R is given by: R=38mm

Output torque Te [Nm] 1 12 R=39mm

R=40mm Output stiffness K [Nm/rad] 0 10 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 ì PPb b Angular deflection, θ[rad] Angular deflection,θ [rad] L = VR= VR VR ï VR 2ph 2 p í (18) Figure 15. Characteristics of torque and stiffness of the second VSA ï 2 2 2 2 RR= + ab'' - cdL - - abcd - = f b îï 0 ( VR ) ( ) 4.4 Mechanical structure solution of the second VSA

where the radius R is a function of angular displacement of The mechanical solution of the second VSA for linear the stiffness motor. By substituting equation (16) and (17) compression spring configuration is shown in Figure 16.

10 Int J Adv Robot Syst, 2015, 12:62 | doi: 10.5772/60580 5. Conceptualcompletely rigid.design T heand allowable analysis angle of the range third of VSA the stiffness motor with respect to the fixed housing is limited5.1 Conceptual by the cablelayout attached of the third to it. VSA The allowable range of the rotation angle of the VSA is defined as ±150°. The third actuator is antagonistically actuated by two quadratic5. Conceptual torsion design springs. and Antagonisticanalysis of the actuation third VSA of two quadratic torsion springs mathematically guarantees perfectly linear5.1 Conceptual behaviour layout with of the adjustable third VSA stiffness. To illustrate this point, consider a quadratic torsion spring, whichThe third can rotateactuator about is antagonistically point O. The characteristicsactuated by two of the (a) Stiffness adjustment mechanism quadraticquadratic torsion springs. are Antagonistic defined by actuationG(θ)=aθ2 .of In two Figure 17(b),quadratic a mirror torsion image springs pair mathematically of these mechanisms guarantees is used together.perfectly Theylinear are behaviour rotated withby an adjustable angle φ andstiffness. attached To to eachillustrate other this as point, shown consider in Figure a quadr 17(c).atic It torsion is noted spring, that the whichequilibrium can rotate positions about of point the assembledO. The characteristic joint do nots of change the 2 withquadratic φ. As shown torsion in spring Figure are 16(d), defined when by the G( θexerted)=aθ . In torque M(β)Figure is 1 applied7(b), a mirror at the image assembled pair of joint, these the mechanisms joint is rotated is by anused angle together. of β. As They is shown are rotated in equation by an angle (20), φ the and applied attachedtorque M(β) to each is a other linear as function shown inof Figureβ and the17(c) corresponding. It is noted stiffnessthat the equilibrium is a linear positions function of of the φ assembled. In this mechanicaljoint do schematic,not change the with allowable φ. As shown operating in Figure region 16(d), of the when assembled the exertedjoint is definedtorque M(β) as β ∈{is applied−φ≤ θ ≤φ at }.the assembled joint, the joint is rotated by an angle of β. As is shown in equation (20), the applied torque M(β) is a linear function of β and the correspondingìM(b) = G( fstiffness + b) -G is( fa -linear b) = func4 a fbtion of φ. In (b) Section view of the second VSA this mechanicalï schematic, the allowable operating region í dM (b ) (20) of theï Kassembled= joint = 4a fis defined as β∈{ −φ≤ θ ≤φ }. Figure 16. CAD assembly of the second actuator db Figure 16. CAD assembly of the second actuator. î The conceptual model is essentially composed of the M( ) G (  ) G (  ) 4 a   drivingThe discconceptual (3) functioning model is as essentially the frame, composed where the of variable the  dM () (20) driving disc (3) functioning as the frame, where the Ka 4  stiffness mechanism components are located. This rotary  d actuatorvariable consists stiffness of the mechanism motor (13) components devoted to areset thelocated. output G() positionThis rotary and actuator the motor consists (16) of usedthe motor to tune (13) devoted the output to stiffness.set the Theoutput (13) position is connected and the to motor the reducer. (16) used The to outputtune G() ofthe the output reducer stiffness. is then The connected (13) is connected to the torque to the sensor reducer. (14) The output of the reducer is then connected to the torque O O and the spur gear. The motion of the rotatable housing (4)  M () issensor realized (14) by and engaging the spur the gear. internal The motion ring gear of the (15) rotatable and the  (a)  (b)  M+() housing (4) is realized by engaging the internal ring gear      spur gear. The (15) and the (3) are fixedly connected with + the(15) (4) through and the thespur screws. gear. The The (15) rotatable and the spring (3) are pedestal fixedly (6) is connected connected with with the the (4) (4) through by the the bearing screws. (5) The and rotatable fixedly connectedspring to pedestal the output (6) is housing connected (9) withby the the set (4) screws. by the The O O bearing (5) and fixedly connected to the output housing (9) is connected with the fixed housing (12) by the bearing (c) (d) (9) by the set screws. The (9) is connected with the fixed (10) and (11). The internal ring gear (17) mounted on the housing (12) by the bearing (10) and (11). The internal Figure 17. Antagonistic set-up of two quadratic torsion springs for an adjustable stiffness joint mechanism (12) is engaged with another small spur gear (18) mounted Figure 17. Antagonistic set-up of two quadratic torsion ring gear (17) mounted on the (12) is engaged with Figure 17. Antagonistic set-up of two quadratic torsion springs for an on the (9). The spur gear (18) is connected with the angular adjustablesprings stiffness for an joint adjustable mechanism stiffness joint mechanism. another small spur gear (18) mounted on the (9). The spur deflection encoder, and the angular deflection θ can be 5.2. Characteristics of torque and stiffness of the third VSA gear (18) is connected with the angular deflection encoder, obtained. In the stiffness regulation mechanism, the motor 5.2 Characteristics of torque and stiffness of the third VSA and the angular deflection θ can be obtained. In the 5.2 Characteristics of torque and stiffness of the third VSA (16) is mounted on the (3) by the bracket. The output of the stiffness regulation mechanism, the motor (16) is As is shown in Figure 18, in the design of mechanical solution, the maximum allowable angular deflection β is defined as reducer of the (16) is connected to the ball screw shaft (1). AsAs is is shownshown in Figure Figure 18, 18, in inth e the design design of mechanical of mechanical mounted on the (3) by the bracket. The output of the ±0.7rad, and the initial angle of rotation of the two quadratic torsion springs are −0.827rad and +0.827rad, respectively. The rotation of the nut (2) is restricted by the connecting solution, tthehe maximummaximum allowable allowable angular angular deflection deflection β is β is reducer of the (16) is connected to the ball screw shaft (1). The mechanically allowable torque range at the elastic transmission of the VSA is from 0Nm to 7.878Nm, and the rod and slider mechanism. A mechanical stop in the spring defined asas ±0.7rad,±0.7rad, and and the the initial initial angle angle of ofrotation rotation of theof the The rotation of the nut (2) is restricted by the connecting maximum elastic torque M(β) will reduce with the increase of the initial rotation angle φ. The mechanically allowable guide sleeve (7) constrains the amount of deformation of twotwo quadratic quadratic torsion torsion springs springs are are − 0.827−0.827radrad and and +0.827 +0.827rad,rad, rod and slider mechanism. A mechanical stop in the stiffness range at the elastic transmission is from 11.25 Nm/rad to 20.78 Nm/rad. the spring. When the working radius R has reached the respectivelyrespectively.. TThehe mechanically mechanically allowable allowable torque torque range range at at spring guide sleeve (7) constrains the amount of maximum allowable value, the VSA is completely rigid. thethe elastic elastic transmission transmission of of the the VSA VSA is from is from 0Nm 0Nmto to deformation of the spring. When the working radius R Characteristic curve of actuating torque: M(β )-β Characteristic curve of output stiffness The allowable angle range of the stiffness motor with 7.878Nm,7.8788 Nm, and and the the maximum maximum elastic elastic 22torque torque M(β) M(β) will will reduce has reached the maximum allowable value, the VSA is respect to the fixed housing is limited by the cable attached withreduc thee with increase the increase of the of initialthe initial rotation20 rotation angle angle φ φ. . The 6 to it. The allowable range of the rotation angle of the VSA mechanically) [Nm] allowable =0.827rad stiffness range at the elastic

β 18 =0.9rad =1rad is defined as ±150°. transmission4 is from 11.25 Nm/rad 16to 20.78 Nm/rad. =1.1rad =1.2rad 14 2 =1.3rad Jishu Guo and Guohui Tian: 11 Conceptual Design and Analysis of Four Types of Variable Stiffness =1.4radActuators Based12 on Spring Pretension Output torque M( =1.5rad Output stiffness K [Nm/rad] 0 10 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Angular deflection [rad] Angular deflection [rad]

Figure 18. Characteristics of torque and stiffness of the third VSA

5.3. Mechanical structure solution of the third VSA

The mechanical solution for the antagonistically actuated VSA is shown in Figure 19. The equivalent torsion spring for two compression spring configurations is used. Two DC motors are used to change the position and stiffness. Changing the position is done by corotation of the two motors while counter rotation of two motors is used to regulate the stiffness. The base frame (1) is used to connect the two frames (2) by the screws. The equivalent torsion spring (4) is housed in the frame (2) by a radial ball bearing. Frame (2) and frame (3) are fixedly connected by screws. The worm (7) and worm wheel (6) are housed in the frame (3) by four angular contact ball bearings (8). The output shaft (5) is driven by the worm wheel (6) and fixedly connected with the input shaft of the equivalent torsion spring (4). The output link (11) is fixed with the housings of the two torsion spring by screws. The ring gear (9) is fixed with the housing of the output link (11). The small spur gear (10) is mounted on the base (1) by the brackets and screws. The rotation shaft of the (10) is connected to a small rotary encoder. The (10) is meshed with the (9) and the angular deflection of the VSA can be obtained by calculation. Since the worm and worm wheel mechanism is used, no energy is required to maintain the output stiffness of the VSA. Due to the mechanical limit, the allowable rotation angle range of the VSA is defined as ±100°. G()

 M ()    +

Figure 17. Antagonistic set-up of two quadratic torsion springs for an adjustable stiffness joint mechanism

5.2. Characteristics of torque and stiffness of the third VSA

As is shown in Figure 18, in the design of mechanical solution, the maximum allowable angular deflection β is defined as ±0.7rad, and the initial angle of rotation of the two quadratic torsion springs are −0.827rad and +0.827rad, respectively. The mechanically allowable torque range at the elastic transmission of the VSA is from 0Nm to 7.878Nm, and the maximum elastic torque M(β) will reduce with the increase of the initial rotation angle φ. The mechanically allowable stiffness range at the elastic transmission is from 11.25 Nm/rad toFigure 20.78 Nm/rad. 19. CAD assembly of the third VSA equilibrium position of the output link and a secondary Characteristic curve of actuating torque: M(β )-β Characteristic curve of output stiffness 8 22 motor to modify the output stiffness of the VSA. The 20 tension spring set with a triangular structure is used in this 6 6. Conceptual design and analysis of the fourth VSA ) [Nm] =0.827rad

β 18 =0.9rad design. Figure 20 illustrates the conceptual design, whose =1rad 4 16 main characteristic is the usage of the tendon-pulley =1.1rad =1.2rad 14 transmission. The output shaft of the main motor is 2 =1.3rad 6.1. Conceptual layout of the fourth VSA =1.4rad 12 attached to the driving pulley (1), and two tendons are Output torque M( =1.5rad Output stiffness K [Nm/rad] 0 10 attached to the (1). In turn, the tendons are attached to the 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Angular deflection [rad] Angular deflection [rad] Inmovable this pedestal section, (3) whose the otherproposed parts are connectedmechanism with comprises a serial configuration, with the main motor determining the the tension spring set. The (3) can be moved along the linear Figure 18. Characteristics of torque and stiffness of the third VSA equilibrium position of the output link and a secondary motor to modify the output stiffness of the VSA. The tension Figure 18. Characteristics of torque and stiffness of the third VSA guides (4). Six linear bearings are used between the linear springguides and set the with movable a triangular parts for low structure friction inis theused in this design. Figure 20 illustrates the conceptual design, whose main 5.35.3. Mechanical Mechanical structure structure solution ofsolution the third VSA of the third VSAcharacteristicmechanical solution. is Thethe movableusage springof the pedestal tend on-pulley (5) is transmission. The output shaft of the main motor is attached to the coupled with the screw transmission mechanism, which The mechanical solution for the antagonistically actuated drivingtransfers the pulley angular (1), motion and of two the secondary tendons motor are toatta a ched to the (1). In turn, the tendons are attached to the movable pedestal (3) VSAThe ismechanical shown in solutionFigure 19. for The the equivalentantagonistically torsion actuated spring VSA is shown in Figure 19. The equivalent torsion spring for linear displacement of the (5) along the linear guides. The fortwo two compression compression spring spring configurations configurations is used.is used. Two Two DC DC motorswhose are used otherto change parts the posi aretion connected and stiffness. with Changing the tension spring set. The (3) can be moved along the linear guides (4). Six linear motorsthe position are usedis done to by change corotation the of position the two andmotors stiffness. while counterbearingsfixed rotation mechanical of aretwo stopmotorsused (2) isandbetween used the to movable regulate the mechanical thelinear stiffness. guides stop and the movable parts for low friction in the mechanical solution. The The base frame (1) is used to connect the two frames (2) by the screws.(6) constrain The equivale the amountnt torsion of spring deformation (4) is housed of the in tension the Changing the position is done by corotation of the two movable spring pedestal (5) is coupled with the screw transmission mechanism, which transfers the angular motion of motorsframe (2) while by a counter radial ball rotation bearing. of twoFrame motors (2) and is usedframe to(3) arespring fixedly set. connect Whened the by (3) screws. is in contact The worm with (7)the and (2) orworm the (6), regulatewheel (6) the are stiffness. housed Thein the base frame frame (3) by(1) fouris used angular to connect contact ballthe bearings springsecondary (8). set The cannot output motor besh aft stretched.to (5) a is lineardriven Furthermore, by displacement the worm the of the (5) along the linear guides. The fixed mechanical stop (2) and the thewheel two (6) frames and fixedly(2) by theconnected screws. with The the equivalent input shaft torsion of the equimovablemaximumvalent torsion allowable mechanical spring displacement (4). Th stope output (6) of link theconstrain (5)(11) along is fixed thethe x amount of deformation of the tension spring set. When the (3) is in contact with the housings of the two torsion spring by screws. The ring directiongear (9) is is fixed greater with than the housingthe maximum of the output amount link of (11). defor‐ spring (4) is housed in the frame (2) by a radial ball bearing. with the (2) or the (6), the spring set cannot be stretched. Furthermore, the maximum allowable displacement of the (5) FrameThe small (2) and spur frame gear (10) (3) areis mounted fixedly onconnected the base by(1) screws.by the bracketsmation and screws.of the spring The rotation set. Therefore, shaft of whenthe (10) the is (3) connected and the (6) Theto aworm small (7) rotary and wormencoder. wheel The (6)(10) are is housed meshed in with the framethe (9) andalongare the in angular contact the xdeflection withdirection each of other,the is VSA greater no can angular be than obtained deflection the by maximum is amount of deformation of the spring set. Therefore, when the (3) and (3)calculation. by four angular Since the contact worm balland wormbearings wheel (8). mechanismThe output is usedthemechanically, no (6) energy are is allowed,in required contact and to maintainthe with actuator each the will output other, switch stiffness intono angularthe deflection is mechanically allowed, and the actuator will switch into the shaftof the (5) VSA. is drivenDue to the by mechanical the worm limit, wheel the (6) allowable and fixedly rotation anglestiff operationrange of the mode VSA from is defined the compliance as ±100°. mode. connected with the input shaft of the equivalent torsion stiff operation mode from the compliance mode. spring (4). The output link (11) is fixed with the housings of the two torsion spring by screws. The ring gear (9) is fixed with the housing of the output link (11). The small spur gear (10) is mounted on the base (1) by the brackets and screws. The rotation shaft of the (10) is connected to a small rotary encoder. The (10) is meshed with the (9) and the angular deflection of the VSA can be obtained by calculation. Since the worm and worm wheel mechanism is used, no energy is required to maintain the output stiffness of the VSA. Due to the mechanical limit, the allowable rotation angle range x of the VSA is defined as ±100°.

Figure 20. Schematic20. Schematic description description of the VSA with seriesof the setup VSA with series setup

When the torque TE is applied on the (1), the spring set is Whenstretched. the This torque elastic deformationTEE is applied leads to on the the angular (1), the spring set is stretched. This elastic deformation leads to the angular deflection θ between θ between the driving the pulleydriving position pulley and theposition and the output link (7). To simplify the calculation, the friction forces output link (7). To simplify the calculation, the friction andforces theand theinertia inertia ofof thethe moving moving bodies bodies are not consid‐ are not considered, the actuator is fixed parallel to the horizontal plane and the effectered, the of actuator gravity is fixed is eliminated.parallel to the horizontal Suppose plane that the relationship between force F(x) (N) and displacement x (mm) of the and the effect of gravity is eliminated. Suppose that the spring set is F(x)=ax22, and the initial amount of displacement of the (5) is Δx. The elastic output torque TEE and the relationship between force F(x) (N) and displacement x torsion(mm) of the stiffness spring set K is F(x)=axof the2 , VSAand the are initial given amount by: of Figure 19. CAD assembly of the third VSA displacement of the (5) is Δx. The elastic output torque TE and the torsion stiffness K of the VSA are2 given by: 6. Conceptual design and analysis of the fourth VSA TFxRaRxR  EJJJ   ì 2 (21) dT T= F( x) R2 = aR D x + R q 6.1 Conceptual layout of the fourth VSA ïE E 2.JJ (  J ) KaRxRí JJ (21)  d dTE 2 In this section, the proposed mechanism comprises a serial ïK = =2 aRJ ( D x + RJq ). î dq configuration, with the main motor determining the

12 Int J Adv Robot Syst, 2015, 12:62 | doi: 10.5772/60580 6.2. Characteristics of torque and stiffness of the fourth VSA 6.2 Characteristics of torque and stiffness of the fourth VSA 6.3 Mechanical structure solution of the fourth VSA

The approximate quadratic tension spring set is used in the The mechanical structure solution of the fourth VSA is mechanical solution. The working radius RJ is 45mm. The shown in Figure 22. The DC motor (1) equipped with the maximum allowable angular deflection θmax is defined as gearbox is fixedly mounted on the output housing (10) by ±0.533rad. Figure 21 shows the characteristics of torque and the brackets (9) and the screws. The screw transmission stiffness of the VSA for different spring configurations. The mechanism (3) is used to drive the spring pedestal (4), and high level of linearity of the stiffness-angular deflection the (4) is moved along the linear guide (2). A potential meter relationship can be achieved by using the tension spring (7) is fixed and connected between the driving shaft (6) and set. The allowable stiffness range at the elastic transmission the rotatable connecting plate (8). The driving pulley (5) is of the VSA is from 0.545 Nm/rad to 13.88 Nm/rad, and the driven by the (6), and the (8) is fixed with the (10) by the maximum elastic output torque T is 3.76Nm. As is Emax flange and screws. Therefore, the angular deflection θ can shown in Figure 21(c), the differences of the actuating be obtained. torque and output stiffness of the VSA between the two different spring configurations are small. For convenience of implementation, the tension spring set can be used to replace the purely quadratic spring in the application. the angular deflection θ can be obtained.

Purely quadratic spring configuration x[mm] Purely quadratic spring configuration x[mm] 4 15

20 20 3 10 15 15 2 10 10 5 1

5 5

Output stiffness K [Nm/rad] K stiffness Output Output elastic torque Te [Nm] Te torque elastic Output 0 0 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Angular deflection θ[rad] Angular deflection θ[rad]

(a) Characteristics of actuating torque and output Figure 22. CAD assembly of the fourth actuator stiffness for the prescribed quadratic spring configuration

7. Conclusion and Future Work Actual tension spring set configuration Actual tension spring set configuration 4 x=0mm 14 Figure 22. CAD assembly of the fourth actuator. x=2mm 12 x=4mm The development of a variable stiffness actuator is an open 3 x=6mm 10 research area. In this paper, four types of conceptual x=8mm 7. Conclusion and Future Work x=10mm 8 2 models of the novel variable stiffness actuators based on x=12mm 6 x=14mm spring pretension were presented. These actuation systems x=16mm 4 The development of a variable stiffness actuator is an 1 x=18mm can simultaneously and independently control the output

x=20mm 2 open research area. In this paper, four types of conceptual Output stiffness K [Nm/rad] K stiffness Output

Output elastic torque Te [Nm] Te torque elastic Output position and the output stiffness. For convenience of torque x=22mm 0 0 models of the novel variable stiffness actuators based on 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 and stiffness control, three types of approximate quadratic Angular deflection θ [rad] Angular deflection θ [rad] spring pretension were presented. These actuation springs are designed and the stiffness regulation mecha‐ systems can simultaneously and independently control nisms are illustrated. In the first VSA, the piecewise linear (b) Characteristics of actuating torque and output the output position and the output stiffness. For stiffness for the actual tension spring set configuration conveniencespring set in of parallel-type torque and stiffnessis used tocontrol, constitute three an types approx‐ of imateapproximate quadratic quadratic spring. spring The screws are designed transmission and the mecha‐ Elastic torque difference curve： Te-θ Stiffness difference curve： K-θ nism and the roller-cam mechanism with self-locking 0.015 x=0mm 0.6 stiffness regulation mechanisms are illustrated. In the first x=2mm function are designed for stiffness regulation. In the second x=4mm VSA, the piecewise linear spring set in parallel-type is 0.01 0.4

x=6mm VSA, the variable working radius mechanism is proposed Te [Nm] Te

K [Nm/rad] K used to constitute an approximate quadratic spring. The

 0.005 x=8mm  x=10mm 0.2 basedscrew on transmissionthe connecting mechanism rod and andslider the mechanism. roller-cam The 0 x=12mm output stiffness is only related to the effective working x=14mm 0 mechanism with self-locking function are designed for -0.005 x=16mm radius for linear spring configuration but not to the exerted x=18mm -0.2 stiffness regulation. In the second VSA, the variable -0.01 x=20mm torque or angular deflection. Moreover, the relationship Torque difference difference Torque x=22mm working radius mechanism is proposed based on the -0.015 difference Stiffness -0.4 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 betweenconnecting the exerted rod and torque slider and mechanism angular deflection. The output is linear. Angular deflection θ [rad] Angular deflection θ [rad] This feature makes the torque control and stiffness regula‐ stiffness is only related to the effective working radius for tion more convenient. For the third VSA, two equivalent (c) The differences of the actuating torque and output linear spring configuration but not to the exerted torque quadratic torsion springs are used. The design method of stiffness between the prescribed and achieved spring or angular deflection. Moreover, the relationship between the equivalent quadratic torsion spring is simple and the configurations the exerted torque and angular deflection is linear. This mechanical structure is compact, which is suitable for feature makes the torque control and stiffness regulation Figure 21. Characteristics of actuating torque and output stiffness of the application. No cable is used in the torsion spring, so it is Figure 21. Characteristics of actuating torque and output more convenient. For the third VSA, two equivalent fourth VSA for the prescribed and achieved spring configurations easy to implement, and the effect of the elongation of the stiffness of the fourth VSA for the prescribed and quadratic torsion springs are used. The design method of achieved spring configurations. the equivalent quadratic torsion spring is simple and the mechanical structure is compact, whichJishu Guo is su anditable Guohui for Tian: 13 Conceptual Design and Analysis of Four Types of Variable Stiffness Actuators Based on Spring Pretension 6.3 Mechanical structure solution of the fourth VSA application. No cable is used in the torsion spring, so it is easy to implement, and the effect of the elongation of the The mechanical structure solution of the fourth VSA is cable is eliminated. The actuating torque of the VSA is shown in Figure 22. The DC motor (1) equipped with the related to the initial amount of deformation of the spring, gearbox is fixedly mounted on the output housing (10) by and it is a linear function of angular deflection. The the brackets (9) and the screws. The screw transmission output stiffness is a linear function of the initial amount mechanism (3) is used to drive the spring pedestal (4), of deformation of the spring. The main novelty of the and the (4) is moved along the linear guide (2). A fourth VSA is the application of the tension springs set. potential meter (7) is fixed and connected between the The approximate quadratic tension spring is easy to driving shaft (6) and the rotatable connecting plate (8). implement and the mechanical structure solution is The driving pulley (5) is driven by the (6), and the (8) is simple and compact. The relationship between the output fixed with the (10) by the flange and screws. Therefore, stiffness and the angular deflection is fairly linear. 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(2012) introduction of compliance, which is an interesting issue The vsaUT-II: a novel rotational variable stiffness worthy of investigating in future research. actuator. In: 2012 IEEE International Conference on Robotics and Automation. 2012 May 14-18; Minne‐ 8. Acknowledgements sota, USA. IEEE. pp. 3355-3360.

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Jishu Guo and Guohui Tian: 15 Conceptual Design and Analysis of Four Types of Variable Stiffness Actuators Based on Spring Pretension